text stringlengths 4 2.78M |
|---|
---
abstract: 'The membership of British political parties has a direct influence on their political effectiveness. This paper applies the mathematics of epidemiology to the analysis of the growth and decline of such memberships. The party members are divided into activists and inactive members, where all activists influence the quality of party recruitment, but only a subset of activists recruit and thus govern numerical growth. The activists recruit for only a limited period, which acts as a restriction on further party growth. This Limited Activist model is applied to post-war and recent memberships of the Labour, Scottish National and Conservative parties. The model reproduces data trends, and relates realistically to historical narratives. It is concluded that the political parties analysed are not in danger of extinction but experience repeated periods of growth and decline in membership, albeit at lower numbers than in the past.'
address: 'School of Computing and Mathematics, University of South Wales, Pontypridd, CF37 1DL, Wales, UK'
author:
- 'Rebecca A. Jeffs'
- John Hayward
- 'Paul A. Roach'
- John Wyburn
bibliography:
- 'JeffsHaywardPoliticalPartyGrowth.bib'
title: Activist Model of Political Party Growth
---
Political Parties ,Social Diffusion ,Differential Equations ,Epidemics ,Population Models ,Sociophysics.
Introduction
============
Political parties play a vital role in the governance of countries. They provide the personnel out of which national and local leaders emerge, a legitimate political identity for those in government, and an arena in which policy can be formed. Even when not in power, parties can provide checks and balances by providing an opposition, both in democratic forums and outside. As such there is keen interest in their growth and decline.
Many theories of political party growth consider the relationship between its membership size and its growth. One of the earliest such theories was Michels’ “Iron Law of Oligarchy” which states that any form of organisation will eventually develop an oligarchy as it grows to the point where real democracy becomes difficult [@Michels:Political]. The reasons given for this oligarchy are that a large complex organisation requires a specialist bureaucracy in order to make efficient day to day decisions, thus inevitably removing the rank and file members from the centre of power. Michels applied this theory, originally published in 1915, to the growing socialist parties of Europe, showing that they evolved into oligarchies as did earlier conservative parties, despite all their ideals. This has clear implications for the growth of a party as the membership suffers diminishing ability to become involved in decision making, rendering membership less attractive.
@Tan:size hypothesised that there would be a direct effect of increasing party size on member participation and growth due to free riding, that is becoming inactive, as proposed by @Olson:Logic, as well as there being an indirect effect due to increasing complexity, as put forward by @Michels:Political. Using data on 23 political parties [@Janda:Political], Tan argued that although party size can directly reduce participation, the indirect effect through increasing complexity can positively influence the participation of members, thus contradicting that aspect of Michel’s law. Although the debate as to the effect of party size on participation is still not resolved it is clear that there is a potential limit to the size of a political party related to its ability to keep members active. This suggests the importance of activists in the growth of the party.
The importance of activists in the interests of a political party has been emphasised by @Seyd:British, noting their role in fund raising, political legitimacy, and as a source of voters, campaigners and potential candidates for parties. However they argue that party decline is due to choice rather than the structural reasons of @Tan:size and @Michels:Political. They further claim that a party chooses to restrict the supply of activists as they can also be a hindrance to the party leadership’s freedom of action. The reduction in the benefits for party membership, and the restriction of activists, is thus deemed to be one of the causes of the decline. In this case party growth comes from the often brief periods during which a party requires activists for electoral purposes.
Further support for belief in the importance of the role of activists in party growth is provided by @Weldon:polity and @Norris:Phoenix who both showed that small parties have a higher degree of activism compared with larger parties due to the lack of funds to support a professional organisational structure. It is thus suggested that larger parties find it harder to grow, as the reduced incentives that can be offered are insufficient to keep activists in the party or to maintain their level of activism.
However a decline in party size does not necessarily increase activism as the party can remain organisationally complex if it lacks the funds to support the change back to a state in which activism could again flourish [@Tan:decline]. By contrast some parties can develop complex structures to make organisation and activity more efficient and effective [@Tan:size]. These phenomena suggest that activism is a natural feature of the smaller and growing party, but not of the larger and diminishing. However, larger parties are capable of sustaining activism, at least during those periods where activism is not contrary to the oligarchy’s aims. In either case, activism is the cause of growth.
It is inevitable in the light of the above that much of the attention on change in party size concentrates on decline [@Whiteley:Dynamics; @Tan:decline; @Dalton:Partisans; @Whiteley:High], with less attention being given to the mechanisms behind their growth. Discussions on growth generally focus on the reasons *why* parties wish to grow and the incentives provided for such increases in membership [@Seyd:British; @Weldon:polity]. There has been less discussion on *how* parties grow but it is clear that activists, those most involved in the life of the party, are key to its growth [@Norris:Phoenix; @Whiteley:High].
The question needs to be asked as to how activists recruit new members to the party. @Jeffreys:History points to very specific recruitment campaigns that dramatically increased the membership of the main UK political parties in the immediate post-war period. These campaigns were largely carried out using door-to-door recruitment by the most active members of the parties. Also a deliberate recruitment strategy by the Labour party from 1994 resulted in it temporarily becoming the fastest growing party in Europe [@Whiteley:High p. 24]. These periods of growth could be explained in terms of a word-of-mouth phenomenon driven by party activists.
This paper proposes a word-of-mouth model of political party growth using ideas similar to mathematical epidemiology. The model divides the party into activists and inactive members. Activists are further divided into those who recruit (the “infectious”) and those who do not recruit. However non-recruiting activists do contribute to the party by discouraging new members from free riding. Thresholds of growth are computed and the results of the model are compared with historical party data from the UK.
Previous Models of Social Diffusion
===================================
Word-of-mouth models for social diffusion and organisational growth are not new. @Burbeck:Riot used an SIR model to investigate the spread of rioting, applying the model to three riots from the 1960s. Similar models with more variables have been applied to riots, public outrage and terrorist groups [@hayward2014model; @nizamani2013public; @camacho2013development].
Other models dealing with the spread of behaviour include language acquisition [@Abrams:language; @wyburn2008future], alcohol consumption [@Manthey:drink; @Sanchez:Drink], cigarette smoking [@Rowe:Smoke] and psychological/social diseases such as bulimia and obesity [@gonzalez2003too; @santonja2010mathematical]. Most of these models employ multiple sub-populations with one or more acting as infectious agents and at least one non-infectious. In contrast, the influential Abram-Strogatz model [@Abrams:language] assumes there is no non-infectious category, a specific feature of language acquisition, making it less applicable to the other behaviours modelled.
Additionally epidemiological ideas have been applied to the spread of rumours [@Kawachi:Rumour; @zhao2011sihr], ideologies [@santonja2008mathematical; @vitanov2010verhulst] and online networks [@cannarella2014epidemiological; @woo2013modeling; @chhabra2014alternative]. @bettencourt2006power modelled the spread in the use of Feynman diagrams throughout the scientific community. Their model allowed for some new recruits to the Feynman methodology to be non-infectious, an exposed class, thus allowing for a weaker growth in recruiters compared with all users of the methodology.
None of the above models involve organisational membership as seen in political parties. However models close to these ideas have been used for the spread of religious affiliation. @Hayward:Church [@Hayward:General] applied SIR type models to church and denominational membership where the infectious church members responsible for recruitment, called “enthusiasts”, were a subset of new recruits. The models were applied to a range of religious denominations which could be categorised as experiencing rapid growth, as stable or as heading for extinction. Like @bettencourt2006power, @Hayward:General also allowed for some new recruits to be non-infectious. By contrast @ochoche2013evaluating [@madubueze2014mathematical] considered the longest serving church members as the recruiters. @mccartney2015three modelled a variety of recruitment mechanisms, predicting that a range of denominations in Northern Ireland faced future extinction.
Additionally there are statistical physics models of religious adherence [@ausloos2007statistical; @ausloos2009statistical; @ausloos2012econophysics] that employ the epidemiological analogy. In particular @Ausloos:econ [@abrams2011dynamics] draw the parallel between the spread of language and religion. The model of @abrams2011dynamics has the same restriction of no inactive members as the Abrams-Strogatz language acquisition model, on which it is based.
One specific epidemiological model related to political party growth was presented by @romero2009epidemiological who modelled the rise of third political parties in a state dominated by two parties. In the model the third party seeks to recruit by changing voter opinion as small start-up parties do not have a voter base from which to recruit. Results illustrate the critical role activists play in sustaining grassroots movements and indentify conditions under which such minority parties can thrive. Although primarily about opinion change, and not directly relevant to the growth of mainline political parties, their model will provide a useful comparison in model construction. Additionally there are opinion dynamics models that link political party size to party characteristics [@galam2000dictatorship], and that model the influence of party activists on opinion change [@qian2015activeness].
Although not exhaustive, the above review is sufficient to justify word-of mouth-modelling being applied to political party growth and decline.
Initial Considerations
======================
To investigate the hypothesis that political party growth is due to the recruitment activities of party activists, the SIR epidemic model is taken as an initial exploratory model. This leads to the following assumptions:
Assumption 1
: The population is partitioned into susceptibles $S$, those who do not belong to the political party, and party members $P$. Party members are further partitioned into activists $A$ and inactive members $M$. This latter group are only inactive in the sense of recruitment. They may still attend party functions and engage in other activities.
Assumption 2
: The activists take on the role of the infectives in a disease and the model assumes that they only remain active for an average time $\tau$.
Assumption 3
: All new recruits become activists with the same average ability to recruit others to the party.
Assumption 4
: The contacts that activists make are limited by how many people they can meet in a given time period, rather than by the density of the population in which they work. This is more in keeping with deliberate door-to-door recruitment campaigns and other word-of-mouth phenomena in a large population. Thus the standard incidence model of contacts is assumed, similar to that used in the spread of sexually transmitted diseases [@hethcote:effects].
The model is intended to be applied to short time periods only, thus births, deaths, leaving and migration are ignored. Thus the equations are: $$\begin{aligned}
\frac {dS} {dt} &=&-\frac{C_{p}}{\tau N} SA \label {standard.1} \\
\frac {dA} {dt} &=&\frac{C_{p}}{\tau N} SA - \frac{A}{\tau}
\label {standard.2} \\
\frac {dM} {dt} &=& \frac{A}{\tau} \label{standard.3}\end{aligned}$$ where $N=S+A+M$ is the total adult population, a constant; homogeneous mixing has been assumed. $C_{p}$ is the recruitment potential, that is the number of susceptibles recruited, or converted, to the party by one activist throughout their entire active period.
The condition for epidemic growth in the party can be derived from setting $\dot{A}>0$ in (\[standard.2\]), giving $S/N>1/C_{p}$. This can be re-arranged to provide a threshold value for $C_p$ over which an epidemic will occur: $ C_{p} > 1/(S/N) \triangleq C_{\mathrm{epi}}$.
In this standard incidence SIR model the epidemic threshold $C_{\mathrm{epi}}$ depends on the fraction of susceptibles alone. The larger the fraction susceptible, the lower the threshold and the less recruitment potential to give epidemic party growth, as also seen in models of disease [@anderson1992infectious]. Thus rapid growth becomes harder to achieve as the pool of susceptibles reduces, and party growth will cease because activists are no longer able to reproduce themselves sufficiently quickly to make up for their losses. Growth ends before all people join the party.
Importantly the threshold $C_{\mathrm{epi}}$ does not depend on the number of activists. One activist is sufficient to start epidemic growth if the recruitment potential is sufficiently high (at least $C_{p}>1$ ) and the proportion of susceptibles in the population is sufficient. This assumes activists can reproduce activists from their recruits with the same ability to recruit. In this case $C_{p}$ is the equivalent of the basic reproduction number $R_{0}$ for the epidemic [@hethcote:effects; @anderson1992infectious].
To test the SIR model, data from the UK Labour party are used from 1947–1953 (Table \[lab44.tab\]), a period of rapid growth resulting from a deliberate door-to-door recruitment campaign by the party with the intention of making every Labour voter a party member, [@Jeffreys:History pp. 93–99]. The party membership, $P=A+M$, in the model (\[standard.1\]–\[standard.3\]) is compared with the data using a Runge-Kutta-Fehlberg method and optimised using least squares. Because of the uncertainty in the fraction of the adult population reached by activists during the period, three optimised runs are produced: (i) $N=33.24$ million, the electorate in the 1945 election, those with a known interest in politics [@uk:political]; (ii) $N=11.995$ million, the number of Labour voters in the 1945 general election, the people worth an activist spending time with [@uk:political]; and (iii) $N=5$ million, a more conservative estimate based on average pre-war union membership, the hard core Labour supporters [@keen2014membership]. It is unclear the extent to which the recruitment campaign was targeted towards likely recruits through, for example, the trade union movement.
The model parameters estimated for the three population sizes, (i)–(iii), are respectively $C_p=$ 1.025, 1.073, 1.194 ; $\tau = 1.1939, 0.0100, 0.0283$ and $A_0/P_0= 0.005\%,$ $ 0.011\%, 0.040\%$. In each case the threshold $C_{\mathrm{epi}}$ exceeds the recruitment potential from about 1949, indicating the slowing down of party growth. Irrespective of the target population size, the durations spent as activist, $\tau$, are very small, merely a matter of weeks at best. While this may be an acceptable duration for a spreader in word-of-mouth diffusion, it is far too short a period for a political activist, some of whom remain active for many years [@Jeffreys:History]. Such a short duration leads to an unacceptably small number of activists, less than 1% of the party at the peak. Estimates of party activists for major political parties range from 30-40% [@Janda:Political ch. 12] [@Norris:Phoenix p. 127]. Thus the SIR model is not suitable as a model of political party growth without modification.
It is proposed that the fault in the modelling lies with the identification of recruiters with activists. Recruitment is only a small part of an activist’s role in the party; most of their time is spent on campaigning, fund raising and political discussion [@Weldon:polity; @Norris:Phoenix; @Whiteley:High]. @pedersen2004sleeping found that only 10% of activists spent more than 5 hours a month on recruitment and electoral duties, indicating that recruitment is very much a minority activity of party activists. Thus it is proposed in the next section that the model be extended to allow for recruiting and non-recruiting activists, giving the latter a distinct dynamical role in the quality, rather than the quantity, of recruitment.
Limited Activist Model Construction
===================================
The model of Section 3 is modified to distinguish between activists who recruit and those who do not. All activists have a role in determining the quality of the party as measured by the proportion who are active. Further the issue as to what fraction of the population is genuinely open to joining the party is resolved by introducing a separate population of people hardened against the party beliefs. The revised model will be referred to as the Limited Activist model.
Assumption 1
: The population is partitioned into party members $P$ and non-party members. Party members are further partitioned into recruiting activists $I$, non-recruiting activists $A$ and inactive members $M$. This latter group are free-riders who do not participate in any party activities. The non-recruiting activists contribute to the political activity of the party but are ineffective in recruitment. The non-party members are partitioned into susceptibles $S$, those open to the party because of their political persuasion, and hardened $H$, those sufficiently opposed to the party that they would never become members.
Assumption 2
: The recruiting activists take on the role of the infectives in the spread of a disease and become non-recruiting activists after an average time $\tau_i$. Non-recruiting activists become inactive after an average time $\tau_a$. It is expected $\tau_a \gg \tau_i$ as recruitment is only carried out by a small minority of activists at any one time.
Assumption 3
: All new recruits may become either type of activist, or inactive members.
Assumption 4
: The contacts that activists make are limited by how many people they can meet in a given time period, rather than by the density of the population in which they work.
Assumption 5
: All activists influence the number of recruits who themselves become active according to the proportion of activists in the party. That is a more politically active party will have more politically active recruits.
Assumption 6
: Inactive members leave the party at a fixed rate $\alpha$ and are open to rejoining, reflecting the ease with which members may lapse through non-renewal of membership.
Following from the above assumptions, the equations of the Limited Activist Model are: $$\begin{aligned}
\frac {dS} {dt} &=&-\frac{C_p}{\tau_i N}SI + \alpha M \label {model.S} \\
\frac{dI}{dt} &=& g\frac{C_p}{\tau_i N}SI - \frac{I}{\tau_i} \label{model.I} \\
\frac {dA} {dt} &=& f(1-g)\frac{C_p}{\tau_i N}SI + \frac{I}{\tau_i} -\frac{A}{\tau_{a}}\label{model.A}\\
\frac {dM} {dt} &=& (1-f)(1-g)\frac{C_p}{\tau_i N}SI +\frac{A}{\tau_{a}} - \alpha M \label{model.M} \end{aligned}$$ where $$f=f(A_T) \triangleq \frac{I+A}{P} \label{model.f}$$ represents the fraction of those new recruits who do not become recruiters, but who nevertheless contribute to the party activism rather than free-riding. The coefficient $g$ is the fraction of recruits who become recruiters, $C_p$ is the recruitment potential, and total active membership $A_T=I+A$.
The total population $N=H+S+I +A+M$; total open population with sympathy for the party $O=N-H$; and total party membership $P=I +A+M$. As the model is intended only for short-term growth and decline, births and deaths are excluded and the hardened $H$ and total population $N$ are assumed constant. Thus $(S,I,A,M)$ are determined by (\[model.S\]–\[model.M\]) with the stated constraints.
Multiple influential populations have been used in structurally similar models of the diffusion of bulimia [@gonzalez2003too], binge drinking [@Manthey:drink], rumours [@zhao2011sihr], public violence [@nizamani2013public], terrorist groups [@camacho2013development] and political opinion change [@romero2009epidemiological]. This mechanism is required to allow for party members in $I$ and $A$ to have different types of activity; in particular many party activists do not recruit [@Weldon:polity; @Norris:Phoenix; @pedersen2004sleeping]. The Limited Activist model proposes that the influence of non-recruiting activists is on the quality of recruits, determined by $f(A_T)$, rather than on the quantity as in epidemiological mass action. However this does increase the number of activists at the expense of the inactive. @galam2000dictatorship have also modelled the influence of party size on the qualitative characteristics of a party, especially the role of activists in that process [@qian2015activeness].
Multiple routes from susceptibles into different populations are used in the diffusion of scientific ideas [@bettencourt2006power], rumours [@zhao2011sihr], religion [@Hayward:General] and terrorist groups [@camacho2013development]. Likewise @romero2009epidemiological have constant proportions entering different voter populations. What is different in the proposed Limited Activist model is that, rather than the proportions of recruits entering the three party member categories being fixed, the proportion becoming activists, $f(A_T)$, can vary, allowing a highly active party to become more active, even if numerical recruitment is moderate. This is required to account for the differing effects of recruitment by parties with different levels of activism. For example @Whiteley:High [p. 126] note the poor quality of recruits in the Labour party in the 1990s, when the party was relatively inactive compared with campaigns in previous generations where activity, not just numbers, was improved.
Multiple susceptible populations with different degrees of susceptibility were proposed by @granovetter1978threshold and have been used in the diffusion of binge drinking [@Manthey:drink], religion [@Hayward:General], terrorist groups [@camacho2013development], riots [@hayward2014model] and knowledge diffusion [@ausloos2015slow]. The Limited Activist model proposes two such populations, one aligned to the party, $S$, and one opposed, $H$. It is assumed that the latter would not change political alignment and join the party on the timescales being considered.
Analysis
========
As with the SIR model the epidemic threshold follows from $\dot{I}>0$. Let $R_p = g C_p$, the reproduction potential, or basic reproductive ratio, which measures how many recruiting activists one recruiting activist reproduces during their recruiting period, assuming the whole population $N$ is susceptible. Then from (\[model.I\]) $$R_{p} > \frac{1}{\bar{S}} \triangleq R_{\mathrm{epi}}
\label{LAMthresh.2}$$ where $R_{\mathrm{epi}}$ is the threshold value for the reproduction potential, above which there is an epidemic of recruiting activists. Barred variables denote the fraction of the total population $N$: $\bar{S} = S/N$.
Setting (\[model.I\]) to zero gives either $I=0$ or $\bar{S} = 1/R_p$. The former is the equivalent of the disease-free equilibrium (DFE) where the party vanishes $(1-\bar{H}, 0,0,0)$.
The other equilibria will be determined from $\bar{S} = 1/R_p$. Applying this to (\[model.S\]) set to zero, with $\bar{A} = 1 - \bar{H}-\bar{S}-\bar{I} -\bar{M}$ gives: $$\bar{A}= \hat{p}-\hat{q}\bar{I} \label{model.Seq}$$ where $$\begin{aligned}
\hat{p} &=& 1 - \bar{H}-\frac{1}{R_p} \label{model.p} \\
\hat{q} &=& \frac{g\alpha \tau_i + 1}{g\alpha \tau_i} \label{model.q} .\end{aligned}$$ Further, $f = 1-\bar{I}/(g\alpha \tau_i \hat{p})$, using $ \hat{p} = \bar{I} +\bar{A}+\bar{M}$. Thus a single equation for $\bar{I}$ can be obtained by setting (\[model.A\]) to zero, and the substitution of $f$ and $\bar{A}$ (\[model.Seq\]), giving: $$a_1 \bar{I}^2 - b_1 \bar{I} + c_1 = 0 \label{model.a1}$$ with $$\begin{aligned}
a_1 &=& \frac{(1-g)}{g^2\alpha \tau_i^2 } \label{model.a2} \\
b_1 &=& \hat{p}\left[\frac{1}{g\tau_i} + \frac{\hat{q}}{\tau_a}\right] \label{model.a3} \\
c_1 &=& \frac{\hat{p}^2}{\tau_a} \label{model.a4} \end{aligned}$$ Thus $$\bar{I}_{\pm} = \frac{b_1 \pm \sqrt{ b_1^2 - 4a_1c_1}}{2a_1} \label{model.a5}$$
Note that $a_1$ and $c_1$ are always positive and the sign of $b_1$ depends on the sign of $\hat{p}$, as $\hat{q}$ is positive. Thus there are two endemic equilibria (EE) with differing recruiting activists, $\bar{I}_{\pm}$ but the same party size $P=N-H-S$, given by $(1/R_p,I_{\pm},\hat{p}-\hat{q}\bar{I}_{\pm},\bar{I}_{\pm}/(g\alpha \tau_i))$. Both roots of (\[model.a1\]) are real as its discriminant reduces to: $$\left[\tau_a-\frac{1}{\alpha} \right]^2+ 2\tau_a g\tau_i+g^2\tau_i^2+2\frac{g\tau_i}{\alpha} + \frac{4g\tau_a}{\alpha }$$ which is always positive (\[model.a2\]–\[model.a4\]). Neither root is physical when $\hat{p}<0$, as from (\[model.a5\]) $\bar{I}_{\pm}<0$. However the one root is never physical as $A_{+}=\hat{p}-\hat{q}\bar{I}_{+}$ is always negative (\[proof.appendix\]). Thus there are only two relevant equilibrium points, the DFE and a single EE given by $I_{-}$ in (\[model.a5\]), the latter only physical when $\hat{p}>0$.
Stability of the two remaining equilibria is deduced from the Jacobian of (\[model.S\]–\[model.M\]): $$J =
\left[
\begin{array}{cccc}
-\frac{C_{p}\bar{I}}{\tau_i} & -\frac{C_{p}\bar{S}}{\tau_i} & 0 & \alpha\\
\frac{gC_{p}\bar{I}}{\tau_i} &
\frac{gC_{p}\bar{S}}{\tau_i}-\frac{1}{\tau_i} &
0 & 0\\
f(1-g)\frac{C_{p}\bar{I}}{\tau_i} &
f(1-g)\frac{C_{p}\bar{S}}{\tau_i} +\frac{1}{\tau_i} +f_{,\bar{I}} Q &-\frac{1}{\tau_a} +f_{,\bar{A}} Q & f_{,\bar{M}} Q\\
(1-f)(1-g)\frac{C_{p}\bar{I}}{\tau_i} & (1-f)(1-g)\frac{C_{p}\bar{S}}{\tau_i} -f_{,\bar{I}} Q & \frac{1}{\tau_a} -f_{,\bar{A}} Q &- \alpha- f_{,\bar{M}} Q
\end{array}
\right]
\label{model.j1}$$ where $f_{,\bar{I}} \triangleq \partial f / \partial \bar{I}$ etc, and $Q = (1-g)C_p\bar{S}\bar{I} $.
For the DFE the eigenvalues $\lambda$ of (\[model.j1\]) are: $$\lambda \left(\frac{gC_{p}\bar{S}}{\tau_i}-\frac{1}{\tau_i}-\lambda\right) \left(\frac{1}{\tau_a} +\lambda\right) \left(\alpha +\lambda\right) = 0 \label{model.j5}.$$
One eigenvalue is zero, as expected for a conserved system, and two are always negative. Thus stability of the DFE is determined by the remaining eigenvalue. As $\bar{S}=1-\bar{H}$ in equilibrium then the DFE is stable when: $ R_{p}(1-\bar{H})/\tau_i < 1/\tau_i$ or $$R_{p}<\frac{1}{1-\bar{H}} \triangleq R_{\mathrm{ext}}\label{model.j6}$$ where $R_{\mathrm{ext}}$ is termed the extinction threshold, and unstable when $R_p > R_{\mathrm{ext}}$. If the reproduction potential is less than this threshold the party becomes extinct. If above the threshold the party will survive and reach the EE, with epidemic growth first if $R_p$ is above the epidemic threshold. Note the extinction threshold is always below the epidemic threshold: $R_{\mathrm{ext}}\le R_{\mathrm{epi}}$.
When the DFE is stable, $R_p < R_{\mathrm{ext}}$ and then, from (\[model.p\]), $\hat{p}<0$ and, as noted earlier, the EE is not physical. By contrast when the reproduction potential is above the extinction threshold then the DFE is unstable, $\hat{p}>0$, and thus the EE is physical and the party settles on a non-zero equilibrium. Thus the model displays a similar transcritical bifurcation as occurs in an SIRS model.
Results
=======
The model is applied to three periods of rapid political party growth in the UK: the Labour Party from post-war and in the 1990s, and the Scottish National Party (SNP) post 2000. In addition the model is applied to the recent decline in the membership of the UK Conservative Party.
UK Labour Party 1944–1953
-------------------------
The data, Table \[lab44.tab\], may be considered in three phases according to the change in numbers and historical evidence [@Jeffreys:History].
Firstly, the rise in membership 1944–1945 is due to the renewal of former members who had disengaged during the Second World War.
The second phase is associated with the short recruitment campaign of 1945–46. This was mainly conducted door to door, and followed on from the enthusiasm due to the Labour general election victory of 1945. This phase is assumed to have involved a high initial number of activists, including recruiting activists. To estimate the total number of activists it is assumed the mid-war low of 219,000 [@keen2014membership] is a lower limit to the number of activists, those keen enough to retain membership during the war. Assume further that some activists re-joined in the first phase. Thus it is estimated that the second phase started with about 250,000 activists.
The Limited Activist model was applied to the rise in data 1945–46 from 487,000 to 645,000, and the post campaign decline to 608,000 in 1947. The most convincing data fit has the reproduction potential under the epidemic threshold with the growth due to a high initial number of recruiting activists of about $I_0=4000$. This leaves about 200,000 total activists in 1947, which corresponds to roughly a third of the party active, as also reported by @Janda:Political and @Norris:Phoenix. This figure is used as the initial value in the third phase of growth, which is examined in greater detail.
The third phase of growth in Labour party membership ran from 1947–1953 and was a more measured campaign conducted through the Trade Union movement as well as door to door. The UK electorate is used as the total population $N$, which is close to the adult population. The total open population $O=S+I+A+M$ is taken from those who voted Labour in 1945, i.e. those already members or open to joining due to their political persuasion. The time as a non-recruiting activist, $\tau_a$ is assumed to be 10 years. This is supported by @Jeffreys:History, who notes that a typical activist will be involved in at least one electoral campaign, implying up to 5 years as an activist. However the average figure will be higher than this as a number of activists remain so for life. The leaving rate $\alpha$ is taken as 5% consistent with the average post-campaign decline to 1960 of 3% per annum. No attempt to fit values 1954–1960 was made as the high volatility in change was assumed to be due to political factors for which a differential equation can only model the average. However, the 5% leaving rate ensured the model was close to the 1961 figure of 751,000. This figure was the last year of decline in this phase [@keen2014membership].
The remaining parameters were obtained by least squares with a value for the fraction of recruits who become recruiters $g$ of 0.5, consistent with @Jeffreys:History and ensuring a high ratio of activist to inactive in the party. The 1951 data point was ignored as Labour lost an election that year and it is assumed the sudden drop was a temporary reaction, followed by rejoining in 1952. The parameter values are given in Table \[lab2.tab\].
[lll]{} Estimated Parameters & Value & Source\
Total population $N$ & 33.24 mill & UK electorate 1945 election [@uk:political]\
Open population $O=1-H$ & 11.995 mill & Labour voters 1945 election [@uk:political]\
Initial party membership $P_0$ & 608,000& Party data [@keen2014membership]\
Initial activists $ A_{T0}=I_0+A_0$ &200,000 & Estimate from mid-war low and\
&& roughly a third of party active [@Janda:Political; @Norris:Phoenix]\
Duration non recruiting activist $\tau_{a}$ &10 years & Estimate anecdotal evidence [@Jeffreys:History]\
Leaving rate $\alpha$&0.05& Estimate from post 1952 decline [@keen2014membership]\
Fraction infected recruited $g$ & 0.5& High ratio of active to inactive recruits\
Recruitment potential $C_{p}$&5.9501 & *Optimised*\
Duration recruiting activist $\tau_{i}$& 0.0144 & *Optimised*\
Initial recruiting Activists $ I_0$ & 139 & *Optimised*\
A good fit is made to party data 1947–53, with the exception of 1951, a year with poor election results for Labour, Fig. \[politicalfig1.fig\](a). The largest share of recruitment is due to the recruiting activists, reflecting the known highly active nature of the party, Fig. \[politicalfig1.fig\](b). Although the recruiting activists peak at about 1949, Fig. \[politicalfig1.fig\](c), the total number of activists does not peak until two years later, rising from a third to 42% of the party. They do not return to the pre-campaign values until 1959, showing that the effect of the campaign on party activity was sustained until long after the epidemic phase was over [@Jeffreys:History]. The reproduction potential, $R_p$ is well above the extinction threshold, Fig. \[politicalfig1.fig\](d), and above the epidemic threshold in 1947. The latter threshold rises due to the shrinking susceptible pool, but by 1960 is back below $R_p$, indicating that another period of growth may have naturally occurred in the 1960s due to shifting recycling of party members who left. It is more likely that political events in the 1960s initiated the post 1961 growth that did occur. The interpretation of periodicity in membership numbers will be discussed in the conclusion.
![Limited Activist Model applied to UK Labour Party. (a) Data fitting 1947–1953; (b) Recruitment rates to party membership sub-populations $I$, $A$, $M$; (c) Party membership $P$ and sub-populations $I$, $I+A$, $M$; (d) Thresholds $R_{\mathrm{epi}}$, $R_{\mathrm{ext}}$, compared with reproduction potential $R_p$.[]{data-label="politicalfig1.fig"}](JeffsHaywardPoliticalPartyGrowth1.eps){height="10cm"}
Not all the parameters could be estimated by data fitting, or from other known values. For example the parameter $g$, the fraction of recruits who become recruiting activists, was set at 0.5. Alternative data fits were obtained for other values of $g=0.2$ and $g=1$. In the first case the number of activists in the party declined too quickly compared with the historical narratives or the time. In the later case the activists rose too high. Thus 0.5 is taken as a good compromise, with the model robust against small variations in this value. A similar argument can be applied to the initial number of activists $A_{T0}=I_0 + A_0$, and to time active $\tau_a$, neither of whose values are critical.
UK Labour Party 1993–1997 {#lab93.sec}
-------------------------
In 1992 the leadership of the Labour Party passed to John Smith who set about a recruitment campaign to rebuild the party in a time of internal disillusionment, after a lengthy period out of electoral office [@Whiteley:High; @Jeffreys:History]. Thus the start of the rise in party numbers is taken from 1993. The intention of Smith, and his successor from 1994 Tony Blair, was to increase membership in order to win the 1997 general election. This they succeeding in doing, with party membership rising to a peak in 1997, Table \[lab93.tab\].
As with Labour in the 1940s the initial population and open population are taken from the electorate and voter base of the election prior to the simulation. The duration non-recruiting activist is set at 10 years for the same reasons as before. The fraction of activists is taken to be lower than the 1947 case, reflecting the poor state of engagement of the party membership in 1993 [@Whiteley:High; @Jeffreys:History]. The leaving rate is set at 15% reflecting the greater nominal party membership in the late 20th century, compared with the 1940s, and from the post 2000 decline. Membership of political parties has to be renewed annually, and thus people can leave by default unless they intentionally rejoin. The fraction recruited to the recruiting activists was taken to be lower than the 1947 case, $g=0.2$, as the party’s intention was to increase its size to gain legitimacy for the 1997 election, rather than to build a permanent active membership [@Whiteley:High].
Data were fitted to the Limited Activist model from 1993–1998 (Table \[lab93.tab\]), the latest year a good fit can be obtained, although the 1999 data point is also reasonable (Table \[lab3.tab\], Fig. \[politicalfig2.fig\](a)). The inactive members have the largest recruitment, in the early part of the campaign, with a more even balance later, Fig. \[politicalfig2.fig\](b). The recruiting activists peak before the end of 1994, not even rising to double their initial number, Fig. \[politicalfig2.fig\](c), yet this is sufficient to drive the large increase in the party and the total activists. The latter peak in 1997, the year of Labour’s general election victory, Fig. \[politicalfig2.fig\](c), is consistent with the known enthusiasm of the party at that time [@Whiteley:High]. The peak value of around a third of the party membership is typical of party activism for a healthy party [@Janda:Political; @Norris:Phoenix].
[lll]{} Estimated Parameters & Value & Source\
Total population $N$ & 43.25 mill & UK electorate 1992 election [@uk:political]\
Open population $O=1-H$ & 11.56 mill & Labour voters 1992 election [@uk:political]\
Initial party membership $P_0$ & 266,000& Party data [@keen2014membership]\
Initial activists $ A_{T0}=I_0+A_0$ &50,000 & Estimate lower than one third of party active\
&& due to prior internal party conflict. [@Janda:Political; @Norris:Phoenix]\
Duration non recruiting activist $\tau_{a}$ &10 years & Estimate similar to Labour 1940s\
Leaving rate $\alpha$&0.15& Estimate from post 1997 decline[@keen2014membership]\
&& and presence of nominal members[@Whiteley:High].\
Fraction infected recruited $g$ & 0.2& Low ratio of active to inactive recruits\
Recruitment potential $C_{p}$&19.2712 & *Optimised*\
Duration recruiting activist $\tau_{i}$& 0.00945 & *Optimised*\
Initial recruiting Activists $ I_0$ & 101 & *Optimised*\
![Limited Activist Model applied to UK Labour Party. (a) Data fitting 1993–1998, showing deviation of model from post 1998 decline; (b) Recruitment rates to party membership sub-populations $I$, $A$, $M$; (c) Party membership $P$ and sub-populations $I$, $I+A$, $M$; (d) Thresholds $R_{\mathrm{epi}}$, $R_{\mathrm{ext}}$, compared with reproduction potential $R_p$, assuming falling $R_p$, and rising leaving rate $\alpha$, to account for post 1998 drop.[]{data-label="politicalfig2.fig"}](JeffsHaywardPoliticalPartyGrowth2.eps){height="10cm"}
The epidemic threshold starts below the reproduction potential, initiating the rapid growth in the party, with growth ceasing as the threshold reaches its peak, Fig. \[politicalfig2.fig\](d). For fixed values of the recruitment potential $C_p$ and leaving rate $\alpha$ it is impossible to mimic the decline after 1998, the post election phase. Indeed had those values remained the party would have started growing again, in the early 2000s. However if the leaving rate is allowed to rise from 1998, and the recruitment potential to fall, perhaps due to the increasing focus of the party on government rather than retention and activism, then the data can be replicated to 2004. The reproduction potential falls, staying under the epidemic threshold and approaching the extinction threshold, perhaps typical of a party giving little attention to its membership, Fig. \[politicalfig2.fig\](d).
Scottish Nationalist Party (SNP) 2003–2013
------------------------------------------
In 2003 the SNP failed to win power in Scotland for a second time in succession. In addition they lost members and chose to change leadership in order to rebuild the party. Alex Salmond became leader in 2004, and under his leadership the party membership increased by 70% between 2003 and 2007, Table \[snp.tab\], achieving power as a minority government in 2007 [@uk:political; @cairney2012scottish]. With the need to gain legitimacy, and thus further members, for the 2011 Scottish parliament elections, the party saw another surge from 2010 to 2013, Table \[snp.tab\], winning the election outright. The data will be considered in these two natural phases, 2003-2010 and 2010-2013.
The initial population is taken from the typical Scottish electorate size 2003–2014, which remained reasonably stable throughout. Because of the increase in popularity of the SNP throughout the period, the open population is taken from an average of the SNP voters through all Scottish elections of the period. It is noted that fewer people vote SNP in UK general elections, which probably does not reflect the true sympathy for the party [@uk:political]. The duration non-recruiting activist and initial fraction activist are taken to be the same as the Labour party in the 1940s, as the SNP pursued a similarly positive recruitment drive [@snp:double]. Likewise the fraction recruited to recruiting activists, and the leaving rate, are taken to be the same as 1940s Labour.
The Limited Activist model was fitted to data from 2003–2010 (the first phase) and then separately for 2010–2013 (the second phase), Table \[snp.tab\]. The party numbers and number of activists were matched at 2010, Fig. \[politicalfig3.fig\](a–b), but a change in the initial recruiting activists was needed for the second phase, Table \[SNP1.tab\]. The recruiting activists have their first peak just before the Scottish election of 2007 and the party growth had almost finished by 2010, Fig. \[politicalfig3.fig\](b).
[lll]{} Estimated Parameters & Value & Source\
Total population $N$ & 4 mill & Scotland electorate typical\
&& value 2003-2013 [@scot:elect]\
Open population $O=1-H$ & 686,000 & Average SNP voters 2003–2011\
&& Scottish elections [@uk:political]\
Initial party membership $P_0$ & 9,500& Party data [@keen2014membership]\
Initial activists $ A_{T0}=I_0+A_0$ &33,000 & Roughly a third of party active [@Janda:Political; @Norris:Phoenix]\
Duration non recruiting activist $\tau_{a}$ &10 years & Estimate similar to Labour 1940s.\
Leaving rate $\alpha$&0.05& Estimate similar to Labour 1940s.\
Fraction infected recruited $g$ & 0.5& Estimate similar to Labour 1940s.\
Recruitment potential $C_{p}$&(i) 11.8842, (ii) 12.002 & *Optimised.*\
Duration recruiting activist $\tau_{i}$& (i) 0.0163, (ii) 0.01 & *Optimised.*\
Initial recruiting activists $ I_0$ & (i) 8, (ii) 21 & *Optimised.*\
![Limited Activist Model applied to SNP. (a) Data fitting in two phases, 2003–2010 and 2010–2013; (b) Recruitment rates to party membership sub-populations $I$, $A$, $M$; (c) Party membership $P$ and sub-populations $I$, $I+A$, $M$; (d) Thresholds $R_{\mathrm{epi}}$, $R_{\mathrm{ext}}$, compared with reproduction potential $R_p$.[]{data-label="politicalfig3.fig"}](JeffsHaywardPoliticalPartyGrowth3.eps){height="10cm"}
In the second phase the party growth of 50% in 3 years is explained by a sudden jump in the number, and effectiveness, of recruiting activists, Fig. \[politicalfig3.fig\](b,d), timed for the 2011 Scottish parliamentary elections. The epidemic period ends early in the party growth, Fig. \[politicalfig3.fig\](d), with most of the party growth after this period.
The model predicts that growth had finished by the time of the 2014 Scottish Independence Referendum campaign, Fig. \[politicalfig3.fig\](a), a long way short of the SNP’s target of doubling their party membership before that date [@snp:double]. Subsequently, however, the party membership doubled within days of the referendum of 18/09/2014 which delivered a “No” result [@snp:monday]. The latest figure available, 92,200 [@snp:ninety], shows an increase of five times the 2010 value which the model does not predict. This indicates a limitation in the applicability of the model, which will be discussed in the conclusion.
UK Conservative Party 2005–2012
-------------------------------
The final application of the limited activism model is the period of sustained decline in the UK Conservative Party that followed their brief rise from 2005 to 2006, Table \[con.tab\]. The hypothesis is explored that the decline is due to a large leaving rate with the recruitment potential remaining the same throughout the growing and declining period.
Due to the weakness of the party following a period in which they had lost two successive general elections, the duration non-recruiting activist is assumed to be shorter than the previous applications, $\tau_a=5$ years, and the recruitment to recruiting activists as being low, $g=0.2$. Likewise the initial number of activists is taken as significantly less than the one third used in the growing party scenarios of the SNP and Labour 1940s. The rapid decline from 2006 indicates a high leaving rate of about $\alpha = 20\%$. Other population parameters are set using the electorate and voter base, as before, Table \[con1.tab\].
The data fit is less precise than the other applications, Fig. \[politicalfig4.fig\](a). The reproduction potential is lower than the epidemic threshold initially, Fig. \[politicalfig4.fig\](d), indicating that the growth to 2006 was due to a large number of activists, rather than to their effectiveness in recruitment. The subsequent decline is not only due to a high leaving rate but also due to virtually no recruitment 2007–2011, Fig. \[politicalfig4.fig\](b), which leads to a party that is proportionally more active due to the loss of inactive members, Fig. \[politicalfig4.fig\](c).
[lll]{} Estimated Parameters & Value & Source\
Total population $N$ & 44.1802 mill & UK electorate 2005 election [@scot:elect]\
Open population $O=1-H$ & 8.7726 mill & Conservative voters 2005 UK general election [@uk:political]\
Initial party membership $P_0$ & 258,000& Party data [@keen2014membership]\
Initial activists $ A_{T0}=I_0+A_0$ &84,000 & Less than third of party active [@Janda:Political; @Norris:Phoenix]\
&& a low value due to weakness of party.\
Duration non recruiting activist $\tau_{a}$ &5 years & Estimate lower than previous data fits\
&& due to party unpopularity.\
Leaving rate $\alpha$&0.2& Estimate higher than previous data fits\
&& to account for rapid decline during period.\
Fraction infected recruited $g$ & 0.2& Low ratio of active to inactive recruits.\
Recruitment potential $C_{p}$&25.769 & *Optimised.*\
Duration recruiting activist $\tau_{i}$& 0.0074 & *Optimised.*\
Initial recruiting Activists $ I_0$ & 182 & *Optimised.*\
![Limited Activist Model applied to UK Conservative Party. (a) Data fitting 2005–2013, extrapolated to 2014; (b) Recruitment rates to party membership sub-populations $I$, $A$, $M$; (c) Party membership $P$ and sub-populations $I$, $I+A$, $M$; (d) Thresholds $R_{\mathrm{epi}}$, $R_{\mathrm{ext}}$, compared with reproduction potential $R_p$.[]{data-label="politicalfig4.fig"}](JeffsHaywardPoliticalPartyGrowth4.eps){height="10cm"}
The reproduction potential is above the extinction threshold, Fig. \[politicalfig4.fig\](d). Thus the model predicts a return to growth for the Conservative party from 2014. Although the party does not release official membership figures, a tentative figure for 2014 of 149,800 [@con:home] is in line with the model’s prediction. While there is a long-term gradual decline in political party membership [@Seyd:British; @Whiteley:Dynamics], the short-term rapid decline of the sort experienced by the Conservative party may be due to the dynamical effects of high leaving rates and moderate reproduction of recruiting activists.
Conclusion
==========
The Limited Activist model was developed to investigate the hypothesis that political parties grow through the action of activists persuading non members to join through word of mouth. The basic SIR construct gave a poor understanding of activists as “infected” spreaders of party membership due to the very short duration as an activist. However the revised format, where activists were compartmentalised into recruiters and non-recruiters was able to give convincing data fits for four different sets of data, with realistic durations for activists. Together with compartmentalising the non-members into those open to persuasion and those closed, each application of the model was able to give a narrative consistent with historical evidence for the party in question. Thus the epidemic analogy as an explanation of political party growth is strongly supported.
In all four application areas the reproduction potentials of the political parties were above the extinction threshold, as might be expected of parties who periodically achieve power. Despite party decline, neither of the three parties studied showed any indication that decline would lead to extinction, the DFE. Instead any decline comes from high reproduction rates with high leaving rates and the periodicity this induces. Such periodicity around the EE would not be expected to be perfect due to high variation in leaving rates from exogenous political factors such as elections and periods of political power. The decline of the Labour Party membership through the early 2000s could only be explained by such an increase in leaving rate, Section \[lab93.sec\]. For constant rates the model explains a return to growth after a period of decline in terms of the recycling of members. Given that in two of the applications studied the regrowth led to a return to political power for the party, it is suggested that electoral success may be connected with the natural recycling of members and its effect on activism.
In order for the concept of activists to have meaning in the model when only a fraction of them were recruiting, and thus contributing to growth, the model allowed the fraction of all activists, $f(A_t)$, to influence the quality of recruitment. Thus the more active the party, the more the recruits became active participants themselves, a form of positive feedback. Improving active participation in the party is a key role of activists, more common than the level of recruitment itself [@Weldon:polity; @Norris:Phoenix; @pedersen2004sleeping]. In each of the four applications $f(A_t)$ had a significant variation. For Labour in the 1940s the fraction rose from a third to over 40% with the fraction of the party active increasing substantially to 1952, Fig. \[politicalfig1.fig\](c). The value of $f(A_t)$ returned to a third by 1955, with the party becoming increasingly inactive. In the case of the SNP an increasing value of this fraction led to almost half the party active by 2012, Fig. \[politicalfig3.fig\][c]{}. Thus this modelling construct is believed to have added to the explanatory power of the model.
It is noted that in all four application areas the duration recruiting activist was short, sometimes only a matter of days, and that at any given time the number of recruiting activists was low, less than 0.1% of the party. Such figures are typical of word-of-mouth models applied to organisations with rapid growth, for example churches [@Hayward:Church], in contrast to more individualised processes [e.g. @Burbeck:Riot; @Manthey:drink; @gonzalez2003too] where the duration influential is longer. Political parties, like churches, have influential recruiters who only engage in recruitment activities for a small portion of a month. @Seyd:British state that most activists devote no more than 10 hours a month to party activities, which would mainly be organised regular party meetings. It would be expected that recruitment activities would likewise be conducted in small but repeated blocks, rather than one continuous period. Thus the short timescales of recruitment activity may reflect the actual time spent recruiting with periods of inactivity removed as aggregate models smooth the behaviour of different individuals over time.
Although the model worked well over periods where there was a sustained recruitment campaign, or lack of one, it was not able to handle a sudden change in membership due to external political events. The massive increase in SNP membership in the days following the Scottish independence referendum could not be predicted by the model. It is likely that a significant realignment towards the SNP took place during the 18-month long referendum campaign, but did not manifest itself in party membership until the referendum result was known. It is possible that the “No” vote mobilised supporters of independence to join the SNP to continue the pressure for more devolved power for Scotland. Such a process could be modelled as a word-of-mouth process, where the closed non-member population in the Limited Activist model was allowed to change through the influence of supporters for independence. Such an extended model would be more complex than the Limited Activist one presented, a complexity unnecessary for most situations of party growth and decline.
The model could also be extended by allowing the activists to have a role in reactivating the long-term inactive members, rather than just the new recruits. Unlike recruitment campaigns it is not clear what the arena for such influence would be. Former active members could be contacted through friendship networks, however inactive members who had never engaged with the party may not be so easily contactable. Such an extension would require the disaggregation of the inactive members, raising further issues with estimating the ratio of the different types of new subpopulations and the extra parameters involved.
The limitation of a purely population model of growth should be noted. The model limits the growth hypothesis to word of mouth. It is known that activists also have a role in building the party’s political legitimacy which in turn enhances growth [@Whiteley:High]. Such legitimacy also influences the engagement of party members, which in the model was approximated by the fraction of the party active $f(A_t)$. It would be interesting to explicitly include such legitimacy in the model as a variable, but this would be a non-population variable which would require care in its construction and use. Such an idea is reserved for a future publication.
It is proposed that the Limited Activist model put forward in this paper has sufficient richness to relate results to historical events, yet is simple enough to avoid over parametrisation. The model is recommended as useful first attempt in modelling political party growth that researchers could use to explore and test on other data.
References {#references .unnumbered}
==========
Party Membership Data
=====================
[\*[12]{}[c]{}]{} Year & 1944 & 1945 & 1946 & 1947 & 1948 & 1949 & 1950 & 1951 & 1952 & 1953\
Membership & 266 & 487 & 645 & 608 & 629 & 730 & 908 & 876 & 1,015 & 1,005\
[\*[13]{}[c]{}]{} Year & 1993 &1994 & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & 2004\
Membership & 266 & 305 & 365 & 400 & 405 & 388 & 361 & 311 & 272 & 248 & 215 & 198\
[\*[13]{}[c]{}]{} Year & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 & 2011 & 2012 & 2013\
Membership & 9.5 & 10.9 & 11.0 & 12.6 & 13.9 & 15.1 & 15.6 & 16.2 & 20.1 & 24.7 & 25.2\
[\*[7]{}[c]{}]{} Year & 2005 & 2006 &2008 & 2010 & 2012 & 2013\
Membership & 258 & 290 & 250 & 177 & 150 & 134\
Proof $A_{+}<0$ given $\hat{p}>0$ {#proof.appendix}
=================================
If $A_+ = \hat{p}-\hat{q}\bar{I_{+}}<0$ then using (\[model.a1\]): $$2a_1-\hat{q}\left(b_1/\hat{p} + \sqrt{ (b_1/\hat{p})^2 - 4a_1c_1/\hat{p}^2} \right)< 0 \label{model.ee5}$$ as $a_1>0$ and $\hat{p}>0$
(\[model.ee5\]) can be re-arranged as: $$2a_1-\hat{q}b_1/\hat{p} < \hat{q}\sqrt{ (b_1/\hat{p})^2 - 4a_1c_1/\hat{p}^2}\label{model.ee7}$$
There are two options, either $2a_1 < \hat{q}b_1/\hat{p}$ in which case the left hand side of (\[model.ee7\]) is negative and the result immediately follows, or $2a_1 > \hat{q}b_1/\hat{p}$. In the latter case (\[model.ee7\]) can be squared as both sides are positive: $ \left(2a_1 -\hat{q}b_1/\hat{p}\right)^2 <\hat{q}^2 \left( (b_1/\hat{p})^2 - 4a_1c_1/\hat{p}^2 \right)$. This simplifies to $0 <g\alpha\tau_i+g $, thus proved.
|
---
abstract: 'The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, that is the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski K [*et al*]{} 1996 [*J. Phys. A*]{} [**29**]{} 4149, we deal with a deformation of the classical phase space related with quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctness of the introduced coherent states is illustrated by an example of the rotator.'
address: |
Department of Theoretical Physics, University of Łódź, ul. Pomorska 149/153,\
90-236 Łódź, Poland
author:
- K Kowalski and J Rembieliński
title: Coherent states for a particle on a sphere
---
8.8in =.5cm =.5cm
Introduction
============
It has become a cliché to say that coherent states abound in quantum physics [@1]. Moreover, it turns out that they can also be applied in the theory of quantum deformations [@2] and even in the theory of classical dynamical systems [@3].
In spite of the fact that the problem of the quantization of a particle motion on a sphere is at least seventy years old, there still remains an open question concerning the coherent states for a particle on a sphere. Indeed, the celebrated spin coherent states introduced by Radcliffe [@4] and Perelomov [@5] are labelled by points of a sphere, i.e., the elements of the configuration space. On the other hand, it seems that as with the standard coherent states, the coherent states for a particle on a sphere should be marked with points of the phase space rather than the configuration space.
The aim of this work is to introduce the coherent states for a quantum particle on the sphere $S^2$, labelled by points of the phase space, that is the cotangent bundle $T^*S^2$. The construction follows the general scheme introduced in [@6] for the case of the motion in a circle, based on the polar decomposition of the operator defining via the eigenvalue equation the coherent states. From the technical point of view our treatment utilizes both the Barut-Girardello [@7] and Perelomov approach [@5]. Namely, as with the Barut-Girardello approach the coherent states are defined as the eigenvectors of some non-Hermitian operators. On the other hand, in analogy to the Perelomov formalism those states are generated from some “vacuum vector”, nevertheless in opposition to the Perelomov group-theoretic construction, the coherent states are obtained by means of the non-unitary action.
In section 2 we recall the construction of the coherent states for a particle on a circle. Sections 3–6 are devoted to the definition of the coherent states for a particle on a sphere and discussion of their most important properties. For an easy illustration of the introduced approach we study in section 7 the case with the free motion on a sphere.
Coherent states for a particle on a circle
==========================================
In this section we recall the basic properties of the coherent states for a particle on a circle introduced in [@6]. Consider the case of the free motion in a circle. For the sake of simplicity we assume that the particle has unit mass and it moves in a unit circle. The classical Lagrangian is $$%<2.1>
L = \hbox{$\scriptstyle1\over2$}\dot \varphi^2,$$ so the angular momentum canonically conjugate to the angle $\varphi$ is given by $$%<2.2>
J = \frac{\partial L}{\partial \dot \varphi}=\dot \varphi,$$ and the Hamiltonian can be written as $$%<2.3>
H = \hbox{$\scriptstyle1\over2$}J^2.$$ Evidently, we have the Poissson bracket of the form $$%<2.4>
\{\varphi,J\} = 1,$$ implying accordingly to the rules of the canonical quantization the commutator $$%<2.5>
[\hat\varphi,\hat J] = i,$$ where we set $\hbar=1$. The operator $\hat\varphi$ does not take into account the topology of the circle and (2.5) needs very subtle analysis. The better candidate to represent the position of the quantum particle on the unit circle is the unitary operator $U$ $$%<2.6>
U = e^{i\hat\varphi}.$$ Indeed, the substitution $\hat\varphi\to\hat\varphi+2n\pi$ does not change $U$, i.e. $U$ preserves the topology of the circle. The operator $U$ leads to the algebra $$%<2.7>
[\hat J,U] = U,$$ where $U$ is unitary. Consider the eigenvalue equation $$%<2.8>
\hat J|j\rangle = j|j\rangle.$$ Using (2.7) and (2.8) we find that the operators $U$ and $U^\dagger $ are the ladder operators, namely $$\begin{aligned}
U|j\rangle &=& |j+1\rangle,\\
U^\dagger |j\rangle &=& |j-1\rangle.\end{aligned}$$ Demanding the time-reversal invariance of representations of the algebra (2.7) we conclude [@6] that the eigenvalues $j$ of the operator $\hat J$ can be only integer (boson case) or half-integer (fermion case).
We define the coherent states $|\xi\rangle$ for a particle on a circle by means of the eigenvalue equation $$%<2.10>
Z|\xi\rangle = \xi|\xi\rangle,$$ where $\xi$ is complex. In analogy to the eigenvalue equation satisfied by the standard coherent states $|z\rangle$ [@8; @9] with complex $z$, of the form $$%<2.11>
e^{i\hat a}|z\rangle = e^{iz}|z\rangle,$$ where $\hat a\sim \hat q+i\hat p$ is the standard Bose annihilation operator and $\hat q$ and $\hat p$ are the position and momentum operators, respectively, we set $$%<2.12>
Z := e^{i(\hat \varphi + i\hat J)}.$$ Hence, making use of the Baker-Hausdorff formula we get $$%<2.13>
Z = e^{-\hat J + \hbox{$\frac{1}{2}$}}U.$$ We remark that the complex number $\xi$ should parametrize the cylinder which is the classical phase space for the particle moving in a circle. The convenient parametrization of $\xi$ consistent with the form of the operator $Z$ such that $$%<2.14>
\xi = e^{-l + i\varphi}.$$ arises from the deformation of the circular cylinder by means of the transformation $$%<2.15>
x=e^{-l}\cos\varphi,\qquad y=e^{-l}\sin\varphi,\qquad z=l.$$ The coherent states $|\xi\rangle$ can be represented as $$%<2.16>
|\xi\rangle = e^{-(\ln\xi) \hat J}|1\rangle,$$ where $$%<2.17>
|1\rangle =
\sum_{j=-\infty}^{\infty}e^{-\frac{j^2}{2}}|j\rangle.$$ The coherent states satisfy $$%<2.18>
\frac{\langle \xi|\hat J|\xi\rangle}{\langle
\xi|\xi\rangle}\approx l,$$ where the maximal error arising in the case $l\to0$ is of order $0.1$ per cent and we have the exact equality in the case with $l$ integer or half-integer. Therefore, $l$ can be identified with the classical angular momentum. Furthermore, we have $$%<2.19>
\frac{\langle \xi|U|\xi\rangle}{\langle \xi|\xi\rangle}
\approx
e^{-\frac{1}{4}}e^{i\varphi}.$$ It thus appears that the average value of $U$ in the normalized coherent state does not belong to the unit circle. On introducing the relative average of $U$ of the form $$%<2.20>
\frac{\langle U\rangle_{\xi}}{\langle U\rangle_{\eta}} :=
\frac{\langle \xi|U|\xi\rangle}{\langle \eta|U|\eta\rangle},$$ where $|\xi\rangle$ and $|\eta\rangle$ are the normalized coherent states, we find $$%<2.21>
\frac{\langle U\rangle_{\xi}}{\langle U\rangle_1}
\approx e^{i\varphi}.$$ From (2.21) it follows that that the relative expectation value $\langle U\rangle_{\xi}/\langle U\rangle_1$ is the most natural candidate to describe the average position of a particle on a circle and $\varphi$ can be regarded as the classical angle.
We remark that the coherent states on the circle have been recently discussed by Gonzáles [*et al*]{} [@10]. In spite of the fact that they formally generalize the coherent states described above, the ambiguity of the definition of those states manifesting in their dependence on some extra parameter, can be avoided only by demanding the time-reversal invariance mentioned earlier, which leads precisely to the coherent states introduced in [@6]. Since the time-reversal symmetry seems to be fundamental one for the motion of the classical particle in a circle and makes the quantization unique, therefore the generalization of the coherent states discussed in [@10] which does not preserve that symmetry is of interest rather from the mathematical point of view.
Having in mind the properties of the standard coherent states one may ask about the minimalization of the Heisenberg uncertainty relations by the introduced coherent states for a particle on a circle. In our opinion, in the case with the compact manifolds the minimalization of the Heisenberg uncertainty relations is not an adequate tool for the definition of the coherent states. A counterexample can be easily deduced from (2.7), (2.8) and (2.9). Indeed, taking into account (2.8) and (2.9) we find that for the eigenvectors $|j\rangle$’s of the angular momentum $\hat J$ the equality sign is attended in the Heisenberg uncertainty relations implied by (2.7) such that $$%<2.22>
(\Delta \hat J)^2\ge\frac{1}{4}\frac{|\langle U \rangle|^2}{1-|\langle U
\rangle|^2}.$$ More precisely, for these states (2.22) takes the form $0=0$. On the other hand, the vectors $|j\rangle$’s are clearly rather poor candidate for the coherent states. In our opinion the fact that the coherent states are “the most classical” ones is better described by the following easily proven formulae: $$\begin{aligned}
(\Delta \hat J)^2&\approx& {\rm const},\\
\frac{\langle U^2\rangle}{\langle U\rangle^2}&\approx&{\rm const},\end{aligned}$$ where the approximations are very good ones. In fact, these relations mean that the quantum variables $\hat J$ and $U$ are at practically constant “distance” from their classical counterparts $\langle \hat J\rangle$ and $\langle U\rangle$, respectively, and therefore the quantum observables and the corresponding expectation values connected to the classical phase space are mutually related. We point out that in the case with the standard coherent states for a particle on a real line we have the exact formulae $$\begin{aligned}
%<2.24>
(\Delta \hat p)^2&=&{\rm const},\\
(\Delta \hat q)^2&=&{\rm const}.\end{aligned}$$ It seems to us that the approximative nature of the relations (2.23) and (2.24) is related to the compactness of the circle.
Unitary representations of the $e(3)$ algebra and quantum mechanics on a sphere
===============================================================================
Our experience with the case of the circle discussed in the previous section indicates that in order to introduce the coherent states we should first identify the algebra adequate for the study of the motion on a sphere. The fact that the algebra (2.7) referring to the case with the circle $S^1$ is equivalent to the $e(2)$ algebra, where $E(2)$ is the group of the plane consisting of translations and rotations, $$%<3.1>
[\hat J,X_\alpha]={\rm i}\varepsilon_{\alpha\beta}X_\beta,
\qquad [X_\alpha,X_\beta]=0,\qquad \alpha,\,\beta=1,\,2,$$ realized in a unitary irreducible representation by Hermitian operators $$%<3.2>
X_1=r(U+U^\dagger)/2,\qquad X_2=r(U-U^\dagger)/2{\rm i},$$ where the Casimir is $$%<3.3>
X_1^2+X_2^2=r^2,$$ and $\varepsilon_{\alpha\beta}$ is the anti-symmetric tensor, indicates that the most natural algebra for the case with the sphere $S^2$ is the $e(3)$ algebra such that $$%<3.4>
[J_i,J_j]={\rm i}\varepsilon_{ijk}J_k,\qquad [J_i,X_j]={\rm i}
\varepsilon_{ijk}X_k,\qquad [X_i,X_j]=0,\qquad i,\,j,\,k=1,\,2,\,3.$$ Indeed, the algebra (3.4) has two Casimir operators given in a unitary irreducible representation by $$%<3.5>
{\bi X}^2=r^2,\qquad {\bi J}\bdot{\bi X}=\lambda,$$ where dot designates the scalar product. Therefore, as with the generators $X_\alpha $, $\alpha=1,\,2$, describing the position of a particle on the circle, the generators $X_i$, $i=1,\,2,\,3$, can be regarded as quantum counterparts of the Cartesian coordinates of the points of the sphere $S^2$ with radius $r$. We point out that unitary irreducible representations of (3.4) can be labelled by $r$ and the new scale invariant parameter $\zeta =\frac{\lambda }{r}$. It is clear that $\zeta $ is simply the projection of the angular momentum ${\bi J}$ on the direction of the radius vector of a particle. Since we did not find any denomination for such an entity in the literature, therefore we have decided to call $\zeta $ the [*twist*]{} of a particle.
Let us now recall the basic properties of the unitary representations of the $e(3)$ algebra. The $e(3)$ algebra expressed with the help of operators $J_3$, $J_\pm=J_1\pm {\rm i}J_2$, $X_3$ and $X_\pm=X_1\pm
{\rm i}X_2$, takes the form $$\begin{aligned}
%<3.6>
[J_+,J_-] &=& 2J_3,\qquad [J_3,J_\pm]=\pm J_\pm,\\
{}[J_\pm ,X_\mp] &=& \pm 2X_3,\qquad [J_\pm,X_\pm]=0,\qquad [J_\pm ,X_3]=\mp X_\pm,\\
{}[J_3,X_\pm] &=& \pm X_\pm,\qquad [J_3,X_3]=0,\\
{}[X_+,X_-] &=& [X_\pm,X_3]=0.\end{aligned}$$ Consider the irreducible representation of the above algebra in the angular momentum basis spanned by the common eigenvectors $|j,m;r,\zeta\rangle$ of the operators ${\bi J}^2=J_+J_-+J_3^2-J_3$, $J_3$, ${\bi X}^2$ and ${\bi J}\bdot{\bi X}/r$ $$\begin{aligned}
%<3.7>
&&{\bi J}^2 |j,m;r,\zeta\rangle = j(j+1) |j,m;r,\zeta\rangle,\qquad J_3
|j,m;r,\zeta\rangle=m|j,m;r,\zeta\rangle,\\
&&{\bi X}^2 |j,m;r,\zeta\rangle=r^2 |j,m;r,\zeta\rangle,\qquad
({\bi J}\bdot{\bi X}/r) |j,m;r,\zeta\rangle=\zeta|j,m;r,\zeta\rangle,\end{aligned}$$ where $-j\le m\le j$. Recall that $$%<3.8>
J_\pm |j,m;r,\zeta\rangle=\sqrt{(j\mp m)(j\pm m+1)}\,|j,m\pm 1;r,\zeta\rangle.$$ The operators $X_\pm$ and $X_3$ act on the vectors $|j,m;r,\zeta\rangle$ in the following way: $$\begin{aligned}
%<3.9>
X_+ |j,m;r,\zeta\rangle
&=&-\frac{r\sqrt{(j+1)^2-\zeta^2}\sqrt{(j+m+1)(j+m+2)}}
{(j+1)\sqrt{(2j+1)(2j+3)}}|j+1,m+1;r,\zeta\rangle\nonumber\\
&&{}+\frac{\zeta r\sqrt{(j-m)(j+m+1)}}{j(j+1)}|j,m+1;r,\zeta\rangle\nonumber\\
&&{}+\frac{r\sqrt{j^2-\zeta^2}\sqrt{(j-m-1)(j-m)}}{j\sqrt{(2j-1)(2j+1)}}
|j-1,m+1;r,\zeta\rangle,\\
X_- |j,m;r,\zeta\rangle
&=&\frac{r\sqrt{(j+1)^2-\zeta^2}\sqrt{(j-m+1)(j-m+2)}}
{(j+1)\sqrt{(2j+1)(2j+3)}}|j+1,m-1;r,\zeta\rangle\nonumber\\
&&{}+\frac{\zeta r\sqrt{(j-m+1)(j+m)}}{j(j+1)}|j,m-1;r,\zeta\rangle\nonumber\\
&&{}-\frac{r\sqrt{j^2-\zeta^2}\sqrt{(j+m-1)(j+m)}}{j\sqrt{(2j-1)(2j+1)}}
|j-1,m-1;r,\zeta\rangle,\\
X_3 |j,m;r,\zeta\rangle
&=&\frac{r\sqrt{(j+1)^2-\zeta^2}\sqrt{(j-m+1)(j+m+1)}}
{(j+1)\sqrt{(2j+1)(2j+3)}}|j+1,m;r,\zeta\rangle\nonumber\\
&&\fl\fl{}+\frac{\zeta rm}{j(j+1)}|j,m;r,\zeta\rangle+
\frac{r\sqrt{j^2-\zeta^2}\sqrt{(j-m)(j+m)}}{j\sqrt{(2j-1)(2j+1)}}|j-1,m;r,\zeta
\rangle.\end{aligned}$$ An immediate consequence of (3.9) is the existence of the minimal $j=j_{\rm min}$ satisfying $$%<3.10>
j_{\rm min}=|\zeta| .$$ Thus, it turns out that in the representation defined by (3.9) the twist $\zeta $ can be only integer or half integer. We finally write down the orthogonality and completeness conditions satisfied by the vectors $|j,m;r,\zeta\rangle$ such that $$\begin{aligned}
%<3.11>
&&\langle j,m;r,\zeta|j',m';r,\zeta\rangle=\delta_{jj'}\delta_{mm'},\\
&&\sum_{j=|\zeta|}^{\infty}\sum_{m=-j}^{j}
|j,m;r,\zeta\rangle\langle j,m;r,\zeta|=I,\end{aligned}$$ where $I$ is the identity operator.
Definition of coherent states for a particle on a sphere
========================================================
Now, an experience with the circle indicates that one should identify by means of the $e(3)$ algebra an analogue of the unitary operator $U$ (2.6), representing the position of a particle on a sphere. To do this, let us recall that a counterpart of the “position” $e^{{\rm
i}\varphi}$ on the circle $S^1$ is a unit length imaginary quaternion which can be represented with the help of the Pauli matrices $\sigma_i$, $i=1,\,2,\,3$, as $$%<4.1>
\eta = {\rm i}{\bi n}\bdot{\bsigma},$$ where ${\bi n}^2=1$. Notice that $\eta$ is simply an element of the $SU(2)$ group and it is related to the $S^2\approx SU(2)/U(1)$ quotient space. Therefore the most natural choice for the “position operator” of a particle on a sphere is to set $$%<4.2>
V=\hbox{$\scriptstyle 1\over r$}\bsigma\bdot{\bi X},$$ where $X_i$, $i=1,\,2,\,3$ obey (3.4) and (3.9) and we have omitted for convenience the imaginary factor i. Furthermore, let us introduce a version of the Dirac matrix operator [@11] $$%<4.3>
K := -(\bsigma\bdot{\bi J}+1).$$ Observe that $$%<4.4>
V^\dagger=V,\qquad K^\dagger=K.$$ Making use of the operators $V$ and $K$ we can write the relations defining the $e(3)$ algebra in the space of the unitary irreducible representation introduced above as $$\begin{aligned}
%<4.5>
({\rm Tr}\bsigma K)^2 &=& 4K(K+1),\\
{}[K,V]_+ &=& {\rm Tr}KV,\\
V^2&=&I,\end{aligned}$$ where ${\rm Tr}A=A_{11}+A_{22}$, and the subscript “+” designates the anti-commutator. In particular, $$%<4.6>
{\rm Tr}KV=-2{\bi J}\bdot{\bi X}/r=-2\zeta .$$ It should also be noted that in view of (4.4) and (4.5[*c*]{}) $V$ satisfies the unitarity condition $V^\dagger V=I$.
We now introduce the vector operator ${\bi Z}$ generating, via the eigenvalue equation analogous to (2.10), the coherent states for a particle on a sphere $S^2$. The experience with the circle (see eq. (2.13)) suggests the following form of the “polar decomposition” for the matrix operator counterpart $Z$ of the operator ${\bi Z}$: $$%<4.7>
Z=e^{-K}V.$$ Indeed, it is easy to see that in the case of the circular motion in the equator defined semiclassically by $J_1=J_2=0$ and $X_3=0$, $Z$ reduces to the diagonal matrix operator with $Z$ given by (2.13) and its Hermitian conjugate on the diagonal. Furthermore, using (4.5[*b*]{}) we find $$%<4.8>
Z-Z^{-1} = 2\zeta K^{-1}\sinh K.$$ Motivated by the complexity of the problem we now restrict to the simplest case of the twist $\zeta=0$ when (4.8) takes the form $$%<4.9>
Z^2=I.$$ In the following we confine ourselves to the case $\zeta=0$. The general case with arbitrary $\zeta\ne0$ will be discussed in a separate work. Besides (4.9) we have also remarkably simple relation (4.5[*b*]{}) referring to $\zeta=0$ such that $$%<4.10>
[K,V]_+ = 0.$$ Notice that the case $\zeta=0$ is the “most classical” one. Indeed, the projection of the angular momentum onto the direction of the radius vector should vanish for the classical particle on a sphere. It should also be noted that in view of (3.10) $j$’s and $m$’s labelling the basis vectors $|j,m;r,\zeta\rangle$ are integer in the case of the twist $\zeta =0$. We finally point out that the condition $\zeta=0$ ensures the invariance of the irreducible representation of the $e(3)$ algebra under time inversions and parity transformations which change the sign of the product ${\bi J}\bdot{\bi X}$. Clearly demanding the time-reversal or the parity invariance when $\zeta\ne0$ one should work with representations involving both $\zeta$ and $-\zeta$.
We now return to (4.7). Making use of (4.10) and the fact that the matrix operator $V$ in view of (4.2) is traceless one we obtain for $\zeta=0$ $$%<4.11>
{\rm Tr}Z=0.$$ Hence, $$%<4.12>
Z = \bsigma\bdot{\bi Z}.$$ Taking into account (4.9) we get from (4.12) $$%<4.13>
{\bi Z}^2=1,$$ and $$%<4.14>
[Z_i,Z_j]=0,\qquad i,j=1,\,2,\,3.$$ As with (4.2) describing in the matrix language the position of a quantum particle on a sphere, the matrix operator (4.12) can be only interpreted as a convenient arrangement of the operators $Z_i$ generating the coherent states, simplifying the algebraic analysis of the problem. Accordingly, we define the coherent states for a quantum mechanics on a sphere in terms of operators $Z_i$, as the solutions of the eigenvalue equation such that $$%<4.15>
{\bi Z} |{\bi z}\rangle = {\bi z} |{\bi z}\rangle,$$ where in view of (4.13) ${\bi z}^2=1$. What is ${\bi Z}$ ? Using (4.7), (4.2), (4.3) and setting $\zeta=0$, we find after some calculation $$\begin{aligned}
%<4.16>
{\bi Z} &=&\left(\frac{e^{\frac{1}{2}}}{\sqrt{1+4{\bi J}^2}}{\rm
sinh}\hbox{$\scriptstyle 1\over2 $}\sqrt{1+4{\bi
J}^2}+e^{\frac{1}{2}}{\rm cosh}\hbox{$\scriptstyle 1\over2 $}
\sqrt{1+4{\bi J}^2}\right){{\bi X}\over r}\nonumber\\
&&{}+{\rm i}\left(\frac{2e^{\frac{1}{2}}}{\sqrt{1+4{\bi J}^2}}{\rm sinh}
\hbox{$\scriptstyle 1\over2 $}\sqrt{1+4{\bi J}^2}\right){\bi
J}\times{{\bi X}\over r}.\end{aligned}$$ We remark that $Z_i$ have the structure resembling the standard annihilation operators. In fact, one can easily check that it can be written as a combination $$%<4.17>
{\bi Z}=a{\bi X}+{\rm i}b{\bi P},$$ of the “position operator” ${\bi X}$ and the “momentum” ${\bi P}$, where the coefficients $a$ and $b$ are functions of ${\bi J}^2$. We finally point out that derivation of the operator ${\bi Z}$ (4.16) without the knowledge of the matrix operator $Z$ seems to be very difficult task.
Construction of the coherent states
===================================
In this section we construct the coherent states specified by the eigenvalue equation (4.15). On projecting (4.15) on the basis vectors $|j,m;r\rangle\equiv|j,m;r,0\rangle$ and using (3.7[*a*]{}), (3.8) and (3.9) with $\zeta=0$ we arrive at the system of linear difference equations satisfied by the Fourier coefficients of the expansion of the coherent state $|{\bi z}\rangle$ in the basis $|j,m;r\rangle$. The direct solution of such system in the general case seems to be difficult task. Therefore, we adopt the following technique. We first solve the eigenvalue equation for ${\bi z}=
{\bi n}_3=(0,0,1)$, and then generate the coherent states from the vector ${\bi n}_3$ using the fact (see (4.16)) that ${\bi Z}$ is a vector operator. As demonstrated in the next section the case with ${\bi z}={\bi n}_3$ refers to ${\bi x}=(0,0,1)$ and ${\bi l}={\bf 0}$, where ${\bi x}$ is the position and ${\bi l}$ the angular momentum, respectively, i.e., the particle resting on the “North Pole” of the sphere. Let us write down the eigenvalue equation (4.15) for ${\bi z}={\bi n}_3$ $$%<5.1>
{\bi Z} |{\bi n}_3\rangle={\bi n}_3 |{\bi n}_3\rangle.$$ Using the following relations which can be easily derived with the help of (4.16), (3.7[*a*]{}), (3.8) and (3.9) with $\zeta =0$: $$\begin{aligned}
%<5.2>
Z_1 |j,m;r\rangle
&=&-\frac{1}{2}e^{-j-1}\sqrt{\frac{(j+m+1)(j+m+2)}{(2j+1)(2j+3)}}
|j+1,m+1;r\rangle\nonumber\\
&&{}+\frac{1}{2}e^j\sqrt{\frac{(j-m-1)(j-m)}{(2j-1)(2j+1)}}
|j-1,m+1;r\rangle\nonumber\\
&&+\frac{1}{2}e^{-j-1}\sqrt{\frac{(j-m+1)(j-m+2)}{(2j+1)(2j+3)}}
|j+1,m-1;r\rangle\nonumber\\
&&{}-\frac{1}{2}e^j\sqrt{\frac{(j+m-1)(j+m)}{(2j-1)(2j+1)}}
|j-1,m-1;r\rangle,\\
Z_2 |j,m;r\rangle
&=&\frac{{\rm i}}{2}e^{-j-1}\sqrt{\frac{(j+m+1)(j+m+2)}{(2j+1)(2j+3)}}
|j+1,m+1;r\rangle\nonumber\\
&&{}-\frac{{\rm i}}{2}e^j\sqrt{\frac{(j-m-1)(j-m)}{(2j-1)(2j+1)}}
|j-1,m+1;r\rangle\nonumber\\
&&+\frac{{\rm i}}{2}e^{-j-1}\sqrt{\frac{(j-m+1)(j-m+2)}{(2j+1)(2j+3)}}
|j+1,m-1;r\rangle\nonumber\\
&&{}-\frac{{\rm i}}{2}e^j\sqrt{\frac{(j+m-1)(j+m)}{(2j-1)(2j+1)}}
|j-1,m-1;r\rangle,\\
Z_3 |j,m;r\rangle
&=&e^{-j-1}\sqrt{\frac{(j-m+1)(j+m+1)}{(2j+1)(2j+3)}}
|j+1,m;r\rangle\nonumber\\
&&{}+e^j\sqrt{\frac{(j-m)(j+m)}{(2j-1)(2j+1)}}|j-1,m;r\rangle,\end{aligned}$$ it can be easily checked that the solution to (5.1) is given by $$%<5.3>
|{\bi
n}_3\rangle=\sum_{j=0}^{\infty}e^{-\frac{1}{2}j(j+1)}\sqrt{2j+1}|j,0;r\rangle.$$ Now, using the commutator $$%<5.4>
[{\bi w}\bdot{\bi J},{\bi Z}]=-{\rm i}{\bi w}\times{\bi Z},$$ where ${\bi w}\in{\Bbb C}^3$, we generate the complex rotation of ${\bi Z}$ $$%<5.5>
e^{{\bi w}\bdot{\bi J}}{\bi Z}e^{-{\bi w}\bdot{\bi J}}=
\cosh\sqrt{{\bi w}^2}\,{\bi Z}-{\rm i}\frac{\sinh\sqrt{{\bi w}^2}}
{\sqrt{{\bi w}^2}}
{\bi w}\times{\bi Z}+\frac{1-\cosh\sqrt{{\bi w}^2}}{{\bi w}^2}{\bi w}
({\bi w}\bdot{\bi Z}).$$ Taking into account (5.5) and (4.15) we find that the coherent states can be expressed by $$%<5.6>
|{\bi z}\rangle = e^{{\bi w}\bdot{\bi J}}|{\bi n}_3\rangle,$$ where ${\bi w}$ is given by $$%<5.7>
{\bi w}=\frac{{\rm arccosh}z_3}{\sqrt{1-z_3^2}}{\bi z}\times{\bi n}_3.$$ It thus appears that the coherent states can be written as $$%<5.8>
|{\bi z}\rangle = \exp\left[\frac{{\rm arccosh}z_3}{\sqrt{1-z_3^2}}
({\bi z}\times{\bi n}_3)\bdot{\bi J}\right]
|{\bi n}_3\rangle.$$ We remark that the discussed coherent states are generated analogously as in the case of the circle described by the equation (2.16). The formula (5.8) can be furthermore written in the form $$%<5.9>
|{\bi z}\rangle = e^{\mu J_-}e^{\gamma J_3}e^{\nu J_+} |{\bi
n}_3\rangle,$$ where $$%<5.10>
\mu =\frac{z_1+{\rm i}z_2}{1+z_3},\qquad \nu=\frac{-z_1+{\rm
i}z_2}{1+z_3},\qquad \gamma =\ln\frac{1+z_3}{2}.$$ Finally, eqs. (5.9), (5.3), (3.7[*a*]{}) and (3.8) taken together yield the following formula on the coherent states: $$%<5.11>
\fl |{\bi z}\rangle =\sum_{j=0}^{\infty}e^{-\frac{1}{2}j(j+1)}
\sqrt{2j+1}\sum_{m=0}^{j}\frac{\nu^m}{m!}\frac{(j+m)!}{(j-m)!}
e^{\gamma m}\sum_{k=0}^{j+m}\frac{\mu^k}{k!}
\sqrt{\frac{(j-m+k)!}{(j+m-k)!}} |j,m-k;r\rangle,$$ where $\mu ,\,\nu$ and $\gamma$ are expressed by (5.10) and ${\bi
z}^2=1$. Taking into account the identities $$%<5.12>
\sum_{s=0}\sp{n}\frac{(s+k)!}{(s+m)!s!(n-s)!}z^s=\frac{k!}{m!n!}\,\,
{}_2F_1(-n,k+1,m+1;-z),$$ $$%<5.13>
C_n^\alpha(x) =\frac{\Gamma(n+2\alpha)}{\Gamma(n+1)\Gamma(2\alpha)}
\,{}_2F_1(-n,n+2\alpha,\alpha+\hbox{$\scriptstyle 1\over2 $};
\hbox{$\scriptstyle 1\over2 $}(1-x)),$$ where ${}_2F_1(a,b,c;z)$ is the hypergeometric function, $C_n^\alpha(x)$ are the Gegenbauer polynomials and $\Gamma(x)$ is the gamma function, we obtain $$%<5.14>
\fl \langle j,m;r|{\bi z}\rangle = e^{-\frac{1}{2}j(j+1)}\sqrt{2j+1}\,
\frac{(2|m|)!}{|m|!}\sqrt{\frac{(j-|m|)!}{(j+|m|)!}}\left(
\frac{-\varepsilon(m)z_1+{\rm i}z_2}{2}\right)^{|m|} C_{j-|m|}^{|m|+\frac{1}{2}}
(z_3),$$ where $\varepsilon(m)$ is the sign of $m$. Let us recall in the context of the relations (5.14) that the polynomial dependence of the projection of coherent states onto the discrete basis vectors, on the complex numbers parametrizing those states is one of their most characteristic properties. Clearly, the polynomials (5.14) should span via the “resolution of the identity operator” the Fock-Bargmann representation. We recall that existence of such representation is one of the most important properties of coherent states. The problem of finding the Fock-Bargmann representation in the discussed case of the coherent states for a particle on a sphere is technically complicated and it will be discussed in a separate work. Finally, notice that the coherent states $|{\bi z}\rangle$ are evidently stable under rotations.
Coherent states and the classical phase space
=============================================
We now show that the introduced coherent states for a quantum particle on a sphere are labelled by points of the classical phase space, that is $T^*S^2$. Referring back to eq. (4.16) and taking into account the fact that the classical limit corresponds to large $j$’s, we arrive at the following parametrization of ${\bi z}$ by points of the phase space: $$%<6.1>
{\bi z}=\cosh|{\bi l}|\,\frac{{\bi x}}{r}+{\rm i}\frac{\sinh|{\bi l}|}
{|{\bi l}|}\,{\bi l}\times \frac{{\bi x}}{r},$$ where the vectors ${\bi l},\,{\bi x}\in{\Bbb R}^3$, fulfil ${\bi x}^2=r^2$ and ${\bi l}\bdot{\bi x}=0$, i.e., we assume that ${\bi l}$ is the classical angular momentum and ${\bi x}$ is the radius vector of a particle on a sphere. In accordance with the formulae (4.15) and (4.13) the vector ${\bi z}$ satisfies ${\bi z}^2=1$. Thus, the vector ${\bi z}$ is really parametrized by the points $({\bi x},{\bi l})$ of the classical phase space $T^*S^2$.
Consider now the expectation value of the angular momentum operator ${\bi J}$ in a coherent state. The explicit formulae which can be derived with the help of (3.7[*a*]{}), (3.8), (3.12) and (5.14) are too complicated to reproduce them herein. From computer simulations it follows that $$%<6.2>
\langle{\bi J}\rangle_{\bi z} =\frac{\langle {\bi z}|{\bi J}|{\bi z}\rangle}{\langle {\bi
z}|{\bi z}\rangle}\approx{\bi l}.$$ Nevertheless, in opposition to the case with the circular motion, the approximate relation (6.2) does not hold for practically arbitrary small $|{\bi l}|$. Namely, we have found that whenever $|{\bi l}|\sim1$, then (6.2) is not valid. Note that returning to dimension entities in the formulae like (3.6) we measure $|{\bi l}|$ in the units of $\hbar$, so in the physical units we deal rather with ${\bi L}=\hbar
{\bi l}$. For $|{\bi l}|\ge10$ the relative error $|(\langle
J_i\rangle_{\bi z}-l_i)/\langle J_i\rangle_{\bi z}|$, $i=1,\,2,\,3$, is small. More precisely, if $|{\bi l}|\sim10$, then $|(\langle
J_i\rangle_{\bi z}-l_i)/\langle J_i\rangle_{\bi z}|\sim$1 per cent. In other words, in the case of the motion on a sphere, the quantum fluctuations are not negligible for $|{\bi L}|\sim$1 $\hbar$ and the description based on the concept of the classical phase space is not adequate one. However, it must be borne in mind that the condition $|{\bi L}|\ge$ 10 $\hbar$, when (6.2) holds is not the same as the classical limit $|{\bi l}|\to\infty$. We only point out that $10\,\hbar\approx 10^{-33}\,{\rm J}\cdot{\rm s}$. It thus appears that the parameter ${\bi l}$ in (6.2) can be identified with the classical angular momentum divided by $\hbar$.
We now study the role of the parameter ${\bi x}$ in (6.1). As with the momentum operator ${\bi J}$ the explicit relations obtained by means of (3.9) with $\zeta=0$, (3.12) and (5.14) are too complicated to write them down herein. The computer simulations indicate that $$%<6.3>
\langle{\bi X}\rangle_{\bi z}=\frac{\langle{\bi z}|{\bi X}|{\bi z}\rangle}
{\langle {\bi z}|{\bi z}\rangle}\approx e^{-\frac{1}{4}}{\bi x}.$$ It seems that the formal resemblance of the formula (6.3) and (2.19) referring to the case with the circular motion is not accidental one. The range of application of (6.3) is the same as for (6.2), i.e., $|{\bi
l}|\ge10$. Because of the term $e^{-\frac{1}{4}}$, it appears that the average value of ${\bi X}$ does not belong to the sphere with radius $r$. Proceeding analogously as in the case of the circle we introduce the relative average value of ${\bi X}$ of the form $$%<6.4>
\langle\!\langle X_i\rangle\!\rangle_{\bi z}=\frac{\langle X_i\rangle_{\bi z}}
{\langle X_i\rangle_{{\bi w}_i}},\qquad i=1,\,2,\,3,$$ where $|{\bi w}_i\rangle$ is a coherent state with $$%<6.5>
{\bi w}_k=\cosh|{\bi l}|{\bi n}_k+{\rm i}\frac{\sinh|{\bi l}|}{|{\bi l}|}
{\bi l}\times{\bi n}_k,\qquad k=1,\,2,\,3,$$ where ${\bi n}_k$ is the unit vector along the $k$ coordinate axis and ${\bi l}$ is the same as in (6.1). In view of (6.3) and (6.4) we have $$%<6.6>
\langle\!\langle {\bi X}\rangle\!\rangle_{\bi z}\approx{\bi x}.$$ Therefore, the relative expectation value $\langle\!\langle {\bi X}\rangle\!
\rangle_{\bi z}$ seems to be the most natural one to describe the average position of a particle on a sphere.
We have thus shown that the parameter ${\bi x}$ can be immediately related to the classical radius vector of a particle on a sphere. As with the case of the circular motion (see formulae (2.18) and (2.21)), we interpret the relations (6.2) and (6.6) as the best possible approximation of the classical phase space. In this sense the coherent states labelled by points of such deformed phase space are closest to the classical ones. The quantum fluctuations which are the reason of the approximate nature of (6.2) and (6.6) are in our opinion a characteristic feature of quantum mechanics on a sphere.
We finally remark that the discussion of the Heisenberg uncertainty relations analogous to that referring to the circle (see section 2) can be performed also in the case with the coherent states for a particle on a sphere. For example a counterpart of the formula (2.22) is $$%<6.7>
(\Delta {\bi J})^2\ge\frac{1}{2}\frac{\frac{1}{2}{\rm Tr}\langle
V\rangle^2}{1-\frac{1}{2}{\rm Tr}\langle V\rangle^2},$$ where according to eq. (4.2) we have $\langle V\rangle=\frac{1}{r}
\bsigma\bdot\langle {\bi X}\rangle$. Such discussion as well as the detailed analysis of the Heisenberg uncertainty relations for the quantum mechanics on a compact manifold will be the subject of a separate paper which is in preparation.
Simple application: the rotator
===============================
We now illustrate the actual treatment by the example of a free twist 0 particle on a sphere, i.e. the rotator. The corresponding Hamiltonian is given by $$%<7.1>
\hat H=\hbox{$\scriptstyle 1\over2 $}{\bi J}^2.$$ By (3.7[*a*]{}) the normalized solution of the Schrödinger equation $$%<7.2>
\hat H |E\rangle = E |E\rangle$$ can be expressed by $$%<7.3>
|E\rangle= |j,m;r\rangle,\qquad E=\hbox{$\scriptstyle 1\over2 $}j(j+1).$$ We now discuss the distribution of the energies in the coherent state. The computer simulations indicate that the function $$%<7.4>
p_{j,m}({\bi x},{\bi l})=\frac{|\langle j,m;r|{\bi z}\rangle|^2}{\langle
{\bi z}|{\bi z}\rangle},\qquad -j\le m\le j,$$ determined by (5.14) and (6.1), which gives the probability of finding the system in the state $|j,m;r\rangle$, when the system is in the normalized coherent state $|{\bi z}\rangle/\sqrt{\langle{\bi
z}|{\bi z}\rangle}$, has the following properties. For fixed integer $m=l_3$ the function $p_{j,m}$ has a maximum at $j_{\rm max}$ coinciding with the integer nearest to the positive root of the equation $$%<7.5>
j(j+1)={\bi l}^2,$$ (see Fig. 1). Thus, it turns out that the parameter $\frac{1}{2}{\bi l}^2$ can be regarded as the energy of the particle. Further, for fixed integer $j$ in $p_{j,m}({\bi
x},{\bi l})$ (see Fig. 2), such that (7.5) holds, the function $p_{j,m}$ has a maximum at $m_{\rm max}$ coinciding with the integer nearest to $l_3$. It thus appears that the parameter $l_3$ can be identified with the projection of the momentum on the $x_3$ axis.
Conclusion
==========
In this work we have introduced the coherent states for a quantum particle on a sphere. An advantage of the formalism used is that the coherent states are labelled by points of the classical phase space. The authors have not found alternative constructions of coherent states for a quantum mechanics on a sphere preserving this fundamental property of coherent states. As pointed out in Sec. 6, the quantum fluctuations arising in the case of the motion on a sphere are bigger than those taking place for the circular motion. This observation is consistent with the appearance of the additional degree of freedom for the motion on a sphere. We remark that as with the particle on a circle, we deal within the actual treatment with the deformation of the classical phase space expressed by the approximate relations (6.2) and (6.6). We also point out that besides (6.2) and (6.6) the quasi-classical character of the introduced coherent states is confirmed by the behaviour of the distribution of the energies investigated in section 7. It seems that the approach introduced in this paper is not restricted to the study of the quasi-classical aspects of the quantum motion on a sphere. For example, the results of this work would be of importance in the theory of quantum chaos. In fact, in this theory the kicked rotator is one of the most popular model systems. Because of the well known difficulties in the analysis of the Heisenberg uncertainty relations occuring in the case with observables having compact spectrum like the position operator ${\bi X}$ satisfying the $e(3)$ algebra (3.4) we have not studied them herein. The analysis of the Heisenberg uncertainty relations as well as the discussion of the case with a nonvanishing twist will be performed in future work.
References {#references .unnumbered}
==========
[VV]{} Klauder J R and Skagerstam B S 1985 [*Coherent States–Applications in Physics and Mathematical Physics*]{} (World Scientific: Singapore) Kowalski K and Rembieliński J 1993 [*J. Math. Phys.*]{} [**34**]{} 2153 Kowalski K 1994 [*Methods of Hilbert Spaces in the Theory of Nonlinear Dynamical Systems*]{} (World Scientific: Singapore) Radcliffe J M 1971 [*J. Phys. A*]{} [**4**]{} 313 Perelomov A M 1972 [*Commun. Math. Phys.*]{} [**26**]{} 222; Perelomov A M 1986 [*Generalized Coherent States and Their Applications*]{} (Springer: Berlin) Kowalski K, Rembieliński J and Papaloucas L C 1996 [*J. Phys. A*]{} [**29**]{} 4149 Barut A O and Girardello L 1971 [*Commun. Math. Phys.*]{} [**21**]{} 41 Glauber R J 1963 [*Phys. Rev*]{} [**130**]{} 2529; [**131**]{} 2766 Klauder J R 1963 [*J. Math. Phys.*]{} [**4**]{} 1055 Gonzáles J A and del Olmo M A 1998 [*J. Phys. A*]{} [**31**]{} 8841 Biedenharn L C and Louck J D 1981 [*Angular Momentum in Quantum Physics. Theory and Application.*]{} (Addison-Wesley: Massachusetts)
|
---
abstract: 'In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then $M$ is biholomorphic to $\C^2$. During the proof, we also discover an interesting gap phenomenon which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.'
---
\[section\] \[Thm\][Lemma]{} \[Thm\][Proposition]{} \[Thm\][Definition]{} \[Thm\][Example]{} \[Thm\][Corollary]{} \[Thm\][Remark]{} \[Thm\][Remarks]{}
Bing–Long Chen\*, Siu–Hung Tang\*\*, Xi–Ping Zhu\*
\* Department of Mathematics, Zhongshan University,\
Guangzhou 510275, P. R. China, and\
Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong\
\*\* Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong\
0.5mm
§1. Introduction {#introduction .unnumbered}
================
One of the most beautiful results in complex analysis of one variable is the classical uniformization theorem of Riemann surfaces which states that a simply connected Riemann surface is biholomorphic to either the Riemann sphere, the complex line or the open unit disc. Unfortunately, a direct analog of this beautiful result to higher dimensions does not exist. For example, there is a vast variety of biholomorphically distinct complex structures on ${\R}^{2n}$ for $n>1$, a fact which was already known to Poincaré (see [@BSW], [@Fe] for a modern treatment). Thus, in order to characterize the standard complex structures for higher dimensional complex manifolds, one must impose more restrictions on the manifolds.
From the point of view of differential geometry, one consequence of the uniformization theorem is that a positively curved compact or noncompact Riemann surface must be biholomorphic to the Riemann sphere or the complex line respectively. It is thus natural to ask whether there is similar characterization for higher dimensional complete Kähler manifold with positive “curvature”. That such a characterization exists in the case of compact Kähler manifold is the famous Frankel conjecture which says that a compact Kähler manifold of positive holomorphic bisectional curvature is biholomorphic to a complex projective space. This conjecture was solved by Andreotti–Frankel [@Fra] and Mabuchi [@Mab] in complex dimensions two and three respectively and the general case was then solved by Mori [@Mor], and Siu–Yau [@SiY] independently. In this paper, we are thus interested in complete noncompact Kähler manifolds with positive holomorphic bisectional curvature. The following conjecture provides the main impetus. (Green–Wu [@GW2], Siu[@Si], Yau [@Y2]) A complete noncompact\
Kähler manifold of positive holomorphic bisectional curvature is biholomorphic to a complex Euclidean space. In contrary to the compact case, very little is known about this conjecture. The first result in this direction is the following isometrically embedding theorem.
(Mok–Siu–Yau[@MSY], Mok[@Mo1]) Let $M$ be a complete noncompact Kähler manifold of nonnegative holomorphic bisectional curvature of complex dimension $n\geq 2$. Suppose there exist positive constants $C_1$, $C_2$ such that for a fixed base point $x_0$ and some $\varepsilon>0$, $$\begin{array}{lll}
\bigbreak
\mbox{(i)} \qquad & \mbox{Vol}\,(B(x_0,r))\geq C_1r^{2n}\
& \qquad \qquad 0\leq r<+\infty \ ,\qquad \qquad \\
\mbox{(ii)} \qquad & R(x)\leq \displaystyle \frac{C_2}
{1+d^{2+\varepsilon}(x_0,x)}\ & \qquad
\qquad \mbox{on}\quad M\ ,\qquad \qquad
\end{array}$$ where $\mbox{Vol}\,(B(x_0,r))$ denotes the volume of the geodesic ball $B(x_0,r)$ centered at $x_0$ with radius $r$, $R(x)$ denotes the scalar curvature and $d(x_0,x)$ denotes the geodesic distance between $x_0$ and $x$. Then $M$ is isometrically biholomorphic to $\C^n$ with the flat metric.
Their method is to consider the Poincaré–Lelong equation $\sqrt{-1}\partial \overline{\partial }u= \mbox{Ric}$. Under the condition (ii) that the curvature has faster than quadratic decay, they proved the existence of a bounded solution $u$ to the Poincaré–Lelong equation. By virtue of Yau’s Liouville theorem on complete manifolds with nonnegative Ricci curvature, this bounded plurisubharmonic function $u$ must be constant and hence the Ricci curvature must be identically zero. This implies that the Kähler metric is flat because of the nonnegativity of the holomorphic bisectional curvature. However, this argument breaks down if the faster than quadratic decay condition (ii) is weaken to a quadratic decay condition. In this case, although we can still solve the Poincaré–Lelong equation with logarithmic growth, the boundedness of the solution can no longer be guaranteed.
In [@Mo1], Mok also developed a general scheme for compactifying complete Kähler manifolds of positive holomorphic bisectional curvature. This allowed him to obtain the following improvement of the above theorem. (Mok[@Mo1]) Let $M$ be a complete noncompact Kähler manifold of complex dimension $n$ with positive holomorphic bisectional curvature. Suppose there exist positive constants $C_1$, $C_2$ such that for a fixed base point $x_0$, $$\begin{array}{lll}
\bigbreak
\mbox{(i)} \qquad & \mbox{Vol}\,(B(x_0,r))\geq C_1r^{2n}\
& \qquad \qquad 0\leq r<+\infty \ ,\qquad \qquad \\
\mbox{(ii)}^{\prime} \qquad & 0<R(x)\leq \displaystyle
\frac{C_2}{1+d^2(x_0,x)}\
& \qquad \qquad \mbox{on} \quad M\ ,\qquad \qquad
\end{array}$$ then $M$ is biholomorphic to an affine algebraic variety. Moreover, if in addition the complex dimension $n=2$ and
(iii)the Riemannian sectional curvature of $M$ is positive,
then $M$ is biholomorphic to $\C^2$.
To the best of our knowledge, the above result of Mok and its slight improvements by To [@T], and Chen-Zhu [@CZ2] are the best results in complex dimension two of the above stated conjecture. Here, we would also like to recall the remark pointed out in [@CZ2] that there is a gap in the proof of Shi [@Sh3] (see [@CZ2] for more explanation) which would otherwise constitute a better result than that of Mok [@Mo1].
In this paper, we consider only the case of complex dimension two. Our principal result is the following Let $M$ be a complete noncompact complex two–dimen- sional Kähler manifold of positive and bounded holomorphic bisectional curvature. Suppose there exists a positive constant $C_1$ such that for a fixed base point $x_0$, we have $$\begin{array}{ll}
\bigbreak
\mbox{(i)}\qquad & \mbox{Vol}(B(x_0,r))\geq C_1r^4\
\qquad \ 0\leq r<+\infty \ , \\
\mbox{(ii)}^{\prime \prime }\qquad & \displaystyle \lim
\limits_{r\rightarrow + \infty }\frac 1{\mbox{Vol}(B(x_0,r))}\int_{B(x_0,r)}R(x)dx=0\ ,
\end{array}$$ then $M$ is biholomorphic to $\C^2$. We remark that the condition $\mbox{(ii)}^{\prime \prime}$ means that the scalar curvature tends to zero at infinity in average sense. In view of the classical Bonnet–Myers theorem, this condition is almost necessary to make sure that the manifold is noncompact under our positive bisectional curvature condition. It is also clear that the pointwise decay condition $$\lim \limits_{d(x,x_0)\rightarrow + \infty }R(x)=0$$ is stronger than $\mbox{(ii)}^{\prime \prime }.$
The proof of the Main Theorem will be divided into three parts. In the first part, we will show that $M$ is a Stein manifold homeomorphic to $\R^4.$ For this, we evolve the Kähler metric on $M$ by the Ricci flow first studied by Hamilton. Note that the underlying complex structure of $M$ is unchanged under the Ricci flow, thus we can replace the Kähler metric in our main theorem by any one of the evolving metric. The advantage is that, in our case, the properties of the evolving metric are improving during the flow. Moreover, we know that the Euclidean volume growth condition (i) as well as the positive holomorphic bisectional curvature condition are perserved by the evolving metric. More importantly, by a blow up and blow down argument as in [@CZ1], we can prove that the curvature of the evolving metric decays linearly in time. This implies that the injectivity radius of the evolving metric is getting bigger and bigger and any geodesic ball with radius less than half of the injectivity radius is almost pseudoconvex. By a perturbation argument as in [@CZ2], we are then able to modify these geodesic balls to a sequence of exhausting pseudoconvex domains of $M$ such that any two of them form a Runge pair. From this, it follow readily that $M$ is a Stein manifold homeomorphic to $\R^4$.
In the second part of the proof, we consider the algebra $P(M)$ of holomorphic functions of polynomial growth on $M$ and we will prove that its quotient field has transcendental degree two over $\C$. For this, we first need to construct two algebraically independent holomorphic functions in the algebra $P(M)$. Using the $L^2$ estimates of Andreotti–Vesentini [@AV] and Hömander [@Ho], it suffices to construct a strictly plurisubharmonic function of logarithmic growth on $M$. Now, if the scalar curvature decays in space at least quadratically, it was known from [@MSY], [@Mo1] that such a strictly plurisubharmonic function of logarithmic growth can be obtained by solving the Poincaré–Lelong equation, as we mentioned before. However, our decay assumption $\mbox{(ii)}^{\prime \prime }$ is too weak to apply their result directly. To resolve this difficulty, we make use of the Ricci flow to verify a new gap phenomenon which was already predicted by Yau in [@Y3]. More explicitly, by using the time decay estimate of evolving metric in the previous part, we prove that the curvature of the initial metric must decay quadratically in space in certain average sense. Fortunately, this turns out to be enough to insure the existence of a strictly plurisubharmonic function of logarithmic growth. Next, by using the time decay estimate and the injectivity radius estimate of the evolving metric, we prove that the dimension of the space of holomorphic functions in $P(M)$ of degree at most $p$ is bounded by a constant times $p^2$. Combining this with the existence of two algebraically independent holomorphic functions in $P(M)$ as above, we can prove that the quotient field $R(M)$ of $P(M)$ has transcendental degree two over $\C$ by a classical argument of Poincaré–Siegel. In other words, $R(M)$ is a finite extension field of some ${\C}(f_1,f_2)$, where $f_1$, $f_2 \in P(M)$ are algebraically independent over $\C$. Then, from the primitive element theorem, we have $R(M)={\C}
(f_1,f_2,g/h)$ for some $g$, $h\in P(M)$. Hence the mapping $F:M\rightarrow {\C
}^4$ given by $F=(f_1,f_2,g,h)$ defines, in an appropriate sense, a birational equivalence between $M$ and some irreducible affine algebraic subvariety $Z$ of $\C^4$.
In the last part of proof, we will basically follow the approach of Mok in [@Mo1] and [@Mo3] to establish a biholomorphic map from $M$ onto a quasi–affine algebraic variety by desingularizing the map $F$. Our essential contribution in this part is to establish uniform estimates on the multiplicity and the number of irreducible components of the zero divisor of a holomorphic function in $P(M)$. Again, the time decay estimate of the Ricci flow plays a crucial role in the arguments. Based on these estimates, we can show that the mapping $F:M\rightarrow Z$ is almost surjective in the sense that it can miss only a finite number of subvarieties in $Z$, and can be desingularized by adjoining a finite number of holomorphic functions of polynomial growth. This completes the proof that $M$ is a quasi–affine algebraic variety. Finally, by combining with the fact that $M$ is homeomorphic to $\R^4$, we conclude that $M$ is indeed biholomorphic to $\C^2$ by a theorem of Ramanujam [@R] on algebraic surfaces.
This paper contains eight sections. From Sections 2 to 4, we study the Ricci flow and obtain several geometric estimates for the evolving metric. In Section 5, we show that the two dimensional Kähler manifold is homeomorphic to $\R
^4$ and is a Stein manifold. Based on the estimates on the Ricci flow, a space decay estimate on the curvature and the existence of a strictly plurisubharmonic function of logarithmic growth are obtained in Section 6. In Section 7, we establish uniform estimates on the multiplicity and the number of irreducible components of the zero divisor of a holomorphic function of polynomial growth. Finally, in Section 8 we construct a biholomorphic map from the Kähler manifold onto a quasi–affine algebraic variety and complete the proof of the Main Theorem.
We are grateful to Professor L. F. Tam for many helpful discussions and Professor S. T. Yau for his interest and encouragement.
§2. Preserving the volume growth {#preserving-the-volume-growth .unnumbered}
================================
Let $(M,g_{\alpha \overline{\beta }})$ be a complete, noncompact Kähler surface (i.e., a Kähler manifold of complex dimension two) satisfying all the assumptions in the Main Theorem. We evolve the metric $g_{\alpha \overline{\beta }}$ according to the following Ricci flow equation $$\label{2.1}\left\{
\begin{array}{ll}
\bigbreak \displaystyle \frac{\partial g_{\alpha \overline{\beta }}(x,t)}{\partial t}=-R_{\alpha \overline{\beta }}(x,t)\ & \qquad x\in M\ \quad
t>0\ , \\
g_{\alpha \overline{\beta }}(x,0)=g_{\alpha \overline{\beta }}(x)\ &
\qquad x\in M\ ,
\end{array}
\right.$$ where $R_{\alpha \overline{\beta }}(x,t)$ denotes the Ricci curvature tensor of the metric $g_{\alpha \overline{\beta }}(x,t)$.
Since the curvature of the initial metric is bounded, it is known from [@Sh1] that there exists some $T_{\max }>0$ such that (\[2.1\]) has a maximal solution on $M\times [0,T_{\max })$ with either $T_{\max
}=+\infty $ or the curvature becomes unbounded as $t\rightarrow T_{\max }$ when $T_{\max }<+\infty $. By using the maximum principle, one knows ( see Mok [@Mo2], Hamilton [@Ha1], or Shi [@Sh4] ) that the positivity of holomorphic bisectional curvature and the Kählerity of $g_{\alpha \overline{\beta }}$ are preserved under the evolution of (\[2.1\]). In particular, the Ricci curvature remains positive.
Our first result for the solution of the Ricci flow (\[2.1\]) is the following proposition. Suppose $(M,g_{\alpha \overline{\beta }})$ is assumed as above. Then the maximal volume growth condition (i) is preserved under the evolution of (\[2.1\]), i.e., $$\label{2.2}
\mbox{Vol}_t(B_t(x,r))\geq C_1r^4\ \qquad \mbox{for all} \quad r>0\ ,\quad
x\in M\ ,$$ with the same constant $C_1$ as in condition (i). Here, $B_t(x,r)$ is the geodesic ball of radius $r$ with center at $x$ with respect to the metric $g_{\alpha \overline{\beta }}(\cdot ,t)$, and the volume $\mbox{Vol}_t$ is also taken with respect to the metric $g_{\alpha \overline{\beta }}(\cdot ,t)$.
Define a function $F(x,t)$ on $M\times [0,T_{\max })$ as follows, $$F(x,t)=\log \frac{\det \left( g_{\alpha \overline{\beta }}(x,t)\right) }{\det \left( g_{\alpha \overline{\beta }}(x,0)\right) }\ .$$ By (\[2.1\]), we have $$\begin{aligned}
\label{2.3}
\frac{\partial F(x,t)}{\partial t} & = &
g^{\alpha \overline{\beta }}(x,t)\cdot \frac \partial
{\partial t}g_{\alpha \overline{\beta }}(x,t)\nonumber \\
& = &-R(x,t) \leq 0 \ ,\end{aligned}$$ which implies that $F(\cdot ,t)$ is nonincreasing in time. Since $R_{\alpha \overline{\beta }}(x,t)\geq 0$, we know from (\[2.1\]) that the metric is shrinking in time. In particular, $$\label{2.4}
g_{\alpha \overline{\beta }}(x,t)\leq g_{\alpha \overline{\beta }}(x,0)\
\qquad \mbox{on} \quad M\times [0,T_{\max })\ .$$ This implies that $$\begin{aligned}
\label{2.5}
e^{F(x,t)}R(x,t)&=&g^{\alpha \overline{\beta }}(x,t)
R_{\alpha \overline{\beta }}(x,t)\cdot
\frac{\det \left( g_{\gamma \overline{\delta }}(x,t)\right) }
{\det \left( g_{\gamma \overline{\delta }}(x,0)\right) } \nonumber \\
& \leq & g^{\alpha \overline{\beta }}(x,0)R_{\alpha \overline{\beta }}(x,t) \nonumber \\
& = & g^{\alpha \overline{\beta }}(x,0)\left( R_{\alpha \overline{\beta }}(x,t)-R_{\alpha \overline{\beta }}(x,0)\right) +R(x,0)\nonumber\\
& = & -\bigtriangleup _0F(x,t)+R(x,0)\ ,\end{aligned}$$ where $\bigtriangleup _0$ denotes the Laplace operator with respect to the initial metric $g_{\alpha \overline{\beta }}(x,0)$ and $R(x,t)$ denotes the scalar curvature of the metric $g_{\alpha \overline{\beta }}(x,t)$.
Combining (\[2.3\]) and (\[2.5\]) gives $$\label{2.6}e^{F(x,t)}\frac{\partial F(x,t)}{\partial t}\geq \bigtriangleup
_0F(x,t)-R(x,0)\ \qquad \mbox{on} \quad M\times [0,T_{\max })\ .$$
Next, we introduce a cutoff function which will be used several times in this paper. Now, as the Ricci curvature of the initial metric is positive, we know from Schoen and Yau (Theorem 1.4.2 in [@ScY]) or Shi [@Sh4] that there exists a positive constant $C_3$ depending only on the dimension such that for any fixed point $x_0\in M$ and any number $0<r<+\infty $, there exists a smooth function $\varphi (x)$ on $M $ satisfying $$\label{2.7}\left\{
\begin{array}{l}
\bigbreak\displaystyle e^{-C_3\left( 1+\frac{d_0(x,x_0)}r\right) }\leq
\varphi (x)\leq e^{-\left( 1+\frac{d_0(x,x_0)}r\right) }\ , \\
\bigbreak \displaystyle \left| \nabla \varphi \right| _0(x)\leq \frac{C_3}r\varphi (x)\ , \\ \displaystyle \left| \bigtriangleup _0\varphi \right|
(x)\leq \frac{C_3}{r^2}\varphi (x)\ ,
\end{array}
\right.$$ for all $x\in M$, where $d_0(x,x_0)$ is the distance between $x$ and $x_0$ with respect to the initial metric $g_{\alpha \overline{\beta }}(x,0)$ and $\left| \cdot \right| _0$ stands for the corresponding $C^0$ norm of the initial metric $g_{\alpha \overline{\beta }}(x,0)$.
Combining (\[2.6\]) and (\[2.7\]), we obtain $$\begin{aligned}
\frac \partial {\partial t}\int_M\varphi (x)e^{F(x,t)}dV_0&\geq&\int_M\left( \bigtriangleup _0F(x,t)-R(x,0)\right) \varphi (x)dV_0\nonumber\\
&\geq&\frac{C_3}{r^2}\int_MF(x,t)\varphi (x)dV_0-\int_MR(x,0)\varphi (x)dV_0\ ,\nonumber\end{aligned}$$ where $dV_0$ denotes the volume element of the initial metric $g_{\alpha
\overline{\beta }}(x,0)$.
Recall that $F(\cdot ,t)$ is nonincreasing in time and $F(\cdot ,0)\equiv 0$. We integrate the above inequality from $0$ to $t$ to get $$\label{2.8}\int_M\varphi (x)\left( 1-e^{F(x,t)}\right) dV_0\leq \frac{C_3t}{r^2}\int_M\left( -F(x,t)\right) \varphi (x)dV_0+t\int_MR(x,0)\varphi
(x)dV_0\ .$$ Since the metric is shrinking under the Ricci flow, we have $$B_t(x_0,r)\supset B_0(x_0,r)\ \qquad \mbox{for} \quad t\geq 0\ ,
\quad 0<r<+\infty \ ,$$ and $$\begin{aligned}
\label{2.9}
\mbox{Vol}_t(B_t(x_0,r))& \geq & \mbox{Vol}_t(B_0(x_0,r)) \nonumber \\
& = & \int_{B_0(x_0,r)}e^{F(x,t)}dV_0 \nonumber \\
& = & \mbox{Vol}_0(B_0(x_0,r))+\int_{B_0(x_0,r)}\left( e^{F(x,t)}-1\right) dV_0\ .\end{aligned}$$ Then by (\[2.7\]) and (\[2.8\]), the last term in (\[2.9\]) satisfies$$\begin{aligned}
\label{2.10}
\int_{B_0(x_0,r)}\left( e^{F(x,t)}-1\right) dV_0&\geq&e^{2C_3}\int_M\left( e^{F(x,t)}-1\right) \varphi (x)dV_0\nonumber\\
&\geq&\frac{C_3e^{2C_3}t}{r^2}\int_MF(x,t)\varphi (x)dV_0\nonumber\\
&&-e^{2C_3}t\int_MR(x,0)\varphi (x)dV_0\ .\end{aligned}$$ To estimate the two terms of the right hand side of (\[2.10\]), we consider any fixed $T_0<T_{\max }$. Since the curvature is uniformly bounded on $M\times [0,T_0]$, it is clear from the equation (\[2.3\]) that $F(x,t)$ is also uniformly bounded on $M\times [0,T_0]$.
Set $$A=\sup \left\{ \left. \left| F(x,t)\right| \right| x\in M\ ,\ t
\in[0,T_0]\right\}$$ and $$M(r)=\sup \limits_{a\geq r}\frac
1{\mbox{Vol}_0\left( B_0(x_0,a)\right) }\int_{B_0(x_0,a)}R(x,0)dV_0\ .$$ Then condition $\mbox{(ii)}^{\prime \prime }$ says that $M(r)\rightarrow 0$ as $r\rightarrow +\infty $. By using the standard volume comparison theorem and (\[2.7\]), we have $$\begin{aligned}
\label{2.11}
\int_MR(x,0)\varphi (x)dV_0&\leq&\int_MR(x,0)e^{-\left(
1+\frac{d_0(x,x_0)}r\right) }dV_0 \nonumber \\
& = & \int_{B_0(x_0,r)}R(x,0)e^{-\left(
1+\frac{d_0(x,x_0)}r\right) }dV_0 \nonumber \\
& & + \sum\limits_{k=0}^\infty \int_{B_0(x_0,2^{k+1}r)
\backslash B_0(x_0,2^kr)}R(x,0)e^{-\left(
1+\frac{d_0(x,x_0)}r\right) }dV_0 \nonumber \\
& \leq & \int_{B_0(x_0,r)}R(x,0)dV_0 + \sum\limits_{k=0}^\infty
e^{-2^k}\left( 2^{k+1}\right) ^4 \cdot \nonumber \\
& & \frac{\mbox{Vol}_0\left( B_0(x_0,r)\right) }{\mbox{Vol}_0
\left( B_0(x_0,2^{k+1}r)\right) }\int_{B_0(x_0,2^{k+1}r)}R(x,0)dV_0\nonumber\\
& \leq & C_4\cdot M(r)\cdot \mbox{Vol}_0\left( B_0(x_0,r)\right) \ ,\end{aligned}$$ and similarly $$\begin{aligned}
\label{2.12}
\int_M\varphi (x)dV_0&\leq&\int_{B_0(x_0,r)}e^{-\left( 1+\frac{d_0(x,x_0)}r\right) }dV_0\nonumber\\
&&+\sum\limits_{k=0}^\infty \int_{B_0(x_0,2^{k+1}r)\backslash B_0(x_0,2^kr)}e^{-\left( 1+\frac{d_0(x,x_0)}r\right) }dV_0\nonumber\\
&\leq&\mbox{Vol}_0\left( B_0(x_0,r)\right) +\sum\limits_{k=0}^\infty e^{-2^k}\left( 2^{k+1}\right) ^4\cdot \mbox{Vol}_0\left( B_0(x_0,r)\right)\nonumber\\
&\leq&C_4\mbox{Vol}_0\left( B_0(x_0,r)\right) \ ,\end{aligned}$$ where $C_4$ is some positive constant independent of $r$.
Substituting (\[2.10\]), (\[2.11\]) and (\[2.12\]) into (\[2.9\]) and dividing by $r^4$, we obtain$$\begin{aligned}
\frac{\mbox{Vol}_t(B_t(x_0,r))}{r^4}&\geq&\frac{\mbox{Vol}_0(B_0(x_0,r))}{r^4}-\frac{C_3e^{2C_3}AT_0}{r^2}\left( C_4\frac{\mbox{Vol}_0(B_0(x_0,r))}{r^4}\right)\nonumber\\
&& -e^{2C_3}T_0\left( C_4M(r)\cdot \frac{\mbox{Vol}_0(B_0(x_0,r))}{r^4}\right)\nonumber\\
&\geq&C_1-\frac{C_3e^{2C_3}AT_0\cdot C_4C_1}{r^2}-e^{2C_3}T_0C_4C_1\cdot M(r)\nonumber\end{aligned}$$ by condition (i). Then letting $r\rightarrow +\infty $, we deduce that $$\lim \limits_{r\rightarrow +\infty }\frac{\mbox{Vol}_t\left( B_t(x_0,r)\right) }{r^4}\geq C_1\ .$$ Hence, by using the standard volume comparison theorem we have $$\mbox{Vol}_t\left( B_t(x,r)\right) \geq C_1r^4\ \quad \mbox{for all}\ x\in M\ ,\ 0\leq
r<+\infty, \,\, t\in [0,T_0]\ .$$ Finally, since $T_0<T_{\max }$ is arbitrary, this completes the proof of the proposition. $\Box$
§3. Singularity Models {#singularity-models .unnumbered}
======================
In this and the next section we will continue our study of the Ricci flow (\[2.1\]). We will use rescaling arguments to analyse the behavior of the solution of (\[2.1\]) near the maximal time $T_{\max }.$
First of all, let us recall some basic terminologies. According to Hamilton ( see for example, definition 16.3 in [@Ha3]), a solution to the Ricci flow, where either the manifold is compact or at each time $t$ the evolving metric is complete and has bounded curvature, is called a singularity model if it is not flat and is of one of the following three types. Here, we have used $Rm$ to denote the Riemannian curvature tensor.
Type I:
: The solution exists for $-\infty <t<\Omega $ for some $
0<\Omega <+\infty $ and $$|Rm|\leq \frac \Omega {\Omega -t}$$ everywhere with equality somewhere at $t=0$;
Type II:
: The solution exists for $-\infty <t<+\infty $ and $$|Rm|\leq 1$$ everywhere with equality somewhere at $t=0$;
Type III:
: The solution exists for $-A<t<+\infty $ for some $0<A<+\infty$ and $$|Rm|\leq \frac A{A+t}$$ everywhere with equality somewhere at $t=0$.
The singularity models of Type I and II are called ancient solutions in the sense that the existence time interval of the solution contains $(-\infty ,0]$.
Next, we recall the local injectivity radius estimate of Cheeger, Gromov and Taylor [@CGT]. Let $N$ be a complete Riemannian manifold of dimension $m$ with $\lambda \leq \mbox{sectional curvature of}\, N \leq \Lambda $ and let $r$ be a positive constant satisfying $r\leq \frac \pi {4\sqrt{\Lambda }}$ if $\Lambda >0$, then the injectivity radius of $N$ at a point $x$ is bounded from below as follows, $$\mbox{inj}_N(x) \geq r\cdot \frac{\mbox{Vol}(B(x,r))}{\mbox{Vol}
(B(x,r))+V^m_{\lambda}(2r)}\ ,$$ where $V^m_{\lambda}(2r)$ denotes the volume of a ball with radius $2r$ in the $m$ dimensional model space $V^m_{\lambda}$ with constant sectional curvature $\lambda .$ In particular, it implies that for a complete Riemannian manifold $N$ of dimension $4$ with sectional curvature bounded between $-1$ and $1$, the injectivity radius at a point $x$ can be estimated as $$\label{3.1}
\mbox{inj}_N(x) \geq \frac 12\cdot \frac{\mbox{Vol}(B(x,\frac 12))}
{\mbox{Vol}(B(x,\frac 12))+V}$$ for some absolute positive constant $V$. Furthermore, if in addition $N$ satisfies the maximal volume growth condition $$\mbox{Vol} \left( B(x,r)\right) \geq C_1r^4\ ,\quad 0\leq r<+\infty \ ,$$ then (\[3.1\]) gives $$\label{3.2}
\mbox{inj}_N(x) \geq \beta >0$$ for some positive constant $\beta $ depending only on $C_1$ and $V.$
Now, return to our setting. Let $(M,g_{\alpha \overline{\beta }})$ be a complete, noncompact Kähler surface satisfying the same assumptions as in the Main Theorem and let $g_{\alpha \overline{\beta }
}(x,t)$ be the solution of the Ricci flow (\[2.1\]) on $M\times [0,T_{\max})$. Denote $$R_{\max }(t)=\sup \limits_{x\in M}R(x,t)\ .$$ We have shown in Proposition 2.1 that the solution $g_{\alpha \overline{ \beta }}(\cdot ,t)$ satisfies the same maximal volume growth condition (i) as the initial metric. Since condition (i) is invariant under rescaling of metrics, by a simple rescaling argument, we get the following injectivity radius estimate for the solution $g_{\alpha \overline{\beta }}(\cdot ,t),$ $$\label{3.3}
\mbox{inj}(M,g_{\alpha \overline{\beta }}(\cdot ,t))\geq \frac \beta {
\sqrt{R_{\max }(t)}} \qquad \mbox{for}\;t \in [0,T_{\max }).$$ Then, by applying a result of Hamilton (see Theorems 16.4 and 16.5 in [@Ha3]), we know that there exists a sequence of dilations of the solution which converges to one of the singularity models of Type I, II or III. We will analyse this limit in the next section.
We conclude this section with the following lemma which will be very useful in our analysis of the Type I and Type II limits.
Suppose $(\widetilde{M},\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t))$ is a complete ancient solution to the Ricci flow on a noncompact Kähler surface with nonnegative and bounded holomorphic bisectional curvature for all time. Then the curvature operator of the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ is nonnegative definite everywhere on $\widetilde{M} \times (-\infty ,0]$.
Choose a local orthonormal coframe $\{\omega _1,\omega _2,\omega _3,\omega _4\}$ on an open set $U\subset \widetilde{M}$ so that $\omega _1+\sqrt{-1}\omega _2$ and $\omega_3+\sqrt{-1}\omega _4$ are $(1,0)$ forms over $U$. Then the self–dual forms $$\varphi _1=\omega _1\land \omega _2+\omega _3\land \omega _4,
\,\,\, \varphi_2=\omega _2\land \omega _3+\omega _1\land \omega _4,
\,\,\, \varphi _3=\omega_3\land \omega _1+\omega _2\land \omega _4$$ and the anti–self–dual forms $$\psi_1=\omega _1\land \omega _2-\omega _3\land \omega _4,
\,\,\, \psi _2=\omega_2\land \omega _3-\omega _1\land \omega _4,
\,\,\, \psi _3=\omega _3\land \omega_1-\omega _2\land \omega _4$$ form a basis of the space of $2$ forms over $U$. In particular, $\varphi _1,\psi _1,\psi _2$ and $\psi _3$ give a basis for the space of $(1,1)$ forms over $U$.
On a Kähler surface, it is well known that its curvature operator has image in the holonomy algebra $u(2) \,(\subset so(4))$ spanned by $(1,1)$ forms. Thus, the curvature operator ${\bf M}$ in the basis $\{\varphi_1,\varphi _2,\varphi _3,\psi _1,\psi _2,\psi _3\}$ has the following form, $${\bf M}=\left(
\begin{array}{cc}
\begin{array}{ccc}
a & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}
&
\begin{array}{ccc}
b_1 & b_2 & b_3 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}
\\
\begin{array}{ccc}
b_1 & 0 & 0 \\
b_2 & 0 & 0 \\
b_3 & 0 & 0
\end{array}
& A
\end{array}
\right) $$ where $A$ is a $3\times 3$ symmetric matrix.
Let $V$ be a real tangent vector of the Kähler surface $\widetilde{M}$. Denote by $J$ the complex structure of the Kähler surface $\widetilde{M}$. It is clear that the complex $2$–plane $V\wedge JV$ is dual to $(1,1)$ form $u\varphi _1+v_1\psi _1+v_2\psi _2+v_3\psi _3$ satisfying the decomposability condition $u^2=v_1^2+v_2^2+v_3^2$. Then after normalizing $u$ to 1 by scaling, we see that the holomorphic bisectional curvature is nonnegative if and only if $$\label{3.4}
a+b\cdot v+b\cdot w+^{\ t}\negthinspace {}vAw\geq 0\ ,$$ for any unit vectors $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$ in ${\bf R^3}$, where $b$ is the vector $(b_1,b_2,b_3)$ in $\bf{M}$.
Denote by $a_1\leq a_2\leq a_3$ the eigenvalues of $A$. Recall that $\mbox{tr}\,A=a $ by the Bianchi identity, so if we choose $v$ to be the eigenvector of $A $ with eigenvalue $a_3$ and choose $w=-v$, (\[3.4\]) gives $$\label{3.5}a_1+a_2\geq 0\ .$$ In particular, we have $a_2\geq 0$.
To proceed further, we need to adapt the maximum principle for parabolic equations on compact manifold in Hamilton [@Ha1] to $\widetilde{M}$. Let $$\left( a_i\right) _{\min }(t)=\inf\limits_{x\in \widetilde{M}}a_i(x,t)\
,\qquad i=1,2,3$$ and $$K=\sup \limits_{(x,t)\in \widetilde{M}\times (-\infty ,0]}\left|
Rm(x,t)\right|.$$ By assumption, the ancient solution $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ has bounded holomorphic bisectional curvature, hence $K$ is finite. Thus, by the derivative estimate of Shi [@Sh1] (see also Theorem 7.1 in [@Ha3]), the higher order derivatives of the curvature are also uniformly bounded. In particular, we can use the maximum principle of Cheng–Yau (see Proposition 1.6 in [@CY]) and then, as observed in [@Ha3], this implies that the maximum principle of Hamilton in [@Ha1] actually works for the evolution equations of the curvature of $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ on the complete noncompact manifold $\widetilde{M}.$ Thus, from [@Ha1], we obtain $$\begin{aligned}
\frac{d\left( a_1\right) _{\min }}{dt}&\geq&\left( \left( a_1\right)_
{\min }\right) ^2+2\left( a_2\right) _{\min }\cdot
\left( a_3\right) _{\min } \nonumber \\
& \geq & 3\left( \left( a_1\right) _{\min }\right) ^2 \nonumber\end{aligned}$$ by (\[3.5\]). Then, for fixed $t_0\in (-\infty ,0)$ and $t>t_0,$ $$\begin{aligned}
\left( a_1\right) _{\min }(t)&\geq&\frac 1{\left( a_1\right)_
{\min }^{-1}\left( t_0\right) -3\left( t-t_0\right) } \nonumber \\
& \geq & \frac 1{-K^{-1}-3\left( t-t_0\right) }\ . \nonumber\end{aligned}$$ Letting $t_0\rightarrow -\infty $, we get $$\label{3.6}
a_1\geq 0\ ,\qquad \mbox{for all} \;\;\; (x,t) \in
\widetilde{M} \times (-\infty ,0]\,$$ i.e. $A \geq 0$.
Finally, to prove the nonnegativity of the curvature operator $\bf{M}$, we recall its corresponding ODE from [@Ha1], $$\frac{d{\bf M}}{dt}={\bf M}^2+\left(
\begin{array}{cc}
0 & 0 \\
0 & A^{\#}
\end{array}
\right) \ ,$$ where $A^{\#} \geq 0$ is the adjoint matrix of $A.$
Let $m_1$ be the smallest eigenvalue of the curvature operator ${\bf M}$. By using the maximum principle of Hamilton ( [@Ha1] or [@Ha3] ) again, we have $$\frac{d\left( m_1\right) _{\min }}{dt}\geq \left( m_1\right) _{\min }^2$$ where $\left( m_1\right) _{\min }(t)=\inf \limits_{x\in \widetilde{M}}m_1(x,t)$. Therefore, by the same reasoning in the derivation of (\[3.6\]), we have $$\label{3.7}
m_1\geq 0\ ,\qquad \mbox{for all} \;\; (x,t)\in \widetilde{M}\times
(-\infty ,0]\ .$$ So ${\bf{M}} \geq 0$ and the proof of the lemma is completed. $\Box $
§4. Time decay estimate on curvature {#time-decay-estimate-on-curvature .unnumbered}
====================================
Let $(M,g_{\alpha \overline{\beta }}(x))$ be a complete noncompact Kähler surface satisfying all the assumptions in the Main Theorem and $(M,g_{\alpha \overline{\beta }}(\cdot ,t))$, $t\in [0,T_{\max })$ be the maximal solution of the Ricci flow (\[2.1\]) with $g_{\alpha \overline{\beta }}(\cdot )$ as the initial metric. Clearly, the maximal solution is of either one of the following types.
[Type I:]{}
: $T_{\max }<+\infty $ and $\sup \left( T_{\max }-t\right) R_{\max }(t)<+\infty $;
[Type II(a):]{}
: $T_{\max }<+\infty $ and $\sup
\left( T_{\max}-t\right) R_{\max }(t)=+\infty $;
[Type II(b):]{}
: $T_{\max }=+\infty $ and $\sup tR_{\max }(t)=+\infty $;
[Type III:]{}
: $T_{\max }=+\infty $ and $\sup tR_{\max }(t)<+\infty $.
In Section 3, we have proved that the maximal solution satisfies the following injectivity radius estimate $$\mbox{inj}(M,g_{\alpha \overline{\beta }}(\cdot ,t))\geq
\frac \beta {\sqrt{R_{\max}(t)}}\ \qquad \mbox{on} \quad [0,T_{\max }),$$ for some $\beta >0$. By applying a result of Hamilton (Theorems 16.4 and 16.5 in [@Ha3]), we know that there exists a sequence of dilations of the solution converging to a singularity model of the corresponding type. Note that since the maximal solution is complete and noncompact, the limit must also be complete and noncompact. The following is the main result of this section which says that this limit must be of Type III or equivalently, the maximal solution must be of Type III. Let $(M,g_{\alpha \overline{\beta }}(x))$ be a complete noncompact Kähler surface as above. Then the Ricci flow (\[2.1\]) with $g_{\alpha \overline{\beta }}(x)$ as the initial metric has a solution $g_{\alpha \overline{\beta }}(x,t)$ for all $t\in [0,+\infty) $ and $x\in M$. Moreover, the scalar curvature $R(x,t)$ of the solution satisfies $$\label{4.1}
0\leq R(x,t)\leq \frac C{1+t}\ \qquad \mbox{on}
\quad M\times [0,+\infty)\ ,$$ for some positive constant $C$.
We prove by contradiction. Thus, suppose the maximal solution is of Type I or Type II and let $(\widetilde{M},\widetilde{g}_{\alpha \overline{\beta }}(x,t))$ be the limit of a sequence of dilations of the maximal solution which is then a singularity model of Type I or Type II respectively. After a study of its properties, we can blow down the singularity model and apply a dimension reduction argument to obtain the desired contradiction.
Now, recall that the maximal solution satisfies the maximal volume growth condition (i) by Proposition 2.1. Since condition (i) is also invariant under rescaling, we see that the singularity model $(\widetilde{M},\widetilde{g}_{\alpha \overline{\beta }}(x,t))$ also satisfies the maximal volume growth condition, i.e. $$\label{4.2}
\mbox{Vol}_t\left( \widetilde{B}_t(x,r)\right) \geq C_1r^4\
\qquad \mbox{for all} \quad 0\leq r< +\infty \quad \mbox{and}
\quad x\in \widetilde{M}\ ,$$ where $\mbox{Vol}_t\left( \widetilde{B}_t(x,r)\right) $ denotes the volume of the geodesic ball $\widetilde{B}_t(x,r)$ of radius $r$ with center at $x$ with respect to the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t).$
It is clear that the limit $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ has nonnegative holomorphic bisectional curvature. Thus, from Lemma 3.1, the curvature operator of the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ is nonnegative definite everywhere.
Denote by $\widetilde{R}(\cdot ,t)$ the scalar curvature of $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ and $\widetilde{d}_t(x,x_0)$ the geodesic distance between two points $x,x_0\in \widetilde{M}$ with respect to the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot,t)$. We claim that at time $t=0,$ we have $$\label{4.3}
\limsup \limits_{\widetilde{d}_0(x,x_0)\rightarrow +\infty }
\widetilde{R}(x,0)\widetilde{d}_0^2(x,x_0)=+\infty $$ for any fixed $x_0\in \widetilde{M}.$
Suppose not, that is the curvature of the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,0)$ has quadratic decay. Now, by applying a result of Shi (see Theorem 8.2 in [@Sh4]), we know that the solution $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ of the Ricci flow exists for all $t\in (-\infty ,+\infty )$ and satisfies $$\label{4.4}
\lim \limits_{t\rightarrow +\infty }\sup \left\{ \left.
\widetilde{R}(x,t)\right| \ x\in \widetilde{M}\right\} =0\ .$$ On the other hand, by the Harnack inequality of Cao [@Cao], we have $$\label{4.5}
\frac{\partial \widetilde{R}}{\partial t}\geq 0\ \qquad \mbox{on}
\quad \widetilde{M}\times (-\infty ,+\infty )\ .$$ Thus, combining (\[4.4\]) and (\[4.5\]), we deduce that $$\widetilde{R}\equiv 0\ ,\qquad \mbox{for all} \quad (x,t)\in
\widetilde{M} \times (-\infty ,+\infty )$$ and hence $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ is flat for all $t\in (-\infty ,+\infty )$. But, by definition, a singularity model cannot be flat. This proves our claim (\[4.3\]).
With the estimate (\[4.3\]), we can then apply a lemma of Hamilton (Lemma 22.2 in [@Ha3]) to find a sequence of positive numbers $\delta _j$, $j=1,2,\cdots$, with $\delta_j\rightarrow 0$ such that
1. $\widetilde{R}(x,0)\leq (1+\delta _j)\widetilde{R}(x_j,0)$ for all $x$ in the ball $\widetilde{B}_0(x_j,r_j)$ of radius $r_j$ centered at $x_j$ with respect to the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot,0)$;
2. $r_j^2\widetilde{R}(x_j,0)\rightarrow +\infty $;
3. if $s_j=\widetilde{d}_0(x_j,x_0)$, then $\lambda
_j=s_j/r_j\rightarrow +\infty $;
4. the balls $\widetilde{B}_0(x_j,r_j)$ are disjoint.
Denote the minimum of the holomorphic sectional curvature of the metric $
\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,0)$ at $x_j$ by $h_j$. We claim that the following holds $$\label{4.6}
\varepsilon _j=\frac{h_j}{\widetilde{R}(x_j,0)}\rightarrow 0\
\qquad \mbox{as} \quad j\rightarrow +\infty \ .$$
Suppose not, there exists a subsequence $j_k\rightarrow +\infty $ and some positive number $\varepsilon >0$ such that $$\label{4.7}
\varepsilon _{j_k}=\frac{h_{j_k}}{\widetilde{R}(x_{j_k},0)}\geq
\varepsilon \ \qquad \mbox{for all} \quad k=1,2,\cdots \ .$$
Since the solution $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ is ancient, it follows from the Harnack inequality of Cao [@Cao] that the scalar curvature $\widetilde{R}(x,t)$ is pointwisely nondecreasing in time. Then, by using the local derivative estimate of Shi [@Sh1] (or see Theorem 13.1 in [@Ha3]) and (a), (b), we have $$\begin{aligned}
\label{4.8}
\sup \limits_{x\in \widetilde{B}_0(x_{j_k},r_{j_k})}\left|
\nabla \widetilde{R}m(x,0)\right| ^2&\leq&C_5\widetilde{R}^2(x_j,0)
\left( \frac 1{r_{j_k}^2}+\widetilde{R}(x_j,0)\right) \nonumber \\
&\leq&2C_5\widetilde{R}^3(x_j,0)\ ,\end{aligned}$$ where $\widetilde{R}m$ is the curvature tensor of $\widetilde{g}_{\alpha \overline{\beta }}$ and $C_5$ is a positive constant depending only on the dimension.
For any $x\in \widetilde{B}_0(x_{j_k},r_{j_k})$, we obtain from (\[4.7\]) and (\[4.8\]) that the minimum of the holomorphic sectional curvature $h_{\min}(x)$ at $x$, satisfies $$\begin{aligned}
\label{4.9}
h_{\min }(x)&\geq&h_{j_k}-\sqrt{2C_5}\widetilde{R}^{3/2}(x_{j_k},0)\widetilde{d}_0(x,x_{j_k})\nonumber\\
&\geq&\widetilde{R}(x_{j_k},0)\left( \varepsilon -\sqrt{2C_5}\cdot \sqrt{\widetilde{R}(x_{j_k},0)}\cdot \widetilde{d}_0(x,x_{j_k})\right)\nonumber\\
&\geq&\frac \varepsilon 2\widetilde{R}(x_{j_k},0)\ \end{aligned}$$ if $$\widetilde{d}_0(x,x_{j_k})\leq \frac{\varepsilon}{2\sqrt{2C_5}\cdot \sqrt{
\widetilde{R}(x_{j_k},0)}}\ .$$ Thus, from (a) and (\[4.9\]), there exists $k_0>0$ such that for any $k\geq k_0$ and $$x \in \widetilde{B}_0(x_{j_k},\frac{\varepsilon}{2\sqrt{2C_5}
\cdot \sqrt{\widetilde{R}(x_{j_k},0)}}),$$ we have $$\label{4.10}
\frac \varepsilon 2\widetilde{R}(x_{j_k},0) \leq
\mbox{holomorphic sectional curvature at $x$} \leq
2\widetilde{R}(x_{j_k},0).$$ We have proved that the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,0)$ has nonnegative definite curvature operator. In particular, the sectional curvature is nonnegative. Then, by the generalized Cohn–Vossen inequality in real dimension 4 [@GW1], we have $$\label{4.11}
\int_{\widetilde{M}}\Theta \leq \chi \left( \widetilde{M}\right)
<+\infty$$ where $\Theta $ is the Gauss–Bonnet–Chern integrand for the metric $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,0)$ and $\chi \left( \widetilde{M}\right)$ is the Euler number of the manifold $\widetilde{M}$ which has finite topology type by the soul theorem of Cheeger–Gromoll.
On the other hand, from the proof of Theorem 1.3 of Bishop–Goldberg [@BG] (see Page 523 of [@BG]), the inequality (\[4.10\]) implies that $$\label{4.12}
\Theta (x)\geq C(\varepsilon)\widetilde{R}^2(x_{j_k},0)\
\qquad \mbox{for all} \;\;\; x\in \widetilde{B}_0(x_{j_k},
\frac{\varepsilon}{2\sqrt{2C_5}\cdot \sqrt{ \widetilde{R}(x_{j_k},0)}}),$$ where $C(\varepsilon)$ is some positive constant depending only on $\varepsilon$. Now, by combining (\[4.2\]), (b), (d), (\[4.11\]) and (\[4.12\]), we get $$\begin{aligned}
+\infty >\chi \left( \widetilde{M}\right)
& \geq & \sum\limits_{k=k_0}^\infty \int_{\widetilde{B}_0(x_{j_k},
\frac{\displaystyle \varepsilon}{2\sqrt{2C_5}\cdot
\sqrt{ \widetilde{R}(x_{j_k},0)}})}\Theta \\
& \geq & C(\varepsilon)\sum\limits_{k=k_0}^\infty \widetilde{R}^2
(x_{j_k},0)\cdot C_1\left( \frac\varepsilon {2\sqrt{2C_5}\cdot \sqrt{\widetilde{R}(x_{j_k},0)}}\right) ^4 \\
& = & C(\varepsilon )\sum\limits_{k=k_0}^\infty
\frac{C_1\varepsilon ^4}{64C_5^2}\\
& = & +\infty\end{aligned}$$ which is a contradiction. Hence our claim (\[4.6\]) is proved.
Now, we are going to blow down the singularity model $(\widetilde{M},
\widetilde{g}_{\alpha \overline{\beta }}(x,t))$. For the above chosen $x_j$, $r_j$ and $\delta _j$, let $x_j$ be the new origin $O$, dilate the space by a factor $\lambda _j$ so that $\widetilde{R}(x_j,0)$ become $1$ at the origin at $t=0$, and dilate in time by $\lambda _j^2$ so that it is still a solution to the Ricci flow. The balls $\widetilde{B}_0(x_j,r_j)$ are dilated to the balls centered at the origin of radii $\widetilde{r}_j=r_j^2
\widetilde{R}(x_j,0)\rightarrow +\infty $ ( by (b) ). Since the scalar curvature of $\widetilde{g}_{\alpha \overline{\beta }}(x,t)$ is pointwise nondecreasing in time by the Harnack inequality, the curvature bounds on $
\widetilde{B}_0(x_j,r_j)$ also give bounds for previous times in these balls. And the maximal volume growth estimate (\[4.2\]) and the local injectivity radius estimate of Cheeger, Gromov and Taylor [@CGT] imply that $$\mbox{inj}_{\widetilde{M}}\left( x_j,\widetilde{g}_{\alpha
\overline{\beta }}(\cdot,0)\right)
\geq \frac \beta {\sqrt{\widetilde{R}(x_j,0)}}\ ,$$ for some positive constant $\beta $ independent of $j.$
So we have everything to take a limit for the dilated solutions. By applying the compactness theorem in [@Ha2] and combining (\[4.2\]), (\[4.6\]), (a) and (b), we obtain a complete noncompact solution, still denoted by $( \widetilde{M},\widetilde{g}_{\alpha \overline{\beta }}(x,t))$, for $t\in
(-\infty ,0]$ such that
1. the curvature operator is still nonnegative;
2. $\widetilde{R}(x,t)\leq 1$, for all $x\in \widetilde{M}$, $
t\in (-\infty ,0]$, and $\widetilde{R}(0,0)=1$;
3. $\mbox{Vol}_t\left( \widetilde{B}_t(x,r)\right) \geq C_1r^4$ for all $x\in \widetilde{M}$, $0\leq r\leq +\infty $;
4. there exists a complex $2$–plane $V\wedge JV$ at the origin $O $ so that at $t=0$, the corresponding holomorphic sectional curvature vanishes.
If we consider the universal covering of $\widetilde{M}$, the induced metric of $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ on the universal covering is clearly still a solution to the Ricci flow and satisfies all of above (e), (f), (g), (h). Thus, without loss of generality, we may assume that $\widetilde{M}$ is simply connected.
Next, by using the strong maximum principle on the evolution equation of the curvature operator of $\widetilde{g}_{\alpha \overline{\beta }}(\cdot ,t)$ as in [@Ha1] (see Theorem 8.3 of [@Ha1]), we know that there exists a constant $K>0$ such that on the time interval $-\infty <t<-K$, the image of the curvature operator of $(\widetilde{M},\widetilde{g}_{\alpha
\overline{\beta }}(\cdot ,t))$ is a fixed Lie subalgebra of $so(4)$ of constant rank on $\widetilde{M}$. Because $\widetilde{M}$ is Kähler, the possibilities are limited to $u(2)$, $so(2)\times so(2)$ or $so(2).$
In the case $u(2)$, the sectional curvature is strictly positive. Thus, this case is ruled out by (h). In the cases $so(2)\times so(2)$ or $so(2)$, according to [@Ha1], the simply connected manifold $\widetilde{M}$ splits as a product $\widetilde{M}=\Sigma _1\times \Sigma _2$, where $\Sigma
_1$ and $\Sigma _2$ are two Riemann surfaces with nonnegative curvature (by (e)), and at least one of them, say $\Sigma _1$, has positive curvature (by (f)).
Denote by $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(\cdot ,t)$ the corresponding metric on $\Sigma _1$. Clearly, it follows from (g) and standard volume comparison that for any $x\in \Sigma _1$, $t\in (-\infty
,-K) $, we have $$\label{4.13}
\mbox{Vol}B_{\Sigma_1}(x,r) \geq C_6 r^2 \qquad \mbox{for} \;\;
0 \leq r < +\infty$$ where both the geodesic ball $B_{\Sigma_1}(x,r)$ and the volume are taken with respect to the metric $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(\cdot ,t)$ on $\Sigma_1$, $C_6$ is a positive constant depending only on $C_1.$ Also as the curvature of $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(x,t)$ is positive, it follows from Cohn–Vossen inequality that $$\label{4.14}
\int_{\Sigma _1}\widetilde{R}^{(1)}(x,t)d\sigma _t\leq 8\pi \ ,$$ where $\widetilde{R}^{(1)}(x,t)$ is the scalar curvature of $(\Sigma _1,
\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(x,t))$ and $d\sigma _t$ is the volume element of the metric $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(x,t).$
Now, the metric $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(x,t)$ is a solution to the Ricci flow on the Riemann surface $\Sigma _1$ over the ancient time interval $(-\infty ,-K)$. Thus, (\[4.13\]) and (\[4.14\]) imply that for each $t\in (-\infty ,-K)$, the curvature of $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(x,t)$ has quadratic decay in the average sense of Shi [@Sh4] and then the apriori estimate of Shi (see Theorem 8.2 in [@Sh4]) implies that the solution $\widetilde{g}_{\alpha \overline{\beta }}^{(1)}(x,t)$ exists for all $t\in (-\infty ,+\infty )$ and satisfies $$\label{4.15}
\lim \limits_{t\rightarrow +\infty }\sup \left\{ \left.
\widetilde{R}^{(1)}(x,t)\right| \ x\in \Sigma _1\right\} =0.$$ Again, by the Harnack inequality of Cao [@Cao], we know that $\widetilde{R}^{(1)}(x,t)$ is pointwisely nondecreasing in time. Therefore, we conclude that $$\widetilde{R}^{(1)}(x,t)\equiv 0\qquad \mbox{on}\quad
\Sigma _1\times (-\infty,+\infty )\ .$$ This contradicts with the fact that $(\Sigma _1,\widetilde{g}_{\alpha
\overline{\beta }}(\cdot ,t))$ has positive curvature for $t<-K.$ Hence we have seeked the desired contradiction and have completed the proof of Theorem 4.1. $\Box $
§5. Topology and Steinness {#topology-and-steinness .unnumbered}
==========================
In this section, we use the estimates obtained in the previous sections to study the topology and the complex structure of the Kähler surface in our Main Theorem. Our result is
Suppose $(M,g_{\alpha \overline{\beta }})$ is a complete noncompact Kähler surface satisfying the assumptions in the Main Theorem. Then $M$ is homeomorphic to $\R^4$ and is a Stein manifold.
The proof of this theorem is exactly the same as in [@CZ2]. For the convenience of the readers, we give a sketch of the arguments and refer to the cited reference for details.
We evolve the metric $g_{\alpha \overline{\beta }}(x)$ by the Ricci flow (\[2.1\]). From Theorem 4.1, the solution $g_{\alpha \overline{\beta }}(x,t)$ exists for all $
t\in [0,+\infty )$ and satisfies $$\label{5.1}
R(x,t)\leq \frac C{1+t}\ \qquad \mbox{on} \quad M\times [0,+\infty )$$ for some positive constant $C$. Also, Proposition 2.1 tells us that the volume growth condition (i) is preserved under the Ricci flow. By using the local injectivity radius estimate of Cheeger–Gromov–Taylor, this implies that $$\label{5.2}
\mbox{inj}\left( M,g_{\alpha \overline{\beta }}(\cdot ,t)\right) \geq
C_7(1+t)^{\frac 12}\ \qquad \mbox{for}\quad t\in [0,+\infty )$$ with some positive constant $C_7$.
Since the Ricci curvature of $g_{\alpha \overline{\beta }}(x,t)$ is positive for all $x\in M$ and $t\geq 0$, the Ricci flow equation (\[2.1\]) implies that the ball $B_t(x_0,\frac{C_7}2(1+t)^{\frac 12})$ of radius $\frac{C_7}2(1+t)^{\frac 12}$ with respect to the metric $g_{\alpha \overline{\beta }}(\cdot ,t)$ contains the ball $B_0(x_0,\frac{C_7}2(1+t)^{\frac 12})$ of the same radius with respect to the initial metric $g_{\alpha \overline{\beta }}(\cdot ,0)$. Combining this with (\[5.2\]), we deduce that $$\pi _p(M,x_0)=0\ \qquad \mbox{for any} \quad p\geq 1$$ and $$\pi _q(M,\infty )=0\ \qquad \mbox{for} \quad 1\leq q\leq 2\ ,$$ where $\pi _q(M,\infty )$ is the $q$th homotopy group of $M$ at infinity.
Thus, by the resolution of the generalized Poincaré conjecture on four manifolds by Freedman [@Fre], we know that $M$ is homeomorphic to $\R^4$.
Next, the injectivity radius estimate (\[5.2\]) also tells us that, for $t$ large enough, the exponential maps provide diffeomorphisms between big geodesic balls $B_t(x_0,\frac{C_7}2(1+t)^{\frac 12})$ of $M$ with big Euclidean balls on $\C^2$. The curvature estimate (\[5.1\]) together with its derivative estimates imply that the difference of the complex structure of $M$ and the standard complex struture of $\C^2$ in those geodesic balls can be made arbitrarily small by taking $t$ large enough. Then, we can use the $L^2$ estimates of Hörmander [@Ho] to modify the exponential map to a biholomorphism between a domain $\Omega (t)$ containing $B_t(x_0,\frac{C_7}4(1+t)^{\frac 12})$ and a Euclidean ball of radius $\frac{C_7}3(1+t)^{\frac 12}$ in $\C^2$. Since the solution $g_{\alpha \overline{\beta }}(\cdot ,t)$ of the Ricci flow is shrinking, the domain $\Omega (t)$ contains the geodesic ball $B_0(x_0,\frac{C_7}4(1+t)^{\frac 12})$. Thus, we can choose a sequence of $t_k\rightarrow +\infty $, such that $$M=\cup _{k=0}^{+\infty }\Omega (t_k)\ ,\qquad \Omega (t_1)\subset \Omega
(t_2)\subset \cdots \subset \Omega (t_k)\subset \cdots \ .$$ Since for each $k$, $\Omega (t_k)$ is biholomorphic to the unit ball of $\C^2$, $(\Omega (t_k),\Omega (t_l))$ forms a Runge pair for any $k,\ l$. Finally, we can appeal to a theorem of Markeo [@Mar] (see also Siu [@Si]) to conclude that $M$ is a Stein manifold. $\Box $
§6. Space decay estimate on curvature and the Poincaré–Lelong equation {#space-decay-estimate-on-curvature-and-the-poincarélelong-equation .unnumbered}
======================================================================
Let $(M,g_{\alpha \overline{\beta }})$ be a complete noncompact Kähler surface satisfying all the assumptions in the Main Theorem. The main purpose of this section is to establish the existence of a strictly plurisubharmonic function of logarithmic growth on $M$. To this end, we first prove a curvature decay estimate at infinity of the metric $g_{\alpha \overline{\beta }}$.
Let $(M,g_{\alpha \overline{\beta }})$ be a complete noncompact Kähler surface as above. Then there exists a constant $C>0$ such that for all $x\in M,\ r>0$, we have $$\label{6.1}
\int_{B(x,r)}R(y)\,\frac 1{d^2(x,y)}\,dy \leq C\log (2+r)\ .$$
Let $g_{\alpha \overline{\beta }}(x,t)$ be the solution of the Ricci flow (\[2.1\]) with $g_{\alpha \overline{\beta }}(x)$ as the initial metric. From Theorem 4.1, we know that the solution exists for all times and satisfies $$\label{6.2}
R(x,t) \leq \frac{C_8}{1+t}\ \qquad \mbox{on} \quad
M\times [0,+\infty )$$ for some positive constant $C_8.$
Let $$F(x,t)=\log \frac{\det \left( g_{\alpha \overline{\beta }}
(x,t)\right) }{\det \left( g_{\alpha \overline{\beta }}(x,0)\right)}$$ be the function introduced in the proof of Proposition 2.1. Since $$-\partial _\alpha \overline{\partial }_\beta \log \frac{\det \left(
g_{\gamma \overline{\delta }}(\cdot ,t)\right) }{\det \left( g_{\gamma
\overline{\delta }}(\cdot ,0)\right) }=R_{\alpha \overline{\beta }}(\cdot
,t)-R_{\alpha \overline{\beta }}(\cdot ,0)\ ,$$ after taking trace with the initial metric $g_{\alpha \overline{\beta }}(\cdot,0)$, we get $$\label{6.3}
R(\cdot ,0)=\bigtriangleup _0F(\cdot ,t)+g^{\alpha \overline{
\beta }}(\cdot ,0)R_{\alpha \overline{\beta }}(\cdot ,t)$$ where $\bigtriangleup _0$ is the Laplace operator of the metric $g_{\alpha \overline{\beta }}(\cdot ,0).$
Since $(M,g_{\alpha \overline{\beta }}(\cdot ,0))$ has positive Ricci curvature and maximal volume growth, it is well known (see [@ScY]) that the Green function $G_0(x,y)$ of the initial metric $g_{\alpha \overline{\beta }}(\cdot ,0)$ exists on $M$ and satisfies the estimates $$\label{6.4}
\frac{C_9^{-1}}{d_0^2(x,y)}\leq G_0(x,y)\leq
\frac{C_9}{d_0^2(x,y)}$$ and $$\label{6.5}
\left| \nabla _yG_0(x,y)\right| _0\leq \frac{C_9}{d_0^3(x,y)}$$ for some positive constant $C_9$ depending only on $C_1.$
For any fixed $\overline{x}_0\in M$ and any $\alpha >0$, we denote $$\Omega _\alpha =\left\{ \left. x\in M\right| \ G_0
(\overline{x}_0,x)\geq \alpha \ \right\} \ .$$ By (\[6.4\]), it is not hard to see $$\label{6.6}
B_0\left( \overline{x}_0,\left( \frac{C_9^{-1}}\alpha \right)
^{\frac 12}\right) \subset \Omega _\alpha \subset B_0\left( \overline{x}
_0,\left( \frac{C_9}\alpha \right) ^{\frac 12}\right) \ .$$ Recall that $F$ evolves by $$\frac{\partial F(x,t)}{\partial t}=-R(x,t)\ \qquad \mbox{on} \quad M\times
[0,+\infty )\ .$$ Combining with (\[6.2\]), we obtain $$\label{6.7}0\geq F(x,t)\geq -C_{10}\log (1+t) \qquad \mbox{on} \quad M\times
[0,+\infty )\ .$$ Multiplying (\[6.3\]) by $G_0(\overline{x}
_0,x)-\alpha $ and integrating over $\Omega _\alpha $, we have $$\begin{aligned}
\label{6.8}
\int_{\Omega _\alpha }R(x,0)\left( G_0(\overline{x}_0,x)-\alpha \right)
dx&=&\int_{\Omega _\alpha }\left( \bigtriangleup _0F(x,t)\right)
\left( G_0(\overline{x}_0,x)-\alpha \right) dx \nonumber \\
& & +\int_{\Omega _\alpha }g^{\alpha \overline{\beta }}(x,0)
R_{\alpha \overline{\beta }}(\cdot ,t)\left( G_0(\overline{x}_0,x)-
\alpha \right) dx \nonumber \\
& = & -\int_{\partial \Omega _\alpha }F(x,t)\frac{\partial G_0
(\overline{x}_0,x)}{\partial \nu }d\sigma -F(\overline{x}_0,t) \nonumber\\
& & +\int_{\Omega _\alpha }g^{\alpha \overline{\beta }}(x,0)
R_{\alpha \overline{\beta }}(\cdot ,t)\left( G_0(\overline{x}_0,x)-\alpha
\right) dx\nonumber\\
& \leq & C_{10}\left( 1+C_9^{\frac 52}\alpha ^{\frac 32}\mbox{Vol}_0
\left( \partial \Omega _\alpha \right) \right) \log (1+t) \nonumber \\
& & +\int_{\Omega _\alpha }g^{\alpha \overline{\beta }}(x,0)
R_{\alpha \overline{\beta }}(\cdot ,t)G_0(\overline{x}_0,x)dx\ ,\nonumber\\\end{aligned}$$ by (\[6.4\]) and (\[6.7\]). Here, we have used $\nu $ to denote the outer unit normal of $\partial \Omega _\alpha .$
From the coarea formula, we have $$\begin{aligned}
\frac 1\alpha \int_\alpha ^{2\alpha }r^{\frac 32}Vol_0\left(
\partial \Omega _r\right) dr&\leq&2^{\frac 32}\alpha ^{\frac 12}
\int_\alpha ^{2\alpha }\int_{\partial \Omega _r}\left| \nabla
G_0(\overline{x}_0,x)\right| _0d\sigma \left| d\nu \right| \nonumber \\
& \leq & 2^{\frac 32}C_9^{\frac 52}\alpha ^2\mbox{Vol}_0\left(
\Omega _\alpha\right) \nonumber \\
& \leq & 2^{\frac 32}C_9^{\frac 52}\alpha ^2\mbox{Vol}_0\left(
B_0\left( \overline{x}_0,\left( \frac{C_9}\alpha \right) ^{\frac 12}\right)
\right)\nonumber\\
& \leq & C_{11} \nonumber\end{aligned}$$ for some positive constant $C_{11}$ by the standard volume comparison. Substitute this into (\[6.8\]) and integrate (\[6.8\]) from $\alpha $ to $2\alpha $, we get $$\begin{aligned}
\label{6.9}
\int_{\Omega _{2\alpha}}R(x,0)\left( G_0(\overline{x}_0,x)-2\alpha
\right) dx&\leq&C_{10}\left( 1+C_9^{\frac 52}C_{11}\right) \log (1+t)
\nonumber\\
& & +\int_{\Omega _\alpha }g^{\alpha \overline{\beta }}(x,0)
R_{\alpha \overline{\beta }}(x,t)G_0(\overline{x}_0,x)dx\ .\nonumber\\\end{aligned}$$ It is easy to see that $$\int_{\Omega _{4\alpha }}R(x,0)G_0(\overline{x}_0,x)dx\leq 2\int_{\Omega
_{2\alpha }}R(x,0)\left( G_0(\overline{x}_0,x)-2\alpha \right) dx$$ and by the equation Ricci flow (\[2.1\]), we also have $$\begin{aligned}
&&\int_0^t\int_{\Omega _\alpha }g^{\alpha \overline{\beta }}(x,0)
R_{\alpha \overline{\beta }}(x,t)G_0(\overline{x}_0,x)dx dt
\nonumber\\
& = & \int_{\Omega _\alpha }g^{\alpha \overline{\beta }}(x,0)
\left( g_{\alpha \overline{\beta }}(x,0)-g_{\alpha \overline{\beta }}
(x,t)\right) G_0(\overline{x}_0,x)dx \nonumber \\
& \leq & 2\int_{\Omega _\alpha }G_0(\overline{x}_0,x)dx\ .\nonumber\end{aligned}$$ Thus by integrating (\[6.9\]) in time from $0$ to $t$ and combining the above two inequalities, we get for any $t>0,$ $$\int_{\Omega _{4\alpha }}R(x,0)G_0(\overline{x}_0,x)dx\leq 2C_{10}\left(
1+C_9^{\frac 52}C_{11}\right) \log (1+t)+\frac 4t\int_{\Omega _\alpha }G_0(
\overline{x}_0,x)dx\ .$$ Finally, substituting (\[6.4\]) and (\[6.6\]) into the above inequality, we see that there exists some positive constant $C_{12}$ such that for any $\overline{x}_0\in M$, $t>0$ and $r>0,$ $$\label{6.10}\int_{B_0(\overline{x}_0,r)}R(x,0)\frac 1{d^2(\overline{x}_0,x)}dx\leq C_{12}\left( \log (1+t)+\frac{r^2}t\right) \ .$$ Choose $t=r^2$, we get the desired estimate. $\Box $
Now we can use the estimate (\[6.1\]) to solve the following Poincaré–Lelong equation on $M$ $$\label{6.11}
\sqrt{-1}\partial \overline{\partial }u= \mbox{Ric} $$ to get the strictly plurisubharmonic function mentioned at the beginning of this section.
As in [@MSY] or [@NST], we first study the corresponding Poisson equation on $M$ $$\label{6.12}\bigtriangleup u=R.$$ After we solve the Poisson equation (\[6.12\]) with a solution of logarithmic growth, we will see that it is indeed a solution of the Poincaré–Lelong equation with logarithmic growth.
To solve (\[6.12\]), we first construct a family of approximate solutions $u_r$ as follows.
For a fixed $x_0\in M$ and $r>0$, define $u_r(x)$ on $B(x_0,r)$ by $$u_r(x)=\int_{B(x_0,r)}\left( G(x_0,y)-G(x,y)\right) R(y)dy$$ where $G(x,y)$ is the Green function of the metric $g_{\alpha \overline{\beta }}$ on $M$. It is clear that $$u_r(x_0)=0 \qquad \mbox{and} \qquad \bigtriangleup u_r(x)=R(x) \quad \mbox{on} \quad B(x_0,r).$$ For $x\in B(x_0,\frac r2)$, we write $$\begin{aligned}
u_r(x) & = & \left( \int_{B(x_0,r)\backslash B(x_0,2d(x,x_0))}+
\int_{B(x_0,2d(x,x_0))}\right) \left( G(x_0,y)-G(x,y)\right)
R(y)dy \nonumber \\
& := &I_1+I_2\ . \nonumber\end{aligned}$$ From (\[6.1\]), we see that $$\label{6.13}
\left| I_2\right| \leq C_{13}\log \left( 2+d(x,x_0)\right)
\qquad \mbox{on} \quad B(x_0,\frac r2)$$ for some positive constant $C_{13}$ independent of $x_0$, $x$ and $r.$
To estimate $I_1$, we get from (\[6.5\]) that for $y\in B(x_0,r)\backslash
B(x_0,2d(x,x_0)),$$$\begin{aligned}
\left| G(x_0,y)-G(x,y)\right| &\leq&d(x,x_0)\cdot \sup
\limits_{z\in B(x_0,d(x,x_0))}\left| \nabla _zG(z,y)\right|
\nonumber \\
& \leq & C_9d(x,x_0)\cdot \sup \limits_{z\in B(x_0,d(x,x_0))}
\frac 1{d^3(z,y)} \nonumber \\
&\leq&8C_9\frac{d(x,x_0)}{d^3(y,x_0)}\ .\nonumber\end{aligned}$$ Thus by (\[6.1\]), we have $$\begin{aligned}
\label{6.14}
\left|I_1\right|&\leq&8C_9d(x,x_0)\int_{B(x_0,r)\backslash B(x_0,2d(x,x_0))}\frac{R(y)}{d^3(y,x_0)}dy \nonumber\\
&\leq&8C_9d(x,x_0)\sum\limits_{k=1}^\infty \frac 1{2^kd(x,x_0)}\cdot \int_{B(x_0,2^{k+1}d(x,x_0))\backslash B(x_0,2^kd(x,x_0))}
\frac{R(y)}{d^2(y,x_0)}dy \nonumber \\
& \leq & 8C_9C\sum\limits_{k=1}^\infty \frac 1{2^k}\log \left( 2+2^{k+1}d(x,x_0)\right)\nonumber\\
& \leq & C_{14}\log \left( 2+d(x,x_0)\right) \end{aligned}$$ for some positive constant $C_{14}.$
Hence, by combining (\[6.13\]) and (\[6.14\]), we deduce $$\label{6.15}
\left| u_r(x)\right| \leq \left( C_{13}+C_{14}\right) \log
\left( 2+d(x,x_0)\right)$$ for any $r\geq 2d(x,x_0).$
On the other hand, by taking the derivative of $u_r(x)$, we get $$\begin{aligned}
\label{6.16}
\left| \nabla u_r(x)\right|&\leq&C_9\int_M\frac{R(y)}
{d^3(x,y)}dy \nonumber \\
& \leq & C_9\left( \int\limits_{B(x,1)}\frac{R(y)}{d^3(x,y)}dy
+\sum\limits_{k=1}^\infty \frac 1{2^{k-1}}\int\limits_{B(x,2^k)
\backslash B(x,2^{k-1})}\frac{R(y)}{d^2(x,y)}dy\right) \nonumber\\
& \leq & C_9\left( C_{15}+\sum\limits_{k=1}^\infty
\frac 1{2^{k-1}}C\log \left( 2+2^k\right) \right) \nonumber \\
& = & C_{16}\ .\end{aligned}$$ Here, we have used (\[6.1\]) and (\[6.5\]), $C_{15}$ and $C_{16}$ are positive constants independent of $r$. Therefore, it follows from the Schauder theory of elliptic equations that there exists a sequence of $r_j\rightarrow +\infty $ such that $u_{r_j}(x)$ converges uniformly on compact subset of $M$ to a smooth function $u$ satisfying $$\label{6.17}
\left\{
\begin{array}{ll}
\bigbreak u(x_0)=0\qquad \mbox{and} \qquad \bigtriangleup
u=R & \mbox{on} \;\; M\ , \\
\bigbreak |u(x)|\leq \left( C_{13}+C_{14}\right) \log \left(
2+d(x,x_0)\right) & \mbox{for} \;\; x\in M\ , \\
|\nabla u(x)|\leq C_{16}\ & \mbox{for} \;\; x\in M\ .
\end{array}
\right.$$ Thus, we have obtained a solution $u$ of logarithmic growth to the Poisson equation (\[6.12\]) on $M$. In the following we prove that $u$ is actually a solution of the Poincaré–Lelong equation (\[6.11\]).
Recall the Bochner identity, with $\bigtriangleup u = R$ $$\begin{aligned}
\label{6.18}
\frac 12\bigtriangleup \left| \nabla u\right| ^2
& = &\left| \nabla ^2u\right| ^2+\left\langle \nabla u,\nabla
R\right\rangle +Ric\left( \nabla u,\nabla u\right) \nonumber \\
& \geq & \left| \nabla ^2u\right| ^2+\left\langle \nabla u,\nabla
R\right\rangle \ .\end{aligned}$$ For any $r>0$ and any $\overline{x}_0\in M$, by multiplying (\[6.18\]) by the cutoff function in (\[2.7\]) and integrating by parts, we get $$\begin{aligned}
\int_M\left| \nabla ^2u\right| ^2\varphi dx
& \leq & \frac 12\int_M\left| \nabla u\right| ^2\cdot \left|
\bigtriangleup \varphi \right| dx+\int_M\left| \nabla ^2u\right|
\cdot R\varphi dx \nonumber \\
& & + \int_M\left| \nabla u\right| \cdot R\cdot \left|
\nabla \varphi \right| dx \nonumber \\
& \leq & \frac{C_{16}^2}2\cdot \frac{C_3}{r^2}\int_M\varphi dx+
\frac 12\int_M\left| \nabla ^2u\right| ^2\varphi dx \nonumber \\
& & +\frac 12\int_MR^2\varphi dx+C_{16}\cdot \left(
\sup \limits_MR\right) \cdot \frac{C_3}r\int_M\varphi dx\ .\nonumber\end{aligned}$$ Thus, $$\label{6.19}
\int_M\left| \nabla ^2u\right| ^2\varphi dx\leq
\left(C_3C_{16}^2\cdot \frac 1{r^2}+2C_{16}C_3\left(
\sup \limits_MR\right) \frac
1r\right) \int_M\varphi dx+\int_MR^2\varphi dx\ .$$ By (\[6.1\]), (\[2.7\]) and the standard volume comparison, we have $$\begin{aligned}
\label{6.20}
\int_MR^2\varphi dx&\leq&\left( \sup \limits_MR\right)
\int_MR(x)e^{-\left( 1+\frac{d(x,\overline{x}_0)}r\right) }dx
\nonumber \\
& \leq & \left( \sup \limits_MR\right)\cdot \nonumber\\
&&\left( \int_{B(\overline{x}_0,r)}R(x)dx+\sum\limits_{k=0}^\infty
e^{-2^{k-1}}\int_{B(\overline{x}_0,2^{k+1}r)\backslash B(\overline{x}_0,2^kr)}R(x)dx\right) \nonumber\\
&\leq&C_{17}r^2\log \left( 2+r\right)\end{aligned}$$ and $$\begin{aligned}
\label{6.21}
\int_M\varphi dx&\leq&\int_Me^{-\left( 1+\frac{d(x,\overline{x}_0)}r
\right) }dx \nonumber\\
& \leq & \int_{B(\overline{x}_0,r)}dx+\sum\limits_{k=0}^\infty
e^{-2^{k-1}}\int_{B(\overline{x}_0,2^{k+1}r)\backslash
B(\overline{x}_0,2^kr)}dx \nonumber\\
& \leq & C_{17}r^4\end{aligned}$$ for some positive constant $C_{17}$ independent of $r$ and $\overline{x}_0.$
Substituting these two inequalities into (\[6.19\]) we have $$\label{6.22}
\frac 1{r^4}\int_{B(\overline{x}_0,r)}\left| \nabla ^2u\right|
^2dx\leq C_{18}\left( \frac 1{r^2}+\frac 1r+\frac{\log (2+r)}{r^2}\right)$$ for some positive constant $C_{18}$ independent of $r$ and $\overline{x}_0.$
Since the holomorphic bisectional curvature of $g_{\alpha \overline{\beta }}$ is positive, it was shown in [@MSY] that the function $\left| \sqrt{-1}\partial \overline{\partial }u-\mbox{Ric}\right| ^2$ is subharmonic on $M$. Then by the mean value inequality and (\[6.20\]), (\[6.22\]), we have $$\begin{aligned}
\left| \sqrt{-1}\partial \overline{\partial }u-Ric\right| ^2(\overline{x}_0)&\leq&\frac{C_{19}}{r^4}\int_{B(\overline{x}_0,r)}
\left| \sqrt{-1}\partial \overline{\partial }u-Ric\right| ^2(x)dx \nonumber\\
& \leq & \frac{2C_{19}}{r^4}\int_{B(\overline{x}_0,r)}\left(
\left| \nabla ^2u\right| ^2+R^2\right) dx \nonumber\\
& \leq & C_{20}\left( \frac 1{r^2}+\frac 1r+\frac{\log (2+r)}{r^2}\right) \nonumber\end{aligned}$$for some positive constants $C_{19},\ C_{20}$ independent of $r$ and $\overline{x}_0.$ Since $\overline{x}_0\in M$ and $r>0$ are arbitrary, by letting $r\rightarrow +\infty $ we know that $$\sqrt{-1}\partial \overline{\partial }u= \mbox{Ric} \qquad
\mbox{on} \quad M\ .$$ In summary, we have proved the following result. Suppose $(M,g_{\alpha \overline{\beta }})$ is a complete noncompact Kähler surface satisfying all the assumptions in the Main Theorem. Then there exists a strictly plurisubharmonic function $u(x)$ on $M$ satisfying the Poincaré–Lelong equation (\[6.11\]) with the estimate $$|u(x)| \leq C\log\left(2+d(x,x_0)\right) \qquad \mbox{for all} \;\; x\in M$$ for some positive constant C.
§7. Uniform estimates on multiplicity and the number of components of an “algebraic” divisor {#uniform-estimates-on-multiplicity-and-the-number-of-components-of-an-algebraic-divisor .unnumbered}
============================================================================================
Let $(M,g_{\alpha \overline{\beta }})$ be a complete noncompact Kähler surface satisfying all the assumptions in the Main Theorem. In this section, we will consider the algebra $P(M)$ of holomorphic functions of polynomial growth on $M$. We first construct $f_1,\ f_2$ in $P(M)$ which are algebraically independent over $\C.$
In the previous section, by solving the Poincaré–Lelong equation, we have obtained a strictly plurisubharmonic function $u$ on $M$ of logarithmic growth. As shown in [@Mo1], the existence of nontrivial functions in the algebra $P(M)$ then follows readily from the $L^2$ estimates of the $\overline{\partial }$ operator on complete Kähler manifold of Andreotti–Vesentini [@AV] and Hörmander [@Ho]. For completeness, we give the proof as follows.
Let $x\in M$ and $\left\{ (z_1,z_2)\,|\ |z_1|^2+|z_2|^2<1\right\}$ be local holomorphic coordinates at $x$ with $z_1(x)=z_2(x)=0$. Let $\eta $ be a smooth cutoff function on $\C^2$ with $\mbox{Supp}\,\eta \subset \subset\left\{ |z_1|^2+|z_2|^2<1\right\}$ and $\eta \equiv 1$ on $\left\{|z_1|^2+|z_2|^2<\frac 14\right\} $. Then the function $$\eta \log |z|=\eta \left( z_1,z_2\right) \log
\left(|z_1|^2+|z_2|^2\right) ^{\frac 12}$$ is globally defined on $M$ and is smooth except at $x$. Furthermore, the $(1,1)$ form $\partial \overline{\partial }\,(\eta \log |z|)$ is bounded from below. Since $u$ is strictly plurisubharmonic, we can choose a sufficiently large positive constant $C$ such that $$v=Cu+ 6 \eta \, \log \left|\, z\right|$$ is strictly plurisubharmonic on $M$. Then, for any nonzero tangent vector $\xi $ of type $(1,0)$ on $M$, we have $$\left\langle \sqrt{-1}\partial \overline{\partial }v+\mbox{Ric},\;
\xi \wedge \overline{\xi } \right\rangle >0\ .$$ Now $\overline{\partial }\,(\eta z_i)$, i=1,2 , is a $\overline{\partial }$ closed $(0,1)$ form on the complete Kähler manifold $M$. Using the standard $L^2$–estimates of $\overline{\partial }$ operator (c.f. Theorem 2.1 in [@Mo1]), there exists a smooth function $u_i$ such that $$\overline{\partial }u_i=\overline{\partial }(\eta z_i)\ \qquad i=1,2$$ and $$\int_M\left| u_i\right| ^2e^{-v}dx\leq \frac 1c\int_M\left| \overline{\partial }(\eta z_i)\right| ^2e^{-v}dx$$ where $c$ is a positive constant satisfying $$\left\langle \sqrt{-1} \partial \overline{\partial }v+ \mbox{Ric},\xi \wedge
\overline{\xi }\right\rangle \geq c|\xi |^2$$ whenever $\xi$ is a tangent vector on $\mbox{Supp}\, \eta $. First of all, this estimate implies that $u_i$ is of polynomial growth as the weight function $v$ is of logarithmic growth. Secondly, because of the singularity of $6\log \left| z\right| $ at $x$, it forces the function $u_i$ and its first order derivative to vanish at $x$. Therefore, the holomorphic functions $f_1=u_1-\eta z_1$ and $f_2=u_2-\eta z_2$ define a local biholomorphism at $x$. Clearly, they are algebraically independent over $\C$. This concludes our construction.
For later use, we also point out here that, as a consequence of the above argument, the algebra $P(M)$ separates points on $M$. In other words, for any $x_1,x_2\in M$ with $x_1 \neq x_2$, there exists $f\in P(M)$ such that $f(x_1) \neq f(x_2).$
Before we can state our main result in this section, we need the following definition. For a holomorphic function $f\in P(M)$, we define the degree of $f$, $\deg (f)$, to be the infimum of all $q$ for which the following inequality holds $$\left| f(x)\right| \leq C(q)\left( 1+d^q(x,x_0)\right) \qquad
\mbox{for all} \; x\in M,$$ where $x_0$ is some fixed point in $M$ and $C(q)$ is some positive constant depending on $q.$
Our main result in this section is the following uniform bound on the multiplicity of the zero divisor of a function $f\in P(M)$ by its degree.
Let $(M,g_{\alpha \overline{\beta }})$ be a complete noncompact Kähler surface as above. For $f\in P(M)$, let $$\left[ V\right] =\frac{\sqrt{-1}}{2\pi }\partial \overline{\partial }\log
\left| f\right| ^2$$ be the zero divisor, counting multiplicity, determined by $f$. Then, there exists a positive constant $C$, independent of $f$, such that $$\mbox{mult} \left( \left[ V\right] ,x\right) \leq C\deg (f)$$ holds for all $x\in M.$
Recall that the Ricci flow (\[2.1\]) with $g_{\alpha \overline{\beta }}(x)$ as initial metric has a solution $g_{\alpha \overline{\beta }}(x,t)$ for all times $t\in [0,+\infty )$ and satisfies the following estimates $$\label{7.1}
R(x,t) \leq \frac C{1+t}$$ and $$\label{7.2}
\mbox{inj} \left( M,g_{\alpha \overline{\beta }}(\cdot ,t)\right) \geq
C_7(1+t)^{\frac 12}$$ on $M\times [0,+\infty )$.
Let $d_t$ be the distance function from an arbitrary fixed point $\overline{x}_0\in M$ with respect to the metric $g_{\alpha \overline{\beta }}(\cdot ,t)$. By the standard Hessian comparison theorem (see [@ScY]), we have, for any unit real vector $v$ orthogonal to the radial direction $\partial /\partial d_t$, $$\frac{\sqrt{\frac{\alpha _1}t}d_t}{\tan \left( \sqrt{\frac{\alpha _1}t}
d_t\right) }\leq \mbox{Hess}\left( d_t^2\right) (v,v)\leq \frac{\sqrt{\frac{\alpha
_1}t}d_t}{\tanh \left( \sqrt{\frac{\alpha _1}t}d_t\right) } \qquad
\mbox{when}\quad d_t\leq \frac \pi 4\sqrt{\frac t{\alpha _1}}\ .$$ Here, $\alpha _1$ is some positive constant depending only on the constants $C $ and $C_7$ in (\[7.1\]) and (\[7.2\]). Hence, for any unit vector $\widetilde{v}$, we have $$\frac{\sqrt{\frac{\alpha _1}t}d_t}{\tan \left( \sqrt{\frac{\alpha _1}t}
d_t\right) }\leq \mbox{Hess}\left( d_t^2\right) (\widetilde{v},\widetilde{v}
) + \mbox{Hess} \left( d_t^2\right) (J\widetilde{v},J\widetilde{v})
\leq \frac{2\sqrt{\frac{\alpha _1}t}d_t}{\tanh \left(
\sqrt{\frac{\alpha _1}t}d_t\right) }$$ whenever $d_t\leq \frac \pi 4\sqrt{\frac t{\alpha _1}}$. Since $M$ is Kähler, the above expression is equivalent to $$\frac{\sqrt{\frac{\alpha _1}t}d_t}{\tan \left( \sqrt{\frac{\alpha _1}t}d_t\right) }\omega _t\leq \sqrt{-1}\partial \overline{\partial }d_t^2\leq
\frac{2\sqrt{\frac{\alpha _1}t}d_t}{\tanh \left( \sqrt{\frac{\alpha _1}t}d_t\right) }\omega _t\ .$$ In particular, we have $$\label{7.3}
\frac 12\omega _t\leq \sqrt{-1}\partial \overline{\partial }
d_t^2\leq 4\omega _t \qquad \mbox{whenever} \quad d_t \leq \frac {\pi} {4}
\sqrt{\frac t{\alpha _1}}\ .$$ Here, $\omega _t$ is the Kähler form of the metric $g_{\alpha \overline{\beta }}(\cdot ,t)$.
We next claim that [^1] $$\label{7.4}\sqrt{-1}\partial \overline{\partial }\log \tan \left( \sqrt{
\frac{\alpha _1}t}\frac{d_t}2\right) \geq 0\ ,\qquad
\mbox{whenever} \quad d_t\leq
\frac \pi 4\sqrt{\frac t{\alpha _1}}\ .$$
In fact, after recaling, we may assume that the sectional curvature of $g_{\alpha \overline{\beta }}(\cdot ,t)$ is less than $1$ and $\sqrt{\frac{\alpha _1}t}=1$. Then by the standard Hessian comparison, we have $$\mbox{Hess} \left( d_t\right) (v,v)\geq \frac 1{\tan d_t}\left(
\left| v\right|_t^2-\left\langle v,\frac \partial
{\partial d_t}\right\rangle _t^2\right)$$ for any vector $v$ and $d_t\leq \frac \pi 4$. Thus, by a direct computation, $$\begin{aligned}
& & \mbox{Hess}\left( \log \tan \left( \frac{d_t}2\right) \right) (v,v)+
\mbox{Hess} \left( \log \tan \left( \frac{d_t}2\right) \right) (Jv,Jv) \nonumber\\
& \geq &\frac 1{\left( \tan d_t\right) \tan \left( \frac{d_t}2\right) }
\left( 1+\tan d_t\right) \left| v\right| _t^2 \nonumber\\
& \geq & 0\ ,\nonumber\end{aligned}$$ which is our claim (\[7.4\]).
Now for any $0<b<a<\frac \pi 8\sqrt{\frac t{\alpha _1}}$, it follows from Stoke’s theorem that $$\begin{aligned}
0&\leq&\sqrt{-1}\int\limits_{\left\{ b\leq d_t\leq a\right\} }\left[ V\right] \wedge\partial\overline\partial \log \tan \left( \sqrt{\frac{\alpha _1}t}\frac{d_t}2\right) \nonumber\\
& = &\sqrt{-1}\int\limits_{\left\{ d_t=a\right\} \ }\left[ V\right]
\wedge \overline{\partial }\log \tan \left( \sqrt{\frac{\alpha _1}t}\frac{d_t}2\right) \nonumber\\
&&-\sqrt{-1}\int\limits_{\left\{ d_t=b\right\} \ }\left[ V\right]
\wedge \overline{\partial }\log \tan \left( \sqrt{\frac{\alpha _1}t}
\frac{d_t}2\right) \ .\nonumber\end{aligned}$$ Then, it is no hard to see that for $0<b<a<\frac \pi 8\sqrt{\frac t{\alpha _1}}$, $$\label{7.5}
\frac{\sqrt{-1}}{a^2}\int\limits_{\left\{ d_t=a\right\} \ \
}\left[ V\right] \wedge \overline{\partial }\left( d_t^2\right) \geq \frac
12\cdot \frac{\sqrt{-1}}{b^2}\int\limits_{\left\{ d_t=b\right\} \ \ }\left[
V\right] \wedge \overline{\partial }\left( d_t^2\right) \ .$$
Using Stoke’s theorem on the right hand side of (\[7.5\]) and letting $b\rightarrow 0$ it follows from the inequality of Bishop–Lelong that $$\label{7.6}
\frac{\sqrt{-1}}{a^2}\int\limits_{\left\{ d_t=a\right\} \ \
}\left[ V\right] \wedge \overline{\partial }\left( d_t^2\right) \geq \alpha
_2\mbox{mult}\,\left( \left[ V\right] ,\overline{x}_0\right) \ ,$$ for some positive absolute constant $\alpha _2$.
Then by (\[7.3\]), (\[7.5\]), (\[7.6\]) and Stoke’s theorem, we have$$\begin{aligned}
\label{7.7}
& &\frac 1{a^2}\int\nolimits_{B_t(\overline{x}_0,a)\backslash
B_t(\overline{x}_0,\frac a2)}\left[ V\right] \wedge \omega _t\nonumber\\
& \geq &\frac 1{4a^2}\int\nolimits_{B_t(\overline{x}_0,a)\backslash
B_t(\overline{x}_0,\frac a2)}\left[ V\right] \wedge \sqrt{-1}\partial
\overline{\partial }\left( d_t^2\right) \nonumber\\
& = &\frac{\sqrt{-1}}4\left( \frac 1{a^2}\int\nolimits_{\{d_t=a\}}
\left[ V\right] \wedge \overline{\partial }\left( d_t^2\right) -
\frac 1{4\cdot \left( \frac a2\right) ^2}\int\nolimits_{\{d_t=\frac a2\}}
\left[ V\right] \wedge \overline{\partial }
\left( d_t^2\right) \right)\nonumber\\
& = &\frac{\sqrt{-1}}8\left( \frac 1{a^2}\int\nolimits_{\{d_t=a\}}
\left[ V\right] \wedge \overline{\partial }\left( d_t^2\right) -
\frac 1{2\cdot \left( \frac a2\right) ^2}\int\nolimits_{\{d_t=\frac a2\}}
\left[ V\right] \wedge \overline{\partial }\left( d_t^2\right) \right)\nonumber\\
& &+\frac{\sqrt{-1}}{8a^2}\int\nolimits_{\{d_t=a\}}\left[ V\right]
\wedge \overline{\partial }\left( d_t^2\right)\nonumber\\
& \geq &\frac{\sqrt{-1}}{8a^2}\int\nolimits_{\{d_t=a\}}\left[ V\right]
\wedge \overline{\partial }\left( d_t^2\right)\nonumber\\
& \geq &\frac{\alpha _2}8\mbox{mult}\,\left( \left[ V\right] ,\overline{x}_0\right)\end{aligned}$$ for $0<a<\frac \pi 8\sqrt{\frac t{\alpha _1}}.$
For the function $f\in P(M)$, let $\widetilde{x}_0$ be a point close to $\overline{x}_0$ such that $f(\widetilde{x}_0)\neq 0$. By definition, for any $\delta >0$, there exists a constant $C(\delta )>0$ such that $$\label{7.8}\left| f(x)\right| \leq C(\delta )\left( 1+d_0^{\deg (f)+\delta
}(x,\widetilde{x}_0)\right) \qquad \mbox{on} \quad M\ .$$ By equation (\[2.1\]) and estimate (\[7.1\]), we have $$\begin{aligned}
\frac{\partial g_{\alpha \overline{\beta }}(\cdot ,t)}
{\partial t}&\geq&-R(\cdot ,t)g_{\alpha \overline{\beta }}(\cdot ,t)
\nonumber \\
& \geq &-\frac C{1+t}g_{\alpha \overline{\beta }}(\cdot ,t)\ ,\nonumber\end{aligned}$$which implies that $$g_{\alpha \overline{\beta }}(\cdot ,0)\leq (1+t)^Cg_{\alpha \overline{\beta }}(\cdot ,t)\qquad \mbox{for any} \quad t>0\ .$$ Hence, (\[7.8\]) becomes $$\label{7.9}
\left| f(x)\right| \leq C(\delta )\left\{ 1+\left[ (1+t)^{\frac
C2}d_t(x,\widetilde{x}_0)\right] ^{\deg (f)+\delta }\right\} \qquad
\mbox{on} \quad M\ .$$
We now fix $t=\frac{\alpha _1}{\pi ^2}4^{K+8}$ for each positive interger $K$. Set $$v_K(x)=\int\nolimits_{B_t(\widetilde{x}_0,2^K)}-G_t^{(K)}(x,y)\bigtriangleup
_t\log \left| f(y)\right| ^2\cdot \omega _t^2(y)\ ,$$ where $G_t^{(K)}$ is the positive Green function with value zero on the boundary $\partial B_t(\widetilde{x}_0,2^K)$ with respect to the metric $g_{\alpha \overline{\beta }}(\cdot ,t)$. The function $\log \left| f\right|^2-v_K$ is then harmonic on $B_t(\widetilde{x}_0,2^K)$. From the maximum principle and (\[7.9\]), we have $$\begin{aligned}
\label{7.10}
\log \left( \left| f(\widetilde{x}_0)\right| ^2\right) -
v_K(\widetilde{x}_0)&\leq&\sup \limits_{x\in \partial B_t
(\widetilde{x}_0,2^K)}\log \left| f(x)\right| ^2 \nonumber\\
&\leq&C_{19}K\left( \deg (f)+\delta \right) +C^{\prime }(\delta )\end{aligned}$$ for some positive constants $C_{19}$, $C^{\prime }(\delta )$ independent of $K,\ f,$ and $\overline{x}_0.$
On the other hand, since the volume growth condition (i) is preserved for all times, by virtue of (\[6.4\]) (c.f. Proposition 1.1 in [@Mo1]), we have $$\begin{aligned}
-v_K(\widetilde{x}_0)
& \geq & \frac 1{C_9}\int\nolimits_{B_t(\widetilde{x}_0,
2^K)}\frac 1{d_t^2(x,\widetilde{x}_0)}
\bigtriangleup _t\log \left| f(x)\right| ^2\cdot \omega _t^2(x)
\nonumber\\
& \geq & \frac 1{C_9}\sum\limits_{j=1}^K\left( \frac 1{2^j}
\right)^2\int\nolimits_{B_t(\widetilde{x}_0,2^j)\backslash
B_t(\widetilde{x}_0,2^{j-1})}\bigtriangleup _t
\log \left| f(x)\right| ^2\cdot \omega _t^2(x)\ . \nonumber\end{aligned}$$ Then, by (\[7.7\]) and the fact that $\widetilde{x}_0$ is arbitrarily close to $\overline{x}_0,$$$\label{7.11}
-v_K(\widetilde{x}_0)\geq C_{20}\,K\,\mbox{mult}\,\left(
\left[ V\right] , \overline{x}_0\right)$$ for some positive constant $C_{20}$ independent of $K,\ f$ and $\overline{x}_0.$
Therefore, by combining (\[7.10\]) and (\[7.11\]) and letting $K\rightarrow +\infty $ and then $\delta \rightarrow 0$, we obtain $$\mbox{mult}\,\left( \left[ V\right] ,\overline{x}_0\right)
\leq C_{21}\deg (f)$$ where $C_{21}$ is some positive constant independent of $f$ and $\overline{x}_0.$ $\Box $
A modified version of the proof of Proposition 7.1 gives the uniform bound on the number of irreducible components of $[V].$
Suppose $(M,g_{\alpha \overline{\beta }})$ is a complete noncompact Kähler surface as assumed in Proposition 7.1. Let $f$ be a holomorphic function of polynomial growth, $$\left[ V\right] =\frac{\sqrt{-1}}{2\pi }
\partial \overline{\partial }\log\left| f\right| ^2$$ be the corresponding zero divisor determined by $f$. Then the number of irreducible components of $[V]$ is not bigger than $C\deg(f)$ for the same positive constant $C$ as in Proposition 7.1.
Let $g_{\alpha \overline{\beta }}(\cdot ,t)$ be the evolving metric to the Ricci flow with $g_{\alpha \overline{\beta }}(\cdot)$ as the initial metric and $[V_1],\ [V_2],\ \cdots ,\ [V_l]$ be any $l$ distinct irreducible components of $\left[ V\right] $. Fix a constant $a>0$ such that the intersection of the smooth points of $\left[ V_i \right]$ with $B_0(\overline{x}_0,a)$ is nonempty for each $0 \leq i \leq l.$
Choose $t=\frac{\alpha _1}{\pi ^2}4^{K+8}a^2$ for each positive integer $K$. As the manifold $M$ is Stein by Theorem 5.1, each $\left[ V_i\right] $ must be noncompact. Hence, for $j=1,2,\cdots ,K$, we have $$\left[ V_i\right] \cap \left( B_t(\overline{x}_0,2^ja)\left\backslash
B_t( \overline{x}_0,2^{j-1}a)\right. \right) \neq \emptyset$$ and there exists a point $x_j\in \left[ V_i\right] $ with $d_t(x_j,\overline{x}_0)=\frac 322^{j-1}a$ in the middle of $B_t(\overline{x}_0,2^ja)\left\backslash B_t(
\overline{x}_0,2^{j-1}a)\right.$. The triangle inequality says $$B_t(x_j, 2^{j-2}a)\subset \left( B_t(\overline{x}_0,2^ja)\left\backslash
B_t(\overline{x}_0,2^{j-1}a)\right. \right) \ .$$ Applying a slight variant of (\[7.7\]) to $\left[ V_i\right] $, we have $$\begin{aligned}
\frac 1{\left( 2^{j-2}a\right) ^2}\int\nolimits_
{B_t(x_j, 2^{j-2}a)}\left[ V_i\right] \wedge \omega _t
& \geq & \frac{\alpha _2}8 \mbox{mult}\,\left(
\left[ V_i\right] ,x_j\right) \nonumber\\
& \geq & \frac{\alpha _2}8\ . \nonumber\end{aligned}$$ Since $\sum\limits_{i=1}^l\left[ V_i\right] $ is only a part of the divisor $\left[ V\right] $, we get $$\frac 1{\left(2^{j-2}a\right) ^2}\int\nolimits_{B_t(\overline{x}
_0,2^ja)\left\backslash B_t(\overline{x}_0,2^{j-1}a)\right. }
\bigtriangleup_t\log \left| f(x)\right| ^2\cdot \omega _t^2(x)
\geq \frac{\alpha _2}8\, l\ .$$
The subsequent argument is then exactly as in the proof of Proposition 7.1. In the end, we have $$C_{20}K\cdot l\leq -\log \left( \left| f\left( \widetilde{x}_0\right)
\right| ^2\right) +C_{19}K\left( \deg (f)+\delta \right) +
C^{\prime }(\delta)\ .$$ Letting $K\rightarrow +\infty $ and then $\delta \rightarrow 0$, we get the desired estimate. $\Box $
§8. Proof of the main theorem {#proof-of-the-main-theorem .unnumbered}
=============================
In this section, we will basically follow the approach of Mok in [@Mo1], [@Mo3] to accomplish the proof of the main theorem. Let $M$ be a Kähler surface as assumed in the Main Theorem. Recall that $P(M)$ stands for the algebra of holomorphic functions of polynomial growth on $M$. Let $R(M)$ be the quotient field of $P(M)$. By an abuse of terminology, we will call it the field of rational functions on $M$.
In the previous section, we showed that there exist two functions $f_1,\ f_2\in P(M)$ giving local holomorphic coordinates at any given point $x \in M$, and that the algebra $P(M)$ separates points on $M$. Moreover, we obtained the following basic multiplicity estimate $$\label{8.1}
\mbox{mult}\,\left( \left[ V\right] ,x\right) \leq C\deg (f)$$ for all $x\in M$ and $f\in P(M)$, where $$\left[ V\right] =\frac{\sqrt{-1}}{2\pi }\partial
\overline{\partial }\log \left| f\right| ^2$$ is the zero divisor of $f$ and $C$ is a constant independent of $f$ and $x$. Thus, by combining these facts with the classical arguments of Poincaré and Siegel, we have (c.f. the proof of Proposition 5.1 in [@Mo1]) $$\label{8.2}
\dim {}_{{\C}}H_p\leq 10^3Cp^2\ ,$$ where $H_p$ denotes the vector space of all holomorphic functions with degree $\leq p$, and the field of rational functions $R(M)$ is a finite extension field over ${\C}(f_1,f_2)$ for some algebraically independent holomorphic functions $f_1,\ f_2\in P(M)$ over $\C$. By the primitive element theorem, we can then write $$R(M)={\C}\left( f_1,f_2,\frac{f_3}{f_4}\right)$$ for some $f_3,\ f_4\in P(M)$.
Now, consider the mapping $F:M\rightarrow {\C}^4 $ defined by $$F=\left( f_1,f_2,f_3,f_4\right) \ .$$ Since $R(M)$ is a finite extension field of ${\C}(f_1\ f_2)$, $f_3$ and $f_4$ satisfy equations of the form $$f_3^p+\sum\limits_{j=0}^{p-1}P_j(f_1,f_2)f_3^j=0\ ,$$ $$f_4^q+\sum\limits_{j=0}^{q-1}Q_j(f_1,f_2)f_4^j=0\ ,$$ where $P_j(w_1,w_2)$, $Q_j(w_1,w_2)$ are rational functions of $w_1,\ w_2$. After clearing denominators, we see that $f_1, f_2, f_3, f_4$ satisfy polynomial equations $$P(f_1,f_2,f_3,f_4) = 0 \qquad \mbox{and} \qquad
Q(f_1,f_2,f_3,f_4) = 0.$$ Let $Z_0$ be the subvariety of $\C^4$ defined by $$Z_0=\left\{ \left( w_1,w_2,w_3,w_4\right)\in \C^4 \left| \
\begin{array}{l}
\bigbreak P(w_1,w_2,w_3,w_4) = 0\ \\
Q(w_1,w_2,w_3,w_4) = 0
\end{array}
\right. \right\} \ ,$$ and let $Z$ be the connected component of $Z_0$ containing $F(M)$. It is clear that $\dim {}_{{\C}}Z=2.$
In the following we will show that $F$ is an “almost injective” and “almost surjective” map to $Z$ and we can desingularize $F$ to obtain a biholomorphic map from $M$ onto a quasi–affine algebraic variety by adjoining a finite number of holomorphic functions of polynomial growth.
First of all, we claim that $Z$ is irreducible and $F$ is “almost injective”, i.e., there exists a subvariety $V$ of $M$ such that $F|_{M\backslash V}:M\backslash V\rightarrow Z$ is an injective locally biholomorphic mapping. Indeed, take $V$ to be the union of $F^{-1}(\mbox{Sing}(Z))$ and the branching locus of $F$, here Sing(Z) denotes the singular set of Z. It is clear that $F$ is locally biholomorphic on $M\backslash V$. That $F$ is also injective there follows from the fact that $P(M)$ separates points and $f_1,...,f_4$ generate $P(M)$. To see the irreducibility of $Z$, note that $M \backslash F^{-1}(\mbox{Sing}(Z))$ is connected and hence $\overline{F(M \backslash F^{-1}(\mbox{Sing}(Z))}$ is irreducible (as its set of smooth points is connected). Since $F(M) \subset \overline{F(M \backslash F^{-1}(\mbox{Sing}(Z))}$, by the definition of $Z$, it must be irreducible. Next, we come to the “almost surjectivity” of $F$, i.e., there exists an algebraic subvariety $T$ of $Z$ such that $F(M)$ contains $Z\backslash T$. The method of Mok[@Mo1] in proving the almost surjectivity of $F$ is to solve an ideal problem for each $x \in Z \backslash T_0$ missed by $F$, where $T_0$ is some fixed algebraic subvariety of $Z$ containing the singular set of $Z$. The solution of the ideal problem gives a holomorphic function $f_{x}\in P(M)$ with degree bounded independent of $x$ which will correspond to a rational function on $\C^4$ with pole set passes through $x$. Then, the almost surjectivity of $F$ follows. Otherwise, one could select an infinite number of linearly independent $f_x$’s contradicting the finite dimensionality of the space of holomorphic functions with polynomial growth of some fixed degree, c.f. (\[8.2\]).
In [@Mo1], Mok used the solution $u$ of the Poincaré–Lelong equation as the weight function in the Skoda’s estimates for solving the ideal problem. In his case, because of his curvature quadratic decay condition, the growth of $u$ is bounded both from above and below by the logarithm of the distance function on $M$. This does not work in our case because we do not have the luxury of the lower bound of $u$. However, thanks to the Steinness of $M$ by Theorem 5.1, we can adapt the argument of Mok in [@Mo3] to choose another weight function by resorting to Oka’s theory of pseudoconvex Riemann domains.
Before we carry out the above procedures in proving the almost surjectivity of $F$. We first need to construct a nontrivial holomorphic $(2,0)$ vector field of polynomial growth on $M$.
Consider the anticanonical line bundle, ${\bf K} ^{-1}$, on $M$ equipped with the induced Hermitian metric, its curvature form $\Omega ({\bf K} ^{-1})$ is then simply the Ricci form of $M$. Let $u$ be the strictly plurisubharmonic function of logarithmic growth obtained in Proposition 6.2. For any given point $\overline{x}_0\in M$, let $\left\{ z_1,z_2\right\} $ be local holomorphic coordinates at $\overline{x}_0$. Choose a smooth cutoff function $\eta $ supporting in this local holomorphic coordinate chart with value $1$ in a neighborhood of $\overline{x}_0$. We study the following $\overline{\partial }$ equation for the sections of ${\bf K} ^{-1}$ on $M$, $$\label{8.3}
\overline{\partial }S=\overline{\partial }\left( \eta \frac
\partial {\partial z_1}\wedge \frac \partial {\partial z_2}\right).$$ Clearly, we can choose $k>0$ large enough such that $$k\sqrt{-1}\partial \overline{\partial }u+\Omega({\bf K} ^{-1})+
3\sqrt{-1}\partial \overline{\partial }\left( \eta \log
\left( |z_1|^2+|z_2|^2\right) \right)>0\ .$$ Then by the standard $L^2$ estimate of $\overline{\partial }$ operator on Hermitian holomorphic line bundles (c.f. Theorem 1.2 in [@Mo3]), equation (\[8.3\]) has a smooth solution $S(x)$ satisfying the estimate $$\begin{aligned}
\label{8.4}
& &\int_M\left| S\right| ^2e^{-ku-3\eta \log \left(
|z_1|^2+|z_2|^2\right) }\omega ^2 \nonumber\\
& \leq & C\int_M\left| \overline{\partial }\left(
\eta \frac \partial {\partial z_1}\wedge \frac \partial
{\partial z_2}\right) \right| ^2e^{-ku-3\eta \log \left(
|z_1|^2+|z_2|^2\right) }\omega ^2\nonumber\\
& < & + \infty\end{aligned}$$ for some positive constant $C$. Recall the Poincaré–Lelong equation for the section $S(x)$ of the anticanonical line bundle $\bf{K}^{-1}$, $$\frac{\sqrt{-1}}{2\pi }\partial \overline{\partial }\log \left| S\right|
^2=[V]-\frac 1{2\pi }\mbox{Ric} \qquad \mbox{on} \quad M\ ,$$ where $\left[ V\right] $ is the zero divisor of $S(x)$ (c.f. [@Mo3]). Thus, $\log \left| S\right| ^2+u$ is subharmonic and so is $\left| S\right|^2e^u=\exp (\log \left| S\right| ^2+u).$ Since $M$ has positive Ricci curvature and maximal volume growth, we can apply the mean value inequality of subharmonic functions, (\[8.4\]) and the fact that $u$ has logarithmic growth to show that $S(x)$ is of polynomial growth. Set $$v=\eta \left( \frac \partial {\partial z_1}\wedge \frac \partial
{\partial z_2}\right) -S\ .$$ Then, $v$ is a nontrivial holomorphic $(2,0)$ vector field over $M$ with polynomial growth we desired.
Now, for any $f_i,f_j\in \{f_1,f_2,f_3,f_4\}$ with $df_i\wedge
df_j\not \equiv 0$, we can choose the point $\overline{x}_0$ in the above construction of $v$ so that the holomorphic function $f_{ij}$ defined by $$\label{8.5}
f_{ij}=\left\langle v,df_i\wedge df_j\right\rangle$$ is a nontrivial holomorphic function of polynomial growth. Here, we have used the fact that $\left\| df_i\wedge df_j\right\| $ grows at most polynomially by the gradient estimate of harmonic functions of Yau [@Y1]. It is obvious that the zero divisor of $df_i\wedge df_j$ is contained in the zero divisor of $f_{ij}$, for which we denote by $V_0$. Since $M$ is Stein, the same is also true for $M \backslash V_0$.
Denote by $\pi_{ij}:Z\rightarrow \C^2$ the projection map given by $(w_1,w_2,w_3,w_4) \mapsto (w_i,w_j)$. Then, the map $$\rho =\pi _{ij}\circ F:M\backslash V_0\rightarrow \C^2$$ realises the Stein manifold $M\backslash V_0$ as a Riemann domain of holomorphy over $\C^2.$ Let $\delta (x)$ be the Euclidean distance to the boundary as in Oka [@O]. Then, $-\log \delta $ is a plurisubharmonic function on $M\backslash V_0$ by a theorem of Oka [@O]. $\delta (x)$ will be used in the weight function of the Skoda’s estimate mentioned above. It is essential to estimate it from below in terms of the intrinsic distance $d(x,x_0)$ on $M$.
There exist positive constants $p$ and $C$ such that $$\delta (x)\geq C\left| f_{ij}(x)\right| ^2\left( d(x,x_0)+1\right) ^{-p}\ .$$
Let $v_i,\ v_j$ be two holomorphic vector fields on $M \backslash V_0$ defined by $$\left\langle v_k,df_l\right\rangle =\delta _{kl}\ ,\qquad k,l=i,j\ .$$ By the Cramer’s rule, we have $$\left| v_k\right| \leq \frac{\left| df_i\right| +\left|
df_j\right| }{\left|df_i\wedge df_j\right| }\leq
\frac{\left| v\right| \left( \left| df_i\right|+\left|
df_j\right| \right) }{\left| f_{ij}\right| }\leq C_{22}
\frac{\left(d(x,x_0)+1\right) ^{k_1}}{\left| f_{ij}(x)\right| }
\quad \mbox{on}\quad M\backslash V_0,$$ for $k=i,j$ and some positive constants $C_{22}$, $k_1$.
Since $f_{ij}$ is of polynomial growth, $|\nabla f_{ij}|$ is also of polynomial growth by the gradient estimate of Yau, i.e.,$$\max \left\{ f_{ij}(x),\left| \nabla f_{ij}(x)\right| \right\} \leq
C_{23}\left( d(x,x_0)+1\right) ^{k_2} \qquad \mbox{on} \quad M\ ,$$ for some positive constants $C_{23}$ and $k_2$. Take $x \in M\backslash V_0$, then for any $y\in B\left( x,\left| f_{ij}(x)\right| \left/ 3C_{23}\left(
d(x,x_0)+1\right) ^{k_2}\right. \right) $, we have $$\begin{aligned}
\label{8.6}
\left| f_{ij}(y)\right|&\geq&\left| f_{ij}(x)\right| -C_{23}
\left( d(x,x_0)+2\right) ^{k_2}\cdot \frac{\left| f_{ij}(x)
\right| }{3C_{23}\left( d(x,x_0)+1\right) ^{k_2}}
\nonumber\\
& \geq & \frac 12\left| f_{ij}(x)\right| \ .\end{aligned}$$ This implies $$B\left( x,\left| f_{ij}(x)\right| \left/ 3C_{23}\left(
d(x,x_0)+1\right) ^{k_2}\right. \right) \subset M\backslash V_0$$ and $$\label{8.7}
\left| v_k(y)\right| \leq 2C_{22}\frac{\left( d(x,x_0)+1\right)
^{k_1}}{\left| f_{ij}(x)\right| }\ ,$$ for all $y\in B\left( x,\left| f_{ij}(x)\right| \left/ 3C_{23}\left(
d(x,x_0)+1\right) ^{k_2}\right. \right) $, $k=i,j$.
By the definition of $\delta (x)$, it suffices to prove $$\begin{aligned}
\label{8.8}
& &\rho \left( B\left( x,\left| f_{ij}(x)\right| \left/ 6C_{23}
\left( d(x,x_0)+1\right) ^{k_2}\right. \right) \right) \nonumber\\
& \supset & B_{\C^2}\left( \rho(x),C_{24}\left| f_{ij}(x)\right|^2
\left/ \left( d(x,x_0)+1\right) ^{k_1+k_2}\right. \right),\end{aligned}$$ for some positive constant $C_{24}$. Here, $B_{{\C^2}}(a,r)$ denotes the Euclidean ball in $\C^2$ with center $a$ and radius $r$.
Consider the real vector field $$\begin{aligned}
\xi & = & \alpha _i\left( 2\mbox{Re}\,\left( v_i\right) \right) +
\alpha _j\left( 2\mbox{Re}\,\left( v_j\right) \right) +
\beta _i\left( 2\mbox{Im}\,\left( v_i\right) \right) +
\beta _j\left( 2\mbox{Im}\,\left( v_j\right) \right) \nonumber\\
& = & \left( \alpha _i-\sqrt{-1}\beta _i\right) v_i+
\left( \alpha _j-\sqrt{-1}\beta _j\right) v_j+
\left( \alpha _i+\sqrt{-1}\beta _i\right) \overline{v}_i \nonumber\\
& & +\left( \alpha _j+\sqrt{-1}\beta _j\right) \overline{v}_j \nonumber\end{aligned}$$ with $\left| \alpha _i\right| ^2+\left| \alpha _j\right| ^2+
\left| \beta_i\right| ^2+\left| \beta _j\right| ^2=1$. Clearly $\xi $ also satisfies (\[8.7\]). Let $\gamma _\xi (\tau )$ be the integral curve in $M$ defined by $\xi$ and passes through $x$, i.e., $$\label{8.9}
\left\{
\begin{array}{l}
\bigbreak \displaystyle \frac{d\gamma _\xi (\tau )}{d\tau }=\xi \\ \gamma
_\xi (0)=x\ .
\end{array}
\right.$$ We have $$\frac{d\left( f_i\circ \gamma _\xi (\tau )\right) }
{d\tau }=\left\langle \xi,df_i\right\rangle =
\alpha _i-\sqrt{-1}\beta _i\ ,$$ $$\frac{d\left( f_j\circ \gamma _\xi (\tau )\right) }
{d\tau }=\left\langle \xi,df_j\right\rangle =
\alpha _j-\sqrt{-1}\beta _j\ ,$$ and $$\label{8.10}
\left| f_i\circ \gamma _\xi (\tau )-f_i(x)\right| ^2+\left|
f_j\circ \gamma _\xi (\tau )-f_j(x)\right| ^2=\tau ^2\ .$$ Note that (\[8.10\]) implies that $\gamma _\xi (\tau )$ cannot always stay in $$B\left(x,\left|f_{ij}(x)\right|\left/6C_{23}\left(d(x,x_0)+1
\right) ^{k_2}\right.\right),$$ otherwise $F=(f_1,f_2,f_3,f_4)$ would become unbounded in this ball. Denote by $\tau_0$ the first time when $\gamma _\xi (\tau )$ touches the boundary $$\partial B\left( x,\left| f_{ij}(x)\right| \left/
6C_{23}\left( d(x,x_0)+1\right) ^{k_2}\right. \right),$$ it is easy to see that $$\begin{aligned}
\frac{\left|
f_{ij}(x)\right| }{6C_{23}\left( d(x,x_0)+1\right) ^{k_2}}
& \leq & \mbox{the length of} \; \gamma _\xi \; \mbox{on} \;
[0,\tau_0] \nonumber\\
& \leq & 2C_{22}\int_0^{\tau _0}\frac{\left( d(x,x_0)+1\right)
^{k_1}}{\left| f_{ij}(x)\right| }d\tau \qquad (\mbox{by} \; (8.7))
\nonumber\\
& = & 2C_{22}\tau _0\frac{\left( d(x,x_0)+1\right) ^{k_1}}{\left|
f_{ij}(x)\right| }\ .\nonumber\end{aligned}$$ Thus, $$\label{8.11}
\tau _0\geq \frac{\left| f_{ij}(x)\right| ^2}{
2C_{22}C_{23}6\left( d(x,x_0)+1\right) ^{k_1+k_2}}\ .$$
Note that the integral curve $\gamma_{\xi}$ projects to straight line passing through $f(x)$ by $\rho$. Thus, when $(\alpha_i, \alpha_j, \beta_i, \beta_j)$ runs through the unit sphere in $\C^2$, the collection of integral curves $\gamma_{\xi}$ inside $$B\left(x,\left|f_{ij}(x)\right|\left/6C_{23}\left(d(x,x_0)+1
\right) ^{k_2}\right.\right)$$ will project, by $\rho$, onto the Euclidean ball $$B_{\C^2}\left( \rho(x),\left| f_{ij}(x)\right|^2
\left/ 2C_22C_26 \left( d(x,x_0)+1\right) ^{k_1+k_2}\right. \right).$$ This proves (\[8.8\]) and hence the lemma. $\Box $
Now, we are ready to prove the almost surjectivity of the holomorphic map $F:M\rightarrow {\C^4}$. For each $1\leq i,j\leq 4$, since $f_{ij}$ is a holomorphic function of polynomial growth and $R(M)$ is generated by $f_1,...,f_4$, we can write $$f_{ij}(x) = H_{ij}\left( f_1(x),f_2(x),f_3(x),f_4(x)\right) \qquad
\mbox{on} \quad M,$$ for some rational function $H_{ij}$ on ${\C^4}$. Let $T_0$ be the union of the singular set of $Z$ and the zero and pole sets of all $H_{ij},\
1\leq i,j\leq 4$. For any $b\in Z\backslash (F(M)\cup T_0)$, there exist fixed $\{i,j\}\subset \{1,2,3,4\}$ such that the projection $\pi _{ij}:Z\rightarrow {\C^2}$ is nondegenerate at $b$. Since $Z$ is algebraic, the number of points contained in $\pi _{ij}^{-1}\circ \pi _{ij}(b)$ is less than some fixed integer $K$ depending only on $Z$. By interpolation, there is a polynomial $h_b$ of degree $\leq K$ on ${\C^4}$ such that $h_b(b)=1$, and $h_b(w)=0$ for all $w\in (\pi _{ij}^{-1}\circ \pi _{ij}(b))\backslash \{b\}$. We now solve on $M\backslash V_0$ the ideal problem with unknown holomorphic functions $g_i$ and $g_j$, $$\label{8.12}
\left( f_i-b_i\right) g_i+\left( f_j-b_j\right) g_j=\left(
h_b\circ F\right) ^4\ ,$$ where $b=(b_1,b_2,b_3,b_4).$
Let $$\psi =-n_1\log \delta +n_2\log (1+|f_i|^2+|f_j|^2),$$ where the integers $n_1,\ n_2 > 0$ will be determined later. Clearly, $\psi $ is a strictly plurisubharmonic function on $M\backslash V_0$. By the estimate of Skoda (c.f. Theorem 1.3 in [@Mo3]), given any $\alpha >1$, there exists a solution $\{g_i,g_j\}$ to (\[8.12\]) such that $$\begin{aligned}
\label{8.13}
& &\int\limits_{M\backslash V_0}\frac{\left( \left| g_i\right|^2+
\left| g_j\right| ^2\right) e^{-\psi }}{\left( \left| f_i-b_i\right|^2+
\left| f_j-b_j\right| ^2\right) ^{2\alpha }}\rho ^{*}dV_E \nonumber\\
& \leq & C_\alpha\int\limits_{M\backslash V_0}\frac{\left( h_b\circ F\right) ^8
e^{-\psi }}{\left( \left| f_i-b_i\right| ^2+\left| f_j-b_j\right| ^2
\right) ^{2\alpha +1}}\rho^{*}dV_E\ ,\end{aligned}$$ provided the right hand side is finite. Recall that $\rho = \pi_{ij} \circ F$ and here $$\rho ^{*}dV_E=\pm \left( \frac{\sqrt{-1}}2\right) ^2df_i\wedge
\overline{df}_i\wedge df_j\wedge \overline{df}_j$$ denotes the pull back of the Euclidean volume element of ${\C^4.}$
Let $\{\zeta _1,\zeta _2,\cdots ,\zeta _m\}=\pi _{ij}^{-1}\circ \pi _{ij}(b)$ $(m<K)$ be the preimages of $\pi _{ij}(b)$ with $\zeta _1=b$. And let $U_k\ ( 1\leq k\leq m )$ be disjoint small neighborhoods of $\zeta _k\
( 1\leq k\leq m )$. The integral on the right hand side of (\[8.13\]) can be decomposed into three parts $$\begin{aligned}
RHS
& = &\left( \int_{F^{-1}(U_1)}+\sum_{k=2}^m\int_{F^{-1}(U_k)}+
\int_{\left( M\backslash V_0\right) \left\backslash
\cup _{k=1}^mF^{-1}(U_k)\right. }\right) \nonumber\\
& &\frac{\left( h_b\circ F\right) ^8e^{-\psi }}{\left(
\left| f_i-b_i\right| ^2+\left| f_j-b_j\right| ^2\right) ^{2\alpha +1}}
\rho ^{*}dV_E \nonumber\\
& = & I_1+I_2+I_3\ .\nonumber\end{aligned}$$
For $I_1$, since $h_b(b)=1$ and $\delta (x)\leq \left( \left| f_i-b_i\right|
^2+\left| f_j-b_j\right| ^2\right) ^{\frac 12}$, we can choose $n_1\geq
2(2\alpha +1)$ and $U_1$ small enough so that the integral $I_1$ is finite.
For $I_2$, since $h_b(\zeta _k)=0$ for $2\leq k\leq m,$ we can choose $\alpha $ such that $2(2\alpha +1)<8$ (e.g. $\alpha =1.4$). Then the integral $I_2$ is also finite.
For $I_3$, we choose $n_2\geq 10+8K+n_1$, where $h_b$ is of degree $\leq K$. Then $I_3$ can be estimated as $$I_3\leq C_{25}\int_{{\C^2}}\frac 1{\left( 1+|w|^2\right) ^{10}}dV_E
<+\infty\ .$$
Hence, we have obtained a solution $\{g_i,g_j\}$ of the ideal problem (\[8.12\]) such that $$\label{8.14}
\int\limits_{M\backslash V_0}\frac{\left( \left| g_i\right|
^2+\left| g_j\right| ^2\right) e^{-\psi }}{\left( \left| f_i-b_i\right|
^2+\left| f_j-b_j\right| ^2\right) ^{2\alpha }}\rho ^{*}dV_E<+\infty \ .$$ Recall from Lemma 8.1 and (\[8.5\]), we have $$\delta (x)\geq C\left| f_{ij}(x)\right| ^2\left( d(x,x_0)+1\right) ^{-p}$$ and$$\begin{aligned}
\rho ^{*}dV_E&=&\pm \left( \frac{\sqrt{-1}}2\right) ^2df_i\wedge df_j\wedge \overline{df}_i\wedge \overline{df}_j\nonumber\\
&\geq&\frac{\left| f_{ij}\right| ^2}{\left| v\wedge \overline{v}\right| }
\omega ^2\nonumber\\
&\geq&C_{26}\frac{\left| f_{ij}(x)\right| ^2}{\left( d(x,x_0)+1\right) ^{k_3}}\omega ^2\nonumber\end{aligned}$$ for some positive constants $C_{26}$ and $k_3$. Substituting these two inequalities into (\[8.14\]) we get $$\label{8.15}
\int\limits_{M\backslash V_0}\frac{\left( \left| g_i\right|
^2+\left| g_j\right| ^2\right) \left| f_{ij}\right| ^{2+2n_1}}{\left(
d(x,x_0)+1\right) ^{k_4}}\omega ^2<+\infty $$ where $k_4$ is some positive constant independent of $b$ and $i,\ j.$ Then, both $g_if_{ij}^{n_1+1}$ and $g_jf_{ij}^{n_1+1}$ are locally square integrable. They can thus be extended holomorphically from $M\backslash V_0$ to $M$. By the mean value inequality of subharmonic functions, we deduce also that they are of polynomial growth with degree bounded by some positive number $k_5$ independent of $b$.
Now, recall that $R(M)={\C}(f{_1,}f{_2,}f{_3/}f{_4)}$, the holomorphic functions $g_if_{ij}^{n_1+1}$ and $g_jf_{ij}^{n_1+1}$ are thus rational functions of $f{_1,\ }f{_2,\ }f{_3}$ and $f{_4}$. Hence, we can regard the equation (\[8.12\]) as an equation on the variety $Z\subset {\C^4}$, namely $$\left( w_i-b_i\right) g_if_{ij}^{n_1+1}+\left( w_j-b_j\right)
g_jf_{ij}^{n_1+1}=H_{ij}^{n_1+1}\cdot h_b^4\ .$$ Since $h_b$ is a polynomial with $h_b(b)=1$ and the point $b$ lies outside of the zero and pole sets of $H_{ij}$, either $g_if_{ij}^{n_1+1}$ or $g_jf_{ij}^{n_1+1}$, when regarded as rational function on $\C^4$, must have a pole at $b$. Denote this function by $G^0$. Thus, $G^0$ is a rational function on $Z$ with $G^0(b)=\infty $ and $G^0\circ F$ is a holomorphic function on $M$ with degree $\leq k_5$. If $Z\backslash (F(M)\cup T_0 \cup \mbox{pole sets of}\, G^0)$ is empty, then we are done. Otherwise, pick any $b_1\in Z\backslash (F(M)\cup T_0 \cup \mbox{pole sets of}\, G^0)$ and repeat the same procedure to obtain a rational function $G^1$ on $Z$ with $G^1(b_1)=\infty $ and $G^1\circ F$ a holomorphic function on $M$ with degree $\leq k_5$. Proceeding this way, we obtain a sequence of points $\{b,b_1,b_2,\cdots \}$ and rational functions $\{G^0,G^1,G^2,\cdots \}$ such that $G^k(b_k)=\infty $ and $G^l$ regular at $b_k$ for $l<k$. So, $\{G^0,G^1,G^2,\cdots \}$ must be linearly independent over ${\C}$. Moreover, all of $G^k\circ F$ are holomorphic functions with degree $\leq k_5$. Hence, by (\[8.2\]), the above procedure must terminate in a finite number of steps. In other words, there exists an algebraic subvariety $T$ of $Z$ such that $F(M)\supset Z\backslash T$.
Moreover, $F$ establishes a quasi embedding from $M$ to a quasi–affine algebraic variety. Indeed, let $W=F^{-1}(T)$. By the definition of $T_0$ and the construction of $T$, we know that $W \supset V$, where $V$ is the union of the branching locus of $F$ and $F^{-1}(\mbox{Sing}(Z))$, and $W$ is the zero divisor of finitely many holomorphic functions of polynomial growth. Therefore, $F$ maps $M\backslash W$ biholomorphically onto $Z\backslash T.$
Finally, to complete the proof of our Main Theorem, we have to show that the mapping $F$ can be desingularized by adjoining a finite number of holomorphic functions of polynomial growth and taking normalization of the image.
We have constructed the mapping $F:M\rightarrow Z$ into an affine algebraic variety which maps $M\backslash W$ biholomorphically onto $Z\backslash T.$ Now, we use normalization of the affine algebraic variety $Z$ to resolve the codimension $1$ singularities of $F.$ Let $\mbox{Reg}(Z)$ denote the Zariski dense subset of $Z$ consisting of its regular points. It is well known that the normalization $\widetilde{Z}$ of $Z$ can be obtained by taking $\widetilde{Z}$ to be the closure of the graph of $\{Q_{1,}Q_2,\cdots ,Q_m\}$ on $\mbox{Reg}(Z)$ where $Q_i$ is a rational function which is holomorphic (or regular in the terminology of algebraic geometry) on $\mbox{Reg}(Z)$. The lifting of $F:M\rightarrow Z$ to $\widetilde{F}:M\rightarrow Z$ is then given by $\{f{_1,}f{_2,}f{_3,}f{_4,}Q_1\circ F,\cdots ,Q_m\circ F\}$ where, as was shown in proposition 8.1 of Mok [@Mo1], for each $i$, $Q_i\circ F$ can be holomorphically extended to the whole manifold $M$ as a holomorphic function of polynomial growth.
Write $F_0=F:M\rightarrow Z$ and denote $\widetilde{F}_0:M\rightarrow \widetilde{Z}$ the normalization of $F_0$. For any smooth point $x$ on the subvariety $W$, by using the $L^2$ estimates of the $\overline{\partial }$ operator as in Section 7, one can find two holomorphic functions $g_x^1,\ g_x^2$ of polynomial growth which give local holomorphic coordinates at $x$. Adding $g_x^1,\ g_x^2$ to the map $\widetilde{F}_0$, we get a new map $F_1=(\widetilde{F}_0,g_x^1,g_x^2):M\rightarrow Z_1\subset {{\C}^{6+m}}$, which is nondegenerate at $x$. Write the normalization of $F_1$ as $\widetilde{F}_1:M\rightarrow \widetilde{Z}_1$ and continue in this way to get holomorphic mappings $F_i:M\rightarrow Z_i$ and their normalizations $\widetilde{F}_i:M\rightarrow \widetilde{Z}_i$ such that$$\widetilde{W}_0\stackrel{\supset }{\neq }\widetilde{W}_1\stackrel{\supset }{\neq }\cdots \stackrel{\supset }{\neq }\widetilde{W}_i\stackrel{\supset }{
\neq }\cdots,$$ where $\widetilde{W}_i$ is the locus of ramification of $\widetilde{F}_i.$
Note that $\widetilde{W}_i$ contains no isolated point because $\widetilde{Z}_i$ is normal. Moreover, by Proposition 7.2, $W$ has only finite number of irreducible components because $W$ is the zero divisor of finitely many holomorphic functions of polynomial growth. This implies that the above procedure must terminate in a finite number of steps, say $l$. Thus, we get a biholomorphism $\widetilde{F}_l$ from $M$ onto its image $\widetilde{F}_l(M)\subset \widetilde{Z}_l$. The argument in our proof of the almost surjectivity shows that $\widetilde{F}_l(M)$ can miss at most finitely many irreducible subvarieties of $\widetilde{Z}_l$, say $\widetilde{T}_1^{(l)},\cdots ,\widetilde{T}_q^{(l)}$. If $\widetilde{F}_l(M)\cap \widetilde{T}_i^{(l)}\neq \emptyset $, then it must intersect $\widetilde{T}_i^{(l)}$ in a nonempty open set because $\widetilde{F}_l$ is open. We arrange $\widetilde{T}_i^{(l)}$ so that $\widetilde{F}_l(M)\cap \widetilde{T}_i^{(l)}=\emptyset $ for $1\leq i\leq p$ and $\widetilde{F}_l(M)\cap \widetilde{T}_i^{(l)}\neq \emptyset $ for $p+1\leq i\leq q$. Note that$\widetilde{F}_l(M)$ is a Stein subset of $\widetilde{Z}_l$ because $M$ is Stein by Theorem 5.1 and $\widetilde{F}_l$ maps $M$ biholomorphically onto its image. By Hartog’s extension theorem, every holomorphic function on $\widetilde{Z}_l\backslash \cup _{1\leq i\leq q}\widetilde{T}_i^{(l)}$ extends to $\widetilde{Z}_l\backslash \cup _{1\leq i\leq p}\widetilde{T}_i^{(l)}$. Hence, we get a biholomorphic map from $M$ onto a quasi–affine algebraic variety. Finally, recall that a classical theorem of Ramanujam [@R] in affine algebraic geometry says that an algebraic variety homeomorphic to ${\R^4}$ is biregular to ${\C^2}$. Combining this result of Ramanujam with Theorem 5.1, we deduce that $M$ is actually biholomorphic to ${\C^2}$. Therefore we have completed the proof of the Main Theorem.
[10]{} Andreotti, A. and Vesentini, E., [*Carleman estimates for the Laplacian–Beltrami operator on complex manifolds*]{}, Publ. Math. Inst. Hantes Études Sci. [**25**]{} (1965), 81-130.
Bishop, R. L., and Goldberg, S. I., [*Some implications of the generalized Gauss–Bennet theorem,*]{} Trans. Amer. Math. Soc. [**112**]{}(1964), 508–535.
Burns, D., Shnider, S. and Wells, R. O., [*On deformations of strictly pseudoconvex domain,* ]{}Invent. Math. [**46**]{} (1978), 237-253.
Cao, H. D., [*On Harnack’s inequalities for the Kähler–Ricci flow,* ]{}Invent. Math. [**109**]{} (1992), 247-263.
Cheeger, J., Gromov, M., and Taylor, M., [*Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds,* ]{}J. Differential Geometry [**17**]{} (1982), 15-53.
Chen, B. L. and Zhu, X. P., [*Complete Riemannian manifolds with pointwise pinched curvature*]{}, Invent. Math. [**140**]{} (2000), 423-452.
Chen, B. L. and Zhu, X. P., [*On complete noncompact Kähler manifolds with positive bisectional curvature*]{}, preprint.
Cheng, S. Y. and Yau, S. T., [*On the existence of a complete Kähler metric on non–compact complex manifolds and the regularity of Fefferman’s equation*]{}, Comm. Pure Appl. Math. [**33**]{} (1980), 507-544.
Fefferman, C., [*The Bergman kernel and biholomorphic mappings of pseudoconvex domains*]{}, Invent. Math. [**26**]{} (1974), 1-65.
Frankel, T., [*Manifolds with positive curvature*]{}, Pacific J. Math. [**11**]{} (1968), 157-170.
Freedman, M. H., [*The topology of four–dimensional manifolds*]{}, J. Differential Geom. [**17**]{} (1982), 357-453.
Greene, R. E., and Wu, H., [*C*]{}$^\infty $[* convex functions and manifolds of positive curvature*]{}, Acta. Math. [**137**]{} (1976), 209-245.
Greene, R. E., and Wu, H., [*Analysis on non–compact*]{} [*Kähler manifolds*]{}, Proc. Symp. Pure. Math. Vol [**30,**]{} Part[**II,**]{} Amer. Math. Soc. (1977).
Hamilton, R. S., [*Four–manifolds with positive curvature operator*]{}, J. Differential Geom. [**24**]{} (1986), 153-179.
Hamilton, R. S., [*A compactness property for solution of the Ricci flow*]{}, Amer. J. Math. [**117**]{} (1995), 545-572.
Hamilton, R. S., [*The formation of singularities in the Ricci flow*]{}, Surveys in Differential Geometry [**2**]{} (1995), 7-136, International Press.
Hörmander, L., [*L*]{}$^2$[*–estimates and existence theorems for the* ]{}$\overline{\partial }$[*–operator*]{}, Acta. Math. [**113**]{} (1965), 89-152.
Mabuchi, T., $^3$[*–actions and algebraic threefolds with ample tangent bundle*]{}, Nagoya Math. J. [**69**]{} (1978), 33-64.
Markeo, A., [*Runge families and increasing unions of Stein spaces*]{}, Bull. Amer. Math. Soc. [**82**]{} (1976), 787-788.
Mok, N., Siu, Y. T. and Yau, S. T., [*The Poincaré–Lelong equation on complete Kähler manifolds*]{}, Compositio Math. [**44**]{} (1981), 183-218.
Mok, N., [*An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebrari varieties*]{}, Bull. Soc. Math. France [**112**]{} (1984), 197-258.
Mok, N., [*The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature*]{}, J. Differential Geom. [**27**]{} (1988), 179-214.
Mok, N., [*An embedding theorem of complete Kähler manifolds of positive Ricci curvature onto quasi–projective varieties*]{}, Math. Ann. [**286**]{} (1990), 377-408.
Mori, S., [*Projective manifolds with ample tangent bundles*]{}, Ann. of Math. [**100**]{} (1979), 593-606.
Ni, L., Shi, Y. G., and Tam, L. F.,[* Poisson equation, Poincaré–Lelong equation and curvature decay on complete Kähler manifolds*]{}, preprint.
Oka, K., [*Domaines finis sans point critique intérieur*]{}, Jap. J. Math. [**27**]{} (1953), 97-155.
Ramanujam, C. P., [*A topological charaterization of the affine plane as an algebraic variety*]{}, Ann. of Math. [**94**]{} (1971), 69-88.
Schoen, R. and Yau, S. T., [*Lectures on differential geometry*]{}, in conference proceedings and Lecture Notes in Geometry and Topology, Volume [**1**]{}, International Press Publications, 1994.
Shi, W. X., [*Deforming the metric on complete Riemannian manifold*]{}, J. Differential Geometry [**30**]{} (1989), 223-301.
Shi, W. X., [*Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature*]{}, Bull. Amer Math. Soc. [**23**]{} (1990), 437-440.
Shi, W. X., [*Ricci deformation of the metric on complete noncompact Kähler manifolds*]{}, Ph. D. thesis Harvard University, 1990.
Shi, W. X., [*Ricci flow and the uniformization on complete noncompact Kähler manifolds*]{}, J. Differential Geom. [**45**]{} (1997), 94-220.
Siu, Y. T., [*Pseudoconvexity and the problem of Levi*]{}, Bull. Amer Math. Soc. [**84**]{} (1978), 481-512.
Siu, Y. T. and Yau, S. T., [*Compact Kähler manifolds of positive bisectional curvature*]{}, Invent. Math. [**59**]{} (1980), 189-204.
To, W. K., [*Quasi–projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions*]{}, Duke Math. J. [**63**]{} (1991), no. [**3,**]{} 745-789.
Yau, S. T., [*Harmonic functions on complete Riemannian manifolds*]{}, Comm. Pure Appl. Math. [**28**]{} (1975), 201-228.
Yau, S. T., [*Problem Section*]{}, Seminar on Diff. Geom. edit by S. T. Yau, Princeton Univ. Press, 1982.
Yau, S. T., [*A review of complex differential geometry*]{}, Proc. Symp. Pure Math., Vol [**52**]{}, Part [**II**]{}, Amer. Math. Soc. (1991).
[^1]: we are grateful to Professor L.F.Tam for this suggestion
|
---
abstract: 'Since the discovery of molecular resonances in $^{12}$C+$^{12}$C in the early sixties a great deal of research work has been undertaken to study $\alpha$-clustering. Our knowledge on physics of nuclear molecules has increased considerably and nuclear clustering remains one of the most fruitful domains of nuclear physics, facing some of the greatest challenges and opportunities in the years ahead. Occurrence of “exotic" shapes and Bose-Einstein Condensates in light $\alpha$-cluster nuclei are investigated. Various approaches of superdeformed/hyperdeformed shapes associated with quasimolecular resonant structures are discussed. The astrophysical reaction rate of $^{12}$C+$^{12}$C is extracted from recent fusion measurements at deep subbarrier energies near the Gamov window. Evolution of clustering from stability to the drip-lines is examined.'
address: |
$^a$Département de Recherches Subatomiques, Institut Pluridisciplinaire Hubert Curien, IN$_{2}$P$_{3}$-CNRS and Université de Strasbourg - 23, rue du Loess BP 28, F-67037 Strasbourg Cedex 2, France\
E-mail: christian.beck@iphc.cnrs.fr\
author:
- 'C. Beck$^a$'
title: Clusters in light stable and exotic nuclei
---
Introduction {#sec:1}
============
In the last decades, one of the greatest challenges in nuclear science is the understanding of the clustered structure of nuclei from both experimental and theoretical perspectives [@Cluster1; @Cluster2; @Cluster3; @Correlations; @Papka; @Oertzen06; @Horiuchi; @Freer18]. Our knowledge on physics of nuclear molecules has increased considerably and nuclear clustering remains one of the most fruitful domains of nuclear physics. Fig. 1 summarizes the different types of clustering [@Correlations]: most of these structures were investigated in an experimental context by using either some new approaches [@Papka] or developments of older methods. The search for resonant structures in the excitation functions for various combinations of light $\alpha$-cluster ($N$=$Z$) nuclei in the energy regime from the Coulomb barrier up to regions with excitation energies of $E_{x}$=20$-$50 MeV remains a subject of contemporary debate.
The question of how nuclear molecules may reflect continuous transitions from scattering states in the ion-ion potential to true cluster states in the compound systems is still unresolved. Clustering in light $\alpha$-like nuclei is observed as a general phenomenon at high excitation energy close to the $\alpha$-decay thresholds [@Correlations]. This exotic behavior has been perfectly illustrated 50 years ago by the famous ”Ikeda-diagram" for $N$=$Z$ nuclei [@Ikeda], which has been modified and recently extended by von Oertzen [@Oertzen06] for neutron-rich nuclei, as shown in the left panel of Fig. 2. Despite the early inception of cluster studies, it is only recently that radioactive ion beams experiments, with great helps from advanced theoretical works, enabled new generation of studies, in which data with variable excess neutron numbers or decay thresholds are compared to predictions with least or no assumptions of cluster cores. Some of the predicted but elusive phenomena, such as molecular orbitals or linear chain structures, are now gradually coming to light.
$^{12}$C nucleus ”Hoyle" state and BEC in light nuclei {#sec:2}
======================================================
The ground state of $^8$Be is the most simple and convincing example of $\alpha$-clustering in light nuclei as suggested by several theoretical models and appears naturally in [*ab initio*]{} calculations [@Correlations; @Freer18]. The picture of the $^8$Be nucleus prediced by the No Core Shell model [@Freer18] as being a dumbbell-shaped configuration of two $\alpha$ particles closely resembles the superdeformed (SD) shapes known to arise in heavier nuclei in the actinide mass region. This dumbbell-like structure gives rise to a rotational band, from which the moment of inertia is found to be commensurate with an axial deformation of 2:1. According to the schematic picture of the ”Ikeda-Diagram“ [@Ikeda] the nuclear cluster structure of $^{12}$C may also induce axial deformation close to 3:1 of a hyperdeformed (HD) shape. The large deformations of light $\alpha$-conjugate nuclei with SD, HD and linear-chain configurations are under discussion.\
The renewed interest in $^{12}$C was mainly focused to a better understanding of the nature of the so called ”Hoyle” state [@Hoyle54; @Freer14], the excited 0$^+$ state at 7.654 MeV that can be described in terms of a bosonic condensate, a cluster state and/or a $\alpha$-particle gas [@Tohsaki01]. The resonant ”Hoyle“ state [@Hoyle54] is regarded as the prototypical $\alpha$-cluster state whose existence is of great importance for the nucleosynthesis of $^{12}$C within stars. Further knowledge of the ”Hoyle” state [@Hoyle54; @Freer14] and its rotational excitations would help not only to understand the debated structure of the $^{12}$C nucleus in the “Hoyle state“, but also to determine the high-temperature (T $\approx$ 1 GK) reaction rate of the triple $\alpha$ process more precisely. The structure of this state has been thoroughly investigated with theoretically modelled with both [*ab initio*]{} and cluster models [@Correlations; @Freer18]. Much experimental progress has been achieved recently as far as the spectroscopy of $^{12}$C near and above the $\alpha$-decay threshold is concerned [@Kirsebom17]. More particularly, the the second 2$^{+}_{2}$ ”Hoyle” rotational excitation in $^{12}$C has been observed [@Zimmerman13]. Another experiment [@Marin14] populates a new state compatible with an equilateral triangle configuration of three $\alpha$ particles. Still, the structure of the ”Hoyle“ state remained controversial as experimental results of its direct decay into three $\alpha$ particles are found to be in disagreement until two experiments provided the most precise picture of how a $^{12}$C excited state decays into three He nuclei [@Kirsebom17; @Aquila17; @Smith17].\
In the study of Bose-Einstein Condensation (BEC), the $\alpha$-particle states in light $N$=$Z$ nuclei [@Tohsaki01], are of great interest. the search for an experimental signature of BEC in $^{16}$O is of highest priority. Furthermore, [*ab initio*]{} calculations [@Correlations; @Freer18] predict that nucleons are arranged in a tetrahedral configuration of $\alpha$ clusters. A state with the structure of the ”Hoyle” state [@Hoyle54] in $^{12}$C coupled to one $\alpha$ particle is predicted in $^{16}$O at about 15.1 MeV (the 0$^{+}_{6}$ state), the energy of which is $\approx$ 700 keV above the 4$\alpha$-particle breakup threshold. However, any state in $^{16}$O equivalent to the ”Hoyle" state [@Hoyle54] in $^{12}$C is most certainly going to decay exclusively by particle emission with very small $\gamma$-decay branches, thus, very efficient particle-$\gamma$ coincidence techniques [@Papka] will have to be used in the near future to search for them.
Nuclear molecules, $^{12}$C+$^{12}$C reaction rate and carbon burning in massive stars {#sec:3}
======================================================================================
The real link between superdeformation/hyperdeformation (SD/HD), nuclear molecules and $\alpha$-clustering [@Correlations] is of particular interest, since nuclear shapes with major-to-minor axis ratios of 2:1–3:1 have the typical ellipsoidal elongation for light nuclei. A further area where electromagnetic transitions would be of great interest in support of cluster models is in the case of the quasi-molecular resonances observed in the $^{12}$C+$^{12}$C reaction. The widths of these resonances were $\approx$ 100 keV, indicating the formation of a $^{24}$Mg intermediate system with a lifetime significantly longer than the nuclear crossing time. These resonances were subsequently interpreted as $^{12}$C+$^{12}$C cluster states. There has been only one valient attempt to directly observe transitions in this reaction [@Correlations] focussing on transitions between 10$^{+}$ and 8$^{+}$ resonant states at a bombarding energy E($^{12}$C) = 32 MeV chosen to populate a known and isolated 10$^+$ resonance. However, the measurement reported only an upper limit (for the radiative partial width of 1.2 $\pm$ 10$^{-5}$) given the extreme challenges of eliminating all background.\
The role of cluster configurations in stellar helium burning is well established and, discussion about the nature and the role of resonance structures that characterize the low-energy cross section of the $^{12}$C+$^{12}$C fusion process is underway in recent experimental investigations [@Spillane07; @Jiang13; @Bucher15; @Jiang18; @Tumino18a; @Zickefoose18; @Heine2018; @Fruet2018]. The resonant structures at very low energies have still been identified as molecular $^{12}$C+$^{12}$C configurations in the $^{24}$Mg compound nucleus [@Correlations; @Spillane07]. However, the reaction rate is calculated on the basis of an average cross section integrating over the molecular resonance components. Indications of possible existence of a pronounced low-energy resonance at E$_{cm}$ = 2.14 MeV [@Spillane07; @Fruet2018] that can only be explained by strong $^{12}$C cluster configurations of the corresponding state in $^{24}$Mg.\
There have been also predictions based on phenomenological considerations of explosive stellar events, such as X-ray superbursts, type Ia supernovae, stellar evolution etc..., that suggest a strong $^{12}$C+$^{12}$C cluster resonance around E$_{cm}$ = 1.5 MeV in $^{24}$Mg that would drastically enhance the energy production and may provide a direct nuclear driver for the superburst phenomenon [@Wiescher; @Kajino]. However, no indication for such a state was reported. Much of the data collected to date [@Spillane07; @Becker; @Aguilera] are shown in Fig. 3 taken from Ref. [@Fruet2018]. First direct $^{12}$C+$^{12}$C measurement [@Spillane07] seemed to indicate such a resonance. Recent measurements were performed at deep subbarrier energies using the newly developed Stella apparatus [@Heine2018] associated with the UK FATIMA detectors [@Fatima] for the exploration of fusion cross sections of astrophysical interest [@Fruet2018]. Gamma-rays have been detected in an array of LaBr$_3$ scintillators whereas proton and $\alpha$-particles were identified in double-sided silicon-strip detectors. A novel rotating target system has been developed in order to be capable to sustain high-intensity carbon beams delivered by the Andomède facility of the University Paris-Saclay and IPN Orsay, France [@Andromede]. The particle-$\gamma$ coincidence technique as well as nanosecond timing conditions have been used in the data analysis in order to minimize the background as much as possible. Our preliminary results [@Fruet2018] obtained with Stella [@Heine2018] confirm the possible occurence of such a resonant structure in the $\alpha$ channel but not in the proton channel.\
At higher energies the $^{12}$C+$^{12}$C cross sections expressed for Stella [@Heine2018] in terms of the modified astrophysical S-factor are typically in fair agreement with those measured at Argonne [@Jiang18] with similar coincidence techniques [@Jiang12]. The comparisons with previous data obtained by Becker et al. [@Becker] (open triangles), E. Aguilera et al. [@Aguilera] (open stars), and Spillane et al. [@Spillane07] (open circles), respectively, show a perfect agreement each other. All sets of chosen data lie in between the Fowler model [@Fowler] (black dotted line) and the hindrance model [@Jiang] (red dashed line).\
Our first conclusions might be summarized such as we confirm the possible fusion hindrance plus persisting resonances near the Gamow energy window. Furthermore, the preliminary Stella S-factors appear to be in qualitative agreement with either classical coupled-channel calculations of Esbensen [@Esbensen; @Notani] or more recent theoretical investigations [@Rowley; @Alexis; @Khoa]. It is not clear that recent studies [@Tumino18a] using the Trojan Horse Method technique (THM) confirmed the cluster level at E$_{cm}$ = 2.1 MeV but rather suggested the existence of the predicted state at E$_{cm}$ = 1.5 MeV for the fusion reaction. Such a discovery of a low energy $^{12}$C+$^{12}$C cluster state would indeed have significant impact on the reaction rate; but some doubts [@Mukha18a; @Mukha18b] have been raised to our attention as far as the validity of the indirect THM is concerned [@Tumino18a; @Tumino18b]. Obviously an experimental confirmation through direct fusion studies would be of utmost importance and several experiments are underway [@Fang2017]. On the other hand, it is expected that if a strong resonance indeed exists around the Gamow energy, then the theoretical structure studies should be able to predict a 0$^+$ excited state of $^{24}$Mg or an L = 0 $^{12}$C+$^{12}$C resonance at this particular energy, which plays about the same igniting role as that of the ”Hoyle" state [@Hoyle54] in the triple-$\alpha$ process of $^{12}$C formation [@Freer14]. A multichannel folding model [@Assuncao13] demonstrates the importance of inelastic channels involving the “Hoyle state" especially in the low-energy range relevant in astrophysics in the vicinity of the Gamow region.\
Clustering in light neutron-rich nuclei {#sec:4}
=======================================
Clustering is a general phenomenon observed also in nuclei with extra neutrons as it is presented in the ”Extended Ikeda-diagram“ [@Ikeda] proposed by von Oertzen [@Oertzen06] (see the left panel of Fig. 2). With additional neutrons, specific molecular structures appear with binding effects based on covalent molecular neutron orbitals. In these diagrams $\alpha$-clusters and $^{16}$O-clusters (as shown by the middle panel of the diagram of Fig. 2) are the main ingredients. Actually, the $^{14}$C nucleus may play similar role in clusterization as the $^{16}$O one since it has similar properties as a cluster: i) it has closed neutron p-shells, ii) first excited states are well above E$^{*}$ = 6 MeV, and iii) it has high binding energies for $\alpha$ particles.\
The possibility of extending molecular structures from dimers (Be isotopes) to trimers [@Oertzen06] has been investigated in detail for C and O isotopes [@Oertzen14]. For C isotopes the neutrons would be exchanged between the three centers ($\alpha$ particles). It is possible that the three $\alpha$-particle configuration can align themselves in a linaer fashion, or alternative collapse into a triangle arrangment - in either case the neutrons being localised across the three centers. Possibly the best case for the linear arrangement is $^{16}$C.\
A general picture of clustering and molecular configurations in light nuclei can also be drawn from the detailed investigation of the light O isotopes [@Oertzen14]. The bands of $^{20}$O [@Oertzen14] compared with the ones of $^{18}$O clearly establishes parity inversion doublets predicted by both the Generator-Coordinate-Method (GCM) and the Antisymmetrized Molecular Dynamics (AMD) [@Horiuchi] calculations for the $^{14}$C–$^6$He cluster and $^{14}$C–2n–$\alpha$ molecular structures. The corresponding moments of inertia are suggesting large deformations for the cluster structures.\
We may conclude that the reduction of the moments of inertia of the lowest bands of $^{20}$O is consistent with the assumption that the strongly bound $^{14}$C nucleus having equivalent properties to $^{16}$O, has a similar role as $^{16}$O in relevant, less neutron rich nuclei. Therefore, the ”Ikeda-Diagram [@Ikeda] and the ”extended Ikeda-Diagram" consisting of $^{16}$O cluster cores with covalently bound neutrons must be further extended to include also the $^{14}$C cluster cores as illustrated in Fig. 2.\
Summary and outlook
===================
The link of $\alpha$-clustering, quasimolecular resonances and extreme deformations (SD, HD etc...) has been discussed. Several examples emphasize the general connection between molecular structure and deformation effects within [*ab initio*]{} models and/or cluster models [@Freer18]. We have also presented the BEC picture of light (and medium-light) $\alpha$-like nuclei that appears to be an alternate way of understanding most of properties of nuclear clusters [@Correlations]. New results regarding cluster and molecular states in neutron-rich oxygen isotopes in agreement with AMD predictions are summarized [@Oertzen14]. Consequently, the ”Extended Ikeda-diagram" has been further modified for light neutron-rich nuclei by inclusion of the $^{14}$C cluster, similarly to the $^{16}$O one. Marked progress has been made in many traditional and novels subjects of nuclear cluster physics and astrophysics (stellar He burning [@Heine2018; @Fruet2018; @Wiescher; @Kajino]).\
The developments in these subjects show the importance of clustering among the basic modes of motion of nuclear many-body systems. All these open questions will require precise coincidence measurements [@Papka] coupled with state-of-the-art theory [@Correlations; @Horiuchi; @Freer18].\
Dedication and acknowledgements
===============================
This written contribution is dedicated to the memory of my friends Alex Szanto de Toledo, Valery Zagrebaev, Walter Greiner and Paulo Gomes who unexpectelly passed away since early 2015. I am very pleased to first acknowledge Walter Greiner for his continuous support of the cluster physics [@Greiner95; @Strasbourg; @Zagrebaev10; @Poenaru10; @Greiner08]. I would like to thank Christian Caron (Springer) for initiating in 2008 the series of the three volumes of *Lecture Notes in Physics* entitled “Clusters in Nuclei” and edited between 2010 and 2014 [@Cluster1; @Cluster2; @Cluster3]. All the 37 authors of the 19 chapters of these volumes are warmly thanked for their fruitfull collaboration during the course of the project which is still in progress [@Papka; @Horiuchi; @Oertzen14; @Zagrebaev10; @Poenaru10; @Gupta10; @Kanada10; @Oertzen10a; @Ikeda10; @Descouvemont12; @Nakamura12; @Baye12; @Adamian12; @Yamada12; @Deltuva14; @Jenkins14; @Zarubin14; @Simenel14; @Kamanin14]. Thanks also to Udo Schroeder for the edition of the volume “Nuclear Particle Correlations and Cluster Physics” that inspired [@Correlations] so much several aspects presented at the EXON2018 Symposium in September 2018 as well as at the ISPUN2017 Symposium held in September 2017 in Halong Bay, Vietnam. Special thanks to all the members of the Stella collaboration [@Heine2018], in particular Sandrine Courtin, Guillaume Fruet, Marcel Heine, Mohamad Moukaddam, Dominique Curien, et al. from the IPHC Strasbourg, Serge Della Negra (Andromède accelerator [@Andromede]) [*et al.*]{} from IPN Orsay, David Jenkins, Paddy Regan (UK FATIMA collaboration [@Fatima]) [*et al.*]{} from the UK and Christelle Stodel from GANIL. The Stella collaboration is supported by the french [*“Investissements d’avenir"*]{} program, the University of Strasbourg [*“IdEx Attractivity"*]{} program and the USIAS, Strasbourg, France. Finally, Dao Khoa, Le Hoang Chien, Alexis Diaz-Torres, Cheng-Lie Jiang, Akram Mukhamedzhanov and Xiadong Tang are acknowledged for their carefull reading of the manuscript.
[0]{}
C. Beck, [*Clusters in Nuclei, Vol. 1*]{}, ed. Springer Verlag Berlin Heidelberg (2010); *Lecture Notes in Physics* [**818**]{} (2010).
C. Beck, [*Clusters in Nuclei, Vol. 2*]{}, ed. Springer Verlag Berlin Heidelberg (2012); *Lecture Notes in Physics* [**848**]{} (2012).
C. Beck, [*Clusters in Nuclei, Vol. 3*]{}, ed. Springer Verlag Berlin Heidelberg (2014); *Lecture Notes in Physics* [**865**]{} (2014).
C. Beck, [*Nuclear Particle Correlations and Cluster Physics*]{}, ed. Wolf-Udo Schroeder (World Scientific Pub.), p.179-202 (2017); C. Beck, arXiv:[**1608.03190**]{} (2016), and references therein.
P. Papka and C. Beck, in [*Clusters in Nuclei, Vol. 1*]{}, ed. Springer Verlag Berlin Heidelberg (2012); P. Papka and C. Beck, *Lecture Notes in Physics* [**848**]{} p.299-353 (2012), and references therein.
W. von Oertzen, M. Freer, and Y. Kanada-En’yo, *Phys. Rep. * [**432**]{}, 43 (2007).
Y. Kanada-En’yo and M. Kimura, in [*Clusters in Nuclei, Vol. 1*]{}, ed. Springer Verlag Berlin Heidelberg (2010); *Lecture Notes in Physics* [**818**]{} p.129-164 (2010), and references therein.
M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U.-G. Meissner, [*Rev. Mod. Phys.*]{} [**90**]{}, 035004 (2018); M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U.-G. Meissner, arXiv:[**1705.06192**]{} (2017), and references therein.
H. Horiuchi and K. Ikeda, *Prog. Theor. Phys.* [**40**]{}, 277 (1968).
F. Hoyle *Astrophys. J. Suppl. Ser.* [**1**]{}, 121 (1954).
M. Freer and H.O.U. Fynbo, [*Prog. Part. Nucl. Phys.*]{} [**71**]{}, 1 (2014).
A. Tohsaki H. Horiuchi, P. Schuck, and G. Roepke, *Phys. Rev. Lett.* [**87**]{}, 192501 (2001); A. Tohsaki, H. Horiuchi, P. Schuck, and G. Roepke, arXiv:[**nucl-th/0110014**]{} (2001).
O. Kirsebom, *Physics* [**10**]{}, 103 (2017).
W. R. Zimmerman, M. W. Ahmed, B. Bromberger, S. C. Stave, A. Breskin, V. Dangendorf, Th. Delbar, M. Gai, S. S. Henshaw, J. M. Mueller, C. Sun, K. Tittelmeier, H. R. Weller, and Y. K. Wu, *Phys. Rev. Lett.* [**110**]{}, 152502 (2014).
D.J. Marin-Lambarri, R. Bijker, M. Freer, M. Gai, Tz. Kokalova, D. J. Parker, and C. Wheldon, *Phys. Rev. Lett.* [**113**]{}, 012502 (2014); D.J. Marin-Lambarri, R. Bijker, M. Freer, M. Gai, Tz. Kokalova, D. J. Parker, and C. Wheldon, arXiv:[**1405.7445**]{} (2014).
D. Dell’Aquila, I. Lombardo, G. Verde, M. Vigilante, L. Acosta, C. Agodi, F. Cappuzzello, D. Carbone, M. Cavallaro, S. Cherubini, A. Cvetinovic, G. D’Agata, L. Francalanza, G. L. Guardo, M. Gulino, I. Indelicato, M. La Cognata, L. Lamia, A. Ordine, R. G. Pizzone, S. M. R. Puglia, G. G. Rapisarda, S. Romano, G. Santagati, R. Spartà ,G. Spadaccini, C. Spitaleri, and A. Tumino, *Phys. Rev. Lett.* [**119**]{}, 132501 (2017); D. Dell’Aquila, I. Lombardo, G. Verde, M. Vigilante, L. Acosta, C. Agodi, F. Cappuzzello, D. Carbone, M. Cavallaro, S. Cherubini, A. Cvetinovic, G. D’Agata, L. Francalanza, G. L. Guardo, M. Gulino, I. Indelicato, M. La Cognata, L. Lamia, A. Ordine, R. G. Pizzone, S. M. R. Puglia, G. G. Rapisarda, S. Romano, G. Santagati, R. Spartà , G. Spadaccini, C. Spitaleri, and A. Tumino, arXiv:[**1705.09196**]{} (2017).
R. Smith, Tz. Kokalova, C. Wheldon, J.E. Bishop, M. Freer, N. Curtis, and D.J. Parker, *Phys. Rev. Lett.* [**119**]{}, 132502 (2017).
T. Spillane, F. Raiola, C. Rolfs, D. Schürmann, F. Strieder, S. Zeng, H.-W. Becker, C. Bordeanu, L. Gialanella, M. Romano, and J. Schweitzer, *Phys. Rev. Lett.* [**98**]{}, 122501 (2007); T. Spillane, F. Raiola, C. Rolfs, D. Schürmann, F. Strieder, S. Zeng, H.-W. Becker, C. Bordeanu, L. Gialanella, M. Romano, and J. Schweitzer, arXiv:[**nucl-ex/0702023**]{} (2007).
C.L. Jiang, B.B. Back, H. Esbensen, R.V.F. Janssens, K.E. Rehm, and R. J. Charity, *Phys. Rev. Lett.* [**110**]{}, 072701 (2013).
B. Bucher, X.D. Tang, X. Fang, A. Heger, S. Almaraz-Calderon, A. Alongi, A.D. Ayangeakaa, M. Beard, A. Best, J. Browne, C. Cahillane, M. Couder, R.J. deBoer, A. Kontos, L. Lamm, Y.J. Li, A. Long, W. Lu, S. Lyons, M. Notani, D. Patel, N. Paul, M. Pignatari, A. Roberts, D. Robertson, K. Smith, E. Stech, R. Talwar, W.P. Tan, M. Wiescher, and S.E. Woosley, *Phys. Rev. Lett.* [**114**]{}, 251102 (2015); B. Bucher, X.D. Tang, X. Fang, A. Heger, S. Almaraz-Calderon, A. Alongi, A.D. Ayangeakaa, M. Beard, A. Best, J. Browne, C. Cahillane, M. Couder, R.J. deBoer, A. Kontos, L. Lamm, Y.J. Li, A. Long, W. Lu, S. Lyons, M. Notani, D. Patel, N. Paul, M. Pignatari, A. Roberts, D. Robertson, K. Smith, E. Stech, R. Talwar, W.P. Tan, M. Wiescher, and S.E. Woosley, arXiv:[**1507.03980**]{} (2015).
C.L. Jiang, D. Santiago-Gonzalez, S. Almaraz-Calderon, K.E. Rehm, B.B. Back, K. Auranen, M.L. Avila, A.D. Ayangeakaa, S. Bottoni, M.P. Carpenter, C. Dickerson, B. DiGiovine, J.P. Greene, C.R. Hoffman, R.V.F. Janssens, B.P. Kay, S.A. Kuvin, T. Lauritsen, R.C. Pardo, J. Sethi, D. Seweryniak, R. Talwar, C. Ugalde, S. Zhu1, D. Bourgin, S. Courtin, F. Haas, M. Heine, G. Fruet, D. Montanari, D.G. Jenkins, L. Morris, A. Lefebvre-Schuhl, M. Alcorta, X. Fang, X.D. Tang, B. Bucher, C. M. Deibel, and S.T. Marley, *Phys. Rev. C* [**97**]{}, 012801(R) (2018).
A. Tumino, C. Spitaleri, M. La Cognata, S. Cherubini, G.L. Guardo, M. Gulino, S. Hayakawa, I. Indelicato, L. Lamia, H. Petrascu, R.G. Pizzone, S.M.R. Puglia, G.G. Rapisarda, S. Romano, M.L. Sergi, R. Spartá, and L. Trache, *Nature*, [**557**]{}, 687 (2018).
J. Zickefoose, A. Di Leva, F. Strieder, L. Gialanella, G. Imbriani, N.De Cesare, C. Rolfs, J. Schweitzer, T. Spillane, O. Straniero, and F. Terrasi, *Phys. Rev. C*, [**97**]{}, 065806 (2018).
M. Heine, S. Courtin, G. Fruet, D.G. Jenkins, L. Morris, D. Montanari, M. Rudigier, P. Adsley, D. Curien, S. Della Negra, J. Lesrel, C. Beck, L. Charles, P. Dené, F. Haas, F. Hammache, G. Heitz, M. Krauth, A. Meyer, Zs. Podolyák, P.H. Regan, M. Richer, N. de Séréville, C. Stodel, *Nucl. Instr. Meth. A*, [**903**]{}, 1 (2018); M. Heine, S. Courtin, G. Fruet, D.G. Jenkins, L. Morris, D. Montanari, M. Rudigier, P. Adsley, D. Curien, S. Della Negra, J. Lesrel, C. Beck, L. Charles, P. Dené, F. Haas, F. Hammache, G. Heitz, M. Krauth, A. Meyer, Zs. Podolyák, P.H. Regan, M. Richer, N. de Séréville, C. Stodel, arXiv:[**1802.07679**]{} (2018).
G. Fruet, [*Structure des Ions Lourds et Nucléosynthèse dans les Etoiles Massives : la Réaction $^{12}$C+$^{12}$C*]{}, Ph.D Thesis (2018), Université de Strasbourg (unpublished).
O.J. Roberts, A.M. Bruce, P.H. Regan, Z. Podolyk, C.M. Townsley, J.F. Smith, K.F. Mulholland, and A. Smith, [*Nucl. Instr. Meth. A*]{} [**748**]{} 91, (2014); R. Shearman, S.M. Collins, G. Lorusso, M. Rudigier, S.M. Judge, J. Bell, Zs.Podolyak, and P.H. Regan, [*Rad. Phys. Chem.*]{}, 140 (2017), p. 475
S. Della Negra, [*Inn. Rev.*]{} [**93**]{}, 38-40 (2016); see also http://ipnwww.in2p3.fr/ANDROMEDE?lang=fr
M. Wiescher, [*Nuclear Particle Correlations and Cluster Physics*]{}, ed. Wolf-Udo Schroeder (World Scientific Pub.), p.202-255 (2017), and references therein.
K. Mori, M.A. Famiano, T. Kajino, M. Kusakabe, and X. Tang, [*Monthly Notice of the Astrophysical Society*]{} MNRAS [**482**]{}, L70 (2019); K. Mori, M.A. Famiano, T. Kajino, M. Kusakabe, and X. Tang, arXiv:[**1810.01025**]{} (2018).
C.L. Jiang, K.E. Rehm, X.D. Tang, M. Alcorta, B.B. Back, B. Bucher, P. Collon, C.D. Deibel, B. DiGiovine, J.P. Greene, D.J. Henderson, R.V.F. Janssens, T. Lauritzen, C.J. Lister, S.T. Marley, R.C. Pardo, D. Seweryniak, C. Ugalde, S. Zhu, and M. Paul, *Nucl. Instr. Meth. A*, [**682**]{}, 12 (2012).
H.W. Becker, H. Lorenz-Wirzba, and C. Rolfs, [*Z. Phys. A*]{} [**303**]{}, 305 (1981).
E.F. Aguilera, R. Rosales, E. Martinez-Quiroz, G. Murillo, M. Fernandez, H. Berdejo, D. Lizcano, A. Gomez-Camacho, R. Policroniades, A. Verala, [*et al.*]{}, [*Phys. Rev. C*]{} [**73**]{}, 064601 (2006).
W.A. Fowler, [*Rev. Mod. Phys.*]{} [**56**]{}, 149 (1984).
C.L. Jiang, K.E. Rehm, B.B. Back, and R.V.F. Janssens, [it Phys. Rev. C]{} [**75**]{}, 015803 (2007).
H. Esbensen, X. Tang, and C.L. Jiang, [*Phys. Rev. C*]{} [**84**]{}, 064613 (2011); H. Esbensen, X. Tang, and C. L. Jiang, arXiv:[**1112.0496**]{} (2011).
M. Notani, H. Esbensen, X. Fang, B. Bucher, P. Davies, C.L. Jiang, L. Lamm, C. J. Lin, C. Ma, E. Martin, K. E. Rehm, W. P. Tan, S. Thomas, X. D. Tang, and E. Brown, [*Phys. Rev. C*]{} [**85**]{}, 014607 (2012).
N. Rowley and K. Hagino, [*Phys. Rev. C*]{} [**91**]{}, 044617 (2015).
A. Diaz-Torres and M. Wiescher, *Phys. Rev. C*, [**97**]{}, 055902 (2018); A. Diaz-Torres and M. Wiescher, arXiv:[**1802:01160**]{} (2018).
Le Hoang Chien, Dao T. Khoa, Do Cong Cuong, and Nguyen Hoang Phuc, *Phys. Rev. C*, [**98**]{}, 064604 (2018); Le Hoang Chien, Dao T. Khoa, Do Cong Cuong, and Nguyen Hoang Phuc, arXiv:[**1810.07887**]{} (2018); Dao T. Khoa, Le Hoang Chien, Do Cong Cuong, and Nguyen Hoang Phuc, [*Nucl. Sci. Tech.*]{} [**29**]{}, 182 (2018).
Akram Mukhamedzanov, Xiaodong Tang, and Daniang Pang, arXiv:[**1806.05921**]{} (2018).
A.M. Mukhamedzanov, X. and P.Y. Pang, arXiv:[**1806.08828**]{} (2018).
A. Tumino, C. Spitaleri, M. La Cognata, S. Cherubini, G.L. Guardo, M. Gulino, S. Hayakawa, I. Indelicato, L. Lamia, H. Petrascu, R.G. Pizzone, S.M.R. Puglia, G.G. Rapisarda, S. Romano, M. L. Sergi, R. Sparta, and L. Trache, arXiv:[**1807.06148**]{} (2018).
X. Fang, B. Bucher, A. Howard, J.J. Kolata, Y.J. Li, A. Roberts, X.T. Fang, and M. Wiecher, *Nucl. Instr. Meth. A*, [**871**]{}, 35 (2017).
M. Assuncao and P. Descouvemont, [*Phys. Lett. B*]{} [**723**]{}, 355 (2013).
W. von Oertzen and M. Milin, in [*Clusters in Nuclei, Vol. 3*]{}, ed. Springer Verlag Berlin Heidelberg (2014); *Lecture Notes in Physics* [**865**]{} p.146-182 (2014), and references therein.
W. Greiner, Y.J. Park, and W. Scheid, [*Nuclear Molecules*]{}, ed. World Scientific (1995).
W. Greiner in Proceedings of the Workshop on the State of the Art in Nuclear Cluster Physics (SOTANCP 2008) 13-16 May 2008, Strasbourg, France, ed. C. Beck, M. Dufour, and P. Schuck; in Special issue of *International Journal of Modern Physics E* [**17**]{} (2008), p.2379-2395.
V. Zagrebaev and W. Greiner, [*Clusters in Nuclei*]{}, Vol.1, p. 267, ed. C. Beck, *Lecture Notes in Physics* [**818**]{}, 267 (2010).
D.N. Poenaru and W. Greiner, [*Clusters in Nuclei*]{}, Vol.1, p.1, ed. C. Beck, [*Lecture Notes in Physics*]{} [**818**]{}, 1 (2010).
W. Greiner, *Int. J. Mod. Phys. E* [**17**]{}, 2379 (2008).
R.K. Gupta, [*Clusters in Nuclei*]{}, Vol.1, p.232, ed. C. Beck, *Lecture Notes in Physics* [**818**]{}, 232 (2010).
H. Horiuchi, [*Clusters in Nuclei*]{}, Vol. 1, p.57, ed. C. Beck, *Lecture Notes in Physics* [**818**]{}, 57 (2010).
W. von Oertzen, [*Clusters in Nuclei*]{}, Vol.1, p.102, ed. C. Beck, *Lecture Notes in Physics* [**818**]{}, 102 (2010); W. von Oertzen, arXiv:[**1004.4247**]{}.
K. Ikeda [*et al.*]{}, [*Clusters in Nuclei*]{}, Vol. 1, p.165, ed. C. Beck, *Lecture Notes in Physics* [**818**]{}, 165 (2010); K. Ikeda [*et al.*]{}, arXiv:[**1007.2474**]{} (2010).
P. Descouvemont and M. Dufour, [*Clusters in Nuclei*]{}, Vol.2, p.1, ed. C. Beck, *Lecture Notes in Physics* [**848**]{}, 1 (2012).
N. Nakamura and Y. Kondo, [*Clusters in Nuclei*]{}, Vol. 2, p.67, ed. C. Beck, *Lecture Notes in Physics* [**848**]{}, 67 (2012).
D. Baye and P. Capel, [*Clusters in Nuclei*]{}, Vol.2, p.121, ed. C. Beck, *Lecture Notes in Physics* [**848**]{}, 121 (2012); D. Baye and P. Capel, arXiv:[**1011.6427**]{} (2010).
G. Adamian, N. Antonenko, and W. Scheid, [*Clusters in Nuclei*]{}, Vol.2, p.165, ed. C. Beck, *Lecture Notes in Physics* [**848**]{}, 165 (2012).
T. Yamada, Y. Funaki, H. Horiuchi, G. Roepke, P. Schuck, and A. Tohsaki, [*Clusters in Nuclei*]{}, Vol.2, p.229, ed. Beck C, *Lecture Notes in Physics* [**848**]{}, 229 (2012); T. Yamada, Y. Funaki, H. Horiuchi, G. Roepke, P. Schuck, and A. Tohsaki, arXiv:[**1103.3940**]{} (2011).
A. Deltuva, A.C. Fonseca, and R. Lazauskas, [*Clusters in Nuclei*]{}, Vol.3, p.1, ed. C. Beck, *Lecture Notes in Physics* [**875**]{}, 1 (2014); A. Deltuva [*et al.,*]{} arXiv:[**1201.4979**]{} (2012).
D.G. Jenkins, [*Clusters in Nuclei*]{}, Vol.3, p.25, ed. C. Beck, *Lecture Notes in Physics* [**875**]{}, 25 (2014).
P.I. Zarubin, [*Clusters in Nuclei*]{}, Vol. 3, p. 183, ed. C. Beck, *Lecture Notes in Physics* [**875**]{}, 51 (2014); P.L. Zarubin, arXiv:[**1309.4881**]{} (2013).
C. Simenel, [*Clusters in Nuclei*]{}, Vol.3, p. 95 ed. C. Beck , *Lecture Notes in Physics* [**875**]{}, 95 (2014); C. Simenel, arXiv:[**1211.2387**]{} (2012).
D. Kamanin and Y. Pyatkov, [*Clusters in Nuclei*]{}, Vol. 3, p. 183, ed. C. Beck, *Lecture Notes in Physics* [**875**]{}, 183 (2014).
|
---
abstract: 'Unimodular gravity is classically equivalent to standard Einstein gravity, but differs when it comes to the quantum theory: The conformal factor is non-dynamical, and the gauge symmetry consists of transverse diffeomorphisms only. Furthermore, the cosmological constant is not renormalized. Thus the quantum theory is distinct from a quantization of standard Einstein gravity. Here we show that within a truncation of the full Renormalization Group flow of unimodular quantum gravity, there is a non-trivial ultraviolet-attractive fixed point, yielding a UV completion for unimodular gravity. We discuss important differences to the standard asymptotic-safety scenario for gravity, and provide further evidence for this scenario by investigating a new form of the gauge-fixing and ghost sector.'
author:
- Astrid Eichhorn
title: On unimodular quantum gravity
---
Introduction
============
In the search for an ultraviolet (UV) complete theory of quantum gravity, it is of particular interest to investigate classically equivalent formulations of gravity that differ at the quantum level. Here we explore two such options, namely standard Einstein gravity and unimodular gravity. In unimodular gravity, the determinant of the metric, $\sqrt{g}$, is not a dynamical variable, i.e., $$\sqrt{g}= \bar{\epsilon}, \label{unimodcond}$$ where $\bar{\epsilon}$ is a constant density. Only the conformal part of the metric is dynamical, i.e., an external notion of local scale exists. This formulation is of particular interest for quantum gravity, as it is equivalent to General Relativity (GR) classically, see [@Finkelstein:2000pg], since the equations of motion agree and each local coordinate patch of a solution to Einstein’s equations admits the introduction of coordinates where $\sqrt{g} = \rm const$. At the quantum level, both theories show crucial differences, while both have only a massless spin-2 excitation as their propagating degree of freedom [@vanderBij:1981ym]. An important motivation to consider unimodular gravity is that the cosmological constant arises as a constant of integration in the equations of motion and not as a coupling in the action. Thus it is not renormalized [@Weinberg:1988cp]. This solves the technical naturalness problem, which consists in the question why quantum fluctuations do not set the cosmological constant to the “natural” value $M_{\rm Planck}^2$. Note that quantum fluctuations, such as those responsible for the Lamb-shift *do* gravitate in this setting, thus unimodular gravity only degravitates the cosmological constant. Furthermore, unimodular gravity differs from GR in that the spectrum of quantum fluctuations around a background $g_{\mu \nu}$ differs, since implies that $h=g^{\mu \nu} \delta g_{\mu \nu}=0$. Thus the quantum fluctuations that must be integrated over in the partition function for quantum gravity differ. This can have crucial consequences for the theory and might in particular imply, that only one of the two classically equivalent theories, GR and unimodular gravity, exists as a quantum theory.
Quantizing unimodular gravity is possible in different ways, see, e.g., [@Smolin:2009ti]: One possibility is to impose the condition in the action and the path-integral, thus reducing the dynamical variables in comparison to GR [@Unruh:1988in]. Also the gauge symmetry consists of local-volume-preserving diffeomorphisms, only. As a second option, the unimodular condition is implemented using a Lagrange multiplier [@Henneaux:1989zc], thus keeping full diffeomorphism symmetry with the full metric being the variable in the path-integral. Evidently the classical equivalence of the two formulations needs not to carry over to the quantum case. Here, we focus on the first option. Possibly, the reduction in the number of unphysical degrees of freedom that are integrated over in the path-integral could reduce the regularization-scheme dependence of Renormalization Group (RG) studies of the path-integral, and thus yield more reliable results already in simpler truncations of the full space of operators.
The difference between GR and the unimodular theory becomes important already in the context of an effective-field theory setting for gravity, where quantum gravity effects are treated perturbatively, see, e.g., [@Alvarez:2005iy] or in semiclassical calculations, see [@Fiol:2008vk]. Here, we go a step further and consider a UV complete theory of GR as well as unimodular gravity. In the context of a continuum path-integral quantization of gravity, such a UV completion relies on the finiteness of running couplings in the effective action. As is well-known, a perturbative treatment reveals the perturbative non-renormalizability, not yielding a UV-complete theory, see [@'tHooft:1974bx; @Goroff:1985sz; @vandeVen:1991gw]. This is due to the negative dimensionality of the Newton coupling in $d=4$ spacetime dimensions and implies that the Gaußian fixed point of the RG flow is ultraviolet repulsive. In a non-perturbative context, this does not imply the breakdown of the theory in the UV, since a nontrivial fixed point of the RG flow, if found, yields a UV-completion for the effective theory, see [@Weinberg:1980gg]. In principle, this scenario can be realized in either standard Einstein gravity or in unimodular gravity. The first option has been studied extensively, see, e.g., [@Lauscher:2001ya; @Litim:2003vp; @Codello:2008vh; @Benedetti:2009rx] following the work in [@Reuter:1996cp] and will be referred to as Quantum Einstein Gravity (QEG) here, for reviews see [@Niedermaier:2006ns; @Percacci:2007sz; @Litim:2008tt; @Reuter:2012id]. Here we explore the second option, Unimodular Quantum Gravity (UQG), for the first time.
We use the functional RG (FRG), where the Wetterich equation [@Wetterich:1993yh] allows to evaluate $\beta$ functions even in the non-perturbative regime, by using an infrared (IR) mass-like regulator $R_k(p)$ that suppresses IR modes (with $p^2 <k^2$) in the generating functional. The $k$-dependent effective action $\Gamma_k$ contains the effect of quantum fluctuations above the scale $k$ only. Its scale-dependence is given by a functional differential equation: $$\partial_t \Gamma_k= \frac{1}{2} {\rm STr} \left(\Gamma_k^{(2)}+R_k \right)^{-1}\partial_t R_k.$$ Herein $\partial_t = k\, \partial_k$, and $\Gamma_k^{(2)}$ is the second functional derivative of $\Gamma_k$ with respect to the fields and is matrix-valued in field space. Adding the regulator and taking the inverse yields the full propagator. The supertrace contains a trace over all indices and summation over all fields with a negative sign for Grassmannian fields. The FRG framework is well-tested in diverse theories; with results agreeing with the universal one-loop $\beta$ functions of dimensionless couplings, see, e.g., [@Reuter:1993kw].
We apply the background field formalism [@Abbott:1980hw], where the metric is split into background and fluctuation field. In contrast to the standard linear split $g_{\mu \nu}= \bar{g}_{\mu \nu}+ h_{\mu \nu}$, we have to adapt this split to the unimodular setting: $$g_{\mu \nu} = \bar{g}_{\mu}^{\kappa} e^{h_{..}}_{\kappa \nu}= \bar{g}_{\mu \nu}+ h_{\mu \nu}+ \frac{1}{2} h_{\mu}^{\kappa}h_{\kappa\nu}+...\label{split}$$ Such a non-linear split includes the unimodularity condition , since, imposing $h=0$ on the fluctuations ensures that ${\rm det}g = {\rm det} \bar{g}$ to all orders in $h$. This departure from the usual linear split of the metric into background and fluctuations will also imply that the spectrum of fluctuations will change, as we will discuss below. Note that this split does not mean that we consider only small fluctuations around, e.g., a flat background. Within the FRG approach we can also access the fully non-perturbative regime. The background-field formalism is used in gravity, since the background metric distinguishes “high-momentum” and “low-momentum” modes by the spectrum of the background covariant Laplacian. Background-independence follows as the $\beta$ functions do not rely on any specific field configuration, excepting possible topological subtleties [@Reuter:2008qx].
Within the FRG framework as applied to quantum gravity, the unimodular theory is particularly interesting for several reasons: Firstly, within the background field formalism, the running of background couplings receives contributions which arise from the background dependence of the regulator function. Here, this problem is reduced as the cosmological constant is not a running coupling, and therefore the scheme dependence of the RG flow is reduced. Secondly, since the conformal factor is non-dynamical, the RG-scale $k$ becomes an external parameter, as in other quantum field theories. This is different from the setting in QEG, where $k$ can be redefined by a redefinition of the metric. Thus the meaning of large and small scales becomes dependent on an auxiliary background metric, and different choices of background correspond to a different order in which quantum fluctuations are integrated out. This differs in UQG, where an external notion of scale exists, bringing the RG flow of UQG closer to the flows of standard matter theories. Thirdly, the conformal-factor instability of the standard Euclidean formulation is absent, as the conformal factor is not a dynamical variable in the path integral, see [@vanderBij:1981ym]. In QEG, the full quantum action presumably contains terms beyond an Einstein-Hilbert term which could stabilize the potential. It has been conjectured that the instability signals a non-trivial vacuum for QEG with a finite expectation value for the conformal factor [@Bonanno:2012dg]. The absence of this instability in UQG implies that the vacuum structure of QEG and UQG could differ.
Construction of the gauge-fixing and ghost sector of UQG
========================================================
In order to construct the Wetterich equation for UQG one must introduce a gauge-fixing term, following the Faddeev-Popov trick. The unimodularity condition implies that unimodular gravity restricts the symmetry to transverse diffeomorphisms, for which $$\delta_D g_{\mu \nu} = \mathcal{L}_v g_{\mu \nu} \mbox{ with } D_{\mu}v^{\mu}=0.$$ Note that in gravity theories which are invariant under transverse diffeomorphism only, there appears an additional scalar mode in the linearized theory. As noted in [@Alvarez:2006uu], this mode is absent in two cases: If the symmetry is enhanced to a full diffeomorphism symmetry, arriving at standard Einstein gravity, or if the metric determinant remains fixed. Then the additional scalar, which plays the role of the determinant, is removed from the theory.
Accordingly we cannot impose a standard linear gauge condition, such as, e.g., harmonic gauge, since it consists of four independent conditions, whereas we can impose only three if we are to respect the transversality of the diffeomorphism. One possibility is to simply project the harmonic gauge onto the transversal part, [@Alvarez:2008zw], using a transverse projector $P_{\mu \nu}= \frac{1}{\bar{D}^2}\left(g_{\mu \nu}\bar{D}^2 - \bar{D}_{\mu}\bar{D}_{\nu}\right)$. Then our background-field gauge condition reads $$\begin{aligned}
F_{\mu}&=& \sqrt{2}\left(\bar{D}^2 \bar{D}_{\kappa}h^{\kappa}_{\mu} - \bar{D}_{\mu}\bar{D}_{\rho}\bar{D}_{\sigma}h^{\rho \sigma}\right),\label{gaugecondition}\end{aligned}$$ such that $$S_{\rm gf}= \frac{1}{2\alpha} \int d^4x\, \bar{\epsilon} \,\bar{g}^{\mu \nu}F_{\mu}F_{\nu},\label{gfaction}$$ where $\alpha$ is a (dimensionfull) gauge parameter, which we will later set to $\alpha=0$. We then construct the Faddeev-Popov ghost sector as $$S_{\rm gh}=- \int d^4 x \,\bar{\epsilon} \, \bar{c}_{\mu}\,\bar{g}^{\mu \nu} \frac{\partial F_{\nu}}{\partial h_{\alpha \beta}} \mathcal{L}_C g_{\alpha \beta}.$$ Crucially, since the gauge-transformations are transverse, this implies that the ghost field also fulfills transversality: $\bar{D}_{\mu}c^{\mu}=0$. This implies that the scalar longitudinal ghost, present in QEG, is missing from the unimodular formulation. This is in complete agreement with the intuition: If the gauge-symmetry is smaller, then fewer ghost fields need to be introduced into the theory to cancel the effect of non-physical metric modes.
Inserting the gauge condition we finally arrive at: $$\begin{aligned}
S_{\rm gh}&=&- \int d^4 x \,\bar{\epsilon}\, \bar{c}_{\mu}\,\bar{g}^{\mu \nu} \Bigl( \bar{D}^2 \bar{D}^{\alpha}\bar{g}^{\beta}_{ \nu} + \bar{D}^2 \bar{D}^{\beta}\bar{g}^{\alpha}_{ \nu} \nonumber\\
&{}&\quad \quad \quad- \bar{D}_{\nu}\bar{D}^{\alpha}\bar{D}^{\beta} - \bar{D}_{\nu}\bar{D}^{\beta} \bar{D}^{\alpha}
\Bigr) g_{\rho \beta} D_{\alpha}c^{\rho}.\label{ghaction}\end{aligned}$$
We use a York decomposition of the fluctuation field $$h_{\mu \nu}\!=\! h_{\mu \nu}^{TT}+ \bar{D}_{\!\mu}v_{\nu}+ \bar{D}_{\!\nu}v_{\mu}+ \bar{D}_{\!\mu}\bar{D}_{\!\nu}\sigma - \frac{\bar{g}_{\mu \nu}}{4} \bar{D}^2\sigma + \frac{\bar{g}_{\mu \nu}}{4}h.$$ Here $\bar{D}^{\mu}h_{\mu \nu}^{TT}=0$, $\bar{g}^{\mu \nu} h_{\mu \nu}^{TT}=0$ and $\bar{D}^{\mu}v_{\mu}=0$. We redefine $v_{\mu} \rightarrow ( -\bar{D}^2 - \frac{\bar{R}}{4})^{-1/2}v_{\mu}$ and exponentiate the $\bar{g}_{\mu \nu}$-dependent Jacobians resulting from the decomposition and redefinition [@Lauscher:2001ya] using auxiliary Grassmann fields.
Here we will also explore a new gauge-fixing sector of QEG for the first time. Usually, a covariant gauge condition corresponding to four independent conditions is chosen. To clearly highlight the differences between UQG and QEG, we choose a different route here: Using the same gauge-fixing of the transversal diffeomorphisms in UQG and QEG, see with a dynamical density $\sqrt{g}$, we are left with one further (scalar) gauge-fixing function to choose in QEG, which we take to be $F=h$. Thus the additional piece of the gauge-fixing for QEG reads $$S_{gf\, 2}= \frac{1}{2\alpha_2} \int d^4x\, \sqrt{\bar{g}}\,F^2,\label{scalargf}$$ where $\alpha_2$ is a second gauge parameter. This choice brings QEG as close to UQG as possible, since it enforces trace fluctuations in the path-integral to vanish. The crucial difference between UQG and QEG is that in UQG these fluctuations are absent from the path integral for any choice of gauge, whereas in the case of QEG we could also choose a different gauge that would keep the trace fluctuations. Note that although the fluctuations $h$ do not contribute to the RG running, they still leave an imprint in the QEG case: The gauge-fixing induces a Faddeev-Popov ghost term with scalar ghosts reading $$S_{\rm scalar\, gh} = - \int d^4x\, \sqrt{\bar{g}}\, \bar{\xi}\,D^2 \xi.\label{scalargh}$$
RG flow in QEG and UQG
======================
Our full truncation reads $$\Gamma_{k\, \rm UQG} = - 2 Z_N \bar{\kappa}^2 \int d^4x \,\bar{\epsilon}\,R + S_{\rm gf} + S_{\rm gh},$$ in the case of UQG and with chief differences for QEG: $$\Gamma_{k\, \rm QEG}\!=\! - 2 Z_N \bar{\kappa}^2\! \int\! d^4x \sqrt{g}\,R+ S_{\rm gf}+ +S_{\rm gf\, 2}+S_{\rm gh}+ S_{\rm scalar\, gh}.$$ Herein $Z_N$ is a wave-function renormalization for the graviton, and $\bar{\kappa}^2 = \frac{1}{32 \pi G_N}$. We define a dimensionless Newton coupling $G=\frac{G_N k^2}{Z_N}$. In more detail, the gauge-fixing and ghost action of the two theories is given by and in the unimodular case and with the additional terms and for QEG.
There are three crucial differences between the quantum fluctuations driving the RG flow in QEG and UQG: Firstly, the trace of the metric fluctuations is not a dynamical degree of freedom due to the unimodularity condition: Since $\sqrt{g} =\rm const $, $g^{\mu \nu} \delta g_{\mu \nu} =h =0$. Thus its quantum fluctuations are not integrated over in the path integral. In the case of QEG, the exclusion of $h$ from the path-integral is a valid choice of partial gauge, but introduces an additional scalar ghost field which is absent in UQG. Secondly, the propagator for the graviton does not receive contributions from fluctuations in the volume factor $\sqrt{g}$: In QEG, a contribution $$(\delta^2 \sqrt{g})R \sim - h^{\mu \nu}h_{\mu \nu} R, \label{volumeflucs}$$ arises, which is absent for UQG, simply because $\sqrt{g}= \bar{\epsilon}$ in this case. We will call this effect the absence of volume fluctuations.
Thirdly, the non-linear split of the metric into background and fluctuation implies that the variation of the Ricci scalar changes: This is simply due to the fact, that $\delta^2 g^{\mu \nu} = h^{\mu \kappa}h_{\kappa}^{\nu}$, whereas in the linear split, this differs by a factor of 2. The second variation of the Christoffel symbol also differs for the two cases, but this difference does not enter the second variation of the Ricci scalar, which in $d$ dimensions reads $$\begin{aligned}
&{}&\int_x \delta^2 R \\
&=& \int_x \Bigl(\zeta h^{\mu}_{\kappa} h^{\kappa \nu}R_{\mu \nu} - h_{\mu \kappa}h^{\mu \kappa}\frac{R}{d(d-1)} + \zeta h^2 \frac{R}{d(d-1)} - h_{\mu}^{ \kappa}D_{\kappa}D_{\sigma}h^{\mu \sigma} +\frac{1}{2} h_{\mu \nu}D^2 h^{\mu \nu} + \zeta \frac{1}{2} h D^2 h\Bigr),\nonumber\end{aligned}$$ where we have discarded total derivatives and have inserted a spherical background. For $\zeta=1$ this yields the standard variation of $R$ for the linear split $g_{\mu \nu} = \bar{g}_{\mu \nu} + h_{\mu \nu}$, whereas $\zeta =0$ yields the unimodular case, where the fluctuations are traceless and are related to the full metric through .
In the following we show our results for UQG and QEG. For the RG flow of $G$, we specialize to a spherical background, where the eigenvalues of the Laplacian are known exactly [@Rubin:1984tc]. Note that since $\sqrt{g}=\rm const$ is a valid choice of coordinates in General Relativity, this implies that the gauge-invariant eigenvalues of the Laplacian do not differ in unimodular gravity. We can then evaluate the trace over the spectrum of fluctuations on the right-hand side of the flow equation by a direct summation over the eigenvalues of the Laplacian, employing the Euler-MacLaurin formula. We employ a regulator of the form $R_k = \left( \Gamma_{k}^{(2)}(k^2)- \Gamma_{k}^{(2)}(-\bar{D}^2)\right)\theta(k^2 - (-\bar{D}^2))$, [@Litim:2001up], which we impose directly on the eigenvalues of the covariant Laplacian, which are being summed over on the right-hand side of the flow equation. This yields the following, evidently nonperturbative result for $\beta_G = \partial_t G$: $$\begin{aligned}
\beta_{G\, \rm UQG}&=&\!2G + G^2 \frac{3 \left(1300-309 \sqrt{13} -325 \sqrt{17} \right)}{ 936 \pi -1625 G}\label{betaUQG},\\
\beta_{G\, \rm QEG}&=&\!2 G+ G^2\frac{2 \left( 9763 -3708\sqrt{13}-975 \sqrt{17}\right)}{ 3744 \pi - 650 G}\label{betaQEG}.\end{aligned}$$
![\[betaplot\] We show the $\beta$ functions for the Newton coupling in UQG (green thick line) and QEG (blue dashed line).](beta_fcts_v4.pdf){width="0.9\linewidth"}
Obviously, both $\beta$ functions agree in the perturbative regime, see fig. \[betaplot\], as they have to due to the dimensionality of $G_N$, which stipulates $\partial_G\beta_G\vert_{G=0}=2$.
Interestingly, both $\beta$ functions show a further, interacting fixed point, which is UV attractive in both cases. Thus we find the first indication that UQG could indeed exist as a UV complete quantum theory of gravity. The critical exponent governing the approach to the non-Gaußian fixed point at $G_{\ast\, \rm UQG}= 0.876$ in UQG, reads $$\theta_{\rm UQG}=-\frac{\partial \beta_{G\,\rm UQG}}{\partial G}\Big|_{G= G_{\ast\, {\rm UQG}}}= 3.878,$$ corresponding to a UV attractive direction.
We also find further evidence for asymptotic safety in QEG, as we find an interacting fixed point at $G_{\ast\,\rm QEG}= 2.65$ with critical exponent $-\frac{\partial \beta_{G\,\rm QEG}}{\partial G}\vert_{G= G_{\ast\rm QEG}}= 2.341$, lying in the range which is observed for this universal quantity in other schemes, see, e.g., [@Litim:2008tt; @Codello:2008vh]. Removing the RG improvement on the right-hand side of the flow equation, i.e., disregarding the terms $\sim \eta_N$ arising from our choice of regulator, the fixed point persists, with numerical changes in its value.
Let us discuss several approximations to the full $\beta$ function, in which the fixed point persists in both cases, which can be read as a sign of stability, and suggests that the fixed point should persist beyond our truncation.
Firstly, we can disregard the terms $\sim \eta_N$, that arise on the right-hand-side of the flow equation due to our choice of regulator. In that case, the fixed point persists, with numerical changes in its value and the critical exponent.
Furthermore, we observe that the only mode not affected by our choice of gauge is the transverse traceless mode. In other words, a different choice of gauge would change the propagators of the other modes, and therefore their contribution to $\beta_G$, but not that of the $TT$ mode. This observation suggests to study a scenario, where all modes but the $TT$ mode are actually dropped from the flow equation, which has been dubbed the ”TT-approximation” in [@Eichhorn:2010tb]. In this setting, the fixed point actually persists with only small numerical changes: $G_{\ast\, \rm UQG} = 0.93$ and $\theta_{\rm UQG} = 4.135$.\
In the case of QEG, the changes are slightly larger, with the fixed point in the TT-approximation at $G_{\ast\, \rm QEG}= 5.41$ and a critical exponent of $\theta_{\rm QEG}= 2.85$. The fact that the change in the case of unimodular gravity is smaller could be understood from the observation that due to the reduction in the symmetry from full diffeomorphisms to transverse diffeomorphisms, a smaller portion of the fluctuation modes is actually unphysical in this case. Therefore the effect of leaving out some of these modes by going to the TT approximation is smaller in the case of UQG.
The difference between UQG and QEG relies on the off-shell character of quantum fluctuations, as the spectrum of quantum fluctuations agrees on shell: The information on the sign of the fixed point is present in the transverse traceless component of the propagator, which agrees if evaluated on shell where $R=0$, since $\Gamma_{k\, TT}^{(2)}\sim -D^2 +R \frac{3\zeta+1}{6}$, where $\zeta=1$ for QEG and $\zeta=0$ for UQG. It is actually interesting to trace the effect of the absence of volume fluctuations and the difference of to the standard case: Discarding the volume fluctuations actually leads to a change of sign for the above coefficient of the $R$ term, and does indeed result in a $\beta$ function with a fixed point at $G_{\ast}<0$. Taking into account the additional difference in the spectrum of quantum fluctuations that arises due to then yields a positive coefficient and a positive fixed-point value. Note also that this analysis is in full accordance with the recent ideas of paramagnetic dominance in asymptotic safety [@Nink:2012vd]: The difference in the sign of the fixed-point value can be traced back to the change of sign in the curvature-dependent (“paramagnetic”) term in the propagator for the transverse traceless graviton mode.
It is interesting to observe that the unimodular modes also carry information about a non-trivial fixed point in gravity, as does the conformal mode. The present study complements previous explorations of QEG in settings with reduced degrees of freedom, see, e.g., [@Reuter:2008wj; @Eichhorn:2010tb].
Conclusions
===========
Here we have examined unimodular gravity and GR, which are classically equivalent, but show crucial differences in the quantum theory: Firstly, the cosmological constant is not renormalized in UQG which solves the technical naturalness problem. Secondly, the spectrum of quantum fluctuations differs in the two theories, yielding crucial differences in the RG flows. We have discussed how to set up an RG equation for the unimodular case, where the gauge-fixing and ghost sector has to be adapted to respect the unimodularity condition . We find that within a simple truncation of the RG flow both theories show an interacting fixed point for the Newton coupling and are therefore asymptotically safe. This is the first evidence that UQG could indeed exist as a UV complete theory. We also provide further evidence for asymptotic safety in QEG, as we investigate a new gauge fixing.
[*Acknowledgements*]{} I thank Martin Reuter for helpful and encouraging discussions. I am also indebted to an anonymous referee for helpful comments on the implementation of the unimodularity condition. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
[99]{}
D. R. Finkelstein, A. A. Galiautdinov and J. E. Baugh, J. Math. Phys. [**42**]{}, 340 (2001).
J. J. van der Bij, H. van Dam and Y. J. Ng, Physica [**116A**]{}, 307 (1982). S. Weinberg, Rev. Mod. Phys. [**61**]{}, 1 (1989). L. Smolin, Phys. Rev. D [**80**]{}, 084003 (2009). W. G. Unruh, Phys. Rev. D [**40**]{}, 1048 (1989).
M. Henneaux and C. Teitelboim, Phys. Lett. B [**222**]{}, 195 (1989).
E. Alvarez, JHEP [**0503**]{}, 002 (2005). B. Fiol and J. Garriga, JCAP [**1008**]{}, 015 (2010).
G. ’t Hooft and M. J. G. Veltman, Annales Poincare Phys. Theor. A [**20**]{}, 69 (1974). M. H. Goroff and A. Sagnotti, Phys. Lett. B [**160**]{}, 81 (1985). A. E. M. van de Ven, Nucl. Phys. B [**378**]{}, 309 (1992). S. Weinberg, [*In \*Hawking, S.W., Israel, W.: General Relativity\*, 790-831*]{}.
O. Lauscher and M. Reuter, Phys. Rev. D [**65**]{}, 025013 (2002). D. F. Litim, Phys. Rev. Lett. [**92**]{}, 201301 (2004). A. Codello, R. Percacci and C. Rahmede, Annals Phys. [**324**]{}, 414 (2009). D. Benedetti, P. F. Machado and F. Saueressig, Mod. Phys. Lett. A [**24**]{}, 2233 (2009).
M. Reuter, Phys. Rev. D [**57**]{}, 971 (1998). M. Niedermaier, Class. Quant. Grav. [**24**]{}, R171 (2007). R. Percacci, arXiv:0709.3851 \[hep-th\].
D. F. Litim, arXiv:0810.3675 \[hep-th\].
M. Reuter and F. Saueressig, arXiv:1202.2274 \[hep-th\]. C. Wetterich, Phys. Lett. B [**301**]{}, 90 (1993).
M. Reuter and C. Wetterich, Nucl. Phys. B [**417**]{}, 181 (1994). L. F. Abbott, Nucl. Phys. B [**185**]{}, 189 (1981). M. Reuter and H. Weyer, Phys. Rev. D [**80**]{}, 025001 (2009) \[arXiv:0804.1475 \[hep-th\]\].
A. Bonanno, F. Guarnieri and F. Guarnieri, Phys. Rev. D [**86**]{}, 105027 (2012). E. Alvarez, D. Blas, J. Garriga and E. Verdaguer, Nucl. Phys. B [**756**]{}, 148 (2006).
E. Alvarez, A. F. Faedo and J. J. Lopez-Villarejo, JHEP [**0810**]{}, 023 (2008).
M. A. Rubin and C. R. Ordonez, J. Math. Phys. [**26**]{}, 65 (1985). D. F. Litim, Phys. Rev. D [**64**]{}, 105007 (2001). A. Nink and M. Reuter, arXiv:1208.0031 \[hep-th\]. M. Reuter and H. Weyer, Phys. Rev. D [**79**]{}, 105005 (2009). A. Eichhorn and H. Gies, Phys. Rev. D [**81**]{}, 104010 (2010).
|
---
abstract: 'The solution of Einstein’s field equations in Cosmological General Relativity (CGR), where the Galaxy is at the center of a finite yet bounded spherically symmetrical isotropic gravitational field, is identical with the unbounded solution. This leads to the conclusion that the Universe may be viewed as a finite expanding white hole. The fact that CGR has been successful in describing the distance modulus verses redshift data of the high-redshift type Ia supernovae means that the data cannot distinguish between unbounded models and those with finite bounded radii of at least $c \tau$. Also it is shown that the Universe is spatially flat at the current epoch and has been at all past epochs where it was matter dominated.'
author:
- |
**John G. Hartnett**\
School of Physics, the University of Western Australia,\
35 Stirling Hwy, Crawley 6009 WA Australia\
*john@physics.uwa.edu.au*
title: Finite bounded expanding white hole universe without dark matter
---
Keywords: Cosmological General Relativity, high redshift type Ia supernovae, dark matter
\[sec:Intro\]Introduction
=========================
In an interview with Scientific American George Ellis once said [@Gibbs1995]
“People need to be aware that there is a range of models that could explain the observations, …For instance, I can construct you a spherically symmetrical universe with Earth at its center, and you cannot disprove it based on observations. …You can only exclude it on philosophical grounds. In my view there is absolutely nothing wrong in that. What I want to bring into the open is the fact that we are using philosophical criteria in choosing our models. A lot of cosmology tries to hide that.”
This paper proposes a model where the Galaxy is at the center of a spherically symmetrical finite bounded universe. It contends that fits to the magnitude-redshift data of the high-$z$ type Ia supernovae (SNe Ia) [@Knop2003; @Riess2004; @Astier2005], are also consistent with this model. That is, providing that the radius of the Universe (a spherically symmetrical matter distribution) is at least $c\tau$ where $c$ is the speed of light and $\tau \approx 4.28 \times 10^{17}$ s (or $13.54 \; Gyr$).[@Oliveira2006] Here $\tau$ is the Hubble-Carmeli time constant, or the inverse of the Hubble constant evaluated in the limits of zero gravity and zero distance.
This model is based on the Cosmological General Relativity (CGR) theory [@Carmeli2002a] but explores the motion of particles in a central potential. In this case the central potential is the result of the Galaxy being situated at the center of a spherically symmetrical isotropic distribution comprising all matter in the Universe.
This paper is preceded by Hartnett [@Hartnett2005] that forms the basis of the work presented here. Also Oliveira and Hartnett [@Oliveira2006] progressed the work by developing a density function for higher redshifts. Those paper assumed the unbounded model. The reader should be familiar with Hartnett [@Hartnett2005] at least before reading this.
\[sec:CGR\]Cosmological General Relativity
------------------------------------------
The metric [@Behar2000; @Carmeli1996; @Carmeli2002a] used by Carmeli (in CGR) in a generally covariant theory extends the number of dimensions of the Universe by the addition of a new dimension – the radial velocity of the galaxies in the Hubble flow. The Hubble law is assumed as a fundamental axiom for the Universe and the galaxies are distributed accordingly. The underlying mechanism is that the substance of which space is built, the vacuum, is uniformly expanding in all directions and galaxies, as tracers, are fixed to space and therefore the redshifts of distant first ranked galaxies quantify the speed of the expansion.
In determining the large scale structure of the Universe the usual time dimension is neglected ($dt = 0$) as observations are taken over such a short time period compared to the motion of the galaxies in the expansion. It is like taking a still snap shot of the Universe and therefore only four co-ordinates $x^{\mu}=(x^{1},x^{2},x^{3},x^{4})=(r,\theta,\phi,\tau v)$ are used – three of space and one of velocity. The parameter $\tau$, the Hubble-Carmeli constant, is a universal constant for all observers.
Here the CGR theory is considered using a Riemannian four-dimensional presentation of gravitation in which the coordinates are those of Hubble, i.e. distance and velocity. This results in a phase space equation where the observables are redshift and distance. The latter may be determined from the high-redshift type Ia supernova observations.
\[sec:phasespace\]Phase space equation
--------------------------------------
The line element in CGR [@Carmeli2002b] $$\label{eqn:lineelement}
ds^2= \tau^{2}dv^{2}-e^{\xi}dr^{2}-R^2(d\theta^2+ sin^2\theta d\phi^2),$$ represents a spherically symmetrical isotropic universe, that is not necessarily homogeneous.
It is fundamental to the theory that $ds=0$. In the case of Cosmological Special Relativity (see chap.2 of [@Carmeli2002a]), which is very useful pedagogically, we can write the line element as $$\label{eqn:CSRlineelement}
ds^2= \tau^{2}dv^{2}-dr^{2},$$ ignoring $\theta$ and $\phi$ co-ordinates for the moment. By equating $ds=0$ it follows from (\[eqn:CSRlineelement\]) that $\tau dv = dr$ assuming the positive sign for an expanding universe. This is then the Hubble law in the small $v$ limit. Hence, in general, this theory requires that $ds =0$.
Using spherical coordinates ($r,\theta, \phi$) and the isotropy condition $d\theta = d\phi = 0 $ in (\[eqn:lineelement\]) then $dr$ represents the radial co-ordinate distance to the source and it follows from (\[eqn:lineelement\]) that $$\label{eqn:4Dmetric}
\tau^{2}dv^{2}-e^{\xi}dr^{2}=0,$$ where $\xi$ is a function of $v$ and $r$ alone. This results in $$\label{eqn:4Dmetricderiv}
\frac{dr} {dv} = \tau e^{-\xi/2},$$ where the positive sign has been chosen for an expanding universe.
\[sec:Centralsoln\]Solution in central potential
================================================
Carmeli found a solution to his field equations, modified from Einstein’s, (see [@Hartnett2005] and [@Behar2000; @Carmeli2002a; @Carmeli2002b]) which is of the form $$\label{eqn:soln1field2}
e^{\xi}= \frac{{R'^2}}{1 + f(r)}$$ with $R' = 1 $, which must be positive. From the field equations and (\[eqn:soln1field2\]) we get a differential equation $$\label{eqn:soln1field3}
f' + \frac{f}{r} = - \kappa \tau^{2} \rho_{eff} r,$$ where $f(r)$ is function of $r$ and satisfies the condition $f(r) + 1 > 0$. The prime is the derivative with respect to $r$. Here $\kappa = 8 \pi G /c^{2} \tau^{2}$ and $\rho_{eff} = \rho -\rho_{c}$ where $\rho$ is the averaged matter density of the Universe and $\rho_{c}=3/8 \pi G \tau^{2}$ is the critical density.
The solution of (\[eqn:soln1field3\]), $f(r)$, is the sum of the solution ($2GM/c^{2}r$) to the homogeneous equation and a particular solution (-$\frac{\kappa}{3} \tau^{2} \rho_{eff} r^{2} $) to the inhomogeneous equation. In [@Carmeli2002a] Carmeli discarded the homogeneous solution saying it was not relevant to the Universe, but the solution of a particle at the origin of coordinates, or in other words, in a central potential.
Now suppose we model the Universe as a ball of dust of radius $\Delta$ with us, the observer, at the center of that ball. In this case the gravitational potential written in spherical coordinates that satisfies Poisson’s equation in the Newtonian approximation is $$\label{eqn:potential1}
\Phi(r) =-\frac{GM}{r}$$ for the vacuum solution, but inside an isotropic matter distribution $$\begin{aligned}
\label{eqn:potential2}
\Phi(r)&&=-G \left (\frac{4\pi \rho}{r} \int_{0}^{r} r'^{2} dr' + 4\pi \rho \int_{r}^{\Delta} r' dr'\right) \nonumber \\
&&= \frac{2}{3} G \pi \rho r^{2} - 2 G \pi \rho \Delta^{2},\end{aligned}$$ where it is assumed the matter density $\rho$ is uniform throughout the Universe. At the origin ($r = 0$) $\Phi(0) = - 2 G \pi \rho_{m} \Delta^{2}$, where $\rho = \rho_{m}$ the matter density at the present epoch. In general $\rho$ depends on epoch. Because we are considering no time development $\rho$ is only a function of redshift $z$ and $\rho_{m}$ can be considered constant.
From (\[eqn:potential2\]) it is clear to see that by considering a finite distribution of matter of radial extent $\Delta$, it has the effect of adding a constant to $f(r)$ that is consistent with the constant $2 G \pi \rho \Delta^{2}$ in (\[eqn:potential2\]), where $f(r)$ is now identified with $-4\Phi/c^{2}$.
Equation (\[eqn:soln1field2\]) is essentially Carmeli’s equation A.19, the solution to his equation A.17 from p.122 of [@Carmeli2002a]. More generally (\[eqn:soln1field2\]) can be written as $$\label{eqn:soln1fieldgeneral}
e^{\xi}= \frac{{R'^{2}}}{1 + f(r) - K},$$ where $K$ is a constant. This is the most general form of the solution of Carmeli’s equation A.17. So by substituting (\[eqn:soln1fieldgeneral\]) into Carmeli’s A.18, A.21 becomes instead $$\label{eqn:generalfield2}
\frac{1}{R R'}(2 \dot{R} \dot{R'} - f') + \frac{1}{R^{2}}(\dot{R}^{2}- f + K) = \kappa \tau^2 \rho_{eff}.$$
Therefore (\[eqn:soln1fieldgeneral\]) is also a valid solution of the Einstein field equations (A.12 - A.18 [@Carmeli2002a]) in this model. Making the assignment $R = r$ in (\[eqn:generalfield2\]) yields a more general version of (\[eqn:soln1field3\]), that is, $$\label{eqn:soln1fieldgen2}
f' + \frac{f - K}{r} = - \kappa \tau^{2} \rho_{eff} r.$$
The solution of (\[eqn:soln1fieldgen2\]) is then $$\label{eqn:fvalue1}
f(r) = -\frac{1}{3}\kappa \tau^{2} \rho_{eff} r^{2}+ K.$$ From a comparison with (\[eqn:potential2\]) it would seem that the constant $K$ takes the form $K = 8 \pi G \rho_{eff}(0) \Delta^{2}/c^{2}$. It is independent of $r$ and describes a non-zero gravitational potential of a finite universe measured at the origin of coordinates. There is some ambiguity however as to which density to use in Carmelian cosmology since it is not the same as Newtonian theory. Here $\rho_{eff}$ is used and evaluated at $r = 0$.
In the above Carmelian theory it initially assumed that the Universe has expanded over time and at any given epoch it has an averaged density $\rho$, and hence $\rho_{eff}$. The solution of the field equations has been sought on this basis. However because the Carmeli metric is solved in an instant of time (on a cosmological scale) any time dependence is neglected. In fact, the general time dependent solution has not yet been found. But since we observe the expanding Universe with the coordinates of Hubble at each epoch (or redshift $z$) we see the Universe with a different density $\rho(z)$ and an effective density $\rho_{eff} (z)$. Carmeli arrived at his solution with the constant density assumption. I have made the implicit assumption that the solution is also valid if we allow the density to vary as a function of redshift, as is expected with expansion.
Now it follows from (\[eqn:4Dmetricderiv\]), (\[eqn:soln1fieldgeneral\]) and (\[eqn:fvalue1\]) that $$\label{eqn:phasespacederiv2}
\frac{dr}{dv}= \tau \sqrt{1+ \left(\frac{1-\Omega}{c^{2} \tau^{2}}\right) r^{2} } ,$$ where $\Omega = \rho/\rho_{c}$. This compares with the solution when the central potential is neglected (i.e. $\Delta \rightarrow 0$). In fact, the result is identical as we would expect in a universe where the Hubble law is universally true.
Therefore (\[eqn:phasespacederiv2\]) may be integrated exactly and yields the same result as Carmeli, $$\label{eqn:phasespacesolnnatural}
\frac {r} {c \tau}= \frac {\sinh (\frac{v}{c} \sqrt{1-\Omega})} {\sqrt{1-\Omega}}.$$
Since observations in the distant cosmos are always in terms of redshift, $z$, we write (\[eqn:phasespacesolnnatural\]) as a function of redshift where $r$ is expressed in units of $c \tau$ and $v/c = ((1+z)^2-1)/((1+z)^2+1)$ from the relativistic Doppler formula. The latter is appropriate since this is a velocity dimension.
What is important to note though is that regardless of the geometry of the Universe, provided it is spherically symmetrical and isotropic on the large scale, (\[eqn:phasespacesolnnatural\]) is identical to that we would get where the Universe has a unique center, with one difference which is explored in the following section. For an isotropic universe without a unique center, one can have an arbitrary number of centers. However if we are currently in a universe where the Galaxy is at the center of the local isotropy distribution this means the Universe we see must be very large and we are currently limited from seeing into an adjacent region with a different isotropy center.
\[sec:GravitationalPotential\]Gravitational Redshift
====================================================
In Hartnett [@Hartnett2005] the geometry in the model is the usual unbounded type, as found in an infinite universe, for example. In a finite bounded universe, an additional effect may result from the photons being received from the distant sources. The gravitational redshift ($z_{grav}$) resulting from the Galaxy sitting at the unique center of a finite spherically symmetrical matter distribution must be considered. In this case we need to consider the difference in gravitational potential between the points of emission and reception of a photon. Now the 00th metric component, the time part of the 5D metric of coordinates $x^k=t, r, \theta, \phi, v$ ($k = 0-4$), is required but it has never been determined for the cosmos in the Carmelian theory. In general relativity we would relate it by $g_{00} = 1-4\Phi/c^2$ where $-4\Phi$ is the gravitational potential. The factor 4 and minus sign arise from the Carmelian theory when (\[eqn:fvalue1\]) and (\[eqn:potential2\]) are compared. So the question must be answered, “What is $g_{00}$ metric component for the large scale structure of the universe in CGR?”
First note from (\[eqn:soln1field2\]) and (\[eqn:soln1field3\]) the $g_{11}$ metric component (considered in an unbounded universe for the moment) $$\label{eqn:g11}
g_{11} = -\left(1+\frac{1-\Omega}{c^2\tau^2}r^2\right)^{-1}$$ in CGR we can write a scale radius $$\label{eqn:scaleradius}
R = \frac{c \tau}{\sqrt{|1-\Omega|}}.$$ Hence we can define an energy density from the curvature $$\label{eqn:curvaturedensity}
\Omega_K = \frac{c^2}{h^2 R^2} = \frac{c^2 \tau^2}{R^2},$$ which, when we use (\[eqn:scaleradius\]), becomes $$\label{eqn:curvaturedensity2}
\Omega_K = 1-\Omega.$$ This quantifies the energy in the curved *spacevelocity*.
In the FRW theory the energy density of the cosmological constant is defined $\rho_{\Lambda} = \Lambda/8 \pi G$ hence $$\label{eqn:lambdadensityFRW}
\Omega_{\Lambda} = \frac{\Lambda}{3 H^2_0}.$$ Even though the cosmological constant is not explicitly used in CGR, it follows from the definition of the critical density that $$\label{eqn:criticaldensityCGR}
\rho_c = \frac{3}{8 \pi G \tau^2} = \frac{\Lambda}{8 \pi G},$$ when the cosmological constant $\Lambda$ is identified with $3/\tau^2$. Therefore in CGR it follows that $$\label{eqn:lambdadensityCGR}
\Omega_{\Lambda} = \frac{\Lambda}{3 h^2} = \Lambda\left(\frac{\tau^2}{3}\right) = 1.$$ This means that in CGR the vacuum energy $\rho_{vac} = \Lambda/8\pi G$ is encoded in the metric via the critical density since $\rho_{eff} = \rho - \rho_c$ principally defines the physics. So $\Omega_{\Lambda} = 1$ identically and at all epochs of time. (The determination of $\Omega_{\Lambda}$ in [@Hartnett2005] was flawed due to an incorrect definition.) Also we can relate $\Omega_{\Lambda}$ to the curvature density by $$\label{eqn:curvaturedadensity3}
\Omega_K = \Omega_{\Lambda} - \Omega,$$ which becomes $$\label{eqn:curvaturedadensity4}
\Omega_k = \Omega_{\Lambda} - \Omega_m,$$ at the present epoch ($z \approx 0$). Here $\Omega=\Omega_m(1+z)^3$ and hence $\Omega_K \rightarrow \Omega_k$ as $z \rightarrow 0$.
Finally we can write for the total energy density, the sum of the matter density and the curvature density, $$\label{eqn:totaldensity}
\Omega_t = \Omega + \Omega_K = \Omega + 1 - \Omega = 1,$$ which means the present epoch value is trivially $$\label{eqn:totaldensity0}
\Omega_0 = \Omega_m + \Omega_k = \Omega_m + 1 - \Omega_m = 1.$$ This means that the 3D spatial part of the Universe is always flat as it expands. This explains why we live in a universe that we observe to be identically geometrically spatially flat. The curvature is due to the velocity dimension. Only at some past epoch, in a radiation dominated universe, with radiation energy density $\Omega_R (1+z)^4$, would the total mass/energy density depart from unity.
Now considering a finite bounded universe, from (\[eqn:fvalue1\]), using $\Omega = \rho/\rho_{c}$, I therefore write $g_{00}$ as $$\label{eqn:g00}
g_{00}(r) = 1 + (1-\Omega_t) r^2 + 3(\Omega_t-1) \Delta^2,$$ where $r$ and $\Delta$ are expressed in units of $c \tau$. Equation (\[eqn:g00\]) follows from $g_{00} = 1-4\Phi/c^2$ where $\Phi$ is taken from the gravitational potential but with effective density, which in turn involves the total energy density because we are now considering *spacetime*.
Clearly from (\[eqn:totaldensity\]) it follows that $g_{00}(r) = 1$ regardless of epoch. Thus from the usual relativistic expression $$\label{eqn:gravredshift}
1 + z_{grav} = \sqrt{\frac{g_{00}(0)}{g_{00}(r)}} =1,$$ and the gravitational redshift is zero regardless of epoch. As expected if the emission and reception of a photon both occur in flat space then we’d expect no gravitational effects.
In an unbounded universe, though no gravitational effects need be considered, the result $g_{00} = 1$ is also the same. Therefore we can write down the full 5D line element for CGR in any dynamic spherically symmetrical isotropic universe, $$\label{eqn:linelement5D}
ds^2 = c^2dt^2 -\left(1+\frac{1-\Omega}{c^2\tau^2}r^2\right)^{-1}dr^2 +\tau^2 dv^2.$$ The $\theta$ and $\phi$ coordinates do not appear due to the isotropy condition $d\theta = d\phi=0$. Due to the Hubble law the 2nd and 3rd terms sum to zero leaving $dt = ds/c$, the proper time. Clocks, co-moving with the galaxies in the Hubble expansion, would measure the same proper time.
Since it follows from (\[eqn:g00\]) that $g_{00}(r) = 1$ regardless of epoch, $g_{00}(r)$ is not sensitive to any value of $\Delta$. This means the above analysis is true regardless of whether the universe is bounded or unbounded. The observations cannot distinguish. In an unbounded or bounded universe of any type no gravitational redshift (due to cosmological causes) in light from distant source galaxies would be observed.
However inside the Galaxy we expect the matter density to be much higher than critical, ie $\Omega_{galaxy} \gg 1$ and the total mass/energy density can be written $$\label{eqn:totaldensitygal}
\Omega_0 |_{galaxy} = \Omega_{galaxy} + \Omega_k \approx \Omega_{galaxy},$$ because $\Omega_k \approx 1$, since it is cosmologically determined. Therefore this explains why the galaxy matter density only is appropriate when considering the Poisson equation for galaxies.[@Hartnett2005b]
As a result inside a galaxy we can write $$\label{eqn:g00Galaxy}
g_{00}(r) = 1 + \Omega_K \frac{r^2}{c^2\tau^2} + \Omega_{galaxy} \frac{r^2}{c^2\tau^2} ,$$ in terms of densities at some past epoch. Depending on the mass density of the galaxy, or cluster of galaxies, the value of $g_{00}$ here changes. As we approach larger and larger structures it mass density approaches that of the Universe as a whole and $g_{00} \rightarrow 1$ as we approach the largest scales of the Universe. Galaxies in the cosmos then act only as local perturbations but have no effect on $\Omega_K$. That depends only on the average mass density of the whole Universe, which depends on epoch ($z$).
Equation (\[eqn:g00Galaxy\]) is in essence the same expression used on page 173 of Carmeli [@Carmeli2002a] in his gravitational redshift formula rewritten here as $$\label{eqn:Cgravz}
\frac{\lambda_2}{\lambda_1} = \sqrt{\frac{1+ \Omega_K r^2_2/c^2\tau^2-R_S/r_2}{1+\Omega_K r^2_1/c^2\tau^2-R_S/r_1}}.$$ involving a cosmological contribution ($\Omega_K r^2/c^2\tau^2$) and $R_S=2GM/c^2$, a local contribution where the mass $M$ is that of a compact object. The curvature ($\Omega_K$) results from the averaged mass/energy density of the whole cosmos, which determines how the galaxies ‘move’ but motions of particles within galaxies is dominated by the mass of the galaxy and the masses of the compact objects within. Therefore when considering the gravitational redshifts due to compact objects we can neglect any cosmological effects, only the usual Schwarzschild radius of the object need be considered. The cosmological contributions in (\[eqn:Cgravz\]) are generally negligible. This then leads back to the realm of general relativity.
\[sec:WhiteHole\]White Hole
===========================
Now if we assume the radial extent of a finite matter distribution at the current epoch is equal to the current epoch scale radius, we can write $$\label{eqn:delta}
\Delta =\frac{1}{\sqrt{\Omega_k}} = \frac{1}{\sqrt{|1-\Omega_m|}},$$ expressed in units of $c \tau$. In such a case, $\Delta = 1.02 \;c \tau$ if $\Omega_m = 0.04$ and $\Delta = 1.01 \; c \tau$ if $\Omega_m = 0.02$.
It is important to note also that in Carmeli’s unbounded model (\[eqn:phasespacesolnnatural\]) describes the redshift distance relationship but there is no central potential. In Hartnett [@Hartnett2005] and in Oliveira and Hartnett [@Oliveira2006] equation (\[eqn:phasespacesolnnatural\]) was curve fitted to the SNe Ia data and was found to agree with $\Omega_m = 0.02-0.04$ without the inclusion of dark matter or dark energy. Therefore the same conclusion must also apply to the finite bounded model suggested here.
In order to achieve a fit to the data, using either the finite bounded or unbounded models, the white hole solution of (\[eqn:soln1field3\]) or (\[eqn:soln1fieldgen2\]) must be chosen. The sign of the terms in (\[eqn:fvalue1\]) means that the potential implicit in (\[eqn:fvalue1\]) is a potential hill, not a potential well. Therefore the solution describes an expanding white hole with the observer at the origin of the coordinates, the unique center of the Universe. Only philosophically can this solution be rejected. Using the Carmeli theory, the observational data cannot distinguish between finite bounded models ($\infty > \Delta \geq c\tau$) and finite ($\Delta = 0$) or infinite ($\Delta = \infty$) unbounded models .
The physical meaning is that the solution, developed in this paper, represents an expanding white hole centered on the Galaxy. The galaxies in the Universe are spherically symmetrically distributed around the Galaxy. The observed redshifts are the result of cosmological expansion alone.
Moreover if we assume $\Delta \approx c \tau$ and $\Omega_m = 0.04$ then it can be shown [@Oliveira2006] that the Schwarzschild radius for the finite Universe $$\label{eqn:Rs}
R_s \approx \Omega_m \Delta = 0.04 \; c \tau.$$ Therefore for a finite universe with $\Delta \approx c \tau$ it follows that $R_s \approx 0.04 \; c \tau \approx 200 \; Mpc$. Therefore an expanding finite bounded universe can be considered to be a white hole. As it expands the matter enclosed within the Schwarzschild radius gets less and less. The gravitational radius of that matter therefore shrinks towards the Earth at the center.
This is similar to the theoretical result obtained by Smoller and Temple [@Smoller2002] who constructed a new cosmology from the FRW metric but with a shock wave causing a time reversal white hole. In their model the total mass behind the shock decreases as the shock wave expands, which is spherically symmetrically centered on the Galaxy. Their paper states in part “...the entropy condition implies that the shock wave must weaken to the point where it settles down to an Oppenheimer Snyder interface, (bounding a finite total mass), that eventually emerges from the white hole event horizon of an ambient Schwarzschild spacetime.”
This result then implies that the earth or at least the Galaxy is in fact close to the physical center of the Universe. Smoller and Temple state [@Smoller2003] that “With a shock wave present, the *Copernican Principle is violated* in the sense that the earth then has a special position relative to the shock wave. But of course, in these shock wave refinements of the FRW metric, there is a spacetime on the other side of the shock wave, beyond the galaxies, and so the scale of uniformity of the FRW metric, the scale on which the density of the galaxies is uniform, is no longer the largest length scale”\[emphasis added\].
Their shock wave refinement of a critically expanding FRW metric leads to a big bang universe of finite total mass. This model presented here also has a finite total mass and is a spatially flat universe. It describes a finite bounded white hole that started expanding at some time in the past.
\[sec:Conclusion\]Conclusion
============================
Since the Carmeli theory has been successfully analyzed with distance modulus data derived by the high-z type Ia supernova teams it must also be consistent with a universe that places the Galaxy at the center of an spherically symmetrical isotropic expanding white hole of finite radius. The result describes particles moving in both a central potential and an accelerating spherically expanding universe without the need for the inclusion of dark matter. The data cannot be used to exclude models with finite extensions $\Delta \geq c\tau$.
[99]{} P. Astier, *et al* “The Supernova Legacy Survey: Measurement of $\Omega_M$, $\Omega_\Lambda$ and $w$ from the first year data set”, *Astron. Astrophys.* (2005) arXiv:astro-ph/0510447 S. Behar, M. Carmeli, “Cosmological relativity: A new theory of cosmology”, *Int. J. Theor. Phys.* **39** (5): 1375–1396 (2000) M. Carmeli, “Cosmological General Relativity”, *Commun. Theor. Phys.* **5**:159 (1996) M. Carmeli, “Is galaxy dark matter a property of spacetime?”, *Int. J. Theor. Phys.* **37** (10): 2621–2625 (1998) M. Carmeli, *Cosmological Special Relativity* (World Scientific, Singapore, 2002) M. Carmeli, “Accelerating Universe: Theory versus Experiment”, \[arXiv: astro-ph/0205396\] (2002) M. Carmeli, J.G. Hartnett, F.J. Oliveira, “The cosmic time in terms of the redshift,” *Found. Phys. Lett.* **19**(3):277–283 (2006) arXiv:gr-qc/0506079 F.J. Oliveira, J.G. Hartnett, “Carmeli’s cosmology fits data for an accelerating and decelerating universe without dark matter nor dark energy,” *Found. Phys. Lett.* **19**(6):519-535 (2006) arXiv: astro-ph/0603500 W.W. Gibbs, “Profile: George F. R. Ellis”, *Scientific American* **273**(4): 28-–29 (1995) J.G. Hartnett, “The distance modulus determined from Carmeli’s cosmology fits the accelerating universe data of the high-redshift type Ia supernovae without dark matter,” *Found. Phys.* 36(6): 839–861 (2006) arXiv:astro-ph/0501526 Hartnett, J.G. “Spiral galaxy rotation curves determined from Carmelian general relativity” *Int. J. Theor. Phys.* **45**(11):2147–2165 (2006) arXiv:astro-ph/0511756 R.A. Knop, *et al*, “New constraints on $\Omega_{M}$, $\Omega_{\Lambda}$ and $w$ from an independent set of 11 high-redshift supernovae observed with the Hubble Space Telescope”, *Ap. J.* **598**: 102–137 (2003) A.G. Riess, *et al*, “Type Ia supernovae discoveries at $z >1$ from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution” *Ap. J.* **607**: 665–687 (2004) J. Smoller and B. Temple, PNAS **100**(20): 11216–11218 (2003) J. Smoller and B. Temple, http://www.math.ucdavis.edu/ temple/articles/temple1234.pdf
|
---
abstract: 'An article usually includes an abstract, a concise summary of the work covered at length in the main body of the article. It is used for secondary publications and for information retrieval purposes. For PRL, the rule of thumb is that the abstract should be less than 8 lines and the text (excluding authors, abstract but including tables, figures and references) should be less than 4 pages (leave about 20 lines empty on page 4) in two-column format. PRL and PRD papers have to have PACS (Phsyics and Astronomy Classification Scheme) numbers. Please see [http://www.aip.org/pacs/]{} for the numbers relevant to your paper. A set of standard references can be found at the end of this example paper.'
title: 'Template for PRL/PRD Papers'
---
*DØ INTERNAL DOCUMENT – NOT FOR PUBLIC DISTRIBUTION*
author\_list.tex
This sample document demonstrates proper use of REVTeX 4 (and ) in mansucripts prepared for submission to APS journals. Further information can be found in the REVTeX 4 documentation included in the distribution or available at <http://publish.aps.org/revtex4/>.
When commands are referred to in this example file, they are always shown with their required arguments, using normal TeX format. In this format, `#1`, `#2`, etc. stand for required author-supplied arguments to commands. For example, in `\section{#1}` the `#1` stands for the title text of the author’s section heading, and in `\title{#1}` the `#1` stands for the title text of the paper.
Line breaks in section headings at all levels can be introduced using \\\\. A blank input line tells TeX that the paragraph has ended. Note that top-level section headings are automatically uppercased. If a specific letter or word should appear in lowercase instead, you must escape it using `\lowercase{#1}` as in the word “via” above.
This file may be formatted in both the `preprint` and `twocolumn` styles. `twocolumn` format may be used to mimic final journal output. Either format may be used for submission purposes; however, for peer review and production, APS will format the article using the `preprint` class option. Hence, it is essential that authors check that their manuscripts format acceptably under `preprint`. Manuscripts submitted to APS that do not format correctly under the `preprint` option may be delayed in both the editorial and production processes.
The `widetext` environment will make the text the width of the full page. The width-changing commands only take effect in `twocolumn` formatting. It has no effect if `preprint` formatting is chosen instead.
To cite bibliography entries, use the `\cite{#1}` command. Most journal styles will display the corresponding number(s) in square brackets: [@d0det]. To avoid the square brackets, use `\onlinecite{#1}`: Refs. and . REVTeX “collapses” lists of consecutive reference numbers where possible. We now cite everyone together [@geant; @pythia; @cteq], and once again (Refs. ). Note that the references were also sorted into the correct numerical order as well.
Footnotes are produced using the `\footnote{#1}` command. Most APS journal styles put footnotes into the bibliography. REVTeX 4 does this as well, but instead of interleaving the footnotes with the references, they are listed at the end of the references. Because the correct numbering of the footnotes must occur after the numbering of the references, an extra pass of LaTeX is required in order to get the numbering correct.
Inline math may be typeset using the `$` delimiters. Bold math symbols may be achieved using the `bm` package and the `\bm{#1}` command it supplies. For instance, a bold $\alpha$ can be typeset as `$\bm{\alpha}$` giving $\bm{\alpha}$. Fraktur and Blackboard (or open face or double struck) characters should be typeset using the `\mathfrak{#1}` and `\mathbb{#1}` commands respectively. Both are supplied by the `amssymb` package. For example, `$\mathbb{R}$` gives $\mathbb{R}$ and `$\mathfrak{G}$` gives $\mathfrak{G}$
In LaTeX there are many different ways to display equations, and a few preferred ways are noted below. Displayed math will center by default. Use the class option `fleqn` to flush equations left.
Below we have numbered single-line equations; this is the most common type of equation in *Physical Review*: $$\begin{aligned}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\{%
\openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
\frac{1\sum^{a}_{b}}{A^2}%
\right\}%
\label{eq:one}.\end{aligned}$$ Note the open one in Eq. (\[eq:one\]).
Not all numbered equations will fit within a narrow column this way. The equation number will move down automatically if it cannot fit on the same line with a one-line equation: $$\left\{
ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
\frac{1\sum^{a}_{b}}{A^2}%
\right\}.$$
When the `\label{#1}` command is used \[cf. input for Eq. (\[eq:one\])\], the equation can be referred to in text without knowing the equation number that TeX will assign to it. Just use `\ref{#1}`, where `#1` is the same name that used in the `\label{#1}` command.
Unnumbered single-line equations can be typeset using the `\[`, `\]` format: $$g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~.$$
Figures may be inserted by using either the `graphics` or `graphicx` packages. These packages both define the `\includegraphics{#1}` command, but they differ in how optional arguments for specifying the orientation, scaling, and translation of the figure. Fig. \[fig:epsart\] shows a figure that is small enough to fit in a single column. It is embedded using the `figure` environment which provides both the caption and the imports the figure file.
![\[fig:epsart\] A figure caption. The figure captions are automatically numbered.](fig_1.ps)
Fig. \[fig:wide\] is a figure that is too wide for a single column, so instead the `figure*` environment has been used.
The heart of any table is the `tabular` environment which gives the rows of the tables. Each row consists of column entries separated by `&`’s and terminates with \\\\. The required argument for the `tabular` environment specifies how data are displayed in the columns. For instance, entries may be centered, left-justified, right-justified, aligned on a decimal point. Extra column-spacing may be be specified as well, although REVTeX 4 sets this spacing so that the columns fill the width of the table. Horizontal rules are typeset using the `\hline` command. The doubled (or Scotch) rules that appear at the top and bottom of a table can be achieved enclosing the `tabular` environment within a `ruledtabular` environment. Rows whose columns span multiple columns can be typeset using the `\multicolumn{#1}{#2}{#3}` command (for example, see the first row of Table \[tab:table3\]).
Tables \[tab:table1\]-\[tab:table4\] show various effects. Tables that fit in a narrow column are contained in a `table` environment. Table \[tab:table3\] is a wide table set with the `table*` environment. Long tables may need to break across pages. The most straightforward way to accomplish this is to specify the `[H]` float placement on the `table` or `table*` environment. However, the standard package `longtable` will give more control over how tables break and will allow headers and footers to be specified for each page of the table. A simple example of the use of `longtable` can be found in the file `summary.tex` that is included with the REVTeX 4 distribution.
There are two methods for setting footnotes within a table (these footnotes will be displayed directly below the table rather than at the bottom of the page or in the bibliography). The easiest and preferred method is just to use the `\footnote{#1}` command. This will automatically enumerate the footnotes with lowercase roman letters. However, it is sometimes necessary to have multiple entries in the table share the same footnote. In this case, there is no choice but to manually create the footnotes using `\footnotemark[#1]` and `\footnotetext[#1]{#2}`. `#1` is a numeric value. Each time the same value for `#1` is used, the same mark is produced in the table. The `\footnotetext[#1]{#2}` commands are placed after the `tabular` environment. Examine the LaTeX source and output for Tables \[tab:table1\] and \[tab:table2\] for examples.
Left[^1] Centered[^2] Right
---------- -------------- -------
1 2 3
10 20 30
100 200 300
: \[tab:table1\]This is a narrow table which fits into a narrow column when using `twocolumn` formatting. Note that REVTeX 4 adjusts the intercolumn spacing so that the table fills the entire width of the column. Table captions are numbered automatically. This table illustrates left-aligned, centered, and right-aligned columns.
$r_c$ (Å) $r_0$ (Å) $\kappa r_0$ $r_c$ (Å) $r_0$ (Å) $\kappa r_0$
---- ----------- ----------- -------------- ---- ----------- ----------- --------------
Cu 0.800 14.10 2.550 Sn 0.680 1.870 3.700
Ag 0.990 15.90 2.710 Pb 0.450 1.930 3.760
Au 1.150 15.90 2.710 Ca 0.750 2.170 3.560
Mg 0.490 17.60 3.200 Sr 0.900 2.370 3.720
Zn 0.300 15.20 2.970 Li 0.380 1.730 2.830
Cd 0.530 17.10 3.160 Na 0.760 2.110 3.120
Hg 0.550 17.80 3.220 K 1.120 2.620 3.480
Al 0.230 15.80 3.240 Rb 1.330 2.800 3.590
Ga 0.310 16.70 3.330 Cs 1.420 3.030 3.740
In 0.460 18.40 3.500 Ba 0.960 2.460 3.780
Tl 0.480 18.90 3.550
: \[tab:table2\]A table with more columns still fits properly in a column. Note that several entries share the same footnote. Inspect the LaTeX input for this table to see exactly how it is done.
----- ------------------- ------------------- ------------------- -------------------
Ion 1st alternative 2nd alternative lst alternative 2nd alternative
K $(2e)+(2f)$ $(4i)$ $(2c)+(2d)$ $(4f)$
Mn $(2g)$[^3] $(a)+(b)+(c)+(d)$ $(4e)$ $(2a)+(2b)$
Cl $(a)+(b)+(c)+(d)$ $(2g)$ $(4e)^{\text{a}}$
He $(8r)^{\text{a}}$ $(4j)^{\text{a}}$ $(4g)^{\text{a}}$
Ag $(4k)^{\text{a}}$ $(4h)^{\text{a}}$
----- ------------------- ------------------- ------------------- -------------------
[ccddd]{} One&Two&&&\
one&two&&&\
He&2& 2.77234 & 45672. & 0.69\
C[^4] &C[^5] & 12537.64 & 37.66345 & 86.37\
*Physical Review* style requires that the initial citation of figures or tables be in numerical order in text, so don’t cite Fig. \[fig:wide\] until Fig. \[fig:epsart\] has been cited.
acknowledgement.tex
[99]{}
Standard DØ detector reference:\
V.M. Abazov [*et al.*]{} (D0 Collaboration), Nucl. Instrum. Methods Phys. Res. A [**565**]{}, 463 (2006).
\*\* New \*\* DØ luminosity reference:\
T. Andeen [*et al.*]{}, FERMILAB-TM-2365 (2007).
Particle Data Group reference:\
W.-M. Yao [*et al.*]{}, Journal of Physics G [**33**]{}, 1 (2006).
reference:\
R. Brun and F. Carminati, CERN Program Library Long Writeup W5013, 1993 (unpublished).
reference:\
T. Sjöstrand [*et al.*]{}, Comput. Phys. Commun. [**135**]{}, 238 (2001).
reference:\
J. Pumplin [*et al.*]{}, JHEP [**0207**]{} 012 (2002) and D. Stump [*et al.*]{}, JHEP [**0310**]{} 046 (2003).
LEP CL$_S$ reference:\
T. Junk, Nucl. Instrum. Methods A [**434**]{}, 435 (1999).
DØ Bayesian reference:\
I. Bertram [*et al.*]{}, FERMILAB-TM-2104 (2000).
DØ cone-jet reference:\
G.C. Blazey [*et al.*]{}, in [*Proceedings of the Workshop: QCD and Weak Boson Physics in Run II,*]{} edited by U. Baur, R.K. Ellis, and D. Zeppenfeld, Fermilab-Pub-00/297 (2000).
[^1]: Note a.
[^2]: Note b.
[^3]: The $z$ parameter of these positions is $z\sim\frac{1}{4}$.
[^4]: Some tables require footnotes.
[^5]: Some tables need more than one footnote.
|
---
abstract: 'We find simple saturated alcohols with the given number of carbon atoms and the minimal normal boiling point. The boiling point is predicted with a weighted sum of the generalized first Zagreb index, the second Zagreb index, the Wiener index for vertex-weighted graphs, and a simple index caring for the degree of a carbon atom being incident to the hydroxyl group. To find extremal alcohol molecules we characterize chemical trees of order $n$, which minimize the sum of the second Zagreb index and the generalized first Zagreb index, and also build chemical trees, which minimize the Wiener index over all chemical trees with given vertex weights.'
author:
- |
**Mikhail Goubko, **Oleg Miloserdov\
*Institute of Control Sciences*\
*of Russian Academy of Sciences, Moscow, Russia*\
`mgoubko@mail.ru, omilos92@gmail.com`****
date: '(Received October 28, 2014)'
title: |
**Simple Alcohols with the Lowest Normal\
Boiling Point Using Topological Indices[^1]**
---
=0.30in
Introduction
============
Consider a collection $\Omega$ of *admissible molecules* (for example, represented with their structural formulas or chemical graphs), each endowed with $k+1$ significant physical or chemical *properties* (e.g., normal density, normal boiling point, refraction coefficient, retention index, or more exotic and problem-specific ones), and let $P_i(G)$, $i=0,...,k$, be the numeric value of the $i$-th property of a molecule $G\in \Omega$ (e.g., the normal boiling point value). A typical problem of *molecular design* is the following *optimization problem*: $$\begin{gathered}
\label{eq_property_optimization}
P_0(G)\rightarrow \min_{G\in \Omega}\,(\max_{G\in \Omega})\\
P_i^{min}\le P_i(G)\le P_i^{max}, i=1,...,k.\nonumber\end{gathered}$$
When the functions $P_i(\cdot)$ are only partially known from the experiment, they are replaced with *predicted* figures, relating a chemical graph $G\in\Omega$ to the predicted value $\tilde{P}_i(G)$ of the $i$-th physical or chemical property ($i=0,...,k$) by virtue of numeric characteristics (known as *molecular descriptors*), which can be calculated on basis of a molecular structure. A typical *quantitative structure-property relation* (QSPR) includes several molecular descriptors, and is presented as $$\tilde{P}_i(G) = \tilde{P}_i(I_1(G),...,I_m(G)),i=0,...,k,$$ where $I_1(G),..., I_m(G)$ are the values of molecular descriptors for the molecular graph $G$. The simplest *linear regression* is just a weighted sum of descriptors: $$\tilde{P}_i(G) = \alpha_{1,i} I_1(G)+...+\alpha_{m,i} I_m(G), i=0,...,k.$$
During recent decades a number of *topological*, *geometrical*, and *quantum-mechanical* molecular descriptors were suggested and studied [@Gutman10.1; @Gutman10.2; @Todeschini00; @Todeschini09]. Below we limit ourselves to topological descriptors only (see, for instance, the handbook [@Balaban00]) to study problem (\[eq\_property\_optimization\]) as a problem of the *extremal graph theory* [@Bollobas04] .
Exhaustive enumeration of all feasible molecules (the *brute force* approach) can only be used to solve this problem when the feasible set is relatively small; for bigger sets mathematical chemistry suggests a variety of limited search techniques. In numerous papers lower and upper bounds of dozens topological indices over various feasible sets were obtained [@Dobrynin01; @Volkmann02; @Furtula13; @Lin13; @SurveyOptimLi08; @ReviewOptimRada14; @Wang08; @ReviewOptXu14; @Zhang08; @SurveyOptimZhou04], and, in many cases, extremal graphs were characterized. At the same time, the problem (\[eq\_property\_optimization\]) of optimization of a composition of indices is still understudied.
In fact, finding lower and upper bounds of individual indices can be a step towards solving problem (\[eq\_property\_optimization\]), as a linear combination of lower bounds is a lower-bound estimate of the combination of indices. This estimate can be used in a branch-and-bound algorithm of limited search. Yet, the quality of the estimate may be considerably poor, resulting in lack of efficient cuts in a branch-and-bound algorithm.
Anyway, the common shortcoming of an algorithmic approach to index optimization is that it does not support the analysis of general characteristics of an extremal molecule (i.e., of a corresponding graph). When available, side information would be of great value on why a certain graph is optimal or not, what shape the extremal graphs have, etc. Such information is revealed using analytical tools of discrete optimization.
In this paper we apply recent results in optimization of degree- and distance-based topological indices to find a simple saturated alcohol with the given molecular weight and minimal boiling point. We reduce the property minimization problem to that of minimization of a weighted linear combination of the generalized first Zagreb index, the second Zagreb index, the vertex-weighted Wiener index, and a simple index caring for the degree of a carbon atom being incident to the hydroxyl group. Then we characterize minimizers of this linear combination of indices (see Fig. \[fig\_BP0\_minimizers\]) and of a couple of simpler regressions (see Fig. \[fig\_C\_minimizers\] and \[fig\_VWWI\_minimizers\]).
Predicting Boiling Points of Simple Alcohols {#section_Huffman_alg}
============================================
The normal boiling point of a liquid is determined by its solvation free energy. The solvation free energy can be predicted with high accuracy from computer simulations (see [@Bren06; @Bren12] for details). The simulation-based approach solves well the “direct problem” of predicting the solvation free energy for a given molecule, but it does not help solving the “inverse problem” of finding the molecule having the minimal solvation free energy (and, consequently, the boiling point). For this reason we predict boiling points of simple saturated alcohols (those having a general formula $\textrm{C}_n\textrm{H}_{2n+1}\textrm{OH}$) with the aid of topological indices.
Alcohols have relatively high boiling points when compared to related compounds due to hydrogen bonds involving a highly polarized hydroxyl group, and branched isomers have lower boiling points than alcohols with the linear structure. Another structural feature affecting the boiling point is the *“oxygen shielding” effect* [@AlcoholPenchev07], when atoms surrounding the hydroxyl group partially shield it preventing formation of hydrogen bonds between molecules and, thus, decreasing the boiling point.
We considered several degree-based topological indices (the first Zagreb index $M_1$ [@GutmanTrin72], the second Zagreb index $M_2$ [@GutmanTrin72], Randić index [@Randic75] and the others), which are known to be good metrics of branchiness, and, finally, the generalized first Zagreb index $C_1$ (see also [@Goubko14]) has shown the best results: $$C_1(G)=\sum_{v\in V(G)}c(d_G(v)),$$ where $V(G)$ is the vertex set of graph $G$, $d_G(v)$ is the degree of the vertex $v\in V(G)$ in graph $G$, and $c(d)$ is a non-negative function defined for degrees from 1 to 4. It can alternatively be written as $$\label{eq_gen1stZI}
C_1(G)=c(1) n_1(G)+c(2) n_2(G)+c(3) n_3(G)+c(4) n_4(G),$$ where $n_i(G)$ is the atoms’ count of degree $i=1,...,4$ in a molecular graph $G$, and $c(1), ..., c(4)$ are regression parameters.
We also employed the classical *second Zagreb index* [@GutmanTrin72] $$M_2(G)=\sum_{uv\in E(G)}d_G(u)d_G(v),$$ where $E(G)$ is the edge set of graph $G$.
Another index used was the *Wiener index*, which had been the first topological index for boiling point prediction [@Wiener47] due to its high correlation with the molecule’s surface area. To account for heterogeneity of atoms we allow each pair of vertices $u,v \in V(G)$ to have unique weight $\mu_G(u,v)$ and calculate the *pair-weighted Wiener index* as $$\label{eq_PWWI}
PWWI(G):=\frac{1}{2}\sum_{u,v\in V(G)}\mu_G(u,v) d_G(u,v),$$ where $d_G(u, v)$ is the distance (the length of the shortest path) between vertices $u$ and $v$ in $G$. For example, we can assign different weights to distances between pairs of carbon atoms and between carbons and the oxygen atom in an alcohol molecule.
Regression tuning has shown the distances between carbon atoms to be irrelevant for the alcohol boiling point, and only distances to the oxygen matter. All such distances are accounted with equal weight, so the pair-weighted Wiener index reduces to the *distance* of the oxygen atom, which was first used for the alcohol boiling point prediction in [@AlcoholPenchev07]: $$\label{eq_WIO}
WI_\textrm{O}(G):=\sum_{u\in V(G)}d_G(u,\textrm{O}).$$
In [@AlcoholPenchev07] a geometrical descriptor has also been suggested to account for oxygen shielding, but we extend the approach by [@AlcoholJanezic06] instead, and introduce a simple topological index $S_i(G)$, which is equal to unity when the carbon atom incident to the hydroxyl group in the alcohol molecule $G$ has degree $i=2,3,4$ (we exclude *methanol* from consideration), and is equal to zero otherwise.
We collected a data set of experimental boiling points under normal conditions for 79 simple saturated alcohols having from 2 to 12 carbon atoms and representing various branchiness. Several data sources [@AlcoholGarcia08; @AlcoholJanezic06; @KompanyZareh03; @AlcoholPenchev07] were combined with priority on Alpha Aesar experimental data to resolve discrepancy. In Table \[tab\_stat\_data\] we present basic statistics about the data sample.
\# carbons \# isomers min chain len. max chain len. min b.p. max b.p.
------------ ------------ ---------------- ---------------- ---------- ----------
2 1 2 2 78 78
3 2 2 3 82.5 97
4 4 2 4 82.4 117.5
5 8 3 5 102 137
6 17 3 6 120 157
7 18 4 7 131.5 175.5
8 10 4 8 147.5 194
9 11 4 9 169.5 215
10 5 5 10 168 231
11 2 10 11 228.5 243
12 1 12 12 259 259
TOTAL 79 2 12 78 259
: Data sample: basic statistics[]{data-label="tab_stat_data"}
Information on boiling points of alcohols including more than 12 carbon atoms is less common and reliable. The complete data set together with the best regressions is available online at [@GoubkoMilosOnline14].
We randomly split the sample into the training set containing 50 cases and the testing set containing 29 cases. Then we examined different linear regressions involving the descriptors mentioned above.[^2] The best performance and predictive power was obtained for the linear combination of the oxygen’s distance cube root $WI_\textrm{O}(G)^\frac{1}{3}$ (with weight $b_1^0$), the generalized first Zagreb index $C_1(G)$ (with weights $c^0(1),...,c^0(4)$), the second Zagreb index $M_2(G)$ (weighted by $b^0_3$), and the simple indicator of the sub-root’s degree $S_2(G)$ (weighted by $b_2^0$): $$\label{eq_bp0}
BP^0=b_0^0+b_1^0 WI_\textrm{O}(G)^\frac{1}{3}+c^0(2) n_2(G)+c^0(3)
n_3(G)+b^0_2 S_2(G)+b^0_3 M_2(G).$$
Below this regression is referred to as the *basic* one. The optimal values of weights $b_1^0, c^0(1),...,c^0(4), b_2^0, b_3^0$ (including the constant term $b_0^0$) calculated with the least squares method are presented in the first column of Table \[tab\_param\]. Parameters $c(1)$ and $c(4)$, which weight variables $n_1(G)$ and $n_4(G)$ respectively, appear to be insignificant and can be set to zero in (\[eq\_gen1stZI\]) when calculating the generalized first Zagreb index. The precision of the basic regression is shown in Table \[tab\_precision\]. It is comparable to the best known relations [@AlcoholGarcia08; @AlcoholJanezic06; @KompanyZareh03; @AlcoholPenchev07]. In the following sections we show how far we can come in analytical and numeric minimization of this combination of indices.
-------------- --------- ------------------------ --------- ------------------------ --------- ------------------------
Corr. SD, $^\circ\textrm{C}$ Corr. SD, $^\circ\textrm{C}$ Corr. SD, $^\circ\textrm{C}$
Training set $0.997$ $2.98$ $0.997$ $3.12$ $0.996$ $3.26$
Testing set $0.996$ $3.23$ $0.995$ $3.58$ $0.994$ $3.99$
-------------- --------- ------------------------ --------- ------------------------ --------- ------------------------
: Precision of regressions: Correlation coefficient and standard deviation (SD)[]{data-label="tab_precision"}
Coefficient Basic regression (“$^0$”) Regression I (“$^\textrm{I}$”) Regression II (“$^{\textrm{II}}$”)
------------- --------------------------- -------------------------------- ------------------------------------
$b_0$ $35.245$ $50.626$ $44.134$
$b_1$ $12.233$ $-$ $3.851$
$b_2$ $9.170$ $11.295$ $10.980$
$b_3$ $1.486$ $1.000$ $-$
$c(1)$ $-$ $-$ $-$
$c(2)$ $9.514$ $14.534$ $17.727$
$c(3)$ $9.380$ $20.172$ $29.673$
$c(4)$ $-$ $17.015$ $36.470$
: Parameters of regressions[]{data-label="tab_param"}
We also considered two simplifications of regression (\[eq\_bp0\]), for which alcohol molecules having minimal predicted boiling point can be characterized analytically. The first one (below referred to as “Regression I”) is obtained by withdrawing $WI_\textrm{O}(G)$ in (\[eq\_bp0\]): $$\label{eq_bpI}
BP^\textrm{I}=b_0^\textrm{I}+c^\textrm{I}(2) n_2(G)+c^\textrm{I}(3) n_3(G)+c^\textrm{I}(4) n_4(G)+b^\textrm{I}_2
S_2(G)+b^\textrm{I}_3 M_2(G).$$
See Table \[tab\_precision\] for the precision figures of regression (\[eq\_bpI\]) under the values of parameters delivering the best approximation to the training set (see the middle column of Table \[tab\_param\]).
The second simplified regression (referred to as “Regression II” below) is obtained by withdrawing $M_2(G)$ in (\[eq\_bp0\]): $$\label{eq_bpII}
BP^\textrm{II}=b_0^\textrm{II}+b_1^\textrm{II}
WI_\textrm{O}(G)^{\frac{1}{3}}+c^\textrm{II}(2)
n_2(G)+c^\textrm{II}(3) n_3(G)+c^\textrm{II}(4)
n_4(G)+b^\textrm{II}_2 S_2(G).$$
In Table \[tab\_precision\] we show its precision under the optimal values of parameters depicted in the last column of Table \[tab\_param\].
The shortcoming of Regression II is that the term $WI_\textrm{O}(G)^{\frac{1}{3}}$ appears to be insignificant after disposal of $M_2(G)$, being responsible of approximately 1 per cent of the residual sum of squares. Nevertheless, we keep this regression for illustration of joint optimization of $C_1(G)$ and of the pair-weighted Wiener index.
Minimization of indices and their combinations {#section_min_comb}
==============================================
In the present paper we find a simple saturated alcohol isomer with $n-1$ carbon atoms having the lowest predicted boiling point. As the regressions introduced in the previous section are tested only for alcohols containing from 2 to 12 carbon atoms, we restrict our attention to $n\le14$, where we can expect some accuracy of the obtained results.
For $n\le 14$ admissible sets of all simple saturated alcohol molecules with $n-1$ carbons are not too extensive, and allow for the brute-force enumeration. Moreover, we are sure that no aid of a computer is needed for an organic chemist to draw a molecule being a good approximation to the boiling point minimizer for all $n\le 14$. However, our aim is to show how analytic optimization techniques formalize the professional intuition and help making general conclusions of verifiable reliability.
Let us characterize chemical trees minimizing indices introduced in the previous section and their combinations.
Degree-based indices {#subsection_degree}
--------------------
For a simple connected undirected graph $G$ denote with $W(G)$ the set of *pendent vertices* (those having degree 1) of the graph $G$, and with $M(G):=V(G)\backslash W(G)$ the set of *internal vertices* (with degree $>1$) of $G$.
A simple connected undirected graph of order $n$ is called a *chemical tree* if it has $n-1$ edges and its vertex degrees do not exceed $4$. Denote with $\mathcal{T}(n)$ the set of all chemical trees of order $n$.
A *pendent-rooted chemical tree* is a chemical tree, in which one pendent vertex is distinguished and called a *root*. A typical pendent-rooted tree is denoted with $T_r$, with $r$ being its root. A vertex being incident to the root in $T_r$ is called a *sub-root* and is denoted as $sub(T_r)$. Denote with $\mathcal{R}(n)$ the set of all pendent-rooted chemical trees of order $n$.
Clearly, the set $\Omega(n-1)$ of all molecules of simple saturated alcohols having $n-1$ carbons coincides with the set $\mathcal{R}(n)$ of pendent-rooted chemical trees of order $n$ (with a root corresponding to the hydroxyl group and the other vertices forming the carbon skeleton of a molecule).
For a topological index $I(\cdot)$ defined on an admissible set of graphs $\mathcal{G}$ introduce the notation $I_\mathcal{G}^*:=\min_{G\in \mathcal{G}}I(G)$ and let $\mathcal{G}_I^*:=\operatorname*{Arg\,min}_{G\in \mathcal{G}}I(G)$ be the set of graphs minimizing $I(\cdot)$ over $\mathcal{G}$. For example, $\mathcal{T}_{M_2}^*(n)$ is the set of chemical trees of order $n$ minimizing the second Zagreb index $M_2(\cdot)$.
Define also the set $\mathcal{R}_i(n):=\{T\in \mathcal{R}(n): d_T(sub(T))=i\}$ of all pendent-rooted trees with a sub-root having degree $i=2,...,4$.
We start with the following obvious statement.
\[lemma\_S\_i\]$S_i(G)$ achieves its minimum at any pendent-rooted chemical tree with sub-root’s degree other than $i$. In other words, $\mathcal{R}_{S_i}^*(n)=\mathcal{R}(n)\backslash \mathcal{R}_i(n)$.
is straightforward, as $S_i(G)=1$ for all $G\in \mathcal{R}_i(n)$, and $S_i(G)=0$ otherwise.
Indices $C_1(G)$ and $M_2(G)$ do not account for heterogeneity of atoms in a molecule, so we can minimize them over the set $\mathcal{T}(n)$ of all chemical trees of order $n$ and then assign the root to an arbitrary pendent vertex of the index-minimizing tree to obtain a pendent-rooted tree, which minimizes the index.
Consider an “ad-hoc” degree-based topological index $$\label{eq_ad_hoc_index}
C(G):=C_1(G)+b_3M_2(G)=\sum_{v\in V(G)}c(d_G(v))+b_3\sum_{uv\in
E(G)}d_G(u)d_G(v),$$ where $b_3$ is an arbitrary real constant (we keep notation $b_3$ for compatibility with equations (\[eq\_bp0\]), (\[eq\_bpI\])).
A chemical tree $T\in \mathcal{T}(n)$ is *extremely branched*, if its internal vertices have degree $4$, except one vertex having degree $2$ when $n\bmod{3}=0$, or one vertex having degree $3$ when $n\bmod{3}=1$.
\[theorem\_C\_sufficient\] Assume the following inequalities hold: $$\begin{gathered}
c(1)+c(4)+18b_3<c(2)+c(3),\label{eq_degree_index_condition23}\\
c(1)+c(3)+8b_3<2c(2),\label{eq_degree_index_condition22}\\
c(2)+c(4)+8b_3<2c(3).\label{eq_degree_index_condition33}\end{gathered}$$ If a chemical tree $T\in \mathcal{T}(n)$ for $n\ge 3$ minimizes $C(\cdot)$ over all chemical trees from $\mathcal{T}(n)$, then $T$ is an extremely branched tree. For $n\le 17$ the inequality *(\[eq\_degree\_index\_condition23\])* can be weakened to $$c(1)+c(4)+17b_3<c(2)+c(3).\label{eq_degree_index_condition23bis}$$
We employ the standard argument of index monotonicity with respect to certain tree transformations. Assume the theorem does not hold, and vertices $u,v \in M(T)$ exist such that $u\neq v$ and $d_T(u),d_T(v)<4$. Four cases are possible.
1. $d_T(u)=d_T(v)=2$. Let $v_1,v_2\in V(T)$ be the vertices incident to $v$ in $T$. Also, let $u_1,u_2\in V(T)$ be the vertices incident to $u$ in $T$, and $u_2$ lies on the path to the vertex $v$ in $T$. Without loss of generality assume that $$\label{eq_case1_ineq}
d_T(u_1)+d_T(u_2)\ge d_T(v_1)+d_T(v_2).$$ Consider a graph $T'\in \mathcal{T}(n)$ obtained from $T$ by replacing the edge $u_1u$ with the edge $u_1v$. It is easy to see that $T'$ is a tree. The degree of the vertex $u$ in $T'$ is decreased by one, the degree of vertex $v$ is increased by one, therefore, if $u_2\neq v$, we have $$C(T')-C(T)=c(1)+c(3)-2c(2)+b_3(d_T(u_1)+d_T(v_1)+d_T(v_2)-d_T(u_2)).$$ From (\[eq\_case1\_ineq\]) we obtain $C(T')-C(T)\le c(1)+c(3)-2c(2)+2b_3d_T(u_1)$. Since vertex degrees $\le4$ in a chemical tree, from (\[eq\_degree\_index\_condition22\]) we have $C(T')-C(T)\le c(1)+c(3)-2c(2)+8b_3<0$, which contradicts the assumption that $T$ minimizes $C(\cdot)$.
If $u_2=v$, in the same manner obtain $C(T')-C(T)\le c(1)+c(3)-2c(2)+7b_3$. From (\[eq\_degree\_index\_condition22\]), it is also negative.
2. $d_T(u)=2$, $d_T(v)=3$. Let $v_1,v_2,v_3\in V(T)$ be the vertices incident to $v$ in $T$. Also, let $u_1,u_2\in V(T)$ be the vertices incident to $u$ in $T$, and assume $u_2$ lies on the path to the vertex $v$ in the tree $T$. Consider a tree $T'\in \mathcal{T}(n)$ obtained from $T$ by replacing the edge $u_1u$ with the edge $u_1v$. By analogy to the previous case, if $u_2\neq v$, we obtain $C(T')-C(T)=c(1)+c(4)-c(2)-c(3)+b_3(2d_T(u_1)+d_T(v_1)+d_T(v_2)+d_T(v_3)-d_T(u_2)).$ Vertex degrees $\le4$ in a chemical tree. Moreover, $d_T(u_2)\ge 2$, since it is an intermediate vertex on the path $u,u_2,...,v$. Therefore, $$C(T')-C(T)\le c(1)+c(4)-c(2)-c(3)+18b_3,$$ and, from (\[eq\_degree\_index\_condition23\]), $C(T')-C(T)<0$, which is a contradiction.
To prove the weaker inequality (\[eq\_degree\_index\_condition23bis\]) we are enough to prove that $C(T')-C(T)<0$ for $n\le 17$, since $d_T(u_1)=d_T(v_1)=d_T(v_2)=d_T(v_3)=4$, and $d_T(u_2)=2$ is possible only in a tree of order 18 or more (an example is depicted in Fig. \[fig\_tree18\]a), and $C(T')-C(T)= c(1)+c(4)-c(2)-c(3)+17b_3$ for the tree $T$ depicted in Fig. \[fig\_tree18\]b.
![To the proof of inequality (\[eq\_degree\_index\_condition23bis\])[]{data-label="fig_tree18"}](tree18.pdf){width="10cm"}
If $u_2=v$, without loss of generality assume that $u=v_3$. Then $C(T')-C(T)=c(1)+c(4)-c(2)-c(3)+b_3(2d_T(u_1)+d_T(v_1)+d_T(v_2)-2)\le c(1)+c(4)-c(2)-c(3)+14b_3$. From (\[eq\_degree\_index\_condition23\]) (or from (\[eq\_degree\_index\_condition23bis\]), if $n\le 17$), it is negative.
3. The case of $d_T(v)=2$, $d_T(u)=3$ is considered in the same manner.
4. $d_T(u)=d_T(v)=3$. Let $v_1,v_2,v_3\in V(T)$ be the vertices incident to $v$ in $T$. Also, let $u_1,u_2,u_3\in V(T)$ be the vertices incident to $u$ in $T$, with $u_1$ not laying on a path to $v$ in $T$. Without loss of generality assume that $$\label{eq_case4_ineq}
d_T(u_1)+d_T(u_2)+d_T(u_3)\ge d_T(v_1)+d_T(v_2)+d_T(v_3).$$ Consider a tree $T'\in \mathcal{T}(n)$ obtained from $T$ by replacing the edge $u_1u$ with the edge $u_1v$. If $uv\notin E(T)$, then from (\[eq\_case4\_ineq\]) and $d_T(u_1)\le 4$ we have $$C(T')-C(T)=c(2)+c(4)-2c(3)+b_3(d_T(u_1)+d_T(v_1)+d_T(v_2)+d_T(v_3)-d_T(u_2)-$$ $$-d_T(u_3))\le c(2)+c(4)-2c(3)+2b_3d_T(u_1)\le c(2)+c(4)-2c(3)+8b_3,$$ which is less than zero due to (\[eq\_degree\_index\_condition33\]), and $T$ cannot minimize $C(\cdot)$. If $uv\in E(T)$, in the same way deduce $C(T')-C(T)\le c(2)+c(4)-2c(3)+7b_3$, which is negative.
The obtained contradictions prove that no more than one internal vertex in $T$ may have degree less than 4.
As $T\in \mathcal{T}(n)$ and $n>1$, the well-known equity holds: $$\label{eq_tree_degrees}
n_1(T)+2n_2(T)+3n_3(T)+4n_4(T)=2(n-1).$$ On the other hand, $$\label{eq_tree_vertices}
n_1(T)+n_2(T)+n_3(T)+n_4(T)=n.$$
Assume that $n_2(T)=1$, so that $n_3(T)=0$. From (\[eq\_tree\_vertices\]) we have $n_1(T)+n_4(T)=n-1$, therefore, (\[eq\_tree\_degrees\]) makes $n=3+3n_4(T)$ and, since $n_4(T)\in \mathbb{N}_0$, $n\bmod{3}=0$.
In the same manner we show that if $n_3(T)=1$ then $n\bmod{3}=1$. If both $n_2(T)$ and $n_3(T)=0$, then $n\bmod{3}=2$, and the proof is complete.
\[corollary\_C1\] Under conditions of Theorem *\[theorem\_C\_sufficient\]*, any tree $T$ minimizing $C(T)=C_1(T)+b_3M_2(T)$ over $\mathcal{T}(n)$ enjoys the same number $n_i$ of vertices of degree $i=1,...,4$. Therefore, $C_1(T) = C_1(T')$ for any pair of trees $T, T'\in \mathcal{T}_C^*(n)$.
\[corollary\_ad\_hoc\] Under conditions of Theorem *\[theorem\_C\_sufficient\]* the sets $\mathcal{T}_C^*(n)$ for $n=4,...,14$ are depicted in Fig. *\[fig\_C\_minimizers\]*. $\mathcal{T}_C^*(n)$ contains the sole tree for $n<14$ , while $\mathcal{T}_C^*(14)$ contains two trees.
From Corollary \[corollary\_C1\] we learn that only the value of $M_2(\cdot)$ may vary within $\mathcal{T}_C^*(n)$.
From Theorem \[theorem\_C\_sufficient\], for $n\in \{5,8,11,14\}$ an optimal tree is a 4-tree (in which all internal vertices have degree 4). Each of $n_1$ *stem* edges (those incident to a pendent vertex) adds 4 to the value of $M_2$, while each of $n_4-1$ edges connecting internal vertices adds 16 to the value of $M_2$. Since $n_1$ and $n_4$ are fixed for fixed $n$, all 4-trees have the same value of $M_2(\cdot)$ (and, therefore, the same value of $C(\cdot)$). Consequently, for for $n = 5,8,11,14$ the set $\mathcal{T}_C^*(n)$ consists of all 4-trees of order $n$ (see Fig. \[fig\_C\_minimizers\]).
If $T\in\mathcal{T}_C^*(n)$, and $n \in \{6,9,12\}$, one internal vertex $u\in M(T)$ has degree $d_T(u)=2$, while all others have degree 4. For $n=6$ only one such tree exists depicted in Fig. \[fig\_C\_minimizers\]. It is easy to check that $M_2(\cdot)$ is minimized if vertex $u$ is incident to two internal vertices. Only one such tree exists for $n=9$ (see Fig. \[fig\_C\_minimizers\]), and the same is true for $n=12$.
For $n \in \{4,7,10,13\}$ any tree $T\in\mathcal{T}_C^*(n)$ has one internal vertex $u\in M(T)$ of degree $d_T(u)=3$, while all other have degree 4. For $n=4$ only one such tree exists depicted in Fig. \[fig\_C\_minimizers\], and the same is true for $n=7$. Again, it is easy to check that, in the context of $M_2(\cdot)$ minimization, vertex $u$ being incident to three internal vertices is strictly preferable to vertex $u$ being incident to one pendent and two internal vertices, which is, in turn, preferred to $u$ having two incident pendent vertices. So, optimal trees for $n=10,13$ are depicted in Fig. \[fig\_C\_minimizers\] (black and white filling of circles is explained below).
The same logic allows continuing the sequence of $C(\cdot)$-minimizers to $n > 14$.
![Chemical trees minimizing the “ad-hoc” index $C(\cdot)$ for $n=4,...,14$ (alcohol molecules minimizing $BP^\textrm{I}(\cdot)$ with possible oxygen positions filled with black)[]{data-label="fig_C_minimizers"}](C_minimizers.pdf){width="12cm"}
Therefore, Theorem \[theorem\_C\_sufficient\] says that, when conditions (\[eq\_degree\_index\_condition23\])-(\[eq\_degree\_index\_condition33\]) hold, chemical trees minimizing $C(\cdot)$ have as many vertices of maximal degree 4 as possible. A similar result can be proved for the modified Wiener index $WI_\textrm{O}(\cdot)$.
Wiener index {#section_wiener}
------------
A simple connected undirected graph $G$ is called *vertex-weighted* if each vertex $v\in V(G)$ is endowed with a non-negaive weight $\mu_G(v)$. With $\mu_G$ we denote the total vertex weight of the graph $G$, and $\mathcal{WT}(n)$ stands for the set of all vertex-weighted trees of order $n$.
Klavžar and Gutman [@Klavzar97] defined the *Wiener index for vertex-weighted graphs* as $$VWWI(G):=\frac{1}{2}\sum_{u,v\in V(G)}\mu_G(u)\mu_G(v) d_G(u,v).$$
Clearly, $VWWI(\cdot)$ is a special case of the pair-weighted Wiener index $PWWI(\cdot)$ (defined with formula (\[eq\_PWWI\])) for $\mu_G(u, v):=\mu_G(u)\mu_G(v)$. The path-weighted Wiener index is poorly studied at the moment, but, fortunately, $WI_\textrm{O}(\cdot)$, which is the point of our current interest, can be reduced to the Wiener index for vertex-weighted graphs.
For every alcohol molecule from $\Omega(n-1)$ (or, equivalently, for every pendent-rooted tree $T_r\in\mathcal{R}(n)$) define a vertex-weighted tree $T(\varepsilon)\in \mathcal{WT}(n)$ by assigning the weight $\mu_{T(\varepsilon)}(v):=\varepsilon$ to each vertex $v\in V(T_r)$ (a carbon atom) except the root $r$, and assigning the weight $\mu_{T(\varepsilon)}(r):=1/\varepsilon$ to the root (the oxygen atom). It is easy to see that under these weights $\lim_{\varepsilon\rightarrow 0} VWWI(T(\varepsilon))=WI_\textrm{O}(T_r)$. Since $WI_\textrm{O}(\cdot)$ is integer-valued, minimizers of $VWWI(\cdot)$ and of $WI_\textrm{O}(\cdot)$ coincide for sufficiently small $\varepsilon$.
In [@Goubko15] the majorization technique suggested by Zhang et al. [@Zhang08] is used to minimize $VWWI(\cdot)$ over the set of trees with given vertex weights and degrees. Below we recall the notation and selected theorems from [@Goubko15]. We use them to find the extremal vertex degrees over the set of all trees of order $n$ with fixed vertex weights.
Consider a vertex set $V$. Let the function $\mu: V \rightarrow
\mathbb{R}_+$ assign a non-negative weight $\mu(v)$ to each vertex $v\in V$, while the function $d: V \rightarrow \mathbb{N}$ assign a natural degree $d(v)$. The tuple $\langle\mu, d\rangle$ is called *a generating tuple* if the following identity holds: $$\label{tree_degrees_identity}
\sum_{v\in V} d(v)=2(|V|-1).$$
Denote with $\overline{\mu}:=\sum_{v\in V}\mu(v)$ the total weight of the vertex set $V$. Let $\mathcal{WT}(\mu):=\{T\in \mathcal{WT}(|V|): V(T)=V, \mu_T(v)=\mu(v) \text{ for all }v\in V\}$ be the set of trees over the vertex set $V$ with vertex weights $\mu(\cdot)$. For the set $\mathcal{WT}(\mu, d):=\{T\in \mathcal{WT}(\mu):
d_T(v)=d(v) \text{ for all }v\in V\}$ we also require vertices to have degrees $d(\cdot)$.
Let $V(\mu, d)$ be the domain of functions of a generating tuple $\langle\mu, d\rangle$. Introduce the set $W(\mu,
d):=\{w \in V(\mu, d): d(w)=1\}$ of *pendent* vertices and the set $M(\mu, d):=V(\mu,d)\backslash W(\mu,d)$ of *internal* vertices.
We will say that in a generating tuple $\langle\mu, d\rangle$ *weights are degree-monotone*, if for any $m, m' \in M(\mu,d)$ from $d(m) < d(m')$ it follows that $\mu(m) \le \mu(m')$, and for any $w\in W(\mu,d)$ we have $\mu(w)>0$.
For a generating tuple $\langle\mu,d\rangle$ the *generalized Huffman algorithm* [@Goubko15] builds a tree $H \in
\mathcal{WT}(\mu,d)$ as follows.
**Setup.** Define the vertex set $V_1 := V(\mu,d)$ and the functions $\mu^1$ and $d^1$, which endow its vertices with weights $\mu^1(v) := \mu(v)$ and degrees $d^1(v) := d(v)$, $v \in V_1$. We start with the empty graph $H$ over the vertex set $V(\mu,d)$.
**Steps $i = 1, ..., q-1$.** Denote with $m_i$ the vertex having the least degree among the vertices of the least weight in $M(\mu^i,d^i)$. Let $w_1,
..., w_{d(m_i)-1}$ be the vertices having $d(m_i)-1$ least weights in $W(\mu^i,d^i)$. Add to $H$ edges $w_1m_i, ...,
w_{d(m_i)-1}m_i$.
Define the set $V_{i+1} := V_i \backslash \{w_1, ..., w_{d(m_i)-1}\}$ and functions $\mu^{i+1}(\cdot), d^{i+1}(\cdot)$, endowing its elements with weights and degrees as follows: $$\begin{gathered}
\mu^{i+1}(v) := \mu^i(v)\text{ for }v \neq m_i, \hspace{20pt} \mu^{i+1}(m_i) := \mu^i(m_i)+\mu^i(w_1)+...+\mu^i(w_{d(m_i)-1}),\nonumber\\
\label{eq_Huffman_tuples}d^{i+1}(v) := d^i(v)\text{ for }v \neq m_i, \hspace{20pt} d^{i+1}(m_i) := 1.\end{gathered}$$
**Step $q$.** Consider a vertex $m_q \in M(\mu^q, d^q)$. By construction, $|M(\mu^q, d^q)|=1$, $|W(\mu^q,
d^q)|=d(m_q)$. Add to $H$ edges connecting all vertices from $W(\mu^q, d^q)$ to $m_q$. Finally, set $\mu_H(v) :=
\mu(v)$, $v \in V(H)$.
\[theorem\_huffman\_vwwi\]***[@Goubko15]*** If weights are degree-monotone in a generating tuple $\langle\mu,d\rangle$, then $T\in \mathcal{WT}_{VWWI}^*(\mu,d)$ if and only if $T\in\mathcal{WT}(\mu,d)$ and $T$ is a Huffman tree. In other words, only a Huffman tree minimizes the Wiener index over the set of trees whose vertices have given weights and degrees.
In the present subsection we study how the value of $VWWI_{\mathcal{WT}}^*(\mu,d)$ changes with degrees $d(\cdot)$. Our results are analogous to those proved by Zhang et al. [@Zhang08] for the “classical” Wiener index. Following [@Goubko15], we reformulate the problem for directed trees.
A (weighted) *directed tree* is a weighted connected directed graph with each vertex except the *terminal vertex*[^3] having the sole outbound arc and the terminal vertex having no outbound arcs.
An arbitrary tree $T \in \mathcal{WT}(n)$ can be transformed into a directed tree by choosing an internal vertex $t\in M(T)$, and replacing all its edges with arcs directed towards (a terminal vertex) $t$. Let us denote with $\mathcal{WD}$ the collection of all directed trees, which can be obtained in such a way, and let $\mathcal{WD}(\mu,
d)$ stand for all directed trees obtained from $\mathcal{WT}(\mu,d)$. Vice versa, in a directed tree from $\mathcal{WD}(\mu,d)$ replacement of all arcs with edges makes some tree from $\mathcal{WT}(\mu, d)$.
If at Step $i=1,...,q$ of the generalized Huffman algorithm we add arcs towards the vertex $m_i$ (instead of undirected edges), we obtain a *directed Huffman tree* with the terminal vertex $m_q$.
For a vertex $v\in V(T)$ of a directed tree $T\in \mathcal{WD}$ define its *subordinate group* $g_T(v)\subseteq V(T)$ as the set of vertices having the directed path to the vertex $v$ in the tree $T$ (the vertex $v$ itself belongs to $g_T(v)$). The *weight $f_T(v)$ of a subordinate group* $g_T(v)$ is defined as the total vertex weight of the group: $f_T(v):=\sum_{u\in g_T(v)}
\mu_T(u)$.
\[note\_positive\]If all pendent vertices in $T$ have positive weights, then $f_T(v)>0$ for any $v\in V(T)$. In particular, it is true for any $T\in \mathcal{WD}(\mu, d)$, if weights in $\langle\mu,d\rangle$ are degree-monotone.
The Wiener index is defined for directed trees by analogy to the case of undirected trees: we simply ignore the arcs’ direction when calculating distances. Therefore, a tree and a corresponding directed tree share the same value of the Wiener index.
The value of the Wiener index for a directed tree $T_t\in
\mathcal{WD}(\mu)$ with a terminal vertex $t\in M(T)$ can be written [@Goubko15] as: $$\label{eq_VWWI_directed}
VWWI(T_t)=\sum_{v\in V(T_t)\backslash
\{t\}}f_{T_t}(v)(\bar{\mu}-f_{T_t}(v))=\sum_{v\in V(T)\backslash
\{t\}}\chi(f_{T_t}(v)),$$ where $\chi(x):=x(\bar{\mu}-x)$, and thus, the problems of Wiener index minimization for vertex-weighted trees and for weighted directed trees are equivalent.
\[def\_vector\] Every directed tree $T$ is associated with the *vector of subordinate groups’ weights* $\mathbf{f}(T):=(f_T(v))_{v\in
V(T)\backslash\{t\}}$, where $t$ is the terminal vertex of $T$. From equation (\[eq\_VWWI\_directed\]) we see that the vector $\mathbf{f}(T)$ completely determines the value of $VWWI(T)$.
**[@Marshall79; @Zhang08]** For the real vector $\mathbf{x}=(x_1,..., x_p)$, $p\in \mathbb{N}$, denote with $\mathbf{x}_\uparrow=(x_{[1]},..., x_{[p]})$ the vector where all components of $\mathbf{x}$ are arranged in ascending order.
**[@Marshall79; @Zhang08]** A non-negative vector $\mathbf{x} = (x_1, ..., x_p)$, $p \in
\mathbb{N}$, *weakly majorizes* a non-negative vector $\mathbf{y} = (y_1, ..., y_p)$ (which is denoted with $\mathbf{x}\succeq \mathbf{y}$) if $$\sum_{i=1}^k x_{[i]} \le \sum_{i=1}^k y_{[i]} \text{ for all }k=1,...,p.$$ If $\mathbf{x}_\uparrow \neq \mathbf{y}_\uparrow$, then $\mathbf{x}$ is said to *strictly weakly majorize* $\mathbf{y}$ (which is denoted with $\mathbf{x}\succ \mathbf{y}$).
We will need the following properties of weak majorization.
\[lemma\_Zhang\_b\]***[@Marshall79; @Zhang08]*** Consider a positive number $b>0$ and two non-negative vectors $\mathbf{x} = (x_1, ..., x_k, y_1, ..., y_l)$ and $\mathbf{y} = (x_1
+ b, ..., x_k + b, y_1 - b, ..., y_l - b)$, such that $0 \le k \le
l$. If $x_i \ge y_i$ for $i = 1, ..., k$, then $\mathbf{x}\prec
\mathbf{y}$.
\[lemma\_Zhang\_xy\]***[@Marshall79; @Zhang08]*** If $\mathbf{x}\preceq \mathbf{y}$ and $\mathbf{x}' \prec
\mathbf{y'}$, then $(\mathbf{x},\mathbf{x'})\prec
(\mathbf{y},\mathbf{y'})$, where $(\mathbf{x},\mathbf{x'})$ means concatenation of vectors $\mathbf{x}$ and $\mathbf{x'}$.
\[lemma\_Zhang\_concave\]***[@Marshall79; @Zhang08]*** If $\chi(x)$ is a increasing concave function, and $(x_1,...,x_p)\succeq (y_1,...,y_p)$, then $\sum_{i=1}^p\chi(x_i)\le
\sum_{i=1}^p\chi(y_i)$, and equality is possible only when $(x_1,...,x_p)_\uparrow\\=(y_1,...,y_p)_\uparrow$.
The following lemma establishes an important property of directed Huffman trees:
\[lemma\_Huffman\_monotonicity\]***[@Goubko15]*** For any directed Huffman tree $H$ $$\label{eq_weights_monotone}
vm,v'm'\in E(H), m \neq m', f_H(v)<f_H(v') \Rightarrow f_H(m) < f_H(m').$$
\[lemma\_major\] Consider generating tuples $\langle\mu, d\rangle$ and $\langle\mu,
d'\rangle$ defined on the same vertex set, and let weights be degree-monotone in $\langle\mu, d\rangle$. Let the values of degree functions $d(\cdot)$ and $d'(\cdot)$ differ only for vertices $u$ and $v$, such that $d(u)\ge d(v)$ and $\mu(u)\ge \mu(v)$, while $d'(u)=d(u)+1$, $d'(v)=d(v)-1$. Then for every directed tree $T\in
\mathcal{WD}(\mu,d)$ there exists such a directed tree $T'\in
\mathcal{WD}(\mu,d')$ that $\mathbf{f}(T')\succ \mathbf{f}(T)$.
By Theorem 2 from [@Goubko15], such a directed Huffman tree $H\in \mathcal{WD}(\mu,d)$ exists, that $\mathbf{f}(H)\succeq
\mathbf{f}(T)$.[^4] Since $H\in \mathcal{WD}(\mu,d)$ and $d'(v)=d(v)-1\ge1$, we know that $d(v)\ge 2$ and $v$ has an incoming arc in $H$ from some vertex $v'\in V$. Weights are degree-monotone in $\langle\mu,d\rangle$, therefore, by construction of a Huffman tree, $f_H(u)\ge f_H(v)$ and, thus, without loss of generality we can assume that $u \notin g_H(v)$.
Assume that $v \in g_H(u)$. Then a path $(v, m_1, ..., m_l, u)$ exists in $H$ from the vertex $v$ to the vertex $u$, where $l\ge0$. Consider a directed tree $T'$ obtained from $H$ by deleting the arc $v'v$ and adding the arc $v'u$ instead. It is clear that $T'\in
\mathcal{WD}(\mu,d')$, and weights of groups subordinated to vertices $v, m_1, ..., m_l$ decrease by $f_H(v')$ (which is positive by Note \[note\_positive\]), while weights of the other vertices do not change. Therefore, by Lemma \[lemma\_Zhang\_b\], $$\mathbf{y} := (f_{T'}(v), f_{T'}(m_1), ..., f_{T'}(m_l))=$$ $$=(f_H(v) -f_H(v'), f_H (m_1) -f_H(v'), ..., f_H (m_l) -f_H(v')) \succ$$ $$\succ (f_H(v), f_H (m_1), ..., f_H(m_l)) =: \mathbf{x}.$$
If one denotes with $\mathbf{z}$ the vector of (unchanged) weights of groups subordinated to all other non-terminal vertices of $H$, then, by Lemma \[lemma\_Zhang\_xy\], $\mathbf{f}(T')=(\mathbf{y},\mathbf{z})\succ
(\mathbf{x},\mathbf{z})=\mathbf{f}(H)$.
Assume now that $v \notin g_H(u)$. Then there are disjoint paths $(u, m_1, ..., m_k, m)$ and $(v, m_1', ..., m_l', m)$ (where $k,l\ge0$) in $H$ from vertices $u$ and $v$ to some vertex $m \in M(H)$.
If $f_H(u)> f_H(v)$, then, applying repeatedly formula (\[eq\_weights\_monotone\]) from Lemma \[lemma\_Huffman\_monotonicity\], we write $f_H(m_i) > f_H(m_i'), i
= 1, ..., \min[k, l]$. It also follows from (\[eq\_weights\_monotone\]) that $k \le l$, since otherwise $f_H(m_{l+1})> f_H(m)$, which is impossible, since $m_{l+1} \in
g_H(m)$.
Consider a directed tree $T'\in\mathcal{WD}(\mu, d')$ obtained from $H$ by deleting the arc $v'v$ and adding the arc $v'u$ instead. In the tree $T'$ weights of the groups subordinated to the vertices $u, m_1, ..., m_k$ increase by $f_H(v')$ (i.e., $f_{T'}(u) = f_H(u) + f_H(v')$, $f_{T'}(m_i) = f_H(m_i) + f_H(v'), i = 1, ..., k$), weights of the groups subordinated to the vertices $v, m_1', ..., m_l'$ decrease by $f_H(v')$ (i.e., $f_{T'}(u) =
f_H(u) - f_H(v')$, $f_{T'}(m_i') = f_H(m_i') - f_H(v'), i = 1, ..., l$), weights of all other vertices (including $m$) do not change. Therefore, by Lemma \[lemma\_Zhang\_b\], $$\mathbf{y} := (f_{T'}(u), f_{T'}(m_1), ..., f_{T'}(m_k), f_{T'}(v), f_{T'}(m_1'), ..., f_{T'}(m_l'))=$$ $$=(f_H(u)+f_H(v'), f_H(m_1)+f_H(v'), ..., f_H(m_k)+f_H(v'),$$ $$f_H(v)-f_H(v'), f_H(m_1')-f_H(v'), ..., f_H (m_l')-f_H(v'))\succ$$ $$\succ (f_H (u), f_H (m_1), ..., f_H (m_k), f_H (v), f_H(m_1'), ..., f_H (m_l')) =: \mathbf{x}.$$
If $\mathbf{z}$ is a vector of (unchanged) weights of groups subordinated to all other non-terminal vertices of $H$, then, by Lemma \[lemma\_Zhang\_xy\], $\mathbf{f}(T')=(\mathbf{y},\mathbf{z})\succ
(\mathbf{x},\mathbf{z})=\mathbf{f}(H)$.
By construction of the Huffman tree, the situation of $f_H(u) = f_H(v)$ is possible only when $d(u)=d(v)$ and $\mu(u)=\mu(v)$. In this case we cannot use formula (\[eq\_weights\_monotone\]) to compare subordinate groups’ weights of elements of both chains, since all possible alternatives of $k = 0$, or $l = 0$, or any sign of the expression $f_H(m_1) - f_H(m_1')$ in case of $k, l \ge 1$ are possible.
On the other hand, if $f_H(m_1)> f_H(m_1')$, then formula (\[eq\_weights\_monotone\]) can be used to show that $k \le
l$, $f_H(m_i) > f_H(m_i')$, $i = 2, ..., k$. In case of the opposite inequality, $f_H(m_1) < f_H(m_1')$, formula (\[eq\_weights\_monotone\]) says that, by contrast, $k \ge l$, $f_H(m_i) < f_H(m_i')$, $i = 2, ..., l$. Repeating this argument through the chain, we see that only two alternatives are possible:
- $0\le p \le k \le l$, $f_H(m_i) = f_H(m_i')$, $i = 1, ..., p$, $f_H(m_i) > f_H(m_i')$, $i = p + 1, ..., k$. In this case, as above, we can show that for the directed tree $T'\in\mathcal{WD}(\mu, d')$ obtained from $H$ by deleting the arc $v'v$ and adding the arc $v'u$ instead, $\mathbf{f}(T')\succ\mathbf{f}(H)$.
- $0\le p \le l \le k$, $f_H(m_i) = f_H(m_i')$, $i = 1, ..., p$, $f_H(m_i) < f_H(m_i')$, $i = p + 1, ..., l$. In this case the same inequality is true for the directed tree $T'\in\mathcal{WD}(\mu, d')$ obtained from $H$ by redirecting to/from vertex $v$ all arcs incident to $u$, and by redirecting to/from vertex $u$ all arcs incident to $v$ except the arc $v'v$.
Therefore, we proved that a tree $T'\in \mathcal{WD}(\mu, d')$ exists such that $\mathbf{f}(T')\succ \mathbf{f}(H)$. As shown above, $\mathbf{f}(H)\succeq \mathbf{f}(T)$, so, finally, $\mathbf{f}(T')\succ \mathbf{f}(T)$.
A directed tree $T \in \mathcal{WD}(\mu, d)$ with a terminal vertex $t$ is called a *proper tree* if for all $m\in M(T)$, $m\neq t$, $f_T(m) \le \bar{\mu}/2$.
\[lemma\_concave\] Let a function $\chi(x)$ be concave and increasing for $x\in [0,\bar{\mu}/2]$. Consider a pair of generating tuples, $\langle\mu,d\rangle$ and $\langle\mu,d'\rangle$, satisfying conditions of Lemma *\[lemma\_major\]*. If $T\in \mathcal{WD}(\mu,d)$ and $T'\in \mathcal{WD}(\mu,d')$ are directed Huffman trees, then $$\sum_{v\in V(T')\backslash\{t'\}}\chi(f_{T'}(v))<\sum_{v\in V(T)\backslash\{t\}}\chi(f_T(v)),$$ where $t\in M(T)$ and $t'\in M(T')$ are terminal vertices of $T$ and $T'$ respectively.
From Lemma \[lemma\_major\], such a tree $T''\in
\mathcal{WD}(\mu,d')$ exists that $\mathbf{f}(T'')\succ
\mathbf{f}(T)$. Theorem 2 from [@Goubko15] says that $\mathbf{f}(T')\succeq \mathbf{f}(T'')$, therefore, $\mathbf{f}(T')\succ \mathbf{f}(T)$. Denote for short $n_1=|V(T)|$, $\mathbf{f}(T)=\mathbf{f}:=(f_1,...,f_{n_1-1})$, $\mathbf{f}(T')=\mathbf{f}':=(f'_1,...,f'_{n_1-1})$.
It is known (see Lemma 19 in [@Goubko15]) that each directed Huffman tree with degree-monotone weights is a proper tree, so, $f_T(w)\le \bar{\mu}/2$, $w\in M(\mu,d)\backslash\{t\}$, and $f_{T'}(w')\le \bar{\mu}/2$, $w'\in M(\mu,d')\backslash\{t'\}$. If a vertex $w\in V(T)$ exists, such that $\mu(w)>\bar{\mu}/2$ (there can be at most one such vertex in $V(T)$), then $w$ cannot be an internal vertex in $T$ and a pendent vertex in $T'$, since then conditions of Lemma \[lemma\_major\] imply that $w=v$ and $\mu(u)\ge\mu(w)$, which is impossible. Therefore, $w$ is either a terminal vertex both in $T$ and in $T'$, or a pendent vertex both in $T$ and in $T'$. In the latter case $f_T(v)=f_{T'}(v)=\mu(v)$.
Consequently, $f_i,f_i'\le\bar{\mu}/2$ for $i=1,...,n_1-2$, and if $f_{n_1-1}>\bar{\mu}/2$, then $f_{n_1-1}=f'_{n_1-1}=\mu(w)$.
If $f_{n_1-1}\le \bar{\mu}/2$, the statement of the lemma follows from Lemma \[lemma\_Zhang\_concave\].
If $f_{n_1-1}> \bar{\mu}/2$, we can write $$\sum_{v\in V(T)\backslash\{t\}}\chi(f_T(v))-\sum_{v\in V(T')\backslash\{t'\}}\chi(f_{T'}(v))=$$ $$=\sum_{i=1}^{n_1-2}\chi(f_i)+\chi(f_{n_1-1})-\sum_{i=1}^{n_1-2}\chi(f'_i)-\chi(f'_{n_1-1})=\sum_{i=1}^{n_1-2}\chi(f_i)-\sum_{i=1}^{n_1-2}\chi(f'_i).$$
Since $\mathbf{f}'\succ \mathbf{f}$ and $f_{n_1-1}=f'_{n_1-1}$, we have $(f'_1,...,f'_{n_1-2})\succ (f_1,...,f_{n_1-2})$, and the statement of the lemma again follows from Lemma \[lemma\_Zhang\_concave\].
\[corollary\_Wiener\_1\] If generating tuples $\langle\mu,d\rangle$ and $\langle\mu,d'\rangle$ satisfy conditions of Lemma *\[lemma\_major\]*, then $VWWI^*_{\mathcal{WT}}(\mu,d')<VWWI^*_{\mathcal{WT}}(\mu,d)$.
Theorem 3 from [@Goubko15] says that $\mathcal{WD}_{VWWI}^*(\mu,d)$ consists of all directed Huffman trees. Consider the directed Huffman trees $T\in \mathcal{WD}_{VWWI}^*(\mu,d)$, $T'\in \mathcal{WD}_{VWWI}^*(\mu,d')$. Function $\chi(x)=x(\bar{\mu}-x)$ in (\[eq\_VWWI\_directed\]) satisfies the conditions of Lemma \[lemma\_concave\], so, from (\[eq\_VWWI\_directed\]), $VWWI(T')<VWWI(T)$. Since every proper directed tree from $\mathcal{WD}(\mu,d)$ has a corresponding tree from $\mathcal{WT}(\mu,d)$, and vice versa, we have $VWWI^*_{\mathcal{WT}}(\mu,d)=VWWI^*_{\mathcal{WD}}(\mu,d)$, and the corollary follows immediately.
In the rest of the section we consider a set $V$ consisting of $n$ vertices with weights $\mu(v)$, $v\in V$. The set $V$ can be thought of as a fixed collection of (heterogeneous) atoms used as building blocks for molecules. All molecules constructed from these building blocks belong to $\mathcal{WT}(\mu)$.
We want to use Corollary \[corollary\_Wiener\_1\] to show that, similar to Theorem \[theorem\_C\_sufficient\], the vertex-weighted Wiener index is minimized by a tree having as many vertices of the maximum degree as possible. We cannot apply Corollary \[corollary\_Wiener\_1\] to the whole collection $\mathcal{WT}(\mu)$ of trees with vertices having fixed weights $\mu(\cdot)$ (unless $\mu(v)\equiv const$), as it inevitably contains trees generated by the tuples with non-degree-monotone weights, for which Corollary \[corollary\_Wiener\_1\] is inapplicable. Therefore, we have to carefully limit a set of admissible trees.
\[def\_extremal\_tuple\] Consider an admissible set $\mathcal{M}\subseteq \mathcal{WT}(\mu)$ and denote with $L:=\bigcap_{T\in\mathcal{M}}W(T)$ the set of vertices, which are pendent in all trees from $\mathcal{M}$. A generating tuple $\langle\mu,\bar{d}\rangle$ is called *extremal* for $\mathcal{M}$, if $M(\mu, \bar{d})$ consists of $\lceil\frac{n-2}{3}\rceil$ vertices having the highest weights in $V\backslash L$ (i.e., if $u\in M(\mu, \bar{d})$ and $v\in V\backslash L$, then $\mu(u)\ge\mu(v)$), at most one vertex $u\in M(\mu, \bar{d})$ has degree $\bar{d}(u)<4$ while others having degree $4$, and, when exists, $u$ has the minimal weight in $M(\mu, \bar{d})$.
There can be several extremal tuples for an admissible set, if for some extremal tuple $\langle\mu,\bar{d}\rangle$ such vertices $w\in W(\mu,\bar{d})\backslash L$ and $m\in M(\mu,\bar{d})$ exist that $\mu(w)=\mu(m)$ (swapping $w$ and $m$ then makes a new extremal tuple). It is clear that weights are degree-monotone in $\langle\mu,\bar{d}\rangle$, and there are only extremely branched trees in $\mathcal{WT}(\mu,\bar{d})$.
\[theorem\_degree\_domination\] If $\langle\mu,\bar{d}\rangle$ is an extremal tuple for an admissible set $\mathcal{M} \subseteq \mathcal{WT}(\mu)$ of **chemical** trees with degree-monotone weights, and $H\in
\mathcal{WT}(\mu,\bar{d})$ is a Huffman tree, then $VWWI(H)\le
VWWI_\mathcal{M}^*$, with equality if and only if $\mathcal{M}$ contains $H$ or any other Huffman tree generated by the extremal tuple.
Consider a vertex-weighted tree $T\in \mathcal{M}$. Weights are degree-monotone in $T$, so, if $d_T(u)>d_T(v)$ and $\bar{d}(u)<\bar{d}(v)$ for some $u,v\in M(T)$, then $\mu(u)=\mu(v)$. Consequently, such a tree $T'\in \mathcal{WT}(\mu)$ exists that $VWWI(T)=VWWI(T')$, $W(T')=W(T)$, and from $u,v\in M(T)$ and $d_{T'}(u)<d_{T'}(v)$ it follows that $\bar{d}(u)\le \bar{d}(v)$ ($T'$ is constructed from $T$ with a permutation of several vertices of equal weight, which does not affect the index value). So, degrees in the tree $T'$ are “compatible” to those in the tuple $\langle\mu,\bar{d}\rangle$.
Assume $T'\in \mathcal{WT}(\mu,d^0)$ for some generating tuple $\langle\mu,d^0\rangle$, and $d^0(\cdot)\not\equiv\bar{d}(\cdot)$. Construct such a sequence $\langle\mu,d^0\rangle, \langle\mu,d^1\rangle, ..., \langle\mu,d^k\rangle$ of generating tuples with degree-monotone weights that $k\ge 1$, $d^k(\cdot)\equiv\bar{d}(\cdot)$, and each pair $\langle\mu,d^i\rangle,\langle\mu,d^{i+1}\rangle$ of sequential tuples meets the requirements of Lemma \[lemma\_major\], $i=0,...,k-1$.
We build the elements of this sequence one by one. For a tuple $\langle\mu,d^i\rangle$ define $M^+(\mu,d^i):=\{w\in V: d^i(w)<\bar{d}(w)\}$ and $M^-(\mu,d^i):=\{w\in V: d^i(w)>\bar{d}(w)\}$. Set $d^{i+1}(u^i)=d^i(u^i)+1$ for a vertex $u^i$ having the highest weight in $M^+(\mu,d^i)$, and $d^{i+1}(v^i)=d^i(v^i)-1$ for a vertex $v^i$ having the least weight in $M^-(\mu,d^i)$, while keeping the degrees of all other vertices.
![Sequence of degree functions converging to $\bar{d}(\cdot)$. A brace shows $L$.[]{data-label="fig_degree_seq"}](degree_seq.pdf){width="11cm"}
The proof is clear from Fig. \[fig\_degree\_seq\], where numbers on arrows show steps of transformation of the initial degree function. By Corollary \[corollary\_Wiener\_1\], $VWWI^*_{\mathcal{WT}}(\mu,\bar{d})<...<VWWI^*_{\mathcal{WT}}(\mu,d^0)\le VWWI(T')=VWWI(T)$. From Theorem \[theorem\_huffman\_vwwi\] we know that if $T\in \mathcal{WT}(\mu,\bar{d})$, then $VWWI(T)=VWWI^*_{\mathcal{WT}}(\mu,\bar{d})$ if and only if $T$ is a Huffman tree. Therefore, for every $T\in \mathcal{M}$ we have $VWWI^*_{\mathcal{WT}}(\mu,\bar{d})\le VWWI(T)$ with equality if and only if $T$ is a Huffman tree for $\langle\mu,\bar{d}\rangle$ or another extremal tuple, and the proof is complete.
\[corollary\_VWWI\_max\_degree\] Trees, where $WI_\textrm{\emph{O}}(\cdot)$ achieves its minimum over the set of pendent-rooted trees of order $n=4,...,14$, are depicted in Fig. *\[fig\_VWWI\_minimizers\]*.
As we already argued, $WI_\textrm{O}(\cdot)$ can be seen a special case of $VWWI(\cdot)$ for the vertex set where all vertices have sufficiently small weight $\varepsilon$ except a root having weight $1/\varepsilon$. Let $r$ be the root in all considered pendent-rooted trees. Then the set of pendent-rooted trees satisfies conditions of Theorem \[corollary\_Wiener\_1\] with $L=\{r\}$, and the statement of the corollary follows from Theorem \[corollary\_Wiener\_1\] and the fact that for a given $n$ the value of the index cannot be improved only for the extremal degree function $\bar{d}(\cdot)$, the one distinct from 1 and 4 at no more than one vertex. The concrete degree function for each $n$ is justified from (\[eq\_tree\_degrees\]), as in Theorem \[theorem\_C\_sufficient\]. From [@Goubko15] we know that, for a given degree function $\bar{d}(\cdot)$, Huffman trees (shown in Fig. \[fig\_VWWI\_minimizers\]) minimize $VWWI(\cdot)$.
![Rooted chemical trees minimizing $WI_\textrm{O}(\cdot)$ for $n=4,...,14$ (alcohol molecules minimizing $BP^\textrm{II}(\cdot)$ with oxygens filled with black)[]{data-label="fig_VWWI_minimizers"}](VWWI_minimizers.pdf){width="12cm"}
The extension of the above results to trees with vertex degrees limited to any $\Delta\ge 3$ is straightforward.
Minimal Boiling Point
=====================
Below we combine the theorems from the previous section to find an alcohol molecule with the minimal (predicted) boiling point. The main tool will be the following obvious proposition.
\[prop\_subset\] For an arbitrary admissible set $\mathcal{M}$ of graphs and a pair of indices $I(G)$ and $J(G)$ defined for all graphs $G\in \mathcal{M}$, if $\mathcal{M}_I^*\cap \mathcal{M}_J^*\neq \emptyset$, then $\mathcal{M}_{I+J}^* = \mathcal{M}_I^*\cap \mathcal{M}_J^*$.
is straightforward, since $\mathcal{M}_I^*\cap \mathcal{M}_J^*$ is a collection of graphs where indices $I(\cdot)$ and $J(\cdot)$ attain their minima simultaneously.
We start with the simplest Regression I (see Equation (\[eq\_bpI\])) being a linear combination of degree-based indices – the “ad-hoc index” $C^\textrm{I}(\cdot)$ defined with Equation (\[eq\_ad\_hoc\_index\]) and $S_2(\cdot)$ index, which penalizes presence of a sub-root of degree 2 in a pendent-rooted tree.
\[prop\_BPI\] The collection $\mathcal{R}_{BP^\textrm{\emph{I}}}^*(n)$ of pendent-rooted trees *(*alcohol molecules*)* with the minimal predicted boiling point $BP^\textrm{\emph{I}}$ for $n = 4,..., 14$ consists of the trees depicted in Fig. *\[fig\_C\_minimizers\]* with black circles representing a possible root *(*oxygen atom*)* positions in a tree *(*alcohol molecule*)*.
Using Table \[tab\_param\] one easily checks conditions (\[eq\_degree\_index\_condition22\])-(\[eq\_degree\_index\_condition23bis\]) of Theorem \[theorem\_C\_sufficient\] to hold for the “ad-hoc index” $C^\textrm{I}(\cdot)$ and $n\le14$. Therefore, by Corollary \[corollary\_ad\_hoc\], trees from $\mathcal{T}_{C^\textrm{I}}^*(n)$ are depicted in Fig. \[fig\_C\_minimizers\]. We know that the “ad-hoc” index does not care for the root position in a tree, and, therefore, $\mathcal{R}_{C^\textrm{I}}^*(n)$ can be obtained from $\mathcal{T}_{C^\textrm{I}}^*(n)$ by assigning the root to any pendent vertex of all optimal trees.
From Lemma \[lemma\_S\_i\], $S_2(\cdot)$ is minimized with pendent-rooted trees whose sub-root has degree 3 or 4. Each tree in Fig. \[fig\_C\_minimizers\] has a pendent vertex being incident to a vertex of degree 3 or 4, so, $\mathcal{R}_{C^\textrm{I}}^*(n)\cap \mathcal{R}_{S_2}^*(n)\neq \emptyset$, and conditions of Proposition \[prop\_subset\] hold for all $n=4, ..., 14$. Therefore, $\mathcal{R}_{BP^\textrm{I}}^*(n)$ consists of the trees from Fig. \[fig\_C\_minimizers\] with the root assigned to a pendent vertex incident to a vertex of degree 3 or 4. Possible roots are filled with black color in Fig. \[fig\_C\_minimizers\].
The similar reasoning can be carried out for Regression II, which is defined with Equation (\[eq\_bpII\]) and adds up from the generalized first Zagreb index $C_1^\textrm{II}(\cdot)$, the cube root of the distance $WI_\textrm{O}(\cdot)^{\frac{1}{3}}$ of the oxygen atom in an alcohol molecule, and, again, from $S_2(\cdot)$ index.
\[prop\_BPII\] The collection $\mathcal{R}_{BP^\textrm{\emph{II}}}^*(n)$ of rooted trees *(*alcohol molecules*)* with the minimal predicted boiling point $BP^\textrm{\emph{II}}$ for $n = 4,..., 14$ consists of the trees depicted in Fig. *\[fig\_VWWI\_minimizers\]* with a black circle representing a root *(*oxygen atom*)* position in a tree *(*alcohol molecule*)*.
Using Table \[tab\_param\] we check that conditions (\[eq\_degree\_index\_condition23\])-(\[eq\_degree\_index\_condition33\]) of Theorem \[theorem\_C\_sufficient\] hold, therefore, from Corollary \[corollary\_C1\], the generalized first Zagreb index $C_1^\textrm{II}(\cdot)$ is minimized with **any** extremely branched tree.
According to Corollary \[corollary\_VWWI\_max\_degree\], $\mathcal{R}_{WI_\textrm{O}}^*(n)$ includes only extremely branched trees depicted in Fig. \[fig\_VWWI\_minimizers\]. Since the cube root is a monotone function, by Proposition \[prop\_subset\], $\mathcal{R}_{WI_\textrm{O}^{1/3}+C_1^\textrm{II}}^*(n)=\mathcal{R}_{WI_\textrm{O}}^*(n)$.
The sub-root of all trees depicted in Fig. \[fig\_VWWI\_minimizers\] has degree 3 or 4. Therefore, $\mathcal{R}_{WI_\textrm{O}}^*(n)\cap \mathcal{R}_{S_2}^*(n)=\mathcal{R}_{WI_\textrm{O}}^*(n)$, and, finally, by Proposition \[prop\_subset\], $\mathcal{R}_{BP^\textrm{II}}^*(n)$ consists of rooted trees from Fig. \[fig\_VWWI\_minimizers\] with roots filled black.
The Basic regression (see Equation (\[eq\_bp0\])) combines the “ad-hoc” index $C^\textrm{0}(\cdot)$, the cube root of the oxygen’s distance $WI_\textrm{O}(\cdot)^{\frac{1}{3}}$, and $S_2(\cdot)$ index. From Corollary \[corollary\_VWWI\_max\_degree\] we know that $WI_\textrm{O}(\cdot)$ is minimized with an extremely branched tree, and so is $WI_\textrm{O}(\cdot)^{\frac{1}{3}}$. At the same time, neither of the inequalities (\[eq\_degree\_index\_condition23\])-(\[eq\_degree\_index\_condition23bis\]) from Theorem \[theorem\_C\_sufficient\] holds for $C^0(\cdot)$, and we cannot be sure that $C^\textrm{0}(\cdot)$ is minimized with an extremely branched tree. Therefore, we cannot prove formally that $BP^0(\cdot)$ achieves its minimum at some extremely branched tree.
Nevertheless, since for both simplified regressions ($BP^\textrm{I}$ and $BP^\textrm{II}$) an extremely branched tree appears to be optimal, optimality of an extremely branched tree for the Basic regression $BP^0(\cdot)$ is believed to be a credible hypothesis, which we state below formally.
\[conjecture\_bp0\] If $BP^0(T)=BP^{0*}_\mathcal{R}(n)$ for some pendent-rooted tree $T\in \mathcal{R}(n)$, then $T$ is an extremely branched tree.
\[prop\_BP0\]If Conjecture *\[conjecture\_bp0\]* is supposed to hold, the collection $\mathcal{R}_{BP^0}^*(n)$ of rooted trees *(*alcohol molecules*)* with the minimal predicted boiling point $BP^0$ for $n = 4,...,14$ consists of the trees depicted in Fig. *\[fig\_BP0\_minimizers\]* with a black circle representing a root *(*oxygen atom*)* position in a tree *(*alcohol molecule*)*.
From Conjecture \[conjecture\_bp0\], $BP^0$-minimizer is an extremely branched tree. From Fig. \[fig\_C\_minimizers\] and \[fig\_VWWI\_minimizers\] we see that for $n=4,5,6,8$ optimal trees for $WI_\textrm{O}(\cdot)$ and for $C^0(\cdot)$ coincide, while for $n=7,11,14$ $\mathcal{R}_{WI_\textrm{O}}^*(n)\subset\mathcal{R}_{C^0}^*(n)$. Since all sub-roots in $\mathcal{R}_{WI_\textrm{O}}^*(n)$ have degree 3 or 4, from Proposition \[prop\_subset\] we conclude that $\mathcal{R}_{BP^0}^*(n)=\mathcal{R}_{WI_\textrm{O}}^*(n)$ for such $n$.
For $n=9,10,12,13$, if Conjecture \[conjecture\_bp0\] is assumed to hold, both $WI_\textrm{O}$-minimizers and $C^0$-minimizers are extremely branched trees depicted in Fig. \[fig\_C\_minimizers\] and \[fig\_VWWI\_minimizers\], but $\mathcal{R}_{WI_\textrm{O}}^*(n)\cap
\mathcal{R}_{C^0}^*(n)=\emptyset$, and we cannot use Proposition \[prop\_subset\]. At the same time, the sets of extremely branched rooted trees of order $n$ are remarkably small for $n=9,10,12,13$, and are completely enumerated in Fig. \[fig\_admissible\_sets\] along with their predicted normal boiling points $BP^0$. The tree with the least boiling point for each $n$ is framed in Fig. \[fig\_admissible\_sets\]. Combining the above findings we obtain Fig. \[fig\_BP0\_minimizers\].
![Pendent-rooted chemical trees (alcohol molecules) minimizing $BP^0(\cdot)$ for $n=4,...,14$ with black circles being roots (oxygen atoms)[]{data-label="fig_BP0_minimizers"}](BP0_minimizers.pdf){width="11cm"}
![Extremely branched pendent-rooted trees of order $n=9,10,12,13$ and their $BP^0$ values (minima are framed)[]{data-label="fig_admissible_sets"}](admissible_sets.pdf){width="14cm"}
Predictions of the minimal boiling point based on the Wiener index slightly differ from those based on the “ad-hoc” index $C^\textrm{I}(\cdot)$. Minimizers of both indices are extremely branched trees. For $n=4,5,6,8$ optimal trees for $WI_\textrm{O}(\cdot)$ and for $C^\textrm{I}(\cdot)$ coincide. For $n=7,11,14$ $\mathcal{R}_{WI_\textrm{O}}^*(n)\subset \mathcal{R}_{C^\textrm{I}}^*(n)$, so, $WI_\textrm{O}(\cdot)$ appears to be a refinement of $C^\textrm{I}(\cdot)$, and Regression II refines Regression I.
For $n=9,10,12,13$ minimizers of $WI_\textrm{O}(\cdot)$ and of $C^\textrm{I}(\cdot)$ differ. By Corollaries \[corollary\_ad\_hoc\] and \[corollary\_VWWI\_max\_degree\], extremely branched trees minimize both indices. They have the only internal vertex $v$ of non-maximal degree, but $C^\textrm{I}(\cdot)$ suggests settling this vertex in the very center of a molecule surrounding it with the other internal vertices, while $WI_\textrm{O}(\cdot)$ says vertex $v$ must be a *stem vertex* incident to only one internal vertex. Therefore, predictions of Regression II differ from those of Regression I.
Basic regression, which contains a weighted linear combination of the Wiener index and the “ad-hoc” index, represents a sort of intermediate behavior between these extremal trends (at least for extremely branched trees under Conjecture \[conjecture\_bp0\]). The weight of the Wiener index in the regression is not enough to move the minimal tree sufficiently from the trees depicted in Fig. \[fig\_C\_minimizers\], and Basic regression becomes a yet another refinement of Regression I.
Conclusion
==========
The focus of this paper is development of optimization techniques for combinations of some well-known and novel topological indices over chemically interesting sets of graphs. We derived conditions under which an extremely branched tree minimizes the sum of the second Zagreb index and of the generalized first Zagreb index. We also found minimizers of the vertex-weighted Wiener index over the set of chemical trees with given vertex weights.
We enumerated index minimizers for moderate (up to 14) non-hydrogen atom count in a molecule, and combined them in several regressions of different complexity to forecast a simple alcohol molecule with the lowest boiling point.
For simpler regressions (Regressions I and II) we managed to obtain a complete analytical characterization of extremal alcohol molecules, while for the most complex (yet the most precise) “basic” regression we had to limit our attention to extremely branched trees (see Conjecture \[conjecture\_bp0\]) and employed the brute-force enumeration to find molecules of low-boiling alcohols.
Forecasts based on different regressions slightly differ, but they all comply with the collected experimental data on normal boiling points of simple alcohols.
Finally, let us sketch several promising directions of future research. An obvious shortcoming of this paper is Conjecture \[conjecture\_bp0\], which is explained but is not proven formally. To justify it, we need to refine sufficiently our optimization techniques, *viz*, to optimize jointly the Wiener index and the Zagreb indices.
On the other hand, there is wide space for investigation of popular combinations of topological indices forecasting important physical and chemical properties of compounds.
=0.23in
[99]{} A. T. Balaban, J. Devillers (Eds.), [*Topological indices and related descriptors in QSAR and QSPAR*]{}, CRC Press, Boca Raton, 2000.
U. Bren, V. Martínek, J. Florián, Decomposition of the Solvation Free Energies of Deoxyribonucleoside Triphosphates Using the Free Energy Perturbation Method, [*J. Phys. Chem. B*]{} [**110**]{} (2006), 12782–12788.
U. Bren, D. Janežič, Individual Degrees of Freedom and the Solvation Properties of Water, [*J. Chem. Phys.*]{} [**137**]{} (2012) 024108.
B. Bollobás, [*Extremal graph Theory*]{}, Academic Press, London, 1978.
A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, [*Acta Applicandae Mathematica*]{} [**66**]{} (2001) 211–249.
M. Fischermann, A. Hoffmann, D. Rautenbach, L. Székely, L. Volkmann, Wiener index versus maximum degree in trees, [*Discrete Applied Mathematics*]{} [**122**]{} (2002) 127–137.
B. Furtula, I. Gutman, H. Lin, More trees with all degrees odd having extremal Wiener index, [*MATCH Commun. Math. Comput. Chem.*]{} [**70**]{} (2013) 293–296.
R. García–Domenech, J. Gálvez, J. V. de Julián–Ortiz, L. Pogliani, Some new trends in chemical graph theory, [*Chem. Rev.*]{} [**108**]{} (2008) 1127–1169.
M. Goubko, Minimizing degree-based topological indices for trees with given number of pendent vertices, [*MATCH Commun. Math. Comput. Chem.*]{} [**71**]{} (2014) 33–46.
M. Goubko, Minimizing Wiener index for vertex-weighted trees with given weight and degree sequences, [*MATCH Commun. Math. Comput. Chem.*]{} [**73**]{} (2015) ??–??.
`http://www.mtas.ru/upload/library/GoubkoMilosOnline14.pdf` (last accessed Oct 30, 2014).
I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons, [*Chem. Phys. Lett.*]{} [**17**]{} (1972) 535–538.
I. Gutman, B. Furtula (Eds.), [*Novel molecular structure descriptors — Theory and Applications I*]{}, Univ. Kragujevac, Kragujevac, 2010.
I. Gutman, B. Furtula (Eds.), [*Novel molecular structure descriptors — Theory and Applications II*]{}, Univ. Kragujevac, Kragujevac, 2010.
D. Janežič, B. Lučić, S. Nikolić, A. Miličević, N. Trinajstić, Boiling points of alcohols — a comparative QSPR study, [*Internet Electronic Journal of Molecular Design* ]{} [**5**]{} (2006), 192–200.
S. Klavžar, I. Gutman, Wiener number of vertex-weighted graphs and a chemical application, [*Discrete Appl. Math.*]{} [**80**]{} (1997) 73–81.
M. Kompany–Zareh, A QSPR study of boiling point of saturated alcohols using genetic algorithm, [*Acta chimica slovenica*]{} [**50**]{} (2003) 259–273.
X. Li, Y. Shi, A survey on the Randic index, [*MATCH Commun. Math. Comput. Chem.*]{} [**59**]{} (2008) 127–156.
H. Lin, Extremal Wiener index of trees with all degrees odd, [*MATCH Commun. Math. Comput. Chem.*]{} [**70**]{} (2013) 287–292.
A. W. Marshall, I. Olkin, [*Inequalities: theory of majorization and its applications, mathematics in science and engineering*]{}, V. 143, Academic Press, New York, 1979.
P. Penchev, N. Kochev, V. Vandeva, G. Andreev, Prediction of boiling points of acyclic aliphatic alcohols from their structure, [*Traveaux Scientifiquesd’Universite de Plovdiv*]{} [**35**]{} (2007) 53–57.
M. Randić, On characterization of molecular branching, [*J. Amer. Chem. Soc.*]{} [**97**]{} (1975) 6609.
J. Rada, R. Cruz, Vertex-degree-based topological indices over graphs, [*MATCH Commun. Math. Comput. Chem.*]{} [**72**]{} (2014) 603–616.
R. Todeschini, V. Consonni, [*Handbook of molecular decriptors*]{}, Wiley-VCH, Weinheim, 2000.
R. Todeschini, V. Consonni, [*Molecular decriptors for chemoinformatic*]{}, Wiley-VCH, Weinheim, 2009.
H. Wang, The extremal values of the Wiener index of a tree with given degree sequence, [*Discrete Appl. Math.*]{} [**156**]{} (2008) 2647–2654.
H. Wiener, Structrual determination of paraffin boiling points, [*J. Am. Chem. Soc.*]{} [**69**]{} (1947) 17–20.
K. Xu, M. Liu, K. C. Das, I. Gutman, B. Furtula, A survey on graphs extremal with respect to distance-based topological indices, [*MATCH Commun. Math. Comput. Chem.*]{} [**71**]{} (2014) 461–508.
X.-D. Zhang, Q.-Y. Xiang, L.-Q. Xu, R.-Y. Pan, The Wiener index of trees with given degree sequences, [*MATCH Commun. Math. Comput. Chem.*]{} [**60**]{} (2008) 623–644.
B. Zhou, I. Gutman, Relations between Wiener, hyper–Wiener and Zagreb indices, [*Chem. Phys. Lett.*]{} [**394**]{} (2004) 93–95.
[^1]: This research is supported by the grant of Russian Foundation for Basic Research, project No 13-07-00389.
[^2]: ChemAxon Instant JChem$^\copyright$ was used for index calculation. Authors would like to thank ChemAxon$^\circledR$ Ltd (http://www.chemaxon.com) for the academic license.
[^3]: Typically it is called a *root*, but we will use an alternative notation to avoid confusion with a root of a pendent-rooted tree introduced in the previous subsection.
[^4]: Please note the different notation in [@Goubko15] ($x\preceq_w y$ is used in [@Goubko15] where we write $x\succeq y$).
|
---
abstract: 'We give algorithms for designing near-optimal sparse controllers using policy gradient with applications to control of systems corrupted by multiplicative noise, which is increasingly important in emerging complex dynamical networks. Various regularization schemes are examined and incorporated into the optimization by the use of gradient, subgradient, and proximal gradient methods. Numerical experiments on a large networked system show that the algorithms converge to performant sparse mean-square stabilizing controllers.'
address: 'The University of Texas at Dallas, Richardson, TX 75080 USA '
author:
- Benjamin Gravell
- Yi Guo
- Tyler Summers
bibliography:
- 'ifacconf.bib'
title: Sparse optimal control of networks with multiplicative noise via policy gradient
---
Optimal control, multiplicative noise, networks, sensor & actuator placement
Introduction
============
Emerging highly distributed networked dynamical systems, such as critical infrastructure for power, water, and transportation, are high-dimensional and increasingly instrumented with new sensing, actuation, and communication technologies. A key problem is to design high performance control architectures that limit the number of actuators, sensors, and actuator-sensor communication links to reduce complexity and cost. Sparse control architectures may be crucial for managing complexity in emerging complex networks, but require solution of extremely difficult mixed combinatorial-continuous optimization problems.
There is a variety of performance metrics and optimization methodology for sparse control architecture design in the recent literature. Examples include structural rank conditions from [@liu2011controllability; @ruths2014control; @olshevsky2014minimal], controllability and observability Gramians from [@Pasqualetti2014c; @summers2014submodularity; @tzoumas2016; @jadbabaie2018deterministic], and optimal and robust control metrics from [@Hassibi1998; @Polyak-LMI_sparse_fb; @jovanovic2016controller; @summers2016actuator; @Taha2017d; @Zare2018CDC], which are optimized via greedy algorithms, convex and mixed-integer optimization, and randomization.
Here we develop methods for sparse optimal control design in dynamical networks with multiplicative noise via policy gradient algorithms with sparsity-inducing regularization. Multiplicative noise arises in many networked systems when the weights of edges connecting nodes are stochastic in time. The noise is thus on the system parameters themselves and has a fundamentally different effect on the state evolution than additive noise, and indeed can lead to dramatic robustness issues. Specifically, a noise-ignorant classical optimal linear-quadratic (LQ) controller may actually *destabilize* a multiplicative noise system in the mean-square sense, even if the system was open-loop mean-square stable. Therefore noise-aware control is imperative to network performance and robustness. Moreover, the policy gradient methods we propose here, which operate directly on policy parameters, facilitate data-driven sparse control design when the model is unknown, a topic we are exploring in ongoing work.
In Section 2 we formulate the problem and discusses a policy gradient approach to optimal control design for linear-quadratic systems with multiplicative noise. In Section 3 we propose several sparse control design methods for sensor and actuator selection and communication network design using gradient, subgradient, and proximal algorithms. In Section 4 we present numerical experiments to illustrate the results. Section 5 concludes.
Problem formulation
===================
Consider the discrete-time linear quadratic regulator with multiplicative noise (LQRm) optimal control problem $$\label{eq:LQRm}
\begin{aligned}
&\underset{{\pi \in \Pi}}{\text{min}} && J(\pi) = \mathds{E}\sum_{t=0}^\infty (x_t^T Q x_t + u_t^T R u_t), \\
&\text{s.t.} && x_{t+1} = (A + \sum_{i=1}^p \delta_{it} A_i) x_t + (B + \sum_{j=1}^q \gamma_{jt} B_j) u_t
\end{aligned}$$ where $x_t \in \mathds{R}^n$ is the system state, $u_t \in \mathds{R}^m$ is the control input, $x_0$ is randomly distributed according to $\mathcal{P}$, expectation is with respect to $x_0,\delta_{it}, \gamma_{jt}$, and $Q\succeq 0$ and $R\succ 0$. The dynamics incorporate multiplicative noise terms modeled by the mutually independent and i.i.d. (over time) zero-mean random variables $\delta_{it}$ and $\gamma_{jt}$, which have variance $\alpha_i$ and $\beta_j$, respectively. The matrices $A_i \in \mathds{R}^{n \times n}$ and $B_i \in \mathds{R}^{n \times m}$ specify how each noise term affects the system dynamics and input matrices. The goal is to determine an optimal closed-loop feedback policy $\pi$ with $u_t = \pi(x_t)$. We assume that the problem data $A$, $B$, $\alpha_i$, $A_i$, $\beta_j$, and $B_j$ are such that the optimal value of the problem exists and is finite. Feasibility of this problem is ensured if the system is mean-square stabilizable.
The system in is stable in the mean-square sense if $\lim_{t \to \infty}\mathds{E}[x_t x_t^T] = 0$ for any given initial covariance $\mathds{E}x_0 x_0^T$.
We are ultimately interested in the problem $$\label{eq:LQRm_sparse}
\begin{aligned}
&\underset{{\pi \in \Pi}}{\text{min}} && J(\pi) + \gamma J_{\text{reg}}(\pi)
\end{aligned}$$ where $J_{\text{reg}}(\pi)$ is a sparsity-promoting regularizer of the policy $\pi$ and $\gamma$ specifies the importance of sparsity. The regularizer ideally would measure the number of actuators, sensors, or actuator-sensor links, but for computational tractability will be replaced by other functions defined later. We begin by discussing the solution for $\gamma=0$.
Optimal control via value iteration
-----------------------------------
Dynamic programming can be used to show that the optimal policy is linear state feedback with $u_t = K^* x_t$ where $K^* \in \mathds{R}^{m \times n}$ and the resulting optimal cost for a fixed initial state is quadratic with $V_{K^*}(x_0) = x_0^T P x_0$ where $P \in \mathds{R}^{n \times n}$ is a symmetric positive definite matrix. When the model parameters are known, there are several known ways to compute the optimal feedback gains and corresponding optimal cost. The optimal cost is given by the solution of the generalized algebraic Riccati equation (ARE) (see, e.g., [@Damm2004]). $$\begin{aligned}
\label{genriccati}
P &= Q + A^T P A + \sum_{i=1}^p \alpha_i A_i^T P A_i \\
&\quad - A^T P B (R + B^T P B + \sum_{j=1}^q \beta_j B_j^T P B_j)^{-1} B^T P A. \nonumber \end{aligned}$$ This can be solved via value iteration, and the optimal gain matrix is $$K^* = - \Big(R + B^T P B + \sum_{j=1}^q \beta_j B_j^T P B_j \Big)^{-1} B^T P A.$$
Optimal control via policy gradient
-----------------------------------
For a fixed mean-square stabilizing linear state feedback policy $u_t = K x_t$, there exists a positive semidefinite cost matrix $P_K$ which characterizes the cost by $$J(K) = \underset{x_0}{\mathds{E}} x_0^T P_{K} x_0$$ and is the solution to the generalized Lyapunov equation $$\begin{aligned}
&P_{K} = Q + K^TRK + (A+BK)^T P_{K} (A+BK) \nonumber \\
&\quad + \sum_{i=1}^p \alpha_i A_i^T P_K A_i + \sum_{j=1}^q \beta_j K^T B_j^T P_K B_j K. \label{eq:gen_lyap_P}\end{aligned}$$ Furthermore, there exists a positive semidefinite infinite-horizon aggregate state covariance matrix $$\begin{aligned}
\Sigma_{K} = \underset{x_0,\delta_{it},\gamma_i}{\mathds{E}} \sum_{t=0}^\infty x_t x_t^T\end{aligned}$$ which is the solution to the generalized Lyapunov equation $$\begin{aligned}
\Sigma_{K} &= \Sigma_0 + (A+BK) \Sigma_{K} (A+BK)^T + \sum_{i=1}^p \alpha_i A_i \Sigma_K A_i^T \nonumber \\
&\quad + \sum_{j=1}^q \beta_j B_j K \Sigma_K K^T B_j^T, \label{eq:gen_lyap_S}\end{aligned}$$ where $\Sigma_{0} = \underset{x_0}{\mathds{E}}\left[x_0x_0^T\right]$. Thus, we have $$\begin{aligned}
J(K) = \operatorname{Tr}((Q+K^T R K) \Sigma_K) = \operatorname{Tr}(P_K \Sigma_0). \label{eq:cost}\end{aligned}$$
This leads to the idea of performing gradient descent on $J$ (i.e., policy gradient) to find the optimal gain matrix: $$K \leftarrow K - \eta \nabla_K J(K),$$ for a fixed step size $\eta$. In this work we consider only the case where the model parameters are known, but the methods presented are immediately usable in the model-unknown case by estimating the gradient from trajectory data. The policy gradient for linear state feedback policies applied to the LQRm problem has the following form:
The LQRm policy gradient is given by $$\begin{aligned}
&\nabla_K J(K) \\
&= 2\bigg[ \Big(R + B^T P_{K} B + \sum_{j=1}^q \beta_j B_j^T P_K B_j \Big) K + B^T P_{K} A\bigg] \Sigma_{K} . \nonumber\end{aligned}$$
The proof is omitted due to space constraints and can be found in our technical report (see [@Gravell2019unpublished]).
Gradient domination
-------------------
It was shown recently by [@Fazel2018] that although the deterministic LQR cost is nonconvex, it is *gradient dominated*, also known as the Polyak-[Ł]{}ojasiewicz inequality originally due to [@Polyak1963]. It is simple to show that if a function has a Lipschitz continuous gradient and satisfies this condition then performing gradient descent with a sufficiently small constant step size will result in asymptotic convergence to the optimal function value at a linear rate (see [@Karimi2016]). For the LQRm problem, so long as the initial controller is stabilizing the LQRm cost is continuously differentiable over the sublevel set associated with the initial controller and thus the gradient possesses a local Lipschitz constant $L$ on this set. Identifying $L$ and the gradient domination constant is necessary for selection of a step size which is guaranteed to give convergence using gradient descent. Quantifying these constants is difficult but possible via lengthy chains of matrix inequalities as demonstrated by [@Fazel2018].
These results extend readily to the LQRm problem with relevant quantities pertaining to Lipschitz continuity of the gradient and the gradient domination conditions modified suitably to accommodate the multiplicative noise. In particular, the effect of the noise is to decrease the maximum step size that can be taken using gradient descent. We now state the relevant lemmas; the proofs are lengthy and can be found in our technical report (see [@Gravell2019unpublished]).
\[lemma:gradient\_dominated\] The LQRm cost $J(K)$ satisfies the gradient domination condition $$\begin{aligned}
J(K) - J(K^*) &\leq \frac{\|\Sigma_{K^*}\|}{4\sigma_{\text{min}}(R) \sigma_{\text{min}}(\Sigma_{0})^2 } \|\nabla_K J(K)\|_F^2
\end{aligned}$$
\[lemma:grad\_exact\_convergence\] \
Using the policy gradient step update $$K^{(k+1)} = K^{(k)} - \eta \nabla_K J(K^{(k)})$$ with step size $0 < \eta \leq c_{pg}$ gives global convergence to the optimal $K^*$ at a linear rate described by $$\frac{ J(K^{(k+1)}) - J(K^*)}{ J(K^{(k)}) - J(K^*)} \leq 1 - \eta \frac{ \sigma_{\text{min}}(R)\sigma_{\text{min}}(\Sigma_{0})^2}{\|\Sigma_{K^*}\|}$$ where $c_{\text{pg}}$ is a constant which is polynomial in the parameters $A$, $B$, $B_j$, $Q$, $R$, $J(K^{(0)})$.
Sparse control design
=====================
Entrywise, row, and column sparsity in $K$ correspond to actuator-sensor communication, actuator, and sensor sparsity respectively. With this in mind, we seek to solve the optimization problem of finding the sparsest set of entries, rows and/or columns of $K$ that achieve some prescribed level of performance in terms of the LQRm cost. However this problem is a nonconvex combinatorial problem which is NP-hard; the number of independent problem instances which must be solved scales factorially with $n$ and/or $m$. We instead turn to regularization as a heuristic to identifying good sparsity patterns.
Insufficiency of naive hard thresholding
----------------------------------------
The most naïve method of inducing sparsity is hard thresholding of the ARE solution as $K_{ij} = 0 \text{ if } |K_{ij}| < r$. However, in general this is not useful since the resulting gains may not be stabilizing. Consider the following example system without multiplicative noise: $$\begin{aligned}
A =
\begin{bmatrix}
0.4 & 0.9 & -0.3 \\
0.7 & -0.3 & -0.4 \\
-0.2 & 0.1 & -0.8
\end{bmatrix}, \quad
B =
\begin{bmatrix}
0.2 & -0.6 \\
-1.3 & -1.6 \\
-0.3 & -1.5
\end{bmatrix}, \\
\mathcal{P} = \mathcal{N}(0,I_3),\quad Q=I_3,\quad R=I_2\end{aligned}$$ where $I_n$ is an $n \times n$ identity matrix. Imposing a hard threshold of $0.4$ on the ARE solution results in $$\begin{aligned}
K =
\begin{bmatrix}
0.503931 & -0.880322 & 0 \\
0 & 0.614382 & -0.677758
\end{bmatrix}\end{aligned}$$ which gives a closed-loop state transition matrix $A+BK$ with an eigenvalue of $1.048223$ outside the unit circle. By contrast, by working with the regularized LQRm cost the optimal gains are always guaranteed to be stabilizing; even in the limit as the regularization weight $\rightarrow \infty$ the sparsity increases until the sparsest stabilizing solution is obtained. In practice, using a small step size helps ensure that each iterate remains inside the domain of $J(K)$.
Regularization
--------------
Certain types of regularization are well-known to be capable of inducing sparsity in the solutions to optimization problems. Perhaps the most basic and well-known is $l_1$-norm regularization which operates on a vector of decision variables; see [@Tibshirani1996] for the seminal LASSO problem for sparse least-squares model selection and [@Hassibi1998] for sparse control design. In the case of a convex objective, increasing the regularization weight tends to increase sparsity by moving the global minimum onto the coordinate axes. Once the regularized problem has been solved, a sparsity pattern can easily be identified from the (near-)zero entries. In the current work we consider only the problem of identifying sparsity patterns, however an additional “polishing” step which involves re-solving the LQRm problem under the sparsity pattern can be performed to further improve the LQRm cost, as in [@Lin2013].
Entrywise sparsity is induced by the vector $l_1$-norm $$\begin{aligned}
\|K\|_{1} = \sum_{i=1}^m \sum_{j=1}^n |K_{ij}|.\end{aligned}$$ Row and column sparsity are induced by using matrix row and column norms respectively defined as $$\begin{aligned}
\|K\|_{r} = \sum_{i=1}^m \|K^{r,i}\|_{\infty}, \ \text{ and } \ \|K\|_{c} = \sum_{i=1}^n \|K^{c,i}\|_{\infty},\end{aligned}$$ where $\|K^{r,i}\|_\infty$ and $\|K^{c,i}\|_\infty$ are the maximum absolute values of the $i^{th}$ row and column respectively of $K$. Row and column sparsity are also induced by the row and column group LASSO $$\begin{aligned}
\|K\|_{glr} = \sum_{i=1}^m \|K^{r,i}\|_{2}, \ \text{ and } \ \|K\|_{glc} = \sum_{i=1}^n \|K^{c,i}\|_{2},\end{aligned}$$ where $\|K^{r,i}\|_\infty$ and $\|K^{c,i}\|_\infty$ are the vector $l_2$-norms of the $i^{th}$ row and column respectively of $K$. Combined row and column sparsity can be induced by the row and column sparse group LASSO $$\begin{aligned}
\|K\|_{sglr} = (1-\mu) \|K\|_{1} + \mu \|K\|_{glr} \\
\|K\|_{sglc} = (1-\mu) \|K\|_{1} + \mu \|K\|_{glc}\end{aligned}$$ or by various other weighted combinations of entrywise, row, and column norms. We refer to $\|K\|_M$ as a generic nondifferentiable sparsity-inducing regularizer.
Stationary point characterization
---------------------------------
Before proceeding, we must point out an important consequence of regularizing the LQRm cost. The sum of a convex function and a gradient dominated function is not gradient dominated in general, and in fact can have multiple local minima. For example, consider the scalar function $$\begin{aligned}
f(x) = x^2 + 4 ((x-8)^2 + 3 \sin^2(x-8))\end{aligned}$$ where $x^2$ is strongly convex and $4((x-8)^2 + 3 \sin^2(x-8))$ is gradient dominated. But $f(x)$ has two local minima at $x=5.372$ and $x=7.459$ and therefore is not gradient dominated.
As a result any local first-order search procedure, such as those used by our algorithms, will not be guaranteed to find the global minimum. We conjecture that for the regularized LQRm problem there are at most two local minima, one associated with the LQRm cost and one associated with the regularization which tends to be more sparse. If this is so then choosing the initial point carefully may help the local search find the desired (sparser) local minimum. For open-loop mean-square systems, this motivates using zero gains as the initial condition. Likewise, in both the open-loop mean-square stable and unstable cases, an effective heuristic is to use the solution to a highly regularized problem instance to “warm start” another nearby problem instance with reduced regularization weight.
Other step directions
---------------------
Promising choices of step directions other than the gradient $\nabla_K J(K)$ are the natural gradient $\nabla_K J(K)\Sigma_{K}^{-1}$ and the Gauss-Newton step $R_{K}^{-1} \nabla_K J(K)\Sigma_{K}^{-1}$ as given by [@Fazel2018]. When $\gamma=0$, these step directions give faster convergence than the gradient step and in fact the convergence proofs are much simpler than that for the gradient step. Unfortunately, adding a regularizer makes these steps more difficult to calculate; it is not simply the sum of the gradient of the regularizer and the unregularized natural gradient or Gauss-Newton step of the LQRm cost. For this reason we restrict our attention to the standard (sub)gradient directions.
Regularized policy subgradient descent
--------------------------------------
In order to use nondifferentiable regularizers we use subgradient methods which take steps in the direction of subgradients. It is known that using a constant step size gives convergence to a bounded neighborhood of the optimum and that a diminishing step size gives asymptotic, albeit slow, convergence (see [@Nesterov2013]). One immediate issue is that subgradients are defined only for convex functions; since the LQRm cost is nonconvex, subgradients do not exist for the regularized LQRm cost. However we simply use the gradient of the LQRm cost plus the subgradient of the regularizer as the step direction. Thus our subgradient descent update is
$$\begin{aligned}
K^{(k+1)} &= K^{(k)}-\eta (\nabla_K J(K^{(k)}) + \gamma \blacktriangledown_K \|K^{(k)}\|_M) \\
K_{\text{best}} &= \text{argmax } \{C(K^{(k+1)}),C(K_{\text{best}})\}
\end{aligned}$$
where $C(K)=J(K) + \gamma \|K\|_M$ and $\blacktriangledown_K$ is a subgradient.
Another issue is that there is no guarantee of feasibility of each next step; it is possible to take a step so large that the next point is a mean-square unstable controller giving infinite objective cost. It is not straightforward to obtain restrictions on the step size to guarantee this feasibility. Gradient descent does not suffer from this problem since the gradient is guaranteed to be a true descent direction so there is always a sufficiently small step size to give a feasible next step. Nevertheless, in practice it is rare for a sufficiently small subgradient step to be infeasible.
Proximal policy gradient
------------------------
Proximal gradient methods have become a preferred way to solve optimization problems of the form $$\begin{aligned}
\underset{x}{\text{min}} \ f(x) + g(x)\end{aligned}$$ where $f(x)$ has a Lipschitz continuous gradient and $g(x)$ is convex and nondifferentiable, as is the case when $g(x)$ is a sparsity-inducing regularizer. The proximal gradient method update is $$\begin{aligned}
x^{(k+1)} =\operatorname{prox}_{\eta g}\Big(x^{(k)}-\eta \nabla f\big(x^{(k)}\big)\Big)\end{aligned}$$ where the proximity operator is defined as $$\begin{aligned}
\operatorname{prox}_{\eta g}(v)=\underset{x}{\operatorname{argmin}}\left\{g(x)+(1 /( 2 \eta))\|x-v\|_{2}^{2}\right\} .\end{aligned}$$ Much of the existing literature examines the case where $f(x)$ is convex, in which case gradient descent is guaranteed to converge. The proximal operator has closed-form expressions for $\|K\|_{1}$ and $\|K\|_{glr}$ called *soft thresholding* and *block soft thresholding* (see [@Parikh2014]). Thus to solve we also use a proximal policy gradient algorithm:
$$\begin{aligned}
K_s^{(k)} &= K^{(k)}-\eta \Delta K^{(k)} \\
K^{(k+1)} &=\operatorname{prox}_{\eta \|K\|_M}\big(K_s^{(k)}\big)
\end{aligned}$$
where $\Delta K^{(k)}$ is a generic step direction.
A result from [@Hassan2018] guarantees convergence at a linear rate to the optimal function value using the proximal gradient method on a function satisfying a proximal gradient domination condition. This condition was shown to be equivalent by one given by [@Karimi2016] and an inequality from [@Kurdyka1998]. However, this condition is not guaranteed to hold when $f(x)$ is gradient dominated and $g(x)$ is convex; the full condition must be checked, which involves interaction between $f(x)$ and $g(x)$. It is nontrivial to verify that the condition is satisfied for the regularized LQRm cost. Empirically it appears that the inequality may be satisfied since the proximal gradient method converged to solutions similar to those from our other two methods.
Regularized policy gradient descent
-----------------------------------
Another algorithm for solving is gradient descent:
$$\begin{aligned}
K^{(k+1)} &= K^{(k)}-\eta \nabla_K (J(K^{(k)}) + \gamma \|K\|_M)
\end{aligned}$$
Here we use differentiable Huber-type losses $\|K\|_{M,h,\phi}$ in place of nondifferentiable regularizers, which replace linear corners with quadratic tips for decision variable values smaller than a specified threshold. Although the solutions produced are not exactly sparse, in practice entries are sufficiently close to zero to identify the sparsity pattern. Furthermore, by iteratively decreasing the threshold the solutions can be made arbitrarily close to truly sparse.
We define the Huber function of a scalar $a$ as $$\begin{aligned}
h_{\phi}(a) = \left\{\begin{array}{ll} {|a|-\frac{1}{2} \phi} & {\text { if } |a| > \phi} \\ {\frac{1}{2 \phi} a^{2}} & {\text { if } |a| \leq \phi}\end{array}\right.\end{aligned}$$ and the $p$-Huber function (like a $p$-norm) of a vector $b$ as $$\begin{aligned}
h_{p,\phi}(b) = \left\{\begin{array}{ll} {\|b\|_p-\frac{1}{2} \phi} & {\text { if } \|b\|_p > \phi} \\ {\frac{1}{2 \phi} \|b\|_p^{2}} & {\text { if } \|b\|_p \leq \phi}\end{array}\right.\end{aligned}$$ We define the vector Huber loss as $$\begin{aligned}
\|K\|_{1,h,\phi} = \sum_{i=1}^m \sum_{j=1}^n h_{\phi}(K_{ij})\end{aligned}$$ the Huber row and column norms as $$\begin{aligned}
\|K\|_{r,h,\phi} = \sum_{i=1}^m h_{\infty,\phi}(K^{r,i}) , \quad \|K\|_{c,h,\phi} = \sum_{i=1}^n h_{\infty,\phi}(K^{c,i})\end{aligned}$$ and the Huber row and column group LASSO as $$\begin{aligned}
\|K\|_{glr,h,\phi} = \sum_{i=1}^m h_{2,\phi}(K^{r,i}) , \quad \|K\|_{glc,h,\phi} = \sum_{i=1}^n h_{2,\phi}(K^{c,i})\end{aligned}$$ Subgradients of two regularizers and the gradients of their differentiable counterparts are given in Table \[table:1\].
[|c | c | c|]{}
------------------------------------------------------------------------
$\|K\|_M$ & $\blacktriangledown_K \|K\|_M$ & $\nabla_K \|K\|_{M,h,\phi}$\
\[0.5ex\]
------------------------------------------------------------------------
$\|K\|_{1,ij}$ & $\operatorname{sgn}(K_{ij})$ & $K_{ij}/\max(|K_{ij}|,\phi)$\
\[0.5ex\]
------------------------------------------------------------------------
$\|K\|_{glr}^{r,i}$ & $K^{r,i}/\|K^{r,i}\|_2$ & $K^{r,i}/\max(\|K^{r,i}\|_2, \phi)$\
\[0.5ex\]
Simulation results
==================
We considered an example system which represents diffusion dynamics on a particular undirected Erdős-Rényi random graph. It is well known that if $p_{ER}=(\log n+c) / n$ for constant $c \in \mathbb{R}$, then $ \lim_{n \rightarrow \infty} P(G(n, p) \text{ connected}) = e^{-e^{-c}}$ so we chose $n=51$, $c=7$ and $p_{ER}=0.2144$ and with probability $P=0.999$ obtained a connected graph (see [@Bollobas2001]). The graph was selected so that it was connected, which ensured controllability. The first row and and column of the graph Laplacian were removed in order to fix the system’s state reference to the first node which removed the zero eigenvalue otherwise present. The continuous time system was discretized using a standard bilinear transform (Tustin’s approximation) which preserves the open-loop mean stability of this system. Two multiplicative noises act each on $A$ and $B$ whose entries were drawn from a Gaussian distribution. The multiplicative noise variances were set at two levels, low and high, so that the system was open-loop mean-square stable and unstable, respectively.
For the subgradient and proximal gradient methods, we stopped iterating after the best iterate had been held for 100 iterations. For the gradient method, we stopped iterating when the Frobenius norm of the gradient of the cost function fell below a small threshold value, $0.1 \times \text{card}(K)$. We swept through a range of sparsity levels by solving a problem with low $\gamma$ then increasing $\gamma$ and resolving the problem using the previous solution as the initial guess. The step size $\eta$ was initialized at $10^{-5}$. For the $l_1$-norm and row group LASSO $\gamma$ was initialized at 10 and 100 respectively. For each successive problem, the regularization weight was multiplied by a ratio $r_\gamma=\sqrt{2}$ and the step size was multiplied by $r_\eta = r_\gamma^{-\sqrt[4]{2}}$. To determine sparsity patterns we considered a value to be sparse if it was less than 5% than the max value in $K$. For the $l_1$-norm the sparsity values were the absolute values of the entries. For the row group lasso norm the sparsity values were the the values are the $l_2$-norms of the rows and columns respectively. Sparsity patterns are presented in Figs. \[fig:sparsity\_vec1\] and \[fig:sparsity\_glr\] with white cells representing near-zero entries. The LQRm costs given in Figs. \[fig:plot\_comparison\_vec1\] and \[fig:plot\_comparison\_glr\] are for the sparse gains without any polishing step applied, which otherwise could significantly reduce the cost. We give the total “wall-clock” computation time in Fig. \[fig:plot\_comparison\_glr\] to capture the aggregate computational expense of each algorithm. The main computational expense came from evaluating the LQRm gradient at each iteration, which required solving a generalized discrete Lyapunov equation.
As seen in Fig. \[fig:plot\_comparison\_glr\], the first iteration had the longest compute time since successive iterations benefited from favorable initial conditions from warm-starting. The compute time increased as the regularization weight was increased and a larger number of smaller steps were required to accommodate the increasing gradient magnitude.
From our empirical studies, the three methods presented all gave very similar results with similar efficacy; arbitrarily entrywise and row sparse mean-square stabilizing solutions were obtained for the low noise setting after a reasonable amount of computation time. Similarly, very sparse solutions for the high noise setting were obtained.
Python code which implements the algorithms and generates the figures reported in this work can be found in the GitHub repository at <https://github.com/TSummersLab/polgrad-multinoise/>.
The code was run on a desktop PC with a quad-core Intel i7 6700K 4.0GHz CPU, 16GB RAM.
Concluding Remarks
==================
We developed three policy gradient algorithms for solving the sparse gain design problem for networked dynamical systems with multiplicative noise. We showed that the regularized LQR cost does not necessarily have a unique local minimum, hampering efforts to guarantee global convergence of the algorithms. Nevertheless, efficacy of the algorithms is demonstrated empirically via computational simulations. Through various regularization functions we identified sparsity patterns for near-optimal actuator, sensor, and actuator-sensor link removal. This paves the way for data-driven control design in the model-free setting for such systems.
Future work will attempt to prove unique local minimization of the regularized LQR cost or provide a set of restrictions under which such a condition holds. A salient issue with policy gradient methods relates to scalability; for large systems the gradient calculation is computationally expensive. Hence we will explore low-rank approximations of the gradient and consequent effects on convergence. We will also extend this work to the unknown-model setting and explore alternative model-based learning schemes.
|
---
author:
- 'L. Origlia'
- 'E. Dalessandro'
- 'N. Sanna'
- 'A. Mucciarelli'
- 'E. Oliva'
- 'G. Cescutti'
- 'M. Rainer'
- 'A. Bragaglia'
- 'G. Bono'
date: 'Received .... ; accepted ...'
subtitle: 'GIANO-B spectroscopy of red supergiants in Alicante 7 and Alicante 10'
title: 'Stellar population astrophysics (SPA) with the TNG[^1]'
---
[The Scutum complex in the inner disk of the Galaxy hosts a number of young clusters and associations of red supergiant stars that are heavily obscured by dust extinction. These stars are important tracers of the recent star formation and chemical enrichment history in the inner Galaxy. ]{} [Within the SPA Large Programme at the TNG, we secured GIANO-B high-resolution (R$\simeq$50,000) YJHK spectra of 11 red supergiants toward the Alicante 7 and Alicante 10 associations near the RSGC3 cluster. Taking advantage of the full YJHK spectral coverage of GIANO in a single exposure, we were able to measure several hundreds of atomic and molecular lines that are suitable for chemical abundance determinations. We also measured a prominent diffuse interstellar band at $\lambda$1317.8 nm (vacuum). This provides an independent reddening estimate. ]{} [The radial velocities, Gaia proper motions, and extinction of seven red supergiants in Alicante 7 and three in Alicante 10 are consistent with them being members of the associations. One star toward Alicante 10 has kinematics and low extinction that are inconsistent with a membership. By means of spectral synthesis and line equivalent width measurements, we obtained chemical abundances for iron-peak, CNO, alpha, other light, and a few neutron-capture elements. We found average slightly subsolar iron abundances and solar-scaled \[X/Fe\] abundance patterns for most of the elements, consistent with a thin-disk chemistry. We found depletion of \[C/Fe\], enhancement of \[N/Fe\], and relatively low $\rm ^{12}C/^{13}C<$15, which is consistent with CN cycled material and possibly some additional mixing in their atmospheres. ]{}
Introduction
============
Stellar population astrophysics experiences a golden era because of the Gaia mission and ongoing and near-future massive photometric, astrometric, and spectroscopic surveys that set the observational framework for an exhaustive description of the structure of the Milky Way (MW) and its satellites. Optical and near-infrared (NIR) high-resolution (HR) spectroscopy is crucial to provide the complementary detailed chemical tagging of selected stellar populations, in order to constrain timescales and overall formation and chemical enrichment scenarios. Only echelle spectra with simultaneous HR and wide spectral coverage might be able to allow measurements of the full set of iron-peak, CNO, alpha, neutron-capture, Li, Na, Al, and other light element abundances with the necessary high accuracy. These different elements are synthesized in stars and supernovae with different mass progenitors and are therefore released into the interstellar medium with different time delays with respect to the onset of the star formation event.
SPA Large Programme
-------------------
The unique combination of the High Accuracy Radial velocity Planet Searcher for the Northern emisphere (HARPS-N) and GIANO-B echelle spectrographs at the Telescopio Nazionale Galileo (TNG), which together cover almost the full optical and NIR range out to the K band, is ideal to sample the luminous stellar populations of the the MW thin disk over almost its entire extension, as seen from the Northern Hemisphere at a spectral resolution R$\ge$50,000. Thus, taking advantage of this HR capability, we proposed a Large Program called the [*SPA - Stellar Population Astrophysics: detailed age-resolved chemistry of the Milky Way disk*]{} (Program ID A37TAC\_13, PI: L. Origlia). Observing time has been granted starting from June 2018. The SPA Large Programme aims at obtaining high-quality spectra of more than 500 stars in the MW thin disk and associated star clusters at different Galactocentric distances, including the poorly explored inner disk. We will observe luminous giant and supergiant stars in young star clusters and associations, luminous Cepheid and Mira variables across the entire thin disk, and main-sequence (MS) and other evolved stars of open clusters in the solar neighborhood.
The proposed observations will allow us to provide a detailed mapping of possible gradients, cosmic spreads, and other inhomogeneities of individual abundances and abundance ratios. We will also be able to answer a number of open questions regarding disk formation, evolution, and chemical enrichment, and we will perform critical tests of stellar evolution and nucleosynthesis. This will maximize the scientific return and the overall legacy value for astrophysics.
Based on this detailed chemical tagging and complementary kinematic and evolutionary information from the Gaia mission and other surveys, we expect in particular to set the framework for a comprehensive chemodynamical modeling of the disk formation and evolution. We hope to be able to distinguish radial migration from in situ formation scenarios [e.g., @dasilva16 and references therein] and the overall chemical enrichment history. In addition, given the expected high quality of the acquired spectra and the target properties, we will be able to perform the following stellar evolution tests and calibrations: i) a first systematic investigation of the dependence of period-luminosity relations of variable stars on their detailed chemical content, ii) a probe of the poorly explored physics and nucleosynthesis of red supergiants (RSGs), and iii) a probe of crucial stellar evolution parameters, such as convection, diffusion, mixing, rotation, and magnetic fields in MS and evolved stars for different stellar ages and evolutionary stages.
High-resolution spectroscopy in the Scutum complex
==================================================
Star RA(J2000) DEC(J2000) J H K T$_{eff}$ $\xi $ EW$_{\lambda 1317.8}$ $A_V$ RV$_{hel}$ RV$_{LSR}$
------------ ------------ ------------- ------ ------ ------ ----------- -------- ----------------------- ------- ------------ ------------
AL07-F1S02 18:44:46.9 -03:31:07.5 8.78 6.71 5.76 3700 3.0 0.224 11.6 63 79
AL07-F1S04 18:44:39.4 -03:30:00.5 8.66 6.76 5.82 3600 3.0 0.211 10.9 107 123
AL07-F1S05 18:44:31.0 -03:30:49.9 9.87 7.93 6.97 3700 3.0 0.252 13.0 82 98
AL07-F1S06 18:44:29.4 -03:30:02.5 8.45 6.39 5.41 3600 4.0 0.247 12.7 79 95
AL07-F1S07 18:44:27.9 -03:29:42.7 8.06 5.91 4.85 3600 4.0 0.233 12.0 84 100
AL07-F1S08 18:44:30.4 -03:28:47.1 7.34 5.20 4.10 3500 4.0 0.219 11.3 73 89
AL07-F1S09 18:44:20.5 -03:28:44.7 9.28 7.24 6.21 3600 3.0 0.229 11.8 75 91
AL10-C06 18:45.17.0 -03:41:26.0 9.56 7.29 6.15 3600 3.0 0.252 13.0 75 91
AL10-C09 18:45:11.8 -03:40:53.1 9.34 6.86 5.62 3600 3.0 0.299 15.4 76 92
AL10-C10 18:45:11.2 -03:39:34.1 8.80 6.47 5.35 3600 4.0 0.242 12.5 74 90
AL10-N03 18:45:36.6 -03:39:19.1 7.19 5.82 5.23 3700 3.0 0.075 4.0 17 33
[**Note**]{}: Identification names, coordinates and 2MASS JHK magnitudes from @neg11 for Alicante 7 and @gon12 for Alicante 10. Effective temperature (T$_{eff})$ in units of Kelvin and microturbulence ($\xi$) in units of km/s from spectral synthesis of GIANO-B spectra. The EW of the DIB feature at $\lambda$1317.8 nm (vacuum) are in units of nanometers and heliocentric RVs in units of km/s, also from the GIANO-B spectra. The derived $A_V$ extinction values from the DIB feature at $\lambda$1317.8 nm are based on the calibration with E(B-V) by @ham15 [@DIB_Hamano] and are converted into $A_V$ by using R=3.1. The LSR RVs were computed using the solar motion from @sch10.\
In the central region of the MW close to the base of the Scutum-Crux arm and the tip of the long bar, at a distance of $\approx$3.5 kpc from the Galactic center, lies a giant region of star formation that is known as the Scutum complex. This region is characterized by three young and massive clusters: RSGC1 [@figer06], RSGC2 [@davies07], and RSGC3 [@clark09; @alexander09]. They host a conspicuous population of RSG stars with stellar masses of $\sim$14-20 $M_{\odot}$ and ages between 12 and 20 Myr. A total mass of 2-4$\rm \times 10^{4} M_{\odot}$ has been estimated for these clusters. Some extended associations of RSGs have also been identified around them: Alicante 8, in the proximity of RSGC1 and RSGC2 [@neg10], and Alicante 7 [@neg11] and Alicante 10 [@gon12], which are associated with RSGC3.
The spectroscopic characterization of the chemical and kinematic properties of the stellar poplulations in the Scutum complex started only recently and used IR spectroscopy because of the huge (A$_V$>10) and patchy extinction that affects the Galactic plane in this direction at optical wavelengths.
First radial velocity (RV) measurements suggested that these young clusters and associations might share a common kinematics. For example, @davies07 and @davies08 derived average local standard of rest (LSR) RVs of +109 and +123 km/s for RSGC2 and RSGC1, respectively, from K-band spectra with medium-high resolution of the CO bandheads. @neg11 obtained average LSR RVs of +102 and +95 km/s for RSGC2 and RSGC3, respectively, from tmeasurement of the Ca II triplet lines.
Some chemical abundances of Fe, C, O, and other alpha elements were derived for stars in RSGC1 and RSGC2 [@davies09b] using NIRSPEC-Keck spectra at a resolution R$\approx$17,000. They suggest slightly subsolar abundances.
RSGs are very luminous NIR sources even in highly reddened environments, such as the inner Galaxy, and they can be spectroscopically studied at HR even with 4 m class telescopes when these are equipped with efficient echelle spectrographs. A few years ago, during the commissioning and science verification of GIANO [see, e.g., @oli12a; @oli12b; @oli13; @giano14], we therefore acquired HR (R$\approx$50,000) YJHK spectra of a few RSGs in the RSGC2 and RSGC3 clusters.
In particular, we observed three RSGs in RSGC2 in July 2012 and five RSGs RSGC3 in October 2013. Detailed RVs and chemical abundances of iron-peak, CNO, alpha, and a few other metals have been published in @ori13 [@ori16]. They confirmed slightly subsolar iron and iron-peak elements, some depletion of carbon and enhancement of nitrogen, and approximately solar-scale \[X/Fe\] abundance ratios for the other metals.
Prompted by the results of these pilot projects, we decided to perform a more systematic investigation of the RSGs in the Scutum complex. As part of the SPA Large Program at the TNG, we are observing a representative sample of luminous RSG stars, candidate members of the young clusters and associations in the Scutum complex, using the refurbished GIANO-B spectrograph. This first paper presents the derived RVs and chemical abundances for RSG candidate members of the Alicante 7 and Alicante 10 associations near the RSGC3 star cluster.
Observations and data reduction
===============================
GIANO [@giano14] is the HR (R$\simeq$50,000) YJHK (950–2450 nm) spectrometer of the TNG telescope. GIANO was designed to receive light directly from a dedicated focus of the TNG. The instrument was provisionally commissioned in 2012 and was used in the GIANO-A configuration. In this position, the spectrometer was positioned on the rotating building and fed through a pair of fibers that were connected to another focal station [@tozzi14]. The spectrometer was eventually moved to the originally foreseen configuration in 2016. This configuration is called GIANO-B [@tozzi16]. It can here also be used in the GIARPS mode for simultaneous observations with HARPS-N.
GIANO-B provides a fully automated online data reduction pipeline based on the GOFIO reduction software [@gofio] that processes all the observed data, from the calibrations (darks, flats, and U-Ne lamps taken in daytime) to the scientific frames. The main feature of the GOFIO data reduction is the optimal spectral extraction and wavelength calibration based on a physical model of the spectrometer that accurately matches instrumental effects such as variable slit tilt and order curvature over the echellogram [@giano_2Dreduction]. The data reduction package also includes bad pixel and cosmic removal, sky and dark subtraction, and flat-field and blaze correction. It outputs the scientific data with different formats, including merged and non-merged echelle orders.
The GIANO guiding system uses the 850-950 nm light of the target itself as reference. When reddening is huge, as in the Scutum complex, the limiting factor therefore becomes the z-band magnitude of the target. For observations with GIANO-B we therefore selected those RSGs from the compilations of @neg11 and @gon12 with typically J$<$10 and A$_V<$16, that is bright enough for both the spectrograph and the guiding camera.
Seven RSGs toward Alicante 7 and four RSGs toward Alicante 10 (see Table \[tab1\]) have been observed with GIANO-B on June-July 2018. For the best subtraction of the detector artifacts and background, the spectra were collected by nodding the star along the slit, that is, with the target alternatively positioned at 1/4 (position A) and 3/4 (position B) of the slit length. Integration time was 5 minutes per A,B position. The nodding sequences were repeated to achieve a total integration time of 40 minutes per target.
The spectra were reduced using the offline version of GOFIO (available at <https://atreides.tng.iac.es/monica.rainer/gofio>).
The telluric absorption features were corrected using the spectra of a telluric standard (O-type star) taken at different airmasses during the same nights. The normalized spectra of the telluric standard taken at low and high airmass values were combined with different weights to match the depth of the telluric lines in the stellar spectra.
Spectral analysis
=================
Accurate (better than 1 km/s) RVs and chemical abundances for the observed RSGs have been obtained by comparing the 1D GIANO-B spectra with suitable synthetic templates. This was previously done in @ori13 [@ori16] for RSGs in RSGC2 and RSGC3.
In order to compute the required grid of synthetic spectra to model the observed RSGs, we used an updated version [@ori02] of the code that was first described in @ori93. The code uses the LTE approximation and the MARCS model atmospheres [@gus08]. Thousands of NIR atomic transitions from the Kurucz database[^2], from @bie73 and @mel99, and molecular data from our [@ori93 and subsequent updates] compilation and from the compilation of B. Plez (private communication) are included. From this database, a comprehensive list of suitable lines for each chemical element that can be measured in the RSG spectra, free from significant blending and without strong wings, was extracted [see also @ori13] and used to compute equivalent widths. We used the @gre98 abundances for the solar reference.
Hyperfine structure splitting was accounted for in the Ni, Sc, and Cu line profile computations. Including them does not significantly affect the abundance estimates, however.
Stellar parameters
------------------
![Portion of the normalized telluric corrected spectra, showing the prominent DIB feature at 1317.8 nm. The dashed purple line is the telluric correction applied to the data.[]{data-label="figure_DIB"}](plot_DIB_AL.jpg){width="\hsize"}
In RSG stars, the strength of the OH and CN molecular lines is especially sensitive to temperature variations. Moreover, CN lines are also sensitive to gravity, and OH lines are sensitive to microturbulence. While the shape and broadening of the $^{12}$CO bandheads mostly depend on microturbulence, their strengths is a quite sensitive thermometer in the 3800-4500 K range, where temperature sets the fraction of molecular versus atomic carbon. At temperatures below 3800 K most of the carbon is in molecular form, which drastically reduces the dependence of the CO band strengths on the temperature itself. At temperatures $\ge$4500 K, CO molecules barely survive, most of the carbon is in atomic form, and the CO spectral features become very weak. A simultaneous reasonable fit of all the molecular lines and bandheads can be obtained only in a narrow range of temperature, gravity, and microturbulence, regardless of the adopted CNO abundances.
Moreover, although significantly different temperatures can be inferred using different scales [@lev05; @dav13], synthetic spectra with temperatures that are significantly different (especially significantly warmer) than those providing the best-fit solutions can barely fit the observed spectra and require very peculiar (unlikely) CNO abundance patterns [see also @ori13; @ori16].
As has been discussed in @ori13 [@ori16], at the GIANO spectral resolution of 50,000, we find that in RSGs variations of $\pm$100K in T$_{eff}$, $\pm$0.5 dex in log g, and $\pm$0.5 km/s in microturbulence have effects on the spectral lines that can also be appreciated by some visual inspection. Variations in temperature, gravity, and microturbulence smaller than the above values are difficult to distinguish because the sensitivity of the lines is limited and the stellar parameters are degenerate. Moreover, such a finer tuning would have negligible effect on the inferred abundances (less than a few hundredths of a dex, i.e., smaller than the measurement errors). The effect of using different assumptions for the stellar parameters on the derived abundances is discussed in Sect. \[abun\].
Chemical abundances {#abun}
-------------------
Chemical abundances were derived by minimizing the scatter between observed and synthetic spectra with suitable stellar parameters, by using spectral synthesis and, as figures of merit, line equivalent widths and statistical tests on both the difference between the model and the observed spectrum and on the $\chi^2$. As discussed in @ori13 and references therein, the simple difference is more powerful for quantifying systematic discrepancies than the classical $\chi ^2$ test, which is instead equally sensitive to [**]{} and [**]{} deviations.
The typical random error of the measured line equivalent widths is $<$10 mÅ. It mostly arises from a $\pm $1-2% uncertainty in the placement of the pseudo-continuum, as estimated by overlapping the synthetic and observed spectra. This error corresponds to abundance variations of $<$0.1 dex, that is, smaller than the typical 1$\sigma $ scatter ($<$0.15 dex) in the derived abundances from different lines. The errors quoted in Table \[tab2\] for the final abundances were obtained by dividing the 1$\sigma $ scatter by the square root of the number of lines we used, typically from a few to a few tens per species. When only one line was available, we assumed an error of 0.10 dex.
As detailed in @ori13, a somewhat conservative estimate of the overall systematic uncertainty in the abundance (X) determination, caused by variations of the atmospheric parameters, can be computed with the following formula: $\rm (\Delta X)^2 = (\partial X/\partial T)^2 (\Delta T)^2 + (\partial X/\partial log~g)^2 (\Delta log~g)^2 + (\partial X/\partial \xi)^2 (\Delta \xi)^2$. We computed test models with variations of $\pm $100 K in temperature, $\pm$0.5 dex in log g, and $\pm $0.5 km/s in microturbulence velocity with respect to the best-fit parameters. We found that these systematic uncertainties in stellar parameters can affect the overall abundances at a level of 0.15-0.20 dex.
Diffuse interstellar bands: extinction and contamination
--------------------------------------------------------
{width="\hsize"}
Element$^a$
-------------------------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ -- ------------ ------------ ------------ ------------
F1S02 F1S04 F1S05 F1S06 F1S07 F1S08 F1S09 C06 C09 C10 N03
Fe(26) -0.18(23) -0.12(17) -0.13(17) -0.23(30) -0.21(23) -0.28(30) -0.12(24) -0.18(15) -0.23(20) -0.28(24) -0.17(21)
$\pm $0.04 $\pm $0.05 $\pm $0.05 $\pm $0.04 $\pm $0.04 $\pm $0.04 $\pm $0.05 $\pm $0.05 $\pm $0.05 $\pm $0.05 $\pm $0.04
C(6) -0.46(8) -0.45(7) -0.50(9) -0.52(9) -0.43(9) -0.49(9) -0.42(8) -0.46(7) -0.49(9) -0.45(8) -0.53(9)
$\pm $0.07 $\pm $0.04 $\pm $0.06 $\pm $0.05 $\pm $0.05 $\pm $0.04 $\pm $0.05 $\pm $0.07 $\pm $0.07 $\pm $0.04 $\pm $0.05
N(7) 0.18 (48) 0.25(44) 0.27(16) 0.20(51) 0.06(74) 0.07(61) 0.27(32) 0.19(37) 0.16(50 0.13(67) 0.04(72)
$\pm $0.03 $\pm $0.03 $\pm $0.04 $\pm $0.03 $\pm $0.02 $\pm $0.03 $\pm $0.04 $\pm $0.03 $\pm $0.03 $\pm $0.03 $\pm $0.02
O(8) -0.20 (19) -0.12(16) -0.20(15) -0.25(17) -0.24(19) -0.28(11) -0.15(15) -0.24(16) -0.19(16) -0.27(17) -0.22(11)
$\pm $0.04 $\pm $0.03 $\pm $0.05 $\pm $0.04 $\pm $0.05 $\pm $0.06 $\pm $0.04 $\pm $0.04 $\pm $0.06 $\pm $0.04 $\pm $0.05
F$^c$(9) -0.24 (1) -0.14(1) -0.29(1) -0.20(1) -0.23(1) -0.33(1) -0.11(1) -0.17(1) -0.09(1) -0.22(1) -0.11(1)
$\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10
Na(11) -0.07(1) -0.11(1) -0.15(1) -0.14(1) -0.17(1) -0.24(1) -0.17(1) -0.15(1) -0.14(1) -0.06(2) -0.11(1)
$\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.16 $\pm $0.10
Mg(12) -0.18(6) -0.08(3) -0.10(3) -0.18(5) -0.17(7) -0.27(8) -0.11(3) -0.12(3) -0.20(4) -0.26(3) -0.17(3)
$\pm $0.06 $\pm $0.13 $\pm $0.09 $\pm $0.11 $\pm $0.11 $\pm $0.06 $\pm $0.12 $\pm $0.11 $\pm $0.15 $\pm $0.04 $\pm $0.03
Al(13) -0.14(4) -0.19(4) -0.14(3) -0.23(5) -0.23(5) -0.32(4) -0.18(3) -0.15(4) -0.25(4) -0.22(4) -0.19(4)
$\pm $0.07 $\pm $0.02 $\pm $0.06 $\pm $0.04 $\pm $0.05 $\pm $0.04 $\pm $0.02 $\pm $0.02 $\pm $0.04 $\pm $0.02 $\pm $0.03
Si(14) -0.17(14) -0.13(11) -0.09(4) -0.18(14) -0.20(19) -0.22(21) -0.11(10) -0.10(5) -0.20(12) -0.18(10) -0.20(13)
$\pm $0.05 $\pm $0.07 $\pm $0.11 $\pm $0.05 $\pm $0.04 $\pm $0.04 $\pm $0.06 $\pm $0.08 $\pm $0.04 $\pm $0.04 $\pm $0.04
P(15) -0.10(2) -0.16(2) -0.17(1) -0.09(2) -0.20(1) -0.22(1) -0.18(1) -0.17(1) -0.07(1) -0.14(2) -0.19(2)
$\pm $0.07 $\pm $0.08 $\pm $0.10 $\pm $0.07 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.02 $\pm $0.02
S(16) -0.11(1) -0.11(1) -0.08(1) -0.18(1) -0.29(1) -0.24(1) -0.12(1) -0.15(1) -0.26(1) -0.28(1) -0.16(1)
$\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10
K(19) -0.03(1) -0.05(2) -0.10(2) -0.05(2) -0.20(2) -0.17(2) -0.08(1) – -0.00(1) -0.24(2) –
$\pm $0.10 $\pm $0.09 $\pm $0.06 $\pm $0.02 $\pm $0.08 $\pm $0.04 $\pm $0.10 $\pm $0.10 $\pm $0.03
Ca(20) -0.21(7) -0.17(8) -0.14(7) -0.26(8) -0.26(8) -0.32(6) -0.15(7) -0.15(5) -0.21(5) -0.21(8) -0.17(5)
$\pm $0.04 $\pm $0.04 $\pm $0.06 $\pm $0.06 $\pm $0.03 $\pm $0.04 $\pm $0.06 $\pm $0.09 $\pm $0.09 $\pm $0.06 $\pm $0.04
Sc(21) -0.15(2) -0.14(2) -0.24(2) -0.27(2) -0.18(4) -0.30(4) -0.10(2) -0.16(2) -0.21(4) -0.31(4) -0.11(1)
$\pm $0.05 $\pm $0.05 $\pm $0.10 $\pm $0.07 $\pm $0.05 $\pm $0.06 $\pm $0.03 $\pm $0.06 $\pm $0.07 $\pm $0.06 $\pm $0.10
Ti(22) -0.18(20) -0.14(17) -0.14(20) -0.18(19) -0.24(24) -0.24(23) -0.17(15) -0.14(18) -0.26(18) -0.26(22) -0.21(17)
$\pm $0.05 $\pm $0.05 $\pm $0.05 $\pm $0.05 $\pm $0.05 $\pm $0.04 $\pm $0.06 $\pm $0.06 $\pm $0.05 $\pm $0.05 $\pm $0.05
V(23) -0.18(1) -0.13(1) -0.18(1) -0.27(1) -0.27(1) -0.30(1) -0.15(1) -0.17(1) -0.26(1) -0.25(1) -0.17(1)
$\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10
Cr(24) -0.18(2) -0.15(4) -0.15(2) -0.16(4) -0.20(3) -0.22(10) -0.14(2) -0.20(2) -0.27(2) -0.30(2) -0.22(2)
$\pm $0.00 $\pm $0.01 $\pm $0.05 $\pm $0.05 $\pm $0.05 $\pm $0.06 $\pm $0.02 $\pm $0.00 $\pm $0.00 $\pm $0.03 $\pm $0.09
Co(27) -0.12(1) -0.09(1) -0.04(1) -0.23(1) -0.13(1) -0.29(1) -0.08(1) -0.11(1) -0.21(1) -0.23(1) -0.22(1)
$\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.10
Ni(28) -0.21(2) -0.13(4) -0.14(3) -0.24(2) -0.18(2) -0.27(3) -0.19(3) -0.20(3) -0.28(2) -0.26(2) -0.20(3)
$\pm $0.02 $\pm $0.11 $\pm $0.04 $\pm $0.00 $\pm $0.05 $\pm $0.07 $\pm $0.05 $\pm $0.11 $\pm $0.02 $\pm $0.00 $\pm $0.05
Cu(29) -0.19(2) -0.18(2) -0.17(2) -0.32(2) -0.26(2) -0.27(2) -0.19(2) -0.22(2) -0.26(2) -0.26(2) -0.24(2)
$\pm $0.03 $\pm $0.06 $\pm $0.05 $\pm $0.06 $\pm $0.04 $\pm $0.03 $\pm $0.05 $\pm $0.12 $\pm $0.02 $\pm $0.09 $\pm $0.01
Y(39) -0.29(5) -0.15(4) -0.28(3) -0.30(3) -0.22(4) -0.33(5) -0.16(4) -0.15(4) -0.23(5) -0.26(4) -0.19(5)
$\pm $0.04 $\pm $0.03 $\pm $0.01 $\pm $0.05 $\pm $0.05 $\pm $0.03 $\pm $0.04 $\pm $0.03 $\pm $0.04 $\pm $0.02 $\pm $0.04
Zr(40) -0.18(2) -0.14(1) -0.11(1) -0.40(1) – – -0.17(2) -0.20(1) -0.31(1) – -0.18(2)
$\pm $0.01 $\pm $0.10 $\pm $0.10 $\pm $0.10 $\pm $0.24 $\pm $0.10 $\pm $0.10 $\pm $0.10
$\rm ^{12}C/^{13}C$ $^d$ 10 13 13 11 13 11 13 12 12 11 9
Notes:\
[**$^a$**]{} Chemical element and corresponding atomic number in parentheses.
[**$^b$**]{} Numbers in parentheses refer to the number of lines used to compute the abundances.
[**$^c$**]{} F abundances are obtained from the HF(1-0) R9 line using the parameters listed in @jon14.
[**$^d$**]{} Typical error in the $\rm ^{12}C/^{13}C$ isotopic ratio is $\pm$1.
The observed stars are affected by a large and differential foreground interstellar absorption. Therefore, the diffuse interstellar band (DIB) absorption features [@DIB_Geballe; @DIB_Cox; @DIB_Hamano] are expected to be visible in our spectra. These broad features may be conveniently used as an independent estimate of the foreground extinction within the limits related to the quite large scatter of the feature strength for stars with similar extinction.
The most prominent DIB feature among those reported in the above references falls at $\lambda$1317.8 nm. At this wavelength the RSG spectra are remarkably free of photospheric lines. This DIB feature can therefore easily be recognized and accurately measured in our spectra (see Figure \[figure\_DIB\]). The measured equivalent widths are reported in Table \[tab1\], together with the corresponding estimate of visual extinction ($A_V$), based on the calibration with E(B-V) reported in @ham15 [@DIB_Hamano]. We find $A_V$ values in the 10-16 mag range, which is fully consistent with the corresponding $E(J-K)$ values quoted by @neg11 and @gon12.
The other DIB features that can be directly measured in our spectra are $\lambda$1078.3, $\lambda$1262.6 and $\lambda$1280.2. They are much weaker, and their relative strengths are compatible with those reported in [@DIB_Hamano].
The other DIB features are much more difficult to recognize in our spectra because they fall at wavelengths where the spectra are crowded with photospheric lines of atomic and molecular species. To measure these lines would require a simultaneous modeling and fitting of the stellar and DIB spectra. This task is well beyond the aims of this paper. Nonetheless, the photospheric lines close to the DIB features should be used carefully. In particular, the region around the strong DIB feature at $\lambda$1527.3 nm should be avoided, whose equivalent width is expected to be similar to that of the feature at $\lambda$1317.8 [@DIB_Cox].
Results
=======
The best-fit values of the stellar parameters and RVs for the observed RSGs in Alicante 7 and Alicante 10 are reported in Table \[tab1\]. We find temperatures in the 3500-3700 K range and microturbulence of 3-4 km/s.
For all the observed stars we adopted a surface gravity of log g=0.0. For the RSGs in RSGC2 and RSGC3 [@ori13; @ori16], the line profiles of the observed stars in Alicante 7 and Alicante 10 are also definitely broader than the instrumental line profile (as determined from telluric lines and from the GIANO-B spectra of standard stars). This additional broadening is likely due to macroturbulence and is normally modeled with a Gaussian profile, as for the instrumental broadening. All the analyzed RSGs show macroturbulence velocity of 9-10 km/s, which is well within the range of values measured in the RSGs of the Galactic center [see, e.g. @ram00; @cun07; @davies09a]. We did not find other appreciable line broadening by stellar rotation.
The derived values of $A_V$ from the DIB feature at 1317.8 nm are consistent with the high extinction toward the Scutum complex for all but star AL10-N03 (see Table \[tab1\]), which has a much weaker DIB feature and likely much lower extinction. Notably, this star also has a significantly lower RV than the other stars in Alicante 10.
The RVs of all the other stars measured in Alicante 7 and Alicante 10 are consistent with them being members of the Scutum complex, although they show lower values on average than the average value of RV$_{\rm hel}$=90 km/s with a dispersion of 2.3 km/s that is measured in the parent cluster RSGC3 [@ori16]. Only the heliocentric RV of star AL07-F1S04 exceeds 100 km/s.
![Derived \[X/H\] chemical abundances and corresponding errors for the observed RSGs candidate members of the Alicante 7 (seven stars) and Alicante 10 (three stars) associations. Dotted lines mark the average \[Fe/H\] values in each association.[]{data-label="figabun"}](figabun.jpg){width="\hsize"}
![Derived average \[X/H\] chemical abundances and corresponding errors for the observed RSGs, candidate members of the Alicante 7 and Alicante 10 associations. Dotted lines mark the average \[Fe/H\] values in each association.[]{data-label="meanabun"}](meanabun.jpg){width="\hsize"}
![Derived average \[X/H\] chemical abundances and corresponding errors for the observed RSGs, candidate members of the Alicante 7 and Alicante 10 associations.[]{data-label="meanratio"}](meanratio.jpg){width="\hsize"}
Proper motions for all the observed RSGs in Alicante 7 and Alicante 10 were measured from data of Gaia DR2 [@gaia] and are plotted in Fig. \[figpm\]. The RVs and PMs of the RSGs in Alicante 7 are consistent with them being members of the association within the errors. The only possible exception is F1S09, whose proper motions are barely consistent with a membership at $\approx3-4\sigma$ level, while its RV is well consistent. In Alicante 10 the three stars whose RVs are consistent with membership also have consistent proper motions, while the RV of N03 clearly is too low, the proper motions are too different, and the extinction is too low for it to likely be member of the association. In the following we only consider as member RSGs of the Alicante 10 association stars C06, C09, and C10.
For all the observed stars, Table \[tab2\] lists the derived chemical abundances and their associated errors of Fe and other iron-peak elements (V, Cr, Co, Ni, and Cu), CNO, and the other alpha elements (Mg, Si, Ca, and Ti), other light (F, Na, Al, P, S, K, and Sc), and a few neutron-capture (Y, Zr) elements. The atomic lines of V and Cu are heavily blended, therefore their derived abundances should be taken with caution. The few measurable Mn and SrII lines in the GIANO-B spectra are too strong or saturated to derive reliable abundances. While for most of the metals abundances were derived from the measurements of atomic lines, for the CNO elements they were obtained from the many molecular lines of CO, OH, and CN in the NIR spectra. $^{12}$C and $^{13}$C carbon abundances were determined both from a few individual roto-vibrational lines as well as from bandheads, because of the high level of crowding and blending of the CO lines in these stars. We found fully consistent results with the two methods. The fluorine abundance was determined from the HF(1-0) R9 line using the revised log $gf$ and excitation potential values of @jon14. A few other HF lines are present in the K-band spectrum of RSGs, but they were to too strongly blended or too faint to provide reliable abundance estimates.
Figure \[figabun\] shows the chemical abundances of the individual RSGs that are likely members of the Alicante 7 (seven stars) and Alicante 10 (three stars) associations. Figures \[meanabun\] and \[meanratio\] show the mean \[X/H\] abundances and \[X/Fe\] abundance ratios for all the sampled chemical elements, as computed by averaging the corresponding values for the seven RSGs in Alicante 7 and the three RSGs in Alicante 10 that are likely members of these associations, according to the measured RVs and high extinction.
Average values of \[Fe/H\]=-0.18 and -0.215 dex and corresponding 1$\sigma$ scatter of 0.06 and 0.05 dex for the RSGs that are likely members of Alicante 7 and Alicante 10, respectively, were measured. All the other chemical elements show about solar-scaled \[X/Fe\] abundance ratios within $\pm$0.1 dex, with the only exception of C and N, which show a depletion of a few tenths of a dex for carbon and and enhancement for nitrogen. The $\rm ^{12}C/^{13}C$ isotopic ratios between 9 and 13 were also measured.
Discussion and conclusions
==========================
For the seven RSGs in Alicante 7 and for three of the four RSGs of Alicante 10, the measured LSR RVs in the 79-123 km/s range suggest a Galactocentric radius of $\approx$4$\pm0.5$ kpc and kinematic near [see, e.g., @rom09] distances in the 5-7 kpc range, suggesting that they are likely located beyond the molecular ring and likely members of the Scutum complex. This is also consistent with the corresponding high extinction measured in their direction (see Table \[tab1\]). For N03 toward Alicante 10 with an RV$_{LSR}$ of 33 km/s, Galactocentric radii of $\approx 6$ kpc and a near distance of $\approx2.3$ kpc can be derived. This suggests that N03 might be located in front of the molecular ring, which also agrees with its significantly lower extinction when compared to the values inferred in the Scutum complex.
All the metals measured in the RSGs of Alicante 7 and Alicante 10 show slightly subsolar abundances and about solar-scale abundance ratios. The only remarkable exceptions are the depletion of C, the enhancement of N, and the relatively low (in the 9-13 range) $\rm ^{12}C/^{13}C$ isotopic ratio, which are consistent with CN burning and other possible additional mixing processes that occur in the post-MS stages of stellar evolution. It is worth mentioning that the inferred \[C/N\] abundance ratios in the $\rm -0.77\ge[C/N]\le-0.49$ range are also consistent with the surface values predicted by stellar evolution calculations for RSGs at the end of their lives [e.g., @dav19]. The rather homogeneous chemistry of Alicante 7 and Alicante 10 is also fully consistent with the abundances measured in a few RSGs of the parent RSGC3 cluster [@ori16]. A more comprehensive chemical and kinematic study of the latter and the overall link with its associations will be presented in a forthcoming paper.
As discussed in @ori16, the slightly subsolar metallicity measured in regions of recent star formation in the Scutum complex is intriguing because metal abundances well in excess of solar have been measured in the thin disk at larger Galactocentric distances [see, e.g., @gen13 and references therein] and were also predicted by the inside-out formation scenario for the Galactic disk [@cesc07]. Interestingly enough, the likely foreground RSG AL10-N03 also has slightly subsolar metallicity as well as pre-MS clusters at Galactocentric distances from 6.7 to 8.7 kpc that were studied by @spi17. The innermost of these clusters are 0.10-0.15 dex less metallic than the majority of the older clusters that are located at similar Galactocentric radii.
This may suggest that recent star formation in the inner disk may have occurred from a gas that was not as enriched as expected. One possible explanation is dilution by metal-poor halo gas driven there by dynamical interactions of the disk with other galactic substructures such as bars, rings, and spiral arms.
A precise description of how this dynamical process could work is beyond the scope of this paper, but our observations have indeed pointed out another complexity in the evolution and chemical enrichment of the inner Galaxy that is to be addressed by future studies. At the completion of the SPA Large Programme, when a more comprehensive chemical and kinematic mapping of the Scutum complex and possibly of other regions of recent star formation will become available, this hypothesis can be verified on more comprehensive grounds.
We thank C. Evans, the referee of our paper, for his useful comments and suggestions.
Alexander, M. J., Kobulnicky, H. A., Clemens, D. P., Jameson, K., Pinnick, A., & Pavel, M., 2009, AJ, 137, 4824 Biemont, E., & Grevesse, N. 1973, [*Atomic Data and Nuclear Data tables*]{}, 12, 221 Cescutti, G., Matteucci, F., François, P., & Chiappini, C. 2007, A&A, 462, 943A&A, 462, 943 Clark, J. S., Davies, B., Najarro, F., MacKenty, J., Crowther, P. A., Messineo, M., & Thompson, M. A., 2009, A&A, 504, 429 Cox, N. L. J.; Cami, J.; Kaper, L.; Ehrenfreund, P.; Foing, B. H.; Ochsendorf, B. B.; van Hooff, S. H. M.; Salama, F.; 2014, A&A, 569, 117 Cunha, K., Sellgren, K., Smith, V.V., Ramirez, S.V., Blum, R.D., & Terndrup, D.M. 2007, ApJ, 669, 1011 Davies, B., Figer, D. F., Kudritzki, R.-P.,MacKenty, J., Najarro, F., & Herrero, A., 2007, ApJ, 671, 781 Davies, B., Figer, D. F., Law, Casey, J., Kudritzki, R.-P., Najarro, F., Herrero, A., & MacKenty, J. W., 2008, ApJ, 676, 1016 da Silva, R.; Lemasle, B.; Bono, G., et al., 2016, A&A, 586, 125 Davies, B., Origlia, L., Kudritzki, R., Figer, D.F., Rich, R.M., & Najarro, F. 2009a, ApJ, 694, 46 Davies, B., Origlia, L. Kudritzki, R.-P., Figer, D. F., Rich, R. N., Najarro, F., Negueruela, I., & Clark, J. Simon, 2009b, ApJ, 696, 2014 Davies, B., Kudritzki, R., Plez, B., Trager, S., Lancon, A., Gazak, Z., Bergemann, M.i, Evans, C., & Chaivassa, A., 2013, ApJ, 767, 3 Davies, B., & Dessart, L., 2019, MNRAS, 483, 887 Figer, D. F., MacKenty, J. W., Robberto, M., Smith, K., Najarro, F., Kudritski, R.-P., & Herrero, A., 2006, ApJ, 643, 1166 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, , 616, A1 Francois, P., Inno, L., Laney, C.D., Matsunaga, N., Pedicelli, S., & Thevenin, F. 2013, A&A, 554, 132 Geballe, T. R.; Najarro, F.; Genovali, K., Lemasle, B., Bono, G., Romaniello, M., Primas, F., Fabrizio, M., Buonanno, R., Figer, D. F.; Schlegelmilch, B. W.; de La Fuente, D.; Nature, 479,7372 Gonzalez-Fernandez, C., & Negueruela, I., 2012, A&A, 539, 100 Grevesse, N. & Sauval, A.J. 1998, Space Sci. Rev., 85, 161 Gustafsson B., Edvardsson B., Eriksson K., Joergensen U.G., Nordlund A., Plez B., 2008, A&A, 486, 951 Hamano, S., Kobayashi, N., Kondo, S., et al., 2015, ApJ, 800, 137 Hamano, S., Kobayashi, N., Kondo, S., Sameshima, H., Nakanishi, K., Ikeda, Y., Yasui, C., Mizumoto, M., Matsunaga, N., Fukue, K., Yamamoto, R., Izumi, N., Mito, H., Nakaoka, T., Kawanishi, T., Kitano, A., Otsubo, S., Kinoshita, M., Kawakita, H., 2016, ApJ, 821, 42 Jönsson, H. et al. 2014, A&A, 564, 122 Levesque, E. M., Massey, P., Olsen, K. A. G., et al. 2005, ApJ, 628, 973 Melendez, J., & Barbuy, B. 1999, ApJS 124, 527 Negueruela, I., Gonzalez-Fernandez, C. , Marco, A., Clark, J. S., & Martinez-Nunez, S., 2010, A&A, 513, 74 Negueruela, I., Gonzalez-Fernandez, C. , Marco, A., & Clark, J. S., 2011, A&A, 528, 59 Oliva, E.; Origlia, L.; Maiolino, R., et al. 2012a, SPIE 8446E, 3TO Oliva, E., Biliotti, V., Baffa, C., et al. 2012b, SPIE 8453E, 2TO Oliva, E. et al., A&A, 555, 78 Oliva, E., Sanna, N., Rainer, M., Massi, F., Tozzi, A., Origlia, L. 2018, SPIE, 1070274 Origlia, L., Moorwood, A.F.M., & Oliva, E. 1993, A&A, 280, 536 Origlia, L., Rich, R.M., & Castro, S. 2002, AJ, 123, 1559 Origlia, L. et al. 2013, A&A, 560, 46 Origlia, L., Oliva, E., Baffa, C., et al. 2014, SPIE, 914461E, 1EO Origlia, L. et al. 2016, A&A, 585, 14 Rainer, M., Harutyunyan, A., Carleo, I., et al. 2018, SPIE, 1070266 Ramírez, R. V., Sellgren, K., Carr, J. S., Balachandran, S., C., Blum, R., Terndrup, D. M., & Steed, A., 2000, AJ, 537, 205 Roman-Duval, G., Jackson, J. M., Heyer, M., Johnson, A., Rathborne, J., Shah, R., & Simon, R, 2009, ApJ, 699, 1153 Sch"onrich, R., Binney, J., & Dehnen, W. 2010, MNRAS, 403, 1829 Spina, L., et al. 2017, A&A, 601, 70 Tozzi, A., Oliva, E., Origlia, L., et al. 2014, SPIE, 91479N Tozzi, A., Oliva, E., Iuzzolino, M., et al. 2016, SPIE, 99086C
[^1]: Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. This study is part of the Large Program titled [*SPA - Stellar Population Astrophysics: the detailed, age-resolved chemistry of the Milky Way disk*]{} (PI: L. Origlia), granted observing time with HARPS-N and GIANO-B echelle spectrographs at the TNG.
[^2]: http://www.cfa.harvard.edu/amp/ampdata/kurucz23/sekur.html
|
---
abstract: 'We show that an information theoretic distance measured by the relative Fisher information between canonical equilibrium phase densities corresponding to forward and backward processes is intimately related to the gradient of the dissipated work in phase space. We present a universal constraint on it via the logarithmic Sobolev inequality. Furthermore, we point out that a possible expression of the lower bound indicates a deep connection in terms of the relative entropy and the Fisher information of the canonical distributions.'
author:
- '[Takuya Yamano]{}'
title: 'Phase space gradient of dissipated work and information: A role of relative Fisher information'
---
Introduction
============
Phase space gradient of thermodynamic quantities in a nonequilibrium process and those limitations by information theoretic ones should bring fundamental insights into the understandings of the system. Among others, dissipation and work are central quantities in thermodynamic operation between equilibrium states. The connecting path between the two distinct equilibrium states in phase space has a diversified range depending on the steps of the procedure and it determines the amount of the mechanical work needed to perform the process. It is well recognized that as long as the process is not quasistatic, the free energy at the end of the process becomes less than that of the initial state plus the work invested from outside [@Landau]. In other words, there is dissipative loss of work into the surroundings of the system – the second law of thermodynamics.\
The dissipated work is thus an indicator of the excess of the injected work into the system, and it is defined in terms of the difference in the equilibrium free energy $\Delta F$ as $W_{diss}(\Gamma,\lambda)= W(\Gamma,\lambda)-\Delta F$, where $\lambda$ is a protocol parameter. $W(\Gamma,\lambda)$ denotes the average work done on the system by the external operator as perturbation and it is a function of the position in phase space. The dissipated work also represents the total change in entropy of the system as a result of the operated transition. The free energy difference between terminal states is also called the reversible work. Instead of taking one instance in the definition, we consider a statistical ensemble of realizations of the process as the consequence of infinitely many repetitions of this process. Therefore, we refer to the dissipated work in the sense of mean $\langle W_{diss}(\Gamma,\lambda)\rangle$ throughout this paper, but for simplicity we omit the angular brackets in the following.\
Any displacement in phase space from a specific equilibrium state to another equilibrium phase point induced by a dynamics has a counterpart process by reversing time. The to-and-fro movement makes us to expect finding a universal relation in the dissipated work in an averaged way. One such intriguing example was discovered as a relation [@JE2; @Kawai] $$\begin{aligned}
\frac{\langle W_{diss} \rangle}{k T}= D_{KL}(\mathcal{P}_F\|\mathcal{P}_B)\label{eqn:Kawai}, \end{aligned}$$ under a sequence of procedure where the system is initially at canonical equilibrium with temperature $T$, and then it is detached from the heat reservoir to let the system evolve according to the Liouville equation and lastly equilibrate again to a new canonical state determined by the protocol parameter by attaching the same bath at the end of the process. The equality is replaced by inequality in case the system is kept contacting with the bath [@Kawai]. $\mathcal{P}_F$ and $\mathcal{P}_B$ are phase densities of the forward and the time-reversed processes at a particular time, respectively. Whether or not the prompt thermalization for each instance at any attained phase space point is realized remains a matter of careful thought, however, it is a standard device for consideration commonly used in the literature. The relation gives a meaning of a degree of time-reversal asymmetry measured by the Kullback-Leibler (KL) relative entropy $D_{KL}(\mathcal{P}_F\|\mathcal{P}_B)$ from $\mathcal{P}_F$ and $\mathcal{P}_B$ [@KL; @KL2]. The left-hand side of Eq.(\[eqn:Kawai\]) is a physical content in units of thermal energy with the Boltzmann constant $k$ and, on the other hand, the right-hand side represents purely information theoretic quantity.\
In this paper, based on the above consideration, we see how the gradient of the dissipated work in phase space is inextricably linked to an information theoretic distance between phase density functions of forward and backward processes. Specifically, we show that a novel constraint on the averaged square of the gradient of the dissipated work $\langle |\nabla W_{diss}|^2\rangle$ can be obtained via the relative Fisher information and the logarithmic Sobolev inequality. We further present a possible inequality expression indicating a deep connection in terms of two kinds of information quantities. These studies are in conformity with a modern approach that bases the construction of statistical thermodynamics upon the concept of information (e.g., [@Ben]).\
In Sec.II, we first present a general relation corresponding to Eq.(\[eqn:Kawai\]) when we employ an alternative to the KL relative entropy in order to recognize a distance notion in more general context in terms of information measure. The main ingredient of our consideration is introduced in Sec.III. We then show a constraint of the obtained relation with the inequality in Sec.IV. We give a concluding summary in Sec. V.
A general relation for dissipated work
======================================
Let $M^n$ be the $n$-dimensional configuration space of the present system. The two participating distributions $\mathcal{P}_F$ and $\mathcal{P}_B$ are compared at equal-time when calculating the general relative entropy, which is specified by a convex function $\chi(\mathcal{P}_F/\mathcal{P}_B)$ with $\chi(1)=0$, $$\begin{aligned}
D_{G}(\mathcal{P}_F\|\mathcal{P}_B):=\int \mathcal{P}_F
\chi\left(\frac{\mathcal{P}_F}{\mathcal{P}_B}\right)d\Gamma.\label{eqn:DG}\end{aligned}$$ This form of the general relative entropy is first introduced in [@Csi; @Morimoto] and it is now well recognized as the Csisz[á]{}r-Morimoto divergence (less known in physics, though). It was employed in the proof of the H-theorem for Markov processes [@Morimoto]. In general, we do not require the asymmetry in the arguments but assume positivity. Let $T^*M^n$ be the cotangent bundle to $M^n$, i.e., phase space. The system whose Hamiltonian is parameter dependent such as $H_{\lambda}(\Gamma_\lambda)$ starts its evolution from a phase space point $\Gamma_0\in T^*M^n$ at time $t_0$ and reaches another point $\Gamma_1\in T^*M^n$ at $t_1$. During this interval of time, the protocol parameter changes from $\lambda(t_0)=\lambda_0$ to $\lambda(t_1)=\lambda_1$. Initially, the system is put in the canonical equilibrium form $\mathcal{P}_F(\Gamma_0)=e^{-\beta H_{\lambda_0}(\Gamma_0)}/Z_0$ with $\beta=(k T)^{-1}$. Similarly, prior to the backward process the system is supposed to have a density function $\mathcal{P}_B(\Gamma_1)=e^{-\beta H_{\lambda_1}(\Gamma_1)}/Z_1$ by preparing the canonical form at the start of the return. The reverse protocol that goes from $\lambda_0$ to $\lambda_1$ corresponds to $\Gamma_1$ and $\Gamma_0$ in the phase space. The free energy difference between the initial and the final states is given by $\Delta F=-\beta^{-1}(\ln Z_1-\ln Z_0)$ with the partition functions $Z_0$ and $Z_1$ for each state. We note that the protocol path in $T^*M^n$ joining $\Gamma_0$ to $\Gamma_1$ that represents the operation does not necessarily proceed along the geodesic associated with it. The geodesic on the surface of the phase space manifold is a significant locus, however, the protocol given by an external agent or by an arbitrary schedule does not always go through the shortest path. Moreover, even if an operation for one realization takes the geodesic, other realizations do not trace exactly the same route at every time. They fluctuate around the geodesic path. That is why the work performed on the system is statistically distributed representing the ensemble of realizations and it is averaged over the ensemble.\
Let $C$ be a curve (path) on a surface of $M^n$ connecting $\Gamma_0$ and $\Gamma_1$. Then, the amount of work the system receives on completion of the process depends not only on the location of the two terminal points $\Gamma_0$ and $\Gamma_1$, but also on the route $C$. It should be precisely denoted as $W^C(\Gamma_0,\Gamma_1)$ and equals to the difference in the Hamiltonian $H_{\lambda_1}(\Gamma_1) - H_{\lambda_0}(\Gamma_0)=\int^{\lambda_1}_{\lambda_0}d\lambda
\partial H_\lambda(\Gamma_\lambda)/\partial \lambda$ (the first law of thermodynamics). For simplicity, we omit both the subscript and arguments, and we denote $W$ in the following.\
We note that for the well-defined distance $D_G$, the relative density $\mathcal{P}_F/\mathcal{P}_B$ must have a finite value on a support (that is, $\mathcal{P}_F$ is absolutely continuous with respect to $\mathcal{P}_B$), otherwise it is defined as infinity. Then, the relative phase density becomes $$\begin{aligned}
\frac{\mathcal{P}_F}{\mathcal{P}_B}=\frac{Z_1}{Z_0}
e^{-\beta\left(H_{\lambda_0}(\Gamma_0) - H_{\lambda_1}(\Gamma_1)\right)}
= e^{\beta (W-\Delta F)}.\label{eqn:reld}\end{aligned}$$ Inserting this into Eq.(\[eqn:DG\]), we obtain the general reformulation of Eq.(\[eqn:Kawai\]) as the following form $$\begin{aligned}
D_{G}(\mathcal{P}_F\|\mathcal{P}_B)=\Big\langle\chi(e^{\beta(W-\Delta F)})
\Big\rangle_{\mathcal{P}_F},\end{aligned}$$ where $\langle \cdot \rangle_{\mathcal{P}_F}$ denotes the average with respect to the distribution $\mathcal{P}_F$. This is the most general expression relating the dissipated work and the distance measure in this setting. We remark that the choice $\chi(x)=\ln x$ in Eq.(\[eqn:DG\]) with $x=\mathcal{P}_F/\mathcal{P}_B$ defines the KL distance and recovers exactly the result Eq.(\[eqn:Kawai\]). It is also worth mentioning another example. If we choose $\chi(x)=1/\sqrt{x}$, we have the overlap distance between $\mathcal{P}_F$ and $\mathcal{P}_B$, i.e., $D_O(\mathcal{P}_F\|\mathcal{P}_B):=\int \sqrt{\mathcal{P}_F\mathcal{P}_B} d\Gamma
= \langle \sqrt{\mathcal{P}_F/\mathcal{P}_B}\rangle_{\mathcal{P}_F}$. If two phase densities are identical, it gives unity. However, $\mathcal{P}_F$ and $\mathcal{P}_B$ can never be identical ($\mathcal{P}_F\neq \mathcal{P}_B$) in our consideration. Substituting Eq.(\[eqn:reld\]) stemming from the canonical forms into the definition of the overlap distance, we have $$\begin{aligned}
D_O(\mathcal{P}_F\|\mathcal{P}_B)&=&
\Big\langle e^{-\frac{\beta}{2}(W-\Delta F)} \Big\rangle_{\mathcal{P}_F}\nonumber\\
&=& e^{\frac{\beta}{2}\Delta F}\Big\langle e^{-\frac{\beta}{2}W}\Big\rangle_{\mathcal{P}_F}\nonumber\\
&\stackrel{\text{J.E.}}{=}& 0. \label{eqn:DO}\end{aligned}$$ The second line follows from the fact that the $\Delta F$ is not a statistically distributed quantity. The last line is due to the nonequilibrium work relation called the Jarzynski equality (J.E.) $\langle e^{-\beta W}\rangle=e^{-\beta \Delta F}$ [@JE1; @Crooks], where the angular brackets denotes an average over a statistical ensemble of realizations of a single process through phase space. It is known that for general circumstances including the Hamilton’s evolution, this average is equivalent to taking an average using the initial equilibrium distribution designated by $\lambda_0$ [@JE3]. Furthermore, the J.E. can be potentially useful in various equilibrium statistical physics, in which the estimation of the free energy differences is difficult to obtain. In these cases, such an equilibrium property is obtained from observations (experiments) of work distributions performed on the system. Note that distinct distributions in phase space do not share the common domain, so that Eq.(\[eqn:DO\]) means that the vanishing overlap between the two equilibrium states is consistent with the fact that the J.E. holds. On the other hand, the J.E. is irrelevant to the derivation of Eq.(\[eqn:Kawai\]), i.e. the case of the KL distance. We also remark that when the system realizes $W=\Delta F$ (reversible process), the overlap $D_o$ vanishes, meaning that it can be a measure of how much the nonequilibrium behavior deviates from the reversibility.
Relative Fisher information and dissipated work
===============================================
In Section II, we recognized that Eq.(\[eqn:Kawai\]) is just one realization of the information theoretic expression of the averaged dissipated work among other possibilities. The choice of $\chi$ is basically at our disposal and there is no legitimate criterion in terms of information theory. For operational purposes such as statistical inferences and estimation of measurement values, relative entropy with parameters can be considered advantageous over the one without them. Such a generalized distance measure with keeping the original properties of the Kullback-Leibler has been proposed [@TY2] (see also e.g., [@Special]). But we do not delve into such possibilities including the overlap distance in this paper. Instead, to advance the understanding in line with previous works, we hereafter set the gauge of the distance measure as KL in our consideration. As we pronounced in Introduction, we are concerned with the quantity $\nabla_{\Gamma} W_{diss}$, where $\nabla_{\Gamma}$ indicates the gradient in phase space, because the dissipated work depends on the location in phase space and its gradient reflects a local structure (geometry) of the dissipation occurred as an excess of work performed on the system.\
The relative Fisher information from $f$ to $g$ is defined as (e.g. [@Villani], p.278) $$\begin{aligned}
D_{\rm RFI}(f\|g):=\int f \Big| \nabla_\Gamma \left(\ln \frac{f}{g}\right)\Big|^2 d\Gamma.
\label{eqn:rFi}\end{aligned}$$ It is non-negative and achieves zero iff $f=g$. Further, it is asymmetric, i.e., $D_{\rm RFI}(f\|g)\neq D_{\rm RFI}(g\|f)$, meaning that it is [*directed*]{} as is the case of the KL distance. This form can be defined independently of whether the distributions possess estimation parameters or not. We can also employ it when the two distributions are parametrized by the same family (say, $\boldsymbol{\theta}$). We can say that this metric reflects a local comparison of the two distributions in that we take the derivative of them, whereas the KL distance returns a coarse-grained quantity by gathering each contribution of the displacement (or difference) between distributions. The same remark mentioned before Eq.(\[eqn:reld\]) also applies to this relative distance: the relative phase density must have finite value on the domain of phase space. To the best of our knowledge, there is no study to give a physical connection to that measure. Now, we show one of the applications below. That is, substituting Eq.(\[eqn:reld\]) into the above definition, we have a relation (see Appendix \[app:rfi\]) $$\begin{aligned}
D_{\rm RFI}(\mathcal{P}_F\|\mathcal{P}_B)=\beta^2\Big\langle \Big| \nabla_\Gamma
W_{diss}(\Gamma)\Big|^2 \Big\rangle_{\mathcal{P}_F}\label{eqn:reF1}.\end{aligned}$$ This identity articulates that an average of the square of the gradient of the dissipated work generated by the transient process, taken over a forward equilibrium state, can be equated with the distance between the forward and backward distributions in the phase space measured by the relative Fisher information.\
Our setting of formulation represented by an operational parameter is in a position of Hamilton equations (differential equations on symplectic manifolds). Then, in order to consider nonequilibrium processes in phase space perspective, it is principally necessary to deal with the trajectory on a manifold (either on symplectic or on Riemannian). It is known that paracompact $C^\infty$-manifolds always have their Riemannian metrics. Since the usual manifold is paracompact, the Riemannian metric is introduced if we regard the phase space as a manifold. Then, the metric expression of the above identity is of concern. To this end, we remark that a gradient vector of a differentiable function $f$ on the Riemannian manifold associated with the covector $df$ can be defined as the contravariant vector, which is given in coordinate as $(\nabla f)^i=\sum_j g^{ij}\partial f/\partial x^j$, where $x^i$ represents the generalized coordinates and the contravariant metric tensor $g^{ij}$ is the inverse of the metric tensor $g_{ij}$ ($(g^{ij})=(g_{ij})^{-1}$). We further recall that, for any vector ${\bf a}$, the differential of a function $df$ is defined by the derivative of $f$ along ${\bf a}$. Therefore, the gradient vector $\nabla f$ satisfies $df({\bf a})=\langle \nabla f, {\bf a} \rangle=\sum_i (\partial f/\partial x^i) a^i$. Then, substituting $(\nabla f)^i$ into this, we find that the square of the modulus of the gradient $f$ can be written as $|\nabla f|^2=df(\nabla f)=\sum_{ij}(\partial_{x^i}f)g^{ij}(\partial_{x^j}f)$. Since we now take $f=W_{diss}(\Gamma)$ in the present consideration, therefore, the right-hand side of Eq.(\[eqn:reF1\]) can be expressed as $$\begin{aligned}
\beta^2 \Big\langle \sum_{i j}\frac{\partial W_{diss}(\Gamma)}{\partial x^i}g^{ij}
\frac{\partial W_{diss}(\Gamma)}{\partial x^j} \Big\rangle_{\mathcal{P}_F}.\end{aligned}$$ We can obtain a more direct physical meaning for the quantity of the Dirichlet form $\langle | \nabla_\Gamma W_{diss}(\Gamma)|^2 \rangle_{\mathcal{P}_F}$ appearing in Eq.(\[eqn:reF1\]) via the gradient flow interpretation as we will see shortly. To this end, we recall again what $W$ exactly expresses. It means that the work done on the system when it starts from $\Gamma_0$ and reaches $\Gamma_1$ by changing the control parameter $\lambda\in [\lambda_0,\lambda_1]$ along a path $C$ on the surface of the phase space manifold. Since $\lambda$ designates the instance of the evolution from any starting point $\Gamma_0$, we rewrite it as $W_\lambda(\Gamma)$ by denoting $\Gamma_0$ as $\Gamma$ on a specific $C$. Accordingly, the dissipated work up to the intermediate value of $\lambda$ can be denoted as $W_{diss}(\Gamma_\lambda)$, where the phase point $\Gamma_\lambda\in T^*M^n$ should be read as $\hat{T}_\lambda\Gamma$ with an evolution operator $\hat{T}_\lambda$ that represents the protocol. Recall now that the gradient flow associated with the velocity vector field $-\nabla(\delta G/\delta \rho):=v(\Gamma_\lambda)$ is defined (Appendix \[app:gradf\]) as $$\begin{aligned}
\frac{\partial \rho}{\partial t}={\rm div} \left(\rho \nabla\frac{\delta G}{\delta \rho}\right),\label{eqn:gradf}\end{aligned}$$ where $G(\rho)$ is an energy functional. In our case, we can take it as $$\begin{aligned}
G(\rho)=\int W_{diss}(\Gamma_\lambda) \rho(\Gamma)d\Gamma.\end{aligned}$$ With abuse of notation above, we have used $\rho(\Gamma)$ as the probability density function in phase space at each time instant $t$. This is the phase space average of the dissipated work in our consideration and the first variation of it is $\delta G/\delta \rho= W_{diss}(\Gamma_\lambda)$.
Therefore, the average kinetic energy $K_\lambda$ accompanied to the dissipation up to the protocol $\lambda$ is found to be just the squared mean of the gradient of the dissipated work evaluated by the initial equilibrium state $\mathcal{P}_F$ that is prepared for the start of the process: $$\begin{aligned}
K_\lambda=\int |v(\Gamma_\lambda)|^2\mathcal{P}_F(\Gamma)d\Gamma
= \Big\langle | \nabla_\Gamma W_{diss}(\Gamma)|^2 \Big\rangle_{\mathcal{P}_F}.\end{aligned}$$ In the whole process, the total kinetic energy becomes $\int^{\lambda_1}_{\lambda_0}K_\lambda d\lambda$.
A lower bound for average dissipated-work gradient
==================================================
In this section, we show that our primarily focused quantity can be bounded by the KL distance via the logarithmic Sobolev inequality (LSI) [@Gross]. In this sense, the identity that connects the relative Fisher information and the quantity $\langle | \nabla_\Gamma W_{diss}(\Gamma)|^2 \rangle_{\mathcal{P}_F}$ derived in the previous section can be regarded as an intermediate result. The LSI has wide range of applications and it takes several mathematically equivalent forms [@Villani]. In this paper, we employ the form (Lemma 6.1 in [@Gross]) $$\begin{aligned}
\int |f|^2 \ln |f|d\mu \leqslant c \int |\nabla f|^2d\mu + \|f\|_2^2\ln \|f\|_2^2,\label{eqn:LSI}\end{aligned}$$ which holds for all functions $f$, whose gradient $\nabla f$ and $f$ itself are square integrable in the domain. $\|f\|_2$ denotes $(\int f^2 d\mu )^{1/2}$ and the constant $c>0$ independent of $f$. The physical interpretation of this constant is not obvious in general circumstances and would depend on the physical model. However, when an equilibrium state satisfies the LSI and the probability density function has the corresponding gradient flow, the constant has a clear meaning of how fast the system equilibrates with it. The corresponding relaxation rate takes the exponential form and it appears as a pre-factor to the KL distance between the initial and the equilibrium density functions. This feature is a direct consequence of the result provided in [@Barron] (see also e.g., [@Villani], p.288 and Appendix \[app:LSI\]). If we substitute $d\mu=\mathcal{P}_B(\Gamma)d\Gamma$ for the probability measure and choosing $f=\sqrt{\mathcal{P}_F/\mathcal{P}_B}$ then multiplying both sides by $2$, we find $$\begin{aligned}
\int \mathcal{P}_F \ln \frac{\mathcal{P}_F}{\mathcal{P}_B}d\Gamma & \leqslant &
2c\int \Big| \nabla \sqrt{\frac{\mathcal{P}_F}{\mathcal{P}_B}}\Big|^2 \mathcal{P}_B d\Gamma \nonumber\\
& = & \frac{c}{2} \int \mathcal{P}_F \Big| \nabla \left(\ln \frac{\mathcal{P}_F}
{\mathcal{P}_B}\right)\Big|^2d\Gamma.\label{eqn:LSI2}\end{aligned}$$ In what follows, we set $c=1$ without impeding our consideration. Combining Eq.(\[eqn:reF1\]) and Eq.(\[eqn:LSI2\]), we find that the mean of the gradient dissipated work is lower bounded by the information theoretic distance between forward and backward phase densities : $$\begin{aligned}
\langle | \nabla_\Gamma W_{diss}(\Gamma)|^2 \rangle_{\mathcal{P}_F} \geqslant
2(kT)^2 D_{KL}(\mathcal{P}_F \| \mathcal{P}_B).\label{eqn:NW}\end{aligned}$$ We further pursue a reformulation of thus obtained constraint from an information point of view, that is, in terms of Fisher information in statistics.\
We begin with recalling the followings. For a family of probability distributions $\rho_{\boldsymbol{\theta}}(\boldsymbol{x})$ parametrized by $\boldsymbol{\theta}$, the Fisher information in estimation theory is defined as $I(\boldsymbol{\theta}):=\langle (\nabla_{\boldsymbol{\theta}} \rho/\rho)^2\rangle_{{\rho}_
{\boldsymbol{\theta}}(\boldsymbol{x})}$. In the same way, it is defined also as $I(\rho(\boldsymbol{x})):=\langle (\nabla_{\boldsymbol{x}} \rho/\rho)^2\rangle_{{\rho}(\boldsymbol{x})}$ for a differentiable distribution $\rho(\boldsymbol{x})$, and it measures how much two neighboring density functions are statistically distinguishable [@Cover]. The former form is invariant against any shift $\boldsymbol{\theta}$, which signifies the diagonal entries of the Fisher information matrix, thereby implying i.i.d. data. Indeed, when we choose as $\rho_{\boldsymbol{\theta}}(\boldsymbol{x})=\rho(\boldsymbol{x}-\boldsymbol{\theta})$, we easily find that due to $\nabla_{\boldsymbol{x}}\rho(\boldsymbol{x}-\boldsymbol{\theta})=-\nabla_{\boldsymbol{\theta}}
\rho(\boldsymbol{x}-\boldsymbol{\theta})$, the Fisher information does not depend on $\boldsymbol{\theta}$ and $I(\boldsymbol{\theta})=I(\rho(\boldsymbol{x}))$ follows.\
There have been comprehensive efforts to understand various physical laws in terms of this information [@Frieden]. It also plays a crucial role to upper bound the entropy production (e.g. [@TY1] and references therein). Since the present canonical equilibrium distribution specified by a protocol parameter satisfies $\nabla\mathcal{P}(\Gamma)=-\beta H(\Gamma)\mathcal{P}(\Gamma)$, it becomes $$\begin{aligned}
I(\mathcal{P}) = \Big\langle \left(\frac{\nabla \mathcal{P}}{\mathcal{P}}\right)^2\Big\rangle_{\mathcal{P}}
= \int \frac{\nabla \mathcal{P}}{\mathcal{P}} (-\beta \nabla H(\Gamma)) \mathcal{P}d\Gamma.\nonumber\\\end{aligned}$$ Integrating by parts, the right-hand side reduces further to $$\begin{aligned}
-\beta \int \nabla H(\Gamma) \nabla \mathcal{P}(\Gamma)d\Gamma
=\beta \langle \Delta H(\Gamma) \rangle_{\mathcal{P}},\nonumber\end{aligned}$$ where we have assumed that the phase density vanishes at the boundary ($\partial S$) of the system, so that $[\mathcal{P}\nabla H(\Gamma)]_{\partial S}=0$, and $\Delta$ denotes the Laplacian in phase space. Therefore, we obtain a striking relation $$\begin{aligned}
I(\mathcal{P}) &=& \beta \langle \Delta H(\Gamma) \rangle_{\mathcal{P}},\label{eqn:Ftmp}\end{aligned}$$ which tells that the inverse temperature of the system can be intimately linked with the information quantity (Upon revision of this manuscript, the author became aware that this relation has derived also in Ref. [@Narayanan].). In general coordinates, we note that $\Delta H(\Gamma)$ can be expressed as $(\sqrt{g})^{-1}\partial_i(\sqrt{g}g^{ij}\partial_jH(\Gamma))$, where $g$ is the determinant of the metric $g=det(g_{ij})$. A way to interpret this relation is that the temperature of the system can be defined by the Fisher information of the system and by the averaged second order differential (curvature) of energy [@Ffl]. As we shall use below, this relation is critical for the present study. Since the system contacts with the same heat bath (with common $\beta$) both at the start and the end of the process as described in Sec. II (or we could also restate it as follows; the heat reservoir is so large compared with the system, so that temperature of the reservoir is not disturbed by the heat discarded by the system), we have readily a relation from Eq.(\[eqn:Ftmp\]) $$\begin{aligned}
\frac{I(\mathcal{P}_F)}{\langle \Delta H(\Gamma) \rangle_{\mathcal{P}_F}}=
\frac{I(\mathcal{P}_B)}{\langle \Delta H(\Gamma) \rangle_{\mathcal{P}_B}}=
\beta.\label{eqn:Iofb}\end{aligned}$$ An immediate but profound implication of this consequence is that the ratio of the Fisher information associated with probability distributions of the forward and backward processes is equivalent to the ratio of the averaged curvature of the Hamiltonian. This is true if the system takes the canonical form in distribution. The former ratio has the origin of the information quantity and the latter has the physical one. Next, substituting Eq.(\[eqn:Iofb\]) into Eq.(\[eqn:NW\]), we readily have an inequality $$\begin{aligned}
\frac{\langle | \nabla_\Gamma W_{diss}(\Gamma)|^2 \rangle_{\mathcal{P}_F}}
{\langle \Delta H(\Gamma) \rangle_{\mathcal{P}_{\gamma}}^2} \geqslant
2\frac{D_{KL}(\mathcal{P}_F \| \mathcal{P}_B)}{[I(\mathcal{P}_\gamma)]^2}.\label{eqn:last}\end{aligned}$$ where the symbol $\gamma$ denotes either $F$ or $B$ representing the forward and the backward equilibrium states. The lower bound on the ratio relevant to the averaged physical quantities (the left-hand side) is nicely bounded from below in terms of the ratio of information-associated quantities only. A further interesting observation can be derived for this relation Eq.(\[eqn:last\]) from the well-known Cramer-Rao inequality in statistical estimation theory [@Cover]. For simplicity’s sake, we consider it for one-dimensional case. The Cramer-Rao inequality tells a tradeoff relation between Fisher information of a distribution $P(X)$ and the variance of the distribution $\sigma^2_X$, i.e., $I(P(X))\geqslant 1/\sigma^2_X$, where $X$ is a random variable. Then, we find that the lower bound in Eq.(\[eqn:last\]) is upper bounded by $2D_{KL}(\mathcal{P}_F \| \mathcal{P}_B) \sigma^4_X$. Now that $P(X)$ is of the canonical form in our setting, the equality can be achieved when the Hamiltonian $H(X)$ is of quadratic form.
Summary
=======
To deepen the understandings of a profound information theoretic relation between work and dissipation in nonequilibrium systems, we have derived a universal relation that connect the gradient of the dissipated work and the relative Fisher information within a framework of the setup repeatedly employed in the previous studies. Considering the gradient of the dissipated work at each point in phase space enables us to get geometric information that cannot be obtained from KL entropy only. The relative Fisher information plays the role. By way of this, we have established the information based lower bound for the quantity relevant to the dissipation in phase space. The instantaneous equilibration was a premise to assure the well-posedness of the free energy difference between the two canonical equilibrium states associated to the forward and backward processes. However, the notion of the nonequilibrium free energy change $\Delta F_{neq}$ has recently considered to refine the dissipation occurring in far from thermodynamic equilibrium, where the dissipated work is defined by average work minus $\Delta F_{neq}$ instead of the equilibrium free energy change $\Delta F$ [@Fneq]. An extension of the present result to such a case surely needed if one is to understand and to gain deeper insights into biological systems.
The author wishes to thank Hiroaki Yoshida for a valuable discussion on the relative Fisher information at the Ochanomizu University in August 2012.
Relative Fisher information $D_{\rm{RFI}}$ {#app:rfi}
==========================================
The concept of the relative Fisher information measured from $f$ to $g$, $D_{\rm{RFI}}(f\|g)$ is less acknowledged in application by the general physics community compared with the relative entropy (Kullback-Leibler divergence). We briefly present here a physical origin of this form that is unrecognized in the literature (see also [@TY3] and references therein for a more general discussion). We consider the time change of the KL divergence between two distribution functions $f=f(\boldsymbol{x},t)$ and $g=g(\boldsymbol{x},t)$ that obey a heat equation ($\partial_tf=\mathscr{D}\nabla^2f$ etc.), where $\mathscr{D}$ is a diffusion constant: $$\begin{aligned}
\frac{d}{dt} D_{KL}(f\|g)=\frac{d}{dt}\int f\ln \frac{f}{g}d\boldsymbol{x} =
\int \dot{f} \ln \frac{f}{g}d\boldsymbol{x} +
\int g\left(\frac{\dot{f}}{g}-\frac{f}{g^2}\dot{g} \right) d\boldsymbol{x},\end{aligned}$$ where $\dot{f}=\partial f/\partial t$ etc. The first term is calculated by integration by parts as $$\begin{aligned}
\mathscr{D}\int (\nabla^2f)\ln \frac{f}{g} d\boldsymbol{x}=-\mathscr{D} \int
(\nabla f)\nabla \left( \ln \frac{f}{g}\right)d\boldsymbol{x},\label{eqn:1stt}\end{aligned}$$ where we have used the boundary conditions that $f$ and $\nabla f$ vanish when $|\boldsymbol{x}|\to 0$. Similarly, under the conditions that $f$, $g$ and $\nabla g$ vanish when $|\boldsymbol{x}|\to 0$, the second term becomes $$\begin{aligned}
\mathscr{D}\int f\left( \frac{\nabla g}{g}\right) \nabla\left( \ln \frac{f}{g}\right) d\boldsymbol{x}.
\label{eqn:2ndt}\end{aligned}$$ Therefore, combining Eq.(\[eqn:1stt\]) and Eq.(\[eqn:2ndt\]), we have $$\begin{aligned}
\frac{d}{dt} D_{KL}(f\|g)&=&-\mathscr{D}\int f\left( \frac{\nabla f}{f} -
\frac{\nabla g}{g}\right) \nabla\left( \ln \frac{f}{g}\right) d\boldsymbol{x}\nonumber\\
&=& -\mathscr{D}\int f \Big|\nabla\left( \ln \frac{f}{g}\right)\Big|^2d\boldsymbol{x}.\label{eqn:dBi}\end{aligned}$$ Except for the diffusion coefficient $\mathscr{D}$, the right-hand side provides $D_{\rm{RFI}}$, which is called the de Bruijn-type identity [@TY3]. The information involves how fast the KL distance changes under the heat equation. In a more general context [@TY3], we have $$\begin{aligned}
\frac{d}{dt} D_{KL}(f\|g)&=& \Big\langle \left( \frac{\boldsymbol{j}_f}{f}-
\frac{\boldsymbol{j}_g}{g}\right)\nabla\left( \ln\frac{f}{g}\right)\Big\rangle_f,\label{eqn:gdBi}\end{aligned}$$ where $\boldsymbol{j}_f$ and $\boldsymbol{j}_g$ are the flows associated, respectively, with $f$ and $g$ in the continuity equation $\partial_t\rho=-\nabla\cdot \boldsymbol{j}$, and $\langle \cdot\rangle_f$ denotes the averaging with respect to $f$.
$D_{\rm{RFI}}$ between two canonical equilibrium distributions
--------------------------------------------------------------
The relative Fisher information between forward and backward phase densities is calculated as $$\begin{aligned}
D_{\rm{RFI}}(\mathcal{P}_F \| \mathcal{P}_B) &=& \int \mathcal{P}_F
\Big| \nabla_\Gamma \left(\ln \frac{\mathcal{P}_F}{\mathcal{P}_B}\right)\Big|^2d\Gamma
\quad (\text{by definition Eq.(\ref{eqn:rFi})})\nonumber\\
&=& \int \mathcal{P}_F \Big| \nabla_\Gamma \left(\ln e^{\beta(W-\Delta F)}\right)\Big|^2 d\Gamma
\quad (\text{by relation Eq.(\ref{eqn:reld})}) \nonumber\\
&=& \beta^2\int \mathcal{P}_F \Big| \nabla_\Gamma (W-\Delta F)\Big|^2 d\Gamma.\end{aligned}$$ Finally, by the definition of the dissipated work, we have the relation Eq.(\[eqn:reF1\]).
Gradient flow {#app:gradf}
=============
An evolution equation of the form $$\begin{aligned}
\frac{\partial \rho}{\partial t}={\rm grad_{W}}G(\rho)\end{aligned}$$ for some functionals $G(\rho)$, is called a gradient flow, where ${\rm grad_{W}}G(\rho)$ is the Wasserstein gradient on the Wasserstein space (e.g., [@Villani]). A wide class of partial differential equations can be understood in light of this approach. The Wasserstein gradient here ${\rm div}(\rho v(\Gamma_\lambda))$ for the potential functional $G$ causes the time change in the density function. In other words, the vector field makes the backward density function $\mathcal{P}_B$ different from the forward one $\mathcal{P}_B$, during which the work dissipation is completed. The variation of $G$ with respect to $\rho$ is the gradient in $L^2$, and corresponds to the dissipated work $W_{diss}(\Gamma_\lambda)$ in the present consideration. To be more convinced, it is appropriate to consider a linear Fokker-Planck equation whose free energy functional $G(\rho)$ can be given by $$\begin{aligned}
G(\rho) = \int u(\boldsymbol{x})\rho(\boldsymbol{x})d\boldsymbol{x} +
T\int \rho(\boldsymbol{x})\ln\rho(\boldsymbol{x})d\boldsymbol{x},\end{aligned}$$ with $u(\boldsymbol{x})$ and $T$ denoting the internal energy and temperature, respectively. Then, the gradient of the functional $\delta G/\delta \rho=u(\boldsymbol{x})+\log\rho(\boldsymbol{x})+1$ leads to the gradient flow $\partial \rho/\partial t={\rm div}(\nabla\rho+\rho\nabla u(\boldsymbol{x}))$. In terms of the continuity equation, it is equivalent to that the system has a flow $\boldsymbol{j}=-(\nabla \rho+\rho\nabla u(\boldsymbol{x}))$. That is, it consists of the Fick’s law plus the gradient of the internal energy. Substituting this into Eq.(\[eqn:gdBi\]), we immediately recover the de Bruijn-type identity Eq.(\[eqn:dBi\]).
Approach to equilibrium controlled by a constant in the logarithmic Sobolev inequality {#app:LSI}
======================================================================================
The constant $c$ in the LSI (Eq.(\[eqn:LSI\]): $\int|f|^2\ln|f|d\mu-(\int |f|^2d\mu)\ln(\int |f|^2d\mu)\leqslant c\int |\nabla f|^2d\mu$) determines the rate at which the system approaches to an equilibrium state when measured with the KL distance. Consider the KL distance between a density function $(f_t)_{t\geqslant 0}$ and an equilibrium one $f_\infty$. Its time derivative is then $$\begin{aligned}
\frac{d}{dt}D_{KL}(f_t\|f_\infty)=\int \dot{f}_t\ln \frac{f_t}{f_\infty}d\boldsymbol{x}.\end{aligned}$$ Since we take $G=D_{KL}(f_t\|f_\infty)$ in the definition (Eq.(\[eqn:gradf\])) and then $\delta G/\delta f_t=1+\log f_t/f_\infty$, the gradient flow associated with it is $$\begin{aligned}
\frac{\partial f_t}{\partial t}=\nabla\cdot\left(f_t\nabla\ln\frac{f_t}{f_\infty}\right).\end{aligned}$$ Substituting this into the above and performing the integration by parts under the vanishing boundary condition, we have $$\begin{aligned}
\frac{d}{dt}D_{KL}(f_t\|f_\infty)=-\int f_t\Big|\nabla \ln\frac{f_t}{f_\infty}\Big|^2d\boldsymbol{x}
=-D_{\rm RFI}(f_t\|f_\infty).\label{eqn:C3}\end{aligned}$$ This is the de Bruijn-type identity under a gradient flow with the equilibrium state $f_\infty$ as a reference density function. On the other hand, choosing the probability measure as $d\mu=f_\infty d\boldsymbol{x}$ and putting $f=\sqrt{f_t/f_\infty}$ in the LSI, we have $$\begin{aligned}
\frac{1}{2}\int \frac{f_t}{f_\infty}\ln\left(\frac{f_t}{f_\infty}\right)f_\infty d\boldsymbol{x}
& \leqslant & c\int \Big| \nabla\sqrt{\frac{f_t}{f_\infty}}\Big|^2f_\infty d\boldsymbol{x}\nonumber\\
&=&\frac{c}{4}\int f_t\Big| \nabla \ln \frac{f_t}{f_\infty}\Big|^2f_\infty d\boldsymbol{x}\nonumber.\end{aligned}$$ The right-hand side equals just to $c/4 D_{\rm{RFI}}(\mathcal{P}_F \| \mathcal{P}_B)$. Therefore, from Eq.(\[eqn:C3\]) we have an inequality $$\begin{aligned}
\frac{d}{dt}D_{KL}(f_t\|f_\infty) \leqslant -\frac{2}{c} D_{KL}(f_t\|f_\infty).\label{eqn:C4}\end{aligned}$$ Recall that the Gronwall’s inequality states that if two continuous functions $w(t)$ and $\gamma(t)$ defined on an interval $I=[a,\infty)$ satisfies a differential inequality $w^\prime(t)\leqslant \gamma(t)w(t)$, then $w(t)$ is bounded as $w(t)\leqslant w(a)\exp(\int^{t}_{a}\gamma(s)ds)$ for $\forall t\in I$. Applying this to Eq.(\[eqn:C4\]) by setting $a=0$, we readily have $$\begin{aligned}
D_{KL}(f_t\|f_\infty)\leqslant D_{KL}(f_0\|f_\infty)\exp(-\int^{t}_{0}\frac{2}{c}ds)=
e^{-\frac{2}{c}t}D_{KL}(f_0\|f_\infty),\end{aligned}$$ where $f_0$ is the initial density function. This indicates explicitly that the distance between the initial and equilibrium density functions converges by the factor $e^{-2t/c}$. The smaller the value of $c$, the faster it converges.
[99]{} L. D. Landau and E. M. Lifshitz, *Statistical Physics Part 1, Course of Theoretical Physics* Vol. 5, §20, (Pergamon Press Ltd.,1980). C. Jarzynski, Phys. Rev. E [**73**]{} 046105 (2006). R. Kawai, J. M. R. Parrondo and C. Van den Broeck, Phys. Rev. Lett. [**98**]{} 080602 (2007). S. Kullback and R. A. Leibler, Ann. Math. Stat. [**22**]{} 79 (1951). S. Kullback, *Information Theory and Statistics* (Wiley, New York, 1959). Arieh Ben-Naim, *A Farewell to Entropy: statistical thermodynamics based on information*, World Scientific, (2008). I. Csisz[á]{}r, Publ. Math. Inst. Humgar. Acad. Sci. [**8**]{} 85 (1963). T. Morimoto, J. Phys. Soc. Jpn. [**18**]{} 328 (1963). C. Jarzynski, Phys. Rev. Lett. [**78**]{} 2690 (1997). G. E. Crooks, J. Stat. Phys. [**90**]{} 1481 (1998). C. Jarzynski, Eur. Phys. J. B [**64**]{} 331 (2008). T. Yamano, J. Math. Phys. [**50**]{} 043302 (2009); Phys. Lett. A [**374**]{} 3116 (2010). Special Issue: Distance in information and statistical mechanics, Entropy, edited by T. Yamano (Basel, Switzerland, 2011), see http:// www.mdpi.com/journal/entropy/ special\_issues/distance-info-stat-physics. C. Villani, *Topics in Optimal Transportation*, Graduate Studies in Mathematics Vol.58 (American Mathematical Society, 2000). L. Gross, Amer. J. Math. [**97**]{} 1061 (1975). A. R. Barron, Ann. Probab. [**14**]{} 336 (1986). B. R. Frieden, *Science from Fisher Information - A Unification* (Cambridge University Press, 2004). T. Cover and J. Thomas, *Elements of Information Theory*, 2nd ed. (Wiley-Interscience, 2006). T. Yamano, J. Math. Phys. [**53**]{} 043301 (2012). It is worth conferring that when we regard $\mathcal{P}$ as a function of $\beta$, then the Fisher information provides the fluctuation of energy for the canonical distribution: $I(\mathcal{P}(\beta))=\int \mathcal{P}(\beta) (d\ln\mathcal{P}(\beta)/d\beta)^2d\beta
=\langle H^2(\Gamma)\rangle_\mathcal{P}-\langle H(\Gamma)\rangle^2_\mathcal{P}$. K. R. Narayanan and A. R. Srinivasa, Phys. Rev. E [**85**]{} 031151 (2012). D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. [**108**]{} 150601 (2012). T. Yamano, Eur. J. Phys. B [**86**]{} 363 (2013).
|
---
abstract: 'The microscopic mechanism of photon detection in superconducting nanowire single-photon detectors is still under debate. We present a simple, but powerful theoretical model that allows us to identify essential differences between competing detection mechanisms. The model is based on quasi-particle multiplication and diffusion after the absorption of a photon. We then use the calculated spatial and temporal evolution of this quasi-particle cloud to determine detection criteria of three distinct detection mechanisms, based on the formation of a normal conducting spot, the reduction of the effective depairing critical current below the bias current and a vortex-crossing scenario, respectively. All our calculations as well as a comparison to experimental data strongly support the vortex-crossing detection mechanism by which vortices and antivortices enter the superconducting strip from the edges and subsequently traverse it thereby triggering the detectable normal conducting domain. These results may therefore help to reveal the microscopic mechanism responsible for the detection of photons in superconducting nanowires.'
author:
- Andreas Engel
- Andreas Schilling
bibliography:
- 'Literature.bib'
title: 'Numerical analysis of detection-mechanism models of SNSPD'
---
There is growing interest in superconducting nanowire single-photon detectors (SNSPDs) fueled by their combination of good detection efficiency, low dark-count rate, very short recovery time and exceptionally small jitter. SNSPD compare well with other competing technologies for the detection of visible and near-infrared photons [@Hadfield09] and have already been used in a wide variety of applications from quantum key distribution to time-of-flight depth ranging [@Natarajan12]. Variations of these detectors have also been used to detect higher energy particles, such as high kinetic-energy molecules in mass spectrometry [@Suzuki08; @Ohkubo12] or x-ray photons with keV-energies [@Inderbitzin12; @Inderbitzin13].
The active element of these detectors consists of a typically square meander of a superconducting NbN film of a few nanometer thickness [@Goltsman01]. Recently alternative superconducting materials have been suggested, such as NbTiN [@Dorenbos08], TaN [@Engel12], or WSi [@Baek11], which may be better suited than NbN for certain applications. The detectors are biased with a constant direct current [ $I_{\mathrm{bias}}$]{} which usually equals about 90% to 95% of the experimental critical current $I_c$. The absorption of a photon of sufficient energy, in combination with a suitably chosen bias current, can trigger a normal conducting cross-section, the subsequent growth of which is determined by the electro-thermal properties of the detector [@Gurevich87]. Estimates[@Semenov05a; @Kerman09] and simulations[@Marsili11] made with realistic material and device parameters for SNSPD show that this normal conducting domain initially grows very fast with a correspondingly large resistance. As a consequence part of the bias current is redirected into the readout line, which effectively acts as a $50~\Omega$ parallel impedance, or an additionally installed parallel ohmic resistance, thus reducing the Joule-heating. Depending on the details of this electro-thermal feedback [@Kerman09], the detector can be operated in the desired self-recovering mode or it latches into a resistive state, when the normal-conducting domain is stabilized by self-heating. This electro-thermal feedback imposes limits on the minimum achievable recovery time and the maximum count rate.
Many further aspects of SNSPD relevant for applications are also well understood. State-of-the-art SNSPD consist of a homogeneous superconducting film and a uniform meander without constrictions. Such devices exhibit a nearly constant detection efficiency for a certain range in parameter space of bias current and photon energy. In this “plateau region” nearly every absorbed photon triggers the formation of a normal conducting domain [@Kerman07] leading to an intrinsic detection efficiency ($IDE$) approaching 100% [@Hofherr10; @Lusche13]. The device detection efficiency ($DDE$) is the product of photon absorptance ($ABS$) of the meander and IDE: $DDE = ABS \times IDE$. $ABS$ itself depends on the absorptance of the thin superconducting film and geometric effects, such as the meander filling factor or a polarization dependent absorptance [@Dorenbos08a] and is limited to $\lesssim 20\%$ for a bare meander. Higher $DDE>50\%$ can be achieved by incorporating the detector into an optical cavity [@Rosfjord06], and optimizations of layer thicknesses and the separation of the meander lines allow for high detection efficiencies even at relatively low filling factors [@Akhlaghi13; @Yamashita13]. The highest reported system detection efficiency of up to $93\%$ reported to date, including coupling losses and photon absorption in the optical fibre, was achieved for an optimized WSi SNSPD [@Marsili13]. For a given bias current there is a minimum threshold energy for photons to be detected with the maximum efficiency. Photons with wavelengths $\lambda$ larger than the corresponding cut-off wavelength $\lambda_c$ can only be detected with a rapidly decreasing probability. Narrower meander lines [@Marsili11a] or superconducting films with a lower $T_c$ [@Baek11; @Dorenbos11; @Engel12] result in an increase of $\lambda_c$ and higher detection efficiencies at long photon wavelengths.
Detector noise or so called dark counts increase approximately exponentially on approaching the experimental critical current. Experimental [@Bartolf10] and theoretical investigations [@Bulaevskii11] have favored magnetic vortices crossing the superconducting strips as the dominant mechanism leading to intrinsic dark-count events. Except near the ends of the straight sections of the superconducting meander the current density in the undisturbed equilibrium situation is homogeneous due to the fact that the strip width $w<<\Lambda=2{ \ensuremath{\lambda_{\mathrm{GL}}}}^2/d$, where $\Lambda$ is the effective 2D magnetic penetration depth, ${ \ensuremath{\lambda_{\mathrm{GL}}}}\gg d$ is the corresponding Ginzburg-Landau (GL) magnetic penetration depth in the bulk material and $d$ the film thickness. Due to a current-crowding effect the current density in the $180^\circ$ turnarounds of the meander structure is no longer homogeneous[@Clem11] and dark-count events most likely originate near these turnarounds [@Engel12a], but may be reduced by a more sophisticated meander design [@Akhlaghi12a].
Despite this remarkable progress in understanding and optimization of SNSPD some open questions remain, particularly in connection with the mechanism that is responsible for triggering the initial resistive cross-section. The first model [@Semenov01] describing the detection mechanism in SNSPD assumed the formation of a normal-conducting hot-spot that diverts the applied bias current into the still superconducting side-walks. The current density in these side-walks will eventually increase beyond the critical current density, thereby leading to the initial normal-conducting cross-section. This model, which we will call hard-core model, already captures some important characteristics of SNSPD, such as the existence of a bias-current dependent minimum photon energy, and because it gives a very vivid description, it has been widely used. However, it fails to describe certain observations, *e.g.* the temperature dependence of $\lambda_c$ [@Engel13], and often leads to inconsistencies. As an example we cite data from a recent analysis of cut-off wavelengths within the hard-core model [@Maingault10]. The corresponding analysis results in a hot-spot diameter of about $12$ nm for a photon energy of $1.24$ eV. Using the device parameters given in Ref. and typical superconductivity parameters for NbN one can estimate the superconducting condensation energy of the corresponding volume to be of the order of $0.1$ eV. In the same paper, the energy conversion efficiency was estimated to $\zeta\approx0.5\%$, which means that it would require a photon energy at least one order of magnitude larger than used in the experiment to drive this volume into the normal conducting state.
One inherent shortcoming of the hard-core model originates from neglecting excess quasi-particles (QP) outside the normal-conducting core in the still superconducting side-walks. An alternative detection model has been suggested several years ago [@Engel05; @Semenov05a] in which the reduction of the depairing critical current in a cross-section of a superconducting strip due to excess QP is taken into account. This QP-model explicitly does not require a normal conducting region to form before the critical current has dropped below the applied bias current. Very recently it has been suggested that the relevant current scale is not the depairing critical current but instead the critical current for vortex crossings [@Bulaevskii12], although there is some controversy about the correct theoretical treatment of a vortex very near the strip edge[@Gurevich12]. Also, more advanced theoretical models based on the time-dependent GL theory are being developed [@Zotova12] with the aim to gain a better understanding of the dynamics of the detection mechanism. A full numerical simulation based on the time-dependent GL theory coupled with heat diffusion and the Maxwell equations [@Ota13] resulted in threshold energies required to trigger a normal conducting domain at least one order of magnitude larger than experimentally observed.
In this paper we present the development of simple model of the QP multiplication and diffusion process that is detailed enough to allow a comparison with experimental results. In Sec. \[Sec.model\] we develop the mathematical model that allows us to numerically determine detection criteria for direct photon detection within the hard-core, QP, and vortex model. The corresponding detection criteria will also be specified. In Sec. \[Sec.parameters\] the material and geometric parameters used in our simulations are defined and we perform some consistency tests to validate our results. This will be followed by the presentation of our numerical results and a comparison with experimental data whenever this is possible.
Development of the physical model\[Sec.model\]
==============================================
Modelling the quasi-particle diffusion
--------------------------------------
We will restrict ourselves to a discussion of the detection of mainly visible and near-infrared photons with energies $h\nu\sim1$ eV. These energies are much larger than the superconducting gap $\Delta\sim 1$ meV of typical SNSPD. Absorption of such a photon results in a large number of excitations in the form of QPs and phonons. Details of this QP multiplication process have been already described in an early publication on SNSPD [@Semenov01] and references therein. In this paper we are not interested in the detailed time-evolution of the number of excited QPs, instead we will resort to a simple analytical approximation. We make the assumption that the time scale of electron-electron interactions is much faster than both the electron-phonon and phonon-phonon time scales[@Semenov01]. This means that the electronic system will have thermalized to a local, (near-)equilibrium state long before the phonon system thermalizes. We will also assume that the increase in the local concentration of QPs equals the local reduction in the concentration of superconducting electrons, in other words, that the electronic system is always in a local, near-equilibrium state, and we neglect the background of thermally excited QPs[^1].
Previous models [@Semenov01; @Semenov05a] have made very similar assumptions and furthermore assumed the QP multiplication and QP diffusion to be independent processes. Instead, we will be considering the highly excited electron after photon absorption, which itself diffuses within the superconducting film, to be the source of QPs as it continuously loses energy, thereby breaking up Cooper-pairs. Assuming that the thermalization process does not influence the diffusion of the excited electron, the probability density $C_e(\vec{r},t)$ to find it at position $\vec{r}$ at time $t$ after the absorption follows the diffusion equation $$\frac{\partial C_e(\vec{r},t)}{\partial t} = D_e\nabla^2 C_e(\vec{r},t),
\label{Eq.PDE_e}$$ with $\nabla^2$ the Laplace-operator and $D_e$ the diffusion coefficient of normal electrons. We set the diffusion coefficient to be constant. In reality it may be a function of the excitation energy of the electron, thus depending on $t$ and the photon energy. Already after $t\lesssim 1$ ps the diffusion length $\sim\sqrt{D_{e}t}$ becomes larger than the typical superconducting film thickness $d\approx5$ nm. It is therefore justified to treat this problem in two dimensions for longer time scales. During the diffusion process the electron thermalizes by losing energy in inelastic scattering events. We simplify this process assuming an exponential decay of the excitation energy with a constant time scale $\tau_{qp}$: $E_e=h\nu\exp\left(-t/\tau_{qp}\right)$. A certain fraction of these scattering events results in the generation of two QPs with an energy $\Delta$ and is proportional to the probability density $C_e(\vec{r},t)$.
The excess QPs themselves undergo diffusion in the superconducting film before they recombine to form Cooper-pairs on a time scale $\tau_r\gg\tau_{qp}$. The concentration of excess QPs $C_{qp}(\vec{r},t)$ can then be described by
$$\frac{\partial C_{qp}(\vec{r},t)}{\partial t} = D_{qp}\nabla^2 C_{qp}(\vec{r},t) - \frac{C_{qp}(\vec{r},t)}{\tau_r} + \frac{\zeta h\nu}{\Delta\tau_{qp}}\exp\left(-\frac{t}{\tau_{qp}}\right)C_e(\vec{r},t),
\label{Eq.PDE_QP}$$
with $D_{qp}\neq D_e$ the QP diffusion coefficient and $0<\zeta\leq1$ the conversion efficiency, which has to be determined experimentally. The last term in Eq. is the source term describing the QP-multiplication process. The recombination process is described by $\frac{C_{qp}(\vec{r},t)}{\tau_r}$, which we include in a linear approximation [^2]. Eqs. and form a set of coupled differential equations describing the evolution of the statistically averaged density of excess QPs.
For the case of an infinitely large 2D-film and making the assumption $D_{qp}= D_e=D$, Eqs. and have an analytical solution (see Appendix \[App.Analytic\]),
$$\begin{aligned}
C_e(r,t) &= \frac{1}{4\pi Dt}\exp\left(-\frac{r^2}{4Dt}\right),\label{Eq.2D_Diffusion}\\
C_{qp}(r,t) &= \frac{\zeta h\nu}{\Delta}\frac{\tau_r}{\tau_r-\tau_{qp}}\left[\exp\left(-\frac{t}{\tau_r}\right)-\exp\left(-\frac{t}{\tau_{qp}}\right)\right]\frac{1}{4\pi Dt}\exp\left(-\frac{r^2}{4Dt}\right),
\label{Eq.AnalyticalSolution}\end{aligned}$$
with $r$ being the distance from the absorption site of the photon. Integration of Eq. over the complete film gives the total number of excess QPs, $$N_{qp}(t) = \int_\infty C_{qp}(\vec{r},t)\mathrm{d}V = \frac{\zeta h\nu}{\Delta}\frac{\tau_r}{\tau_r-\tau_{qp}}\left[\exp\left(-\frac{t}{\tau_r}\right)-\exp\left(-\frac{t}{\tau_{qp}}\right)\right].
\label{Eq.totalQPnumber}$$
This last equation also holds for the more realistic case of narrow superconducting strips and $D_e \neq D_{qp}$ as long as $\tau_{qp}$ is independent of $C_{qp}$. However, the concentration $C_{qp}(\vec{r},t)$ itself has to be calculated numerically in this more general situation. In the following we will solve this set of differential equations and for a rectangle of width $w$ and length $L\gg w$, see Fig. \[Fig.Schematic\]. For the side-walls we use Neumann boundary conditions $\partial C(\vec{r},t)/\partial y = 0$, *i.e.* no loss of QP through the side-walls, and perfectly absorbing walls at the beginning and end of the strip. We assume the photon to be absorbed in the center of the strip. The numerical solution to Eqs. and is found using the finite element method (FEM) and Matlab$^\mathrm{TM}$ software. The mesh on which the solutions are calculated was created with a high density of nodes around the absorption site. For the subsequent analysis of the trigger models the results are transformed onto a cartesian grid with a resolution of $0.5$ nm in $x$- and $y$-direction. A finer grid of $0.2$ nm resolution was used in a central region $\pm5\xi(0)$ around the absorption site in $x$-direction and over the full width of the strip. The parameters $\xi$, [ $\lambda_{\mathrm{GL}}$]{}, $\Delta$ and $D_{qp}\neq D_e$ are in general assumed to be temperature dependent (see Appendix \[App.T-dependence\] for details).
![Schematic drawing of meander section with the photon absorption site at its center, not to scale. Indicated is the $\xi$-slab that defines the minimum volume that has to switch into the normal conducting state. Also plotted are Gaussian-profiles (dark and light blue) of the QP concentration according to Eq. at an arbitrary time $t$.\[Fig.Schematic\]](Schematic){width="\columnwidth"}
Detection criteria
------------------
Based on the computed $C_{qp}(\vec{r},t)$, different detection criteria can be formulated and compared with each other. In the following we will consider three models that explain the formation of the initial normal conducting cross-section. The first and original model describing photon detection in SNSPD [@Semenov01] assumes QP-concentrations high enough to completely suppress superconductivity in the vicinity of the absorption site. For this hard-core model we use a condition similar to Ref. , namely we define the extension of the normal conducting core of the hot-spot by $C_{qp}(\vec{r},t)\geq n_{se}^{2D}=n_{se}d$, with $n_{se}$ the equilibrium density of superconducting electrons[^3] in the film being twice the Cooper-pair density, and we require this normal-conducting area to have an extension of at least the coherence length $\xi$ in the direction of the applied current. The current-density inside the normal conducting area is assumed to be zero, thus leading to an enhanced current-density in the side-walks as required by the continuity equation. If the current density in the side-walks exceeds the depairing critical current [ $I_{c,\mathrm{dep}}$]{}, the whole cross-section becomes normal conducting leading to the detection of the absorbed photon. The minimum transversal extension $2{ \ensuremath{R_{hc}}}$ of the normal conducting core for photon detection, the strip width $w$ and the reduced bias current ${ \ensuremath{I_{\mathrm{bias}}}}/{ \ensuremath{I_{c,\mathrm{dep}}}}$ are then related to each other defining the detection criterion $$2{ \ensuremath{R_{hc}}}\geq w\left(1-\frac{{ \ensuremath{I_{\mathrm{bias}}}}}{{ \ensuremath{I_{c,\mathrm{dep}}}}}\right).
\label{Eq.HardCoreCriterium}$$ If one assumes the conversion efficiency $\zeta\lesssim 1$, near-infrared photons with $h\nu\lesssim1$ eV would produce normal conducting areas large enough to be roughly consistent with experimental results. However, more plausible values [@Ilin00; @Semenov05a; @Engel12] for the conversion efficiency are on the order of $0.1$, in which case correspondingly higher photon energies are required.
The most important conceptual short-coming of the hard-core model is the restriction of the QPs to be strictly confined to the potentially present normal conducting volume. For very high photon [@Inderbitzin12] or particle energies [@Suzuki11] the contribution of excess QPs outside the normal conducting volume is probably negligible, but they may become significant or even dominating for smaller excitation energies. This was first realized in the QP-model [@Semenov05a] which does not require a normal conducting volume. Instead, an excess number of QPs results in a reduction of the effective critical current. Under the assumption that the number of excess QPs equals the reduction in the number of superconducting electrons, the effect on the critical current can be easily calculated applying the continuity equation for the applied bias current and the phase coherence of the superconducting electrons. The minimum volume that must reach the normal state to trigger a photon count has to have a length in the direction of the applied current of at least the coherence length and span the complete cross-section, the “$\xi$-slab”, see also Fig. \[Fig.Schematic\]. As a condition for the nucleation of such a normal conducting cross-section one obtains [@Semenov05a] $$\frac{N_{qp}^\mathrm{slab}(t)}{N_{se}} \geq 1-\frac{{ \ensuremath{I_{\mathrm{bias}}}}}{{ \ensuremath{I_{c,\mathrm{dep}}}}},
\label{Eq.CriticalCurrentReduction}$$ where $N_{se}=n_{se}wd\xi$ is the equilibrium number of superconducting electrons in the $\xi$-slab. The number of excess QPs in the $\xi$-slab is computed from $$N_{qp}^\mathrm{slab}(t)=\int_\mathrm{\xi-slab}C_{qp}(\vec{r},t)\mathrm{d}V.
\label{}$$
As a third possibility that could lead to the formation of a normal conducting domain, we consider the photon-assisted crossing of a vortex [@Bulaevskii12]. In the case of a homogeneous current density ${ \ensuremath{j_{\mathrm{bias}}}}={ \ensuremath{I_{\mathrm{bias}}}}/w=\mathrm{const.}$ across the strip, it is straightforward to calculate the energy barrier prohibiting the entry and subsequent crossing of vortices [@Bulaevskii11; @Clem11]. Contrary to the original publication[@Bulaevskii12], where the authors assume a uniformly reduced order parameter, we consider an inhomogeneous current density due to the expanding cloud of QPs after photon absorption. We assume the current redistribution to be instantaneous, justified by an estimate of the GL relaxation time $\tau_\mathrm{GL}\lesssim1$ ps[@Kopnin01]. We can then calculate the local, time-dependent current density proportional to the local density of superconducting electrons, $n_{se}^{2D}-C(\vec{r},t)$, and require current continuity[^4] and $\operatorname{div} \vec{j}=0$. This leads to an enhancement of the current density near the strip edges and a corresponding decrease of the energy barrier for vortex entry very similar to the situation near the meander turn-arounds [@Clem11]. We calculate the forces on a vortex as a function of its position inside the strip taking into account the inhomogeneous current distribution as well as the increase of the effective penetration depth $\Lambda$ due to the reduced density of superconducting electrons and obtain the vortex potential by numerical integration (see Appendix \[App.VortexPotential\]). For a given reduced bias current ${ \ensuremath{I_{\mathrm{bias}}}}/I_{c,v}$ the energy barrier vanishes for a minimum photon energy, giving us the cut-off wavelength $\lambda_c$ in this vortex-model. It is important to note that the current scale in the vortex model is $I_{c,v}<{ \ensuremath{I_{c,\mathrm{dep}}}}$, *i.e.* the current for which the energy barrier is reduced to zero [@Bulaevskii11].
The model considered by Zotova and Vodolazov[@Zotova12] is closely related to this vortex model. These authors assume the formation of a normal-conducting core similar to the hard-core model, but take into account a non-homogeneous current distribution around the normal-conducting area due to a current-crowding effect[@Clem11]. In such a situation the highest current density is expected at a point very close to this normal-conducting area inside the strip, favoring the creation of a vortex-antivortex pair. In case of a continuous variation of the Cooper-pair density, we expect this current-crowding effect to be much less pronounced. Unfortunately, there is, to our knowledge, no simple method available to calculate the exact current distribution in this more complicated situation. The method we used to calculate the current densities does not lead to an enhanced current density close to the photon absorption site, and we therefore do not consider the creation of vortex-antivortex pairs in our analysis.
Before discussing our results for the three models, we state that our numerical model contains a number of simplifications in addition to our general assumptions discussed at the beginning of this section, the most important ones we would like to mention here. We assume the superconducting gap $\Delta$ to be independent of the density of excess QP, but it is known that the presence of QPs leads to a certain reduction of $\Delta$ [@Gilabert90], the magnitude of which depends on the details of the thermalization process. This leads to an underestimation of the effects that an absorbed photon causes in the superconducting strip, and thus to an overestimation of the minimum photon energy required to trigger a detection event in all three detection models, but in general to a different degree. A presumably smaller error is introduced by neglecting the reduction of $\Delta$ due to ${ \ensuremath{I_{\mathrm{bias}}}}$. When comparing our simulation results with experimental measurements, we can partially correct these effects by assuming a higher effective conversion efficiency $\zeta$, which is an adjustable parameter with no well established theoretical estimates. Further significant simplifications are made for the QP multiplication process, for example, the assumption of a constant time-scale $\tau_{qp}$ for the QP-multiplication. Our assumption of a nearly unchanged concentration of Cooper-pairs is also not strictly fulfilled during the early stages of the QP multiplication and diffusion process near the absorption site, since our simulations indicate high concentrations of QPs for $t\lesssim1$ ps and correspondingly small Cooper-pairs concentrations even for photon wavelengths much longer than $\lambda_c$.
Simulation parameters and consistency checks {#Sec.parameters}
============================================
In Table \[Tab.Material\] we summarize typical values of material parameters for high-quality films of TaN [@Engel12] and NbN [@Bartolf10] as we used them in the calculations. The parameters $\Delta$, $\xi$, [ $\lambda_{\mathrm{GL}}$]{}and $D_{qp}$ are assumed to be temperature dependent (see Appendix \[App.T-dependence\] for details). The temperature dependent value of $D_{qp}$ is calculated from $D_e$ of the normal-conducting electrons at $T_c$ as detailed in Appendix \[App.T-dependence\]. For the current report, we have set $T/T_c=0.05$. The general temperature-dependent behavior of SNSPD is the subject of ongoing investigations and will be presented in a future publication. The time constant $\tau_r$ has been chosen as an average value for the whole $T$-range [@Semenov97], and as it turns out, our results are not sensitive to the choice of $\tau_r$. The time constant $\tau_{qp}$ is related to the thermalization time $\tau_{th}$, which has been measured for NbN to be $\approx7$ ps [@Ilin98]. With the chosen time constants the total maximum number of excess QPs is reached at $t=\tau_{th}\approx10.3$ ps, as calculated using Eq. , and reaches $\gtrsim98$% of this maximum number of QPs at $t=7$ ps. Table \[Tab.Geometry\] lists the geometrical parameters of the simulated superconducting strip.
$\Delta$ (meV) $\xi$ (nm) [ $\lambda_{\mathrm{GL}}$]{} (nm) $D_e$ (nm$^2$ ps$^{-1}$) $D_{qp}$ (nm$^2$ ps$^{-1}$) $\zeta$ $\tau_{qp}$ (ps) $\tau_r$ (ps) $N_0$ (nm$^{-3}$eV$^{-1}$)
----- ---------------- ------------ ----------------------------------- -------------------------- ----------------------------- --------- ------------------ --------------- ----------------------------
TaN 1.3 5.3 520 60 8.2 0.25 1.6 1000 48
NbN 2.3 4.3 430 52 7.1 0.25 1.6 1000 51
-------------------------------- -----------
Length $L$ $1\ \mu$m
Width $w$ $100$ nm
Thickness $d$ $5$ nm
rough grid $\Delta x,\Delta y$ $0.5$ nm
fine grid $\Delta x,\Delta y$ $0.2$ nm
-------------------------------- -----------
: \[Tab.Geometry\] Geometric parameters used for the simulations. The FEM solutions have been transformed from the simulation mesh to a cartesian grid for subsequent analyses. Over the whole strip a rough grid was applied, and in an area $\pm5\xi(0)$ around the absorption site and spanning the width of the strip a finer grid was used.
We verified the validity of our numerical calculations by comparing the results to those using analytical expressions for an infinite film based on Eq. . Fig. \[Fig.Nvont\] shows the temporal evolution of the number of QPs in the complete strip as well as in the $\xi$-slab on a double logarithmic scale. Calculations were done for a photon with wavelength $\lambda=1000$ nm absorbed in a TaN-film. The solid red line is the calculated number of QPs $N_{qp}(t)$ according to Eq. and the black squares represent the same quantity obtained by numeric integration of $C_{qp}(\vec{r},t)$ over the complete strip. The numeric results agree very well with the analytical expression (better than $1$% for all $t<=1$ ns) and the thermalization time $\tau_{th}=10.5$ ps also agrees with the analytic result of $10.3$ ps within the temporal resolution of the simulation ($\pm0.5$ ps for $3~\text{ps}\leq t\leq12$ ps).
In the same figure we also plotted the numerically calculated number of QPs in the $\xi$-slab $N_{qp}^\mathrm{slab}(t)$ (black circles). It shows a pronounced maximum at $t_{max}\approx2.6$ ps after absorption of the photon, significantly before the total number of excess QPs have reached their maximum at $\tau_{th}$. According to Eq. $t_{max}$ corresponds to the moment when a certain minimum bias current can still trigger the formation of the initial normal conducting cross-section in the QP-model. An analytic solution can be found using Eq. and approximating the integration in the $x$-direction by $\tanh(\xi/\sqrt{4\pi Dt})$. The approximate number of QPs in the $\xi$-slab then becomes $$N_{qp}^\mathrm{slab}(t) \approx N_{qp}(t)\tanh\left(\frac{\xi}{\sqrt{4\pi Dt}}\right).
\label{Eq.N_xislab}$$ In Fig. \[Fig.Nvont\] $N_{qp}^\mathrm{slab}(t)$ according to Eq. is plotted for $D=D_{qp}$ (blue symbols) and $D=D_e$ (green symbols), respectively. For $t\gtrsim100$ ps the analytic solution with $D=D_{qp}$ asymptotically approaches the numerical solution. For smaller $t$ neither solution gives an adequate description of the numerical results, thus demonstrating the necessity of numerical calculations. The physical reason is the distinction between the diffusion coefficients for normal electrons and QPs, respectively. At the beginning of the multiplication process QPs are generated proportional to $C_e(r,t)$ with $D_e>D_{qp}$. However, the QPs immediately start to diffuse with $D_{qp}$. The effective diffusion coefficient becomes $t$-dependent and approaches $D_{qp}$ once the generation of additional QPs has stopped for $t\gg\tau_{qp}$.
![Double-logarithmic plot of the number of QPs in the complete superconducting strip $N_{qp}(t)$ (black squares and solid red line) and within the $\xi$-slab $N_{qp}^\mathrm{slab}(t)$ (black, blue and green) as functions of time, calculated for a photon wavelength with $\lambda=1000$ nm absorbed in a TaN-film. Numeric results are compared to analytical approximations Eqs. and . Arrows indicate the times of maximum total number of QPs, $\tau_{th}$, maximum number of QPs in the $\xi$-slab, $t_{max}$, and QP multiplication time scale, $\tau_{qp}$. Also indicated is the asymptotic maximum number of QPs $\zeta h\nu/\Delta$ for the theoretical case of no recombination of QPs into Cooper-pairs (dashed horizontal black line). \[Fig.Nvont\]](Nvont){width="\columnwidth"}
Finally we compare our numerically obtained potential energies of single vortices as a function of position $y$ across the strip with the analytical expression for the case of a homogeneous current density [@Bulaevskii12] $$U(y,I,T)/\varepsilon_0 = \ln\left[\frac{2w}{\pi\xi(T)}\cos\left(\frac{\pi y}{w}\right)\right] - \frac{I}{I_{c,v}}\frac{2(y+\frac{w}{2})}{\exp(1)\xi(T)},
\label{Eq.VortexPotential}$$ where $\varepsilon_0=\Phi_0^2/(2\pi\mu_0\Lambda)$ is the characteristic vortex energy, $\Phi_0=h/2e$ the magnetic flux quantum and $\mu_0$ the permeability of free space. Eq. has been derived for the condition $d\ll w\ll\Lambda$, which is typically fulfilled for SNSPD. We follow here Ref. and set the first term on the right hand side in Eq. \[Eq.VortexPotential\] to zero at $y=(\xi(T)-w)/2$ and set $U(y,I,T)=0$ for $y<\xi-w/2(T)$ and $y>w/2-\xi(T)$. We plot in Fig. \[Fig.VortexPotEquilibrium\] the analytical and numerical results for $T/T_c=0.05$ and four values of ${ \ensuremath{I_{\mathrm{bias}}}}/I_{c,v}$ as indicated. Again, numerically and analytically obtained values agree with each other to within a few percent.
![Potential energies of single vortices, as functions of the position across the strip, calculated numerically (open symbols) and according to Eq. (solid lines) for different bias currents as indicated. Calculations were done for homogeneous current densities and $T/T_c=0.05$.\[Fig.VortexPotEquilibrium\]](VortexPot0){width="\columnwidth"}
Simulation results and comparison with experimental data
========================================================
Temporal evolution of QP density, current distribution and vortex edge-barrier
------------------------------------------------------------------------------
The primary goal of our simulations is to reveal the temporal evolution of the excess QP density in the superconducting strip. In Fig. \[Fig.QPdensity\] we show typical results for the case when superconductivity is not completely suppressed in the hot-spot core. These calculations were done with material parameters for TaN at a reduced temperature $T/T_c=0.05$ and an incident photon with $\lambda=1000$ nm absorbed in the center of the strip. At the very early stages after photon absorption ($t=1$ ps, panel (a) in Fig. \[Fig.QPdensity\]) the QPs are highly concentrated near the absorption site, leading to a significant suppression of superconductivity in a very small volume. Already at $t=t_{max}=2.6$ ps the maximum number of QPs in the $\xi$-slab is reached (compare to Fig. \[Fig.Nvont\]), see the situation shown in panel (b) of Fig. \[Fig.QPdensity\]. Despite the relatively low diffusion coefficient $D_{qp}$ at this low temperature, a significant number of QPs has diffused out of the $\xi$-slab, and although the total number of excess QPs continues to increase until $t=t_{th}\approx10.5$ ps, the concentration of QPs in the $\xi$-slab drops more quickly, resulting in a decreasing $N_{qp}^\mathrm{slab}(t)$.
![QP density for different times after absorption of a $1000$ nm photon at $T/T_c=0.05$ in the center of the superconducting TaN strip. In panel (b) the number of QPs reaches its maximum in the $\xi$-slab ($t=t_{max}=2.6$ ps), which is indicated by the vertical dashed lines. Panel (d) shows the situation when the maximum number of QPs for the complete strip is reached at $t\approx10.5$ ps. However, at this point diffusion has led to a significant reduction of QP in the $\xi$-slab. Panels (a) and (c) depict the situation at times before and after $t_{max}$.\[Fig.QPdensity\]](QPdensity){width="\columnwidth"}
As outlined above in Sec. \[Sec.model\] we used the calculated distributions of QPs to obtain the distribution of the bias current by requiring superconducting phase coherence, current continuity and $\operatorname{div}\vec{j}=0$. The resulting relative current distributions are shown in Fig. \[Fig.Currdensity\] for the same conditions as the QP density in Fig. \[Fig.QPdensity\]. Although the current suppression in the center of the QP cloud is strongest right after absorption of the photon ($t\approx1$ ps, Fig. \[Fig.Currdensity\](a)), the maximum current density in the side-walks is again reached for $t\approx t_{max}=2.6$ ps. This fact becomes even clearer in Fig. \[Fig.CurrentProfiles\], where we plot the relative current densities as a function of time after the absorption of a photon for two different positions in the strip. The center position is the photon absorption site, and the edge position is one coherence length away from the geometrical edge of the strip. For the considered situation we obtained a maximum current increase of about $16$% near the strip edges. Again, diffusion dominates over QP-multiplication for $t>t_{max}$, and the current distribution becomes more homogeneous as time progresses.
![Calculated relative current densities after the photon absorption, based on the QP densities shown in Fig. \[Fig.QPdensity\]. The vertical dashed lines outline the $\xi$-slab, and the black lines are streamlines of the bias current.\[Fig.Currdensity\]](Currdensity){width="\columnwidth"}
![Relative current densities as a function of time for two different positions extracted from the current distributions shown in Fig. \[Fig.Currdensity\]. The center position is the absorption site of the photon, and the edge position is one coherence length away from the geometric edge of the strip, at the same $x=0$ as the center position. The maximum current-density increase near the edge is realized at $t\approx2.6$ ps after the absorption of the photon. For clarity, the time axis is plotted on a logarithmic scale.\[Fig.CurrentProfiles\]](CurrDensityTemporal){width="\columnwidth"}
These inhomogeneous current distributions in turn affect the entry barrier for vortices at the edges. In Fig. \[Fig.Barrier3D\] we show a three dimensional representation of the potential energy landscape for a vortex near the edge of the strip. It has been calculated based on the current distribution in Fig. \[Fig.Currdensity\](b) at $t=2.6$ ps after photon absorption and for an applied bias current $I/I_{c,v}=0.85$. For this particular situation the barrier remains positive and a vortex would still need additional thermal energy to enter the strip and trigger the formation of the initial normal conducting cross-section.
 caused by the absorption of a $1000$ nm photon at the center.\[Fig.Barrier3D\]](VortexBarrier3D){width="\columnwidth"}
In order to show the temporal evolution of the energy barrier, we plot in Fig. \[Fig.Barrier\] the potential energy for a vortex near the strip edge as a function of position across the strip for the cross section containing the photon absorption site at $y=0$. We observe again that the strongest reduction of the barrier occurs at around $t=2.6$ ps (red dots) after photon absorption when the number of excess QPs in the $\xi$-slab reaches its maximum. For comparison we have also plotted the energy barrier before the photon absorption (gray dots). The maximum of the curves in Fig. \[Fig.Barrier\] corresponds to the saddle-point of the 3D-energy landscape in Fig. \[Fig.Barrier3D\]. The criterion for a photon-detection event based on the vortex entry mechanism corresponds to a saddle-point value $\leq0$.
![Temporal evolution of the vortex-entry barrier along the cross-section containing the photon absorption site. The strongest influence of the excess QPs is again realized at $t\approx2.6$ ps after the photon absorption when the number of excess QPs in the $\xi$-slab reaches its maximum. For comparison the energy barrier in the undisturbed situation before photon absorption is also shown (gray dots).\[Fig.Barrier\]](VortexBarrier){width="\columnwidth"}
Photon detection as functions of photon energy and bias current
---------------------------------------------------------------
Next we evaluate the minimum (or threshold) bias-current $I_{th}$ for the detection of the photon in all three detection models for a given photon energy. First of all, it is important to realize that the critical current in the “vortex model” is the current for which the edge barrier vanishes, whereas in the other two models the relevant current scale is the depairing current. Within the vortex model developed in Refs. and , $I_{c,v}\approx0.826{ \ensuremath{I_{c,\mathrm{dep}}}}$. For the remainder, we will express all reduced bias currents scaled with the depairing critical current ${ \ensuremath{I_{\mathrm{bias}}}}/{ \ensuremath{I_{c,\mathrm{dep}}}}$.
From the detection criterions in the hard-core model and QP model, Eqs. and , respectively, one can derive explicit relations between photon energy and threshold current. In the latter model, the number of QPs in the $\xi$-slab $N_{qp}^\mathrm{slab}(t)$ is directly proportional to the photon energy, thus we expect $h\nu\propto1-I_{th}/{ \ensuremath{I_{c,\mathrm{dep}}}}$. In the hard-core model one usually assumes a cylindrical normal conducting volume $\pi { \ensuremath{R_{hc}}}^2 d\propto h\nu$. Insertion in Eq. results in $\sqrt{h\nu}\propto1-I_{th}/{ \ensuremath{I_{c,\mathrm{dep}}}}$. In order to check these relations we plot in Fig. \[Fig.Threshold\] the quantity $1-I_{th}/{ \ensuremath{I_{c,\mathrm{dep}}}}$ for all three models as a function of the photon energy up to energies $\approx 12.4$ eV, corresponding to $\lambda=100$ nm. The solid lines are least-squares fits to the corresponding data (green and red) confirming the expectations, namely a linear relationship for the QP model and a square-root behavior for the hard-core model. However, for high photon energies, $h\nu\gtrsim3$ eV, the simulated threshold currents in the QP model deviate systematically from the expected linear behavior. The reason for this discrepancy is the appearance of a normal conducting core at these high photon energies, which is neglected in the derivation of the linear relation in that model.
![Threshold current plotted as $1-I_{th}/{ \ensuremath{I_{c,\mathrm{dep}}}}$ *vs.* the photon energy $h\nu$ for all three detection models as indicated. The temperature was set to $T=0.05\,T_c$. Solid lines are least-squares fits to the simulation data obtained for the hard-core and QP-model as explained in the text. For the QP-model only data points with $h\nu<1.9$ eV have been considered for the fit. In the vortex model $I_{th}$ appears also to follow a linear dependence on $h\nu$ (blue line) up to photon energies that lead to the formation of a normal-conducting core. The inset shows the result of a least-squares fit of the vortex-model simulation data (blue dashed line) to experimental data from a TaN SNSPD (black circles) at $T=0.61~\text{K}\approx0.07\,T_c$ with the conversion efficiency $\zeta=0.24$ as the only adjustable parameter.\[Fig.Threshold\]](Threshold){width="\columnwidth"}
For the vortex model we are not aware of any analytical approximation describing the threshold current dependence on photon energy, except for the gross approximation of a uniform density of excess QPs [@Lusche13a]. Our simulation results (blue data in Fig. \[Fig.Threshold\]) suggest a similar linear dependence as for the QP-model at low photon energies, but shifted upwards along the $y$-axis as a consequence of the different current scales. Once the photon energy is high enough to allow for the formation of the normal conducting core, the simulated data deviate from the linear extrapolation in a similar way as in the QP-model.
Comparing the results of the three models with each other one can easily see that for a given bias current the vortex model requires the lowest photon energy and the hard-core model the highest energy. Photons with a fixed energy, on the other hand, are detected at the lowest bias currents in the vortex model, and at the highest current in the hard-core model. In fact, in the latter model photons may be absorbed without a detection event if the photon energy is too small, even in the hypothetical case of bias currents approaching the depairing critical current. With the simulation parameters given in Tab. \[Tab.Material\] and \[Tab.Geometry\] the minimum detectable photon energy would be $h\nu\approx1.9$ eV corresponding to $\lambda\approx650$ nm. These numbers, however, strongly depend on the choice of the conversion efficiency $\zeta$.
Since in all three models the maximum number of excess QPs ${ \ensuremath{N_{\mathrm{max}}}}\propto\zeta h\nu$, our results obtained for the particular value $\zeta=0.25$ can be easily recalculated for any value of $\zeta$ by an appropriate re-scaling of the photon energy $h\nu$. This has been explicitly verified by running simulations with different values for $\zeta$. This allows us to directly compare our results with experimental data obtained for the threshold current as a function of photon energy. Corresponding data measured on a TaN SNSPD with very similar parameters as used in our calculations [@Engel12] are plotted in the inset of Fig. \[Fig.Threshold\]. These measurements were done at $T=0.61$ K $\approx0.07\,T_c$. The original data were plotted as $1-I_{th}/I_c$, with $I_c$ the experimental critical current. We calculated the theoretical depairing critical current using the two-fluid temperature-dependence and the GL approximation (see Appendix \[App.T-dependence\]) resulting in $I_c/{ \ensuremath{I_{c,\mathrm{dep}}}}\approx0.85$, and we re-scaled the experimental threshold currents accordingly. A satisfactory description of the experimental data is clearly only possible with the simulation results from the vortex model. Using the method of least-squares we fitted the calculated threshold currents to the experimental data by re-scaling the photon energy and obtain a conversion efficiency $\zeta\approx0.24$ with a conservative error estimate of $\pm0.04$. The corresponding best fit is shown as the blue dotted line in the inset of Fig. \[Fig.Threshold\]. The linear relation between threshold currents and excitation energy have also been found in experiments with a variant of an SNSPD[@Renema13a] over a much larger range of energies. The fact that a deviation from the linear behavior is not seen in Ref. , even at very high excitation energies, may be a consequence of multi-photon absorption instead of absorbing a single photon with the same energy.
Threshold currents as a function of photon energy have also been calculated with NbN material parameters. A comparison with the results for TaN reveals qualitatively the same dependence of $I_{th}$ on the photon energy, but shifted to higher current values or higher photon energies, respectively. In a typical experimental situation, in which one chooses a fixed bias current and determines the minimum energy for direct photon detection, this energy turns out to be a factor $\approx 2.5$ larger in all there detection models for using NbN as detector material as compared to TaN. This comparison alone does not allow to distinguish between the considered detection models, but it confirms the experimentally observed differences between the cut-off wavelengths of TaN and NbN SNSPDs [@Engel12].
As mentioned in the introduction, it is not clear at present, how to correctly treat the vortex potential energy when the vortex resides within $\approx\xi$ of the strip edge. At the current stage this leaves some uncertainty, for example, with respect to the critical current for a vanishing vortex barrier. In terms of the photon detection, an improved vortex model could result in a small shift up or down of the blue data points in Fig. \[Fig.Threshold\], but we do not expect any fundamental change of the results presented here.
Trigger times as functions of photon energy
-------------------------------------------
![Time delay between absorption of a photon and trigger of the initial normal-conducting cross-section as a function of photon wavelength (lines are guides, only). Shown are simulation results for three different bias currents. The temperature was set to $T=0.05\,T_c$. The vortex mechanism always leads to a detection event well before the QP-model. For the chosen parameters, no detection event would be registered in the hard-core model.\[Fig.Trigger\]](Triggertime){width="\columnwidth"}
For certain combinations of bias current and photon energy, two or all three models allow for the direct detection of the photon. For $1-I_b/{ \ensuremath{I_{c,\mathrm{dep}}}}=0.2$ and $h\nu=3.13$ eV ($400$ nm), for example, the vortex and the QP-model result in the trigger of the initial normal conducting cross-section, but not necessarily at the same time after photon absorption. For a given bias current we evaluate the spatial QP-distributions as functions of time and wavelength and determine the instant $\tau_\mathrm{trigger}$ when the detection criterion is fulfilled in the three detection models. The results are plotted in Fig. \[Fig.Trigger\] for three different bias currents $I_b/{ \ensuremath{I_{c,\mathrm{dep}}}}=0.70,\ 0.74\ \text{and}\ 0.76$, corresponding to $I_b/I_{c,v}=0.85,\ 0.90\ \text{and}\ 0.925$, for the vortex (filled symbols) and QP-models (open symbols). Results for the hard-core model are not shown in Fig. \[Fig.Trigger\], because at the conversion efficiency $\zeta=0.25$ and the shortest considered photon wavelength $\lambda=300$ nm even higher bias-currents $I_b/{ \ensuremath{I_{c,\mathrm{dep}}}}>0.8$ or $I_b/I_{c,v}>0.97$ would be required to fulfill the detection criterion. Even if one assumes bias currents so close to $I_{c,v}$, it turns out that the detection criterion in the hard-core model will be reached for even longer delays after photon absorption than in the QP-model.
In the vortex model, at these high bias currents relative to the critical current for a vanishing vortex-entry barrier shown in Fig. \[Fig.Trigger\], only a very small number of excess QPs is necessary to alter the current distribution sufficiently to suppress the remaining energy barrier. Accordingly, the detection criterion in the vortex model is reached almost immediately after photon absorption. For longer wavelengths closer to the cut-off wavelength, the time delay increases towards $\approx 2.6$ ps, the time at which the maximum suppression of the energy barrier is expected. The trigger times in the vortex model are all smaller than $2.6$ ps in Fig. \[Fig.Trigger\], because of the fixed values of $\lambda$. For all currents and wavelengths investigated, the trigger times in the QP-model are significantly longer than in the vortex model. We may therefore conclude that the detection via the entry of a magnetic vortex is the primary mechanism to trigger the formation of a normal-conducting cross section, not only as a function of photon energy and bias current, but also temporally in situations when the other detection models might allow for a direct, but delayed, detection event.
Conclusions
===========
model $I_{th}$ $\lim\limits_{h\nu\rightarrow0} I_{th}$ $\tau_\mathrm{trigger}$
----------- -------------------------- ------------------------------------------------ ----------------------------------
hard-core $\propto\sqrt{h\nu-E_0}$ not detected $\tau_{hc}>\tau_{qp}>\tau_{v}$
QP $\propto h\nu$ $={ \ensuremath{I_{c,\mathrm{dep}}}}$ $\tau_v<\tau_{qp}\lesssim2.6$ ps
vortex $\propto h\nu$ $=I_{c,v}<{ \ensuremath{I_{c,\mathrm{dep}}}}$ $\lesssim2.6$ ps
: \[Tab.Summary\]Comparison of the results for the different detection models. Note on $\lim_{h\nu\rightarrow0} I_{th}$: The minimum photon energy that can possibly be detected equals to $2\Delta$.
We have presented a simple numerical model based on the diffusion of QPs generated after photon absorption in a thin superconducting film. The results were applied to predict the detection of an absorption event in SNSPD by comparing three currently considered detection mechanisms. All our results summarized in Table \[Tab.Summary\] are in favor of a vortex assisted detection mechanism, whereby the excess QPs lead to the suppression of the edge barrier for vortex entry and the subsequent dissipative crossing of a vortex. This process triggers the initial normal conducting domain in the superconducting strip. Competing mechanisms require higher photon energies and occur at a later stage of the QP multiplication and diffusion process.
We also compared our numerical results with experimental data. We obtain good agreement for the dependence of the threshold energy on the bias current and the different cut-off wavelengths for NbN and TaN SNSPD.
At the current stage our numerical model is by no means complete. Its aim is to capture the most important processes in the detection event of a photon in SNSPD and it still contains a number of simplifications and assumptions. Despite this simplicity it allows us to obtain a better understanding of the first stages of the detection process in SNSPD, which are otherwise difficult to probe experimentally. Further refinements of this model may allow for a better understanding of additional aspects of the detection process and for the development of a more complete theoretical description.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research received support from the Swiss National Science Foundation grant No. 200021\_135504/1 and 200021\_146887/1. We thank A. Semenov, J. Renema and D. Y. Vodolazov for fruitful discussions.
Analytic solution of partial differential equations\[App.Analytic\]
===================================================================
The set of partial differential equations and can be solved analytically for the case of a two-dimensional film and making the assumption $D_e=D_{qp}=D$. We then have
$$\begin{aligned}
\frac{\partial C_e(\vec{r},t)}{\partial t} &= D\nabla^2 C_e(\vec{r},t),\label{Eq.App.PDE_e}\\
\frac{\partial C_{qp}(\vec{r},t)}{\partial t} &= D\nabla^2 C_{qp}(\vec{r},t) - \frac{C_{qp}(\vec{r},t)}{\tau_r} + \frac{\zeta h\nu}{\Delta\tau_{qp}}\exp\left(-\frac{t}{\tau_{qp}}\right)C_e(\vec{r},t).\label{Eq.App.PDE_qp}\end{aligned}$$
Eq. has the well-known solution Eq. $$C_e(r,t) = \frac{1}{4\pi Dt}\exp\left(-\frac{r^2}{4Dt}\right),$$ and inserted into results in $$\frac{\partial C_{qp}(\vec{r},t)}{\partial t} = D\nabla^2 C_{qp}(\vec{r},t) - \frac{C_{qp}(\vec{r},t)}{\tau_r} + \frac{\zeta h\nu}{\Delta\tau_{qp}}\frac{1}{4\pi Dt}\exp\left(-\frac{t}{\tau_{qp}}\right)\exp\left(-\frac{r^2}{4Dt}\right),\label{Eq.App.inhomogeneous}$$
which is an inhomogeneous differential equation. It can be analytically solved yielding Eq. .
Temperature dependence of material parameters\[App.T-dependence\]
=================================================================
Calculations were done at a very low temperature $T\ll T_c$ to eliminate or at least reduce thermal effects. However, we could not assume $T=0$, because the diffusion coefficient of QPs $D_{qp}=0$ for $T\rightarrow0$. We calculated values for $T$-dependent parameters using the following expressions and methods.
The BCS $T$-dependence of the superconducting gap can be well approximated by the simple formula $$\Delta(\tau) = \Delta\left(1-\tau^2\right)^{0.5}\left(1+\tau^2\right)^{0.3},\label{Eq.App.Delta}$$ with $\tau=T/T_c$ the reduced temperature and $\Delta=\alpha k_BT_c$, where the BCS-value[@Tinkham96] $\alpha=1.764$ has been used for TaN and $\alpha=2$ for NbN[@Romestain04; @Henrich12a].
For the coherence length $\xi(\tau)$ we use an interpolation formula [@Bartolf10] $$\xi(\tau) = \xi\left(1-\tau\right)^{-0.5}\left(1+\tau\right)^{-0.25},\label{Eq.App.coherence}$$ with $\xi=\sqrt[4]{2}\xi_{GL}$ and $\xi_{GL}$ the extrapolated GL-coherence length at $T=0$. Eq. smoothly interpolates between the GL-result near $T_c$ and the estimated zero-temperature value for a dirty type-II superconductor at $T=0$[@Werthamer66].
The magnetic penetration depth in the dirty limit is given by [@Tinkham96] $$\lambda(\tau) = { \ensuremath{\lambda_{\mathrm{GL}}}}\left(\frac{\Delta(\tau)}{\Delta}\tanh\left[\frac{\Delta(\tau)}{2k_B\tau T_c}\right]\right)^{-0.5},$$ with $k_B$ the Boltzman constant.
The theoretical depairing critical current is calculated using the two-fluid temperature dependence and the GL approximation $${ \ensuremath{I_{c,\mathrm{dep}}}}=\frac{\Phi_0 wd}{3\pi\sqrt{3}\mu_0{ \ensuremath{\lambda_{\mathrm{GL}}}}^2\xi}\left(1-\tau^2\right)\left(1-\tau^4\right)^{0.5}.$$
A simple analytical formula for the temperature dependence of the normalized diffusion coefficient for QPs in the superconducting state can be derived in the limiting case of $T\ll T_c$ ($\tau\lesssim0.1$) that resembles the diffusion of an ideal gas. For the thermal conductivity we use the approximation [@Abrikosov88] $$\kappa_s \approx 2 N_0DT\left(\frac{\Delta}{k_B T}\right)^2\exp\left(-\frac{\Delta}{k_B T}\right),$$ which, divided by $\kappa_n(T_c)$, results in $$\tilde{\kappa}(\tau) \approx \left(\frac{\Delta}{k_B T_c}\right)^2\frac{6}{\pi^2\tau}\exp\left(-\frac{\Delta}{k_B T_c\tau}\right).$$
An expression for the normalized specific heat at $T\ll T_c$ has already been given in Ref. ,
$$\tilde{C}(\tau)=\frac{C_s(T)}{C_n(T_c)}\approx \frac{6}{\pi^2}\sqrt{\frac{\pi}{2}}\left(\frac{\Delta}{k_B T_c}\right)^{2.5}\tau^{-1.5}\exp\left(-\frac{\Delta}{k_B T_c\tau}\right),$$
where we have already replaced the modified Bessel-functions by their first order approximation for large arguments $\frac{\Delta}{k_B T_c\tau}\gg1$. It is then easy to see that $$\frac{\tilde{\kappa}(\tau)}{\tilde{C}(\tau)} = \frac{D_{qp}(\tau)}{D_e(T_c)} = \sqrt{\frac{2}{1.764\pi}}\sqrt{\tau}\approx 0.6\sqrt{\tau}.\label{Eq.App.lowTApprox}$$
Calculation of single-vortex potential for inhomogeneous current distribution\[App.VortexPotential\]
====================================================================================================
Eq. is the potential energy of a single-vortex inside a current-carrying superconducting strip with $d\ll w\ll\Lambda$ and a homogeneous current density. One may calculate the force on a vortex inside the strip, $\vec{F}=-\vec{\nabla}U$, which by symmetry has only a component in the $y$-direction, $$F_y(y) = \varepsilon_0\left[\frac{\pi}{w}\tan\left(\frac{\pi y}{w}\right) + \frac{2w j_x}{I_{c,v}\exp(1)\xi}\right],\label{Eq.App.Force_homo}$$ with $j_x=I/w$ the thin-film current-density in the $x$-direction. The first term on the right-hand side of is the Lorentz-force $j\Phi_0$ on a vortex caused by the current-density from the chain of image vortices and antivortices, necessary to fulfill the boundary condition at the strip edges, namely that the current density can only have a longitudinal component at the edges. The second term is the Lorentz-force exerted by the applied bias current.
After absorption of a photon and the creation of the diffusing QP-cloud, the current density becomes a function of $x$ and $y$, $j_x(x,y)$, and the energy scale $\varepsilon_0$ becomes also a function of vortex position. It may be expressed as $$\varepsilon_0(x,y) = \frac{\Phi_0^2}{2\pi\mu_0\Lambda} = \frac{\Phi_0^2 e^2n_{se}^{2D}(x,y)}{4\pi m_e}\label{Eq.App.SelfEnergy},$$ where we have used the definition of the London penetration depth in the last expression, with $\left|e\right|$ the elementary charge and $m_e$ the electron mass. Replacing the $j_x$ and $\varepsilon_0$ by their position dependent counterparts in Eq. and dividing it by the equilibrium value $\varepsilon_0$, the force on a vortex becomes
$$\frac{F_y(x,y)}{\varepsilon_0} = \frac{n_{se}^{2D}(x,y)}{n_{se}^{2D}}\left[\frac{\pi}{w}\tan\left(\frac{\pi y}{w}\right) + \frac{2w j_x(x,y)}{I_{c,v}\exp(1)\xi}\right].\label{Eq.App.Force_inhom}$$
The potential energy $U(x,y)/\varepsilon_0$ can then be calculated from by numerical integration over $y$ for fixed $x$: $$\frac{U(x,y)}{\varepsilon_0} = \frac{\pi}{w}\int_{\frac{\xi-w}{2}}^{\frac{w-\xi}{2}}\frac{n_{se}^{2D}(x,y)}{n_{se}^{2D}}\tan\left(\frac{\pi y}{w}\right)\mathrm{d}y + \frac{2w}{I_{c,v}\exp(1)\xi}\int_{-\frac{w}{2}}^{\frac{w}{2}}\frac{n_{se}^{2D}(x,y)}{n_{se}^{2D}}j_x(x,y)\mathrm{d}y,$$ where the integration limits have been chosen to obtain the same normalization of $U(x,y)$ as used in Ref. .
[^1]: Calculations were done at $T/T_c=0.05$ resulting in a very low concentration of thermal QPs
[^2]: It has to be expected that the recombination is proportional to $C_{qp}(\vec{r},t)^2$, since two QPs are required. Because $\tau_r\gg\tau_{qp}$ recombination has a minor influence on the detection process which happens on a time-scale $\sim\tau_{qp}$. Therefore we use the linear approximation resulting in a set of linear differential equations.
[^3]: The density of superconducting electrons $n_{se}$ has been estimated from the London penetration depth $\lambda_L^2=m_e/(\mu_0n_{se}e^2)$.
[^4]: If a normal conducting core forms, we set the local density of superconducting electrons equal zero inside the core.
|
---
abstract: 'We present a theoretical analysis that associates the resonances of extraordinary acoustic Raman (EAR) spectroscopy \[Wheaton et al., Nat Photon 9, 68 (2015)\] with the collective modes of proteins. The theory uses the anisotropic elastic network model to find the protein acoustic modes, and calculates Raman intensity by treating the protein as a polarizable ellipsoid. Reasonable agreement is found between EAR spectra and our theory. Protein acoustic modes have been extensively studied theoretically to assess the role they play in protein function; this result suggests EAR as a new experimental tool for studies of protein acoustic modes.'
author:
- Timothy DeWolf and Reuven Gordon
bibliography:
- 'nmaPaper.bib'
title: Theory for the Acoustic Raman Modes of Proteins
---
The central dogma of molecular biology involves one-way information transfer from DNA to protein, a process that directly and reliably associates a particular three-dimensional structure with a given amino acid sequence. Each structure is associated with a function (or functions); often, for example, a particular structure catalyzes a chemical reaction with remarkable selectivity. The vibrational modes of a protein reflect its structure and conformation, and are thought to facilitate allostery and conformational change [@nmafunc; @nmabook; @Yang2007920; @Dobbins29072008; @coupl; @Tama01012001]. Of these modes, those with the lowest frequency are termed acoustic modes, and represent the largest thermal fluctuations of the protein.
Whereas many spectroscopic methods can probe localized resonances in a protein [@ref:proteinnmr; @ref:prinInstrumentalAnalysisBook], delocalized collective modes and their role in biological function have been historically difficult to measure [@thzOke]. In the gigahertz (GHz) to low terahertz (THz) spectral window, electromagnetic absorption experiments have to deal with high solvent absorption and dielectric mixtures [@thzAbs1; @thzAbs2; @ghzdielecspec]. Other experimental techniques for studying acoustic protein modes include inelastic incoherent neutron scattering (IINS) [@iinsprotein] and optical Kerr-effect (OKE) spectroscopy [@thzOke].
{width=".05\textwidth"} {width=".05\textwidth"}
We recently reported extraordinary acoustic Raman (EAR) spectroscopy as a way to measure resonances of optically trapped nanoparticles [@natsky]. In EAR, the $\sim$10 to 100 GHz beating of two trapping lasers creates increased RMS fluctuation when the beat frequency matches a Raman-active particle resonance. The frequency of vibrational modes in single polystyrene nanospheres were shown to fit with Lamb’s theory. The EAR spectra of several proteins were measured, however the remaining challenge is “...to associate the observed \[protein\] resonances with specific motions..." [@natreply]. Here we propose a theory that assigns the measured EAR modes to low-frequency Raman-active protein modes.
Our theory uses elastic network model (ENM) normal mode analysis; elastic network models reproduce the essential dynamics of low-frequency protein modes to good accuracy [@PhysRevLett.77.1905]. We use the ENM known as the anisotropic network model (ANM) [@anm2; @Atilgan2001505], as implemented in ProDy [@Bakan01062011].
ANM represents the potential surface of an $N$ atom protein (excluding hydrogen) using a network of springs with spring constant $k$. ANM analyses are often done using a reduced set of atoms, namely the $C_\alpha$ atoms along the protein backbone; we use an all-atom approach (excluding hydrogen) to build the elastic network. Each spring connects a pair of atomic coordinates, but only atoms within cutoff radius $r_c$ are connected. The matrix of second derivatives (taken with respect to the Cartesian coordinates of each atom) of this potential, known as the Hessian, is then computed; it is an $N\times N$ matrix of $3 \times 3$ (the protein coordinates are in $\mathbb{R}^3$) super-elements and has the units of $k$. The diagonalization of this matrix yields $3N-6$ non-zero eigenvalues $\lambda_i$ and eigenvectors $Q_i$ that correspond to the frequency $\omega_i = \sqrt{ \lambda_i / m }$ and the displacement from equilibrium of each mode $i$. $m$ is the mass of an atom; we use 13.2 amu for all atoms (a weighted average). The six zero-valued eigenvalues correspond to rotational and translational degrees of freedom. Fig. \[fig:mode\] shows what one of these eigenvectors $Q_i$ looks like for a protein. At this point, we have a set of mode frequencies, as shown in Fig. \[fig:eigvals\].
To calculate the intensity of each mode, we need to compute the Raman intensity of a given ANM mode from positive and negative coordinate displacements $\vec{r}_{i,\pm}=\vec{r}_0\pm \beta \vec{Q}_i$. The equilibrium coordinates are $\vec{r}_0$; $\vec{Q}_i$ is a unit vector in $\mathbb{R}^{3N}$ and $\beta$ is a small scaling parameter. We construct a quantity closely related to the inertia tensor, and diagonalize it to find the semi-principle axes (unit vectors) and lengths $a_{i,\pm},b_{i,\pm},c_{i,\pm}$ of two best-fitting ellipsoids [@0004-637X-548-1-68], one for each of the stretched protein coordinates. Dielectric polarizability tensors $\alpha_{i,\pm}$ for these best-fitting ellipsoids are then computed, using analytic expressions [@sihvola1999electromagnetic; @doi:10.1080/14786444508521510] that require only the semi-principle axis lengths and the internal and external relative dielectric permittivity. We take the two permittivity values to be $\epsilon_i=n^2=1.6^2$ (protein) [@MCMEEKIN1962151] and $\epsilon_e=1.33^2$ (water).
**PDB ID**
------------------------------ ------- ------------------------- ------ --
pancreatic trypsin inhibitor 6.6 5PTI [@Wlodawer1984301] 1.51
carbonic anhydrase I 29.7 1CRM [@NYAS:NYAS49] 1.28
streptavidin 52.8 3RY2 [@3RY2] 1.29
ovotransferrin 76.2 1OVT [@Kurokawa1995196] 0.79
cyclooxygenase-2 274.4 5COX [@Kurokawa1995197] 1.15
The Raman polarizability $\alpha'_i = ({\partial}\alpha_i / {\partial}Q_i)_0$ measures the change in polarizability due to the difference between the protein coordinates $\vec{r}_{i,\pm}$ [@1970raman]. We calculate this as $\alpha'_i \sim \alpha_{i,+} -\alpha_{i,-}$. We also account for the possibility that the two best-fit ellipsoids have rotated under the action of the pair of mode displacements by rotating one of the polarizability tensors as a rank two tensor, using the rotation matrix formed using the pair of best-fit semi-principle axes [@hand1998analytica]. With this, the Raman intensity $I_i$ of ANM mode $i$ is given by [@1970raman] [$$\label{ramani} I_i \sim 45 \bar{\alpha}'^2_i + 4\gamma'^2_i$$]{} where [$$\begin{aligned}
\bar{\alpha}_i' &= \frac{1}{3} (\alpha'_{i,xx} + \alpha'_{i,yy} + \alpha'_{i,zz} ) \\
\gamma'^2_i &= \frac{1}{2} ( (\alpha'_{i,xx} - \alpha'_{i,yy} )^2 + (\alpha'_{i,yy} - \alpha'_{i,zz} )^2{\nonumber \\}& + (\alpha'_{i,zz} - \alpha'_{i,xx} )^2 +6(\alpha'^2_{i,xy}+\alpha'^2_{i,yz}+\alpha'^2_{i,zx} ) ). \end{aligned}$$]{} The mean value $\bar{\alpha}'_i$ measures change in polarizability of a particular mode due to linear stretching; $\gamma'_i$ gives the anisotropic contribution. Spectra (see for example Fig. \[fig:ramansel\]) are constructed by centering Lorentzian functions at the frequency position $\omega_i$ of each mode, with mode heights proportional to the Raman intensities $I_i$ (and plotting the summation of these curves). We choose a constant Lorentzian linewidth for each spectrum. We compare our theory with previously published experimental data [@natsky] for the five proteins listed in Table \[proteins\], and list the Research Collaboratory for Structural Bioinformatics Protein Data Bank (RCSB PDB) [@Berman01012007] structures used in computation. (The streptavidin data was not published but was acquired during the same period.) It is assumed that the PDB crystal coordinates are close to the potential minimum (i.e. that the crystal coordinates approximately give $\vec{r}_0$). By matching the atomic mean-square fluctuations predicted by ANM (calculated using ProDy) with the crystallographic isotropic temperature factors included with the PDB crystal data, we associate a $k$ with each protein [@Atilgan2001505]. These $k$ values are shown in Table \[proteins\]. Details of these ANM, spring constant, ellipsoid fitting and Raman calculations are given in the supporting information.
**EXPERIMENT THEORY** {width=".93\textwidth"} {width=".93\textwidth"} {width=".93\textwidth"} {width=".93\textwidth"} {width=".93\textwidth"}
The ANM cutoff distance $r_c=7.9$ Å$ $ was selected by hand for best overall agreement between theory and experiment for the five proteins. At values near this $r_c$, EAR mode frequencies $\tilde{\omega}_i$ and ANM frequencies $\omega_i$ are approximately linearly proportional: $\tilde{\omega}_i = \zeta \omega_i$. The proportionality constant $\zeta$ is a free parameter in our theory; a $\zeta$ is chosen for each protein (see supporting information). It is the fine spectral resolution of EAR that allows us to directly fit the spectral modes for each protein; in prior works obtaining a Gaussian-distributed density of states has been used as a criterion for selecting physical values for $r_c$ [@Atilgan2001505].
The computed spectra along with our previously obtained experimental EAR spectra are shown in Fig. \[fig:theoryexpPlots\]. The visual agreement between theory (right side) and experiment (on the left) in Fig. \[fig:theoryexpPlots\] is, in our opinion, quite remarkable. The intensity and frequency placement of the major peaks, as well as some of the minor peaks, agree with the experimental data. We have made many approximations, including the use of a “spring network" potential in place of a more realistic potential map (e.g. the semiempirical potentials employed in molecular dynamics) and the representation of protein polarizability by the polarizability of a dielectric ellipsoid. The normal modes could also be computed in the time domain, by combining molecular dynamics simulation with principal component analysis to accurately capture the low-frequency modes [@mdstructurefunction]. Wider bandwidth Raman intensity spectra for each protein are given in the supporting information. As can be seen in the extended spectra, the EAR data shown here have captured most of the major Raman-active collective modes in these five proteins.
In the light of our theory, we make some comments regarding the experiment. A past optical trapping work by our group [@doi:10.1021/nl203719v] reported on a special type of conformational change, the N-F transition, found in bovine serum albumin (BSA); this conformation change can be viewed as a type of denaturation as it involves a reversible unfolding of BSA domain III [@rosenoer2014albumin; @bsad]. BSA can also be irreversibly denatured [@FEBS:FEBS469]. We assume here, as we did in the EAR experiment [@natsky], that the trapped proteins have not been irreversibly denatured. We also assume that the proteins are not being reversibly unfolded or deformed by the optical forces, so that the equilibrium coordinates given by the x-ray crystal data will be good approximations of the optically trapped protein coordinates. Past works (e.g. [@anha1; @anha2]) have suggested that anharmonic effects play a role in the low frequency modes of proteins, whereas our ANM-based theory is a purely harmonic model of protein motion. In a Duffing oscillator, for example, the absence of nonlinear effects such as jumping, hysteresis, and bistability can be related to the fact that harmonic driving force is sufficiently weak [@enns2012nonlinear Chapter 7]. Thus an explanation for the seeming unimportance of anharmonicity in our theory is that the amplitude of the driving force is low enough that the protein response is linear.
There has hitherto been relatively little experimental evidence for the existence of protein collective modes—accomplishments in protein THz spectroscopy have not been able to conclusively connect measurements with biologically relevant collective protein motions [@thzOke]. There has also been difficulty in assigning physical frequency units to ENM. These results directly connect ENM mode analysis to EAR. They provide another way to validate ENM results, and suggest EAR as a new tool for future experimental studies of low-frequency protein collective modes. They also suggest that EAR may provide a way to improve ENMs.
We would like to acknowledge the use of the computational resources of WestGrid (www.westgrid.ca) and Compute Canada (www.computecanada.ca). This work was supported in part by an NSERC Discovery Grant and funding from the Faculty of Graduate Studies at the University of Victoria. The protein renderings were prepared using PyMOL [@PyMOL] and POV-Ray [@pov].
|
---
abstract: 'We propose a new mechanism called “real CP violation" to originate spontaneous CP violation. Starting with a CP conserving theory with scalar fields in the adjoint representation of a global or local non-abelian symmetry, we show that even though the VEV’s of such scalars are real they give rise to a spontaneous violation of CP. We provide an illustrative example of how this new mechanism of CP violation can give rise to physically significant phases which produce a complex CKM mixing matrix. This mechanism may prove useful in string models with moduli in the adjoint representation as well as in tackling the strong CP problem.'
address:
- 'SISSA – ISAS, Trieste and INFN, Sez. Trieste, Italy'
- 'Department of Physics and RESCEU, Univ. of Tokyo, Tokyo 113-0033, Japan'
author:
- 'A. Masiero'
- 'T. Yanagida'
title: Real CP Violation
---
Since its experimental discovery in 1964, many mechanisms to originate CP violation in K physics have been proposed. They can be grouped into two classes: explicit and spontaneous CP violation. In the former case the Lagrangian describing electroweak interactions contains some terms which are not CP invariant. For instance, some Yukawa couplings may be complex giving rise to a complex CKM matrix after diagonalization of the fermion mass matrices. On the contrary, in the spontaneous option one starts with a CP invariant Lagrangian, but the vacuum of the theory is not CP invariant [@lee]. Typically one has some scalar fields developing complex vacuum expectation values (VEV’s) with some phases remaining after exploiting all the invariances of the theory. These physical phases appear in the quark mass matrices giving rise once again to a complex CKM matrix.
The possibility that CP is broken spontaneously is quite attractive. Still lacking the underlying theory explaining the origin of the Yukawa couplings, in the explicit case we introduce CP violation “by hand" in these complex couplings. Moreover if one decides that CP is not a good symmetry of the theory since the beginning, one may expect arbitrarily large violations of CP also in the strong interactions due to the presence of the $\theta$ term in the QCD Lagrangian [@rebbi]. On the other hand, the spontaneous breaking of CP by the vacuum of the theory is more linked to the “dynamics" of the theory itself and, if CP is a good symmetry to start with, the $\theta$ term has to be vanishing in the initial Lagrangian [@strongcp]. Obviously this fact does not imply by itself that the strong CP problem [@peccei] is solved since the subsequent spontaneous violation of CP with phases in the quark mass matrices in general gives rise to an effective $\theta$ which is too large.
Here we come back to the idea that CP is broken only spontaneously. We propose a new mechanism for this breaking which does not entail the request of having complex VEV’s of the scalar fields. For this reason we call it “real" CP violation and we show that it can be generally applied in theories with non-abelian global or gauge symmetries. The key-ingredient is to have a set of scalars sitting in the adjoint representation of these symmetries. Then, even if these scalars have real VEV’s (which is generally the case for the real fields in the adjoint), CP is broken by the vacuum of the theory.
Apart from the interest in itself of this new mechanism for spontaneous CP violation, we think that there are potentially relevant applications. In particular it may give rise to a source of CP violation in string theories with moduli in the adjoint representation [@bachas] and it can be relevant for the solution of the long-standing problem of strong CP violation [@peccei]. We will elaborate more on this latter aspect in the second part ot this Letter.
First we introduce the mechanism of “real" CP violation. The way one defines CP transformations in the presence of a non-abelian symmetry presents an important difference with respect to the usual way CP is defined in the abelian case, say in QED. For simplicity, consider an SU(2) fermionic current coupled to the triplet of vector bosons $W_i$, $i=1,2,3$. The demand that this interaction lagrangian be invariant under CP entails that $W_3$ and $W_1$ transform into themselves, while $W_2$ has to go into $-W_2$ under a CP transformation. This is equivalent to say that, having defined $W^+$ and $W^-$ in terms of $W_1$ and $W_2$ in the usual way, CP interchanges $W^+$ and $W^-$. Consider now that we replace the $W$ vector bosons with an SU(2) triplet of real scalar fields $\phi$. Once again the presence of $\tau_2$ in the SU(2) generators with $(\tau_2)^T=-(\tau_2)$, implies that under a CP transformation the second component $\phi_2$ of the scalar triplet has to be odd if the interaction respects CP invariance. Hence, a VEV of this scalar component, although it is obviously real, leads to a spontaneous breaking of CP. The key-point is that in the non-abelian case some of the generators are anti-symmetric and the corresponding scalar components of the adjoint representation have to be odd under CP if we want to find a consistent definition of CP to have the interaction lagrangian invariant under it.
We now come to the second task of this Letter, namely we show that the abovementioned mechanism of “real” CP violation can produce physical phases which show up at the level of the CKM quark mixing matrix. To this goal, we provide an illustrative example based on a horizontal $SU(3)_H$ symmetry which may be global or gauged. We introduce three scalar octets, that we generically denote with $\phi$ and a singlet $\phi_0$. As for fermions, consider the 2 vector-like triplets $U_{(L,R)}$ and $D_{(L,R)}$ which are singlets under the $SU(2)$ of the standard model (SM) and triplets of the colour $SU(3)$ symmetry. They can get a direct large mass $M_U$ and $M_D$, respectively. The enforcement of CP violation ensures that these masses are real. Let us now make the connection to the low-energy part of the model with the usual $u$ and $d$ SM quarks. Also $u$ and $d$ are triplets under $SU(3)_H$. Hence we can write the Yukawa couplings of the right-handed components of $u$ and $d$ with the left-handed components of the corresponding $U$ and $D$ and the above $\phi$ fields. Since we ask for CP conservation all these couplings are real. Notice that $u_R$ and $d_R$ have the same quantum numbers of $U_R$ and $D_R$. Since we want to avoid that the previous Yukawa terms put into communication also the right-handed components of $U$ and $D$ with their left-handed counterparts, we impose a discrete symmetry under which $u_R$, $d_R$ and all the $\phi$ fields are odd, while $U$ and $D$ are even. Finally we introduce also the usual SM Higgs doublet $H$. We now have the new couplings of H with $U_R$, $D_R$ and $u_L$, $d_L$. Then, the tree level exchange of D gives rise to the effective interactions :
$$L_{eff} = \frac{\bar{d_R} (g_d \phi^a \lambda^a + g_d^{'}\phi_0)q_LHh_d}{M_D},
\label{eq:eff}$$
where $g_d$, $g_d^{'}$ and $h_d$ denote the Yukawa couplings with $\phi$, $\phi_0$ and $H$, respectively, $\lambda^a$ are the Gell-Mann matrices of $SU(3)_H$, $M_D$ is the direct mass of $D$ and, finally, $q_L$ is the usual doublet of the left-handed up- and down-quarks. Analogous contributions to the up quark sector arise with the different Yukawa couplings $g_u$, $g_u^{'}$ and $h_u$ . When the scalar fields get a VEV, the above $L_{eff}$ produce mass matrices for the up- and down-quarks which are hermitian. The presence of three $\phi$ octets assure that all components, in particular those related to the antisymmetric Gell-Mann matrices, get a nonvanishing VEV. Hence the quark mass matrices possess three phases. It is easy to see that one combination of them can never be reabsorbed by redefining the quark fields. Thus the CKM phase appears.
The fact that the quark mass matrices although complex are hermitean suggests that the “real" CP violation may prove useful in tackling the strong CP problem. Actually for the $\theta$ problem we need the full quark matrix involving both the ordinary and the heavy new quarks $U$ and $D$. For instance, if we consider the down sector, we have the following renormalizable interactions and mass matrix:
$$(\bar{d_R} \bar{D_R})
\left( \begin{array}{cc}
0 & \phi + \phi_0 \\
H & M_D
\end{array} \right)
(d_L D_L)^T,
\label{eq:mass}$$
where the integration of the heavy $D_{R,L}$ fields induces $L_{eff}$ in eq. (\[eq:eff\]). $H$, $\phi$ and $\phi_0$ denote the mass terms coming from the VEV’s of $H$, $\phi$ and $\phi_0$, respectively. Notice that the VEV of $H$ can always be made real by performing a $U(1)$ hypercharge rotation on $H$. Given the hermiticity of the matrix block $\phi$, we conclude that the determinant of the above mass matrix is real.
The $\theta$ term of the QCD lagrangian vanishes because of the initial CP invariance of the theory, while the contribution to the effective $\theta$ arising from the rotation of quark fields to bring them to the physical basis vanishes at the tree level since it is proportional to the argument of the determinant of the above quark mass matrix. The point now is that the phenomenologically required smallness of $\theta$ requires the quark mass matrix hermiticity to be spoiled by very tiny effects [@strongcp]. This computation would require the detailed formulation of a model which is beyond the scope of this Letter. Here we limit ourselves to a few comments on the possible suppression of the contributions giving rise to a nonvanishing effective $\theta$.
The dangerous corrections spoiling the hermiticity of the quark mass matrices arise from loop contributions involving the presence of quartic terms in the $\phi$ fields as well as from terms of the kind ${\phi}^2
H H^*$. Having the scale of $SU(3)_H$ breaking large compared to the electrowek scale, the couplings of the latter terms have to be small not to create a hierarchy problem for the $H$ mass. If we ask also for the quartic couplings to be small, we get relatively low masses for the $\phi$ scalars, for instance in the TeV region, where they can become accessible in next hadronic accelerators. Notice that the demand that the coefficients of the $\phi$ quartic terms be small may be naturally accomplished if the scale of the new physics is large enough to allow for a substantial running of such couplings.
After eliminating the danger represented by the above corrections proportional to the quartic scalar terms, we are left with only the Yukawa couplings for the fermions. However, such couplings at the loop level can induce only the renormalization of the fermion wave functions. Calling $Z$ and $Z'$ such wave function renormalizations of the right- and left-handed fields in eq. (\[eq:mass\]), we obtain:
$$(\bar{d_R} \bar{D_R})Z
\left( \begin{array}{cc}
0 & \phi+\phi_0 \\
H & M_D
\end{array} \right)
Z'(d_L D_L)^T.
\label{eq:Z}$$
$Z$ and $Z'$ have to be hermitean. Hence the determinant of the whole matrix in eq. (\[eq:Z\]) remains real and, thus, there is no contribution to a non-vanishing $\theta$ from these terms.
Another possibility for suppressing the hermiticity-breaking corrections could be to supersymmetrize the proposed scheme. Then the radiative corrections to the fermion mass matrices would be suppressed by at least two powers of the ratio of the scale of low-energy SUSY breaking to the large scale of the theory. Corrections leading to fermionic wave function renormalization do not enjoy such a kind of protection, however, following the above mentioned argument, we conclude that they do not give rise to a non-vanishing $\theta$. However, in SUSY theories the chiral supermultiplets are complex even if they belong to the adjoint representation and, hence, they may have complex VEV’s in general spoiling the hermiticity of the mass matrices. We need some dynamical reason for them to take only real VEV’s.
In conclusion, we have proposed the new mechanism of “real" CP violation to account for spontaneous breaking of CP in models with scalar fields in the adjoint representation of some global or local non-abelian symmetry. The mechanism allows for spontaneous CP violation even though no complex VEV occurs. The resulting CP violating phases leak to the fermionic mixing sector giving rise to a welcome complex CKM matrix. We pointed out that this idea may find interesting applications in those string theories with moduli in the adjoint representation as well as in tackling the strong CP problem in the context of the spontaneous CP proposals. The illustrative example that we offered shows that it may be of interest to pursue in this direction to build a complete model of real CP violation in the non-SUSY or SUSY contexts.
We thank W. Buchmueller for interesting discussions and the DESY Theory Group for its kind hospitality. The work of A.M. is partially supported by the EEC TMR Project “BSM”, Contract Number ERBFMRX CT96 0090.
T.D. Lee, [*Phys. Rev.* ]{}[**D8**]{} (1973) 1226;\
R. Mohapatra and J. Pati, [*Phys. Rev.* ]{}[**D11**]{} (1975) 566.
G. ’t Hooft, [*Phys. Rev. Lett.* ]{}[**37**]{} (1976) 8;\
R. Jackiw and C. Rebbi, [*Phys. Rev. Lett.* ]{}[**37**]{} (1976) 172;\
C. Callan, R. Dashen and D. Gross, [*Phys. Lett.* ]{}[**63B**]{} (1976) 334.
M. Beg and H. Tsao, [*Phys. Rev. Lett.* ]{}[**41**]{} (1978) 278;\
H. Georgi, [*Had. J.* ]{}[**1**]{} (1978) 155;\
R. Mohapatra and G. Senjanovic, [*Phys. Lett.* ]{}[**126B**]{} (1978) 283;\
G. Segre and H. Weldon, [*Phys. Rev. Lett.* ]{}[**42**]{} (1978) 1191;\
S. Barr and P. Langacker, [*Phys. Rev. Lett.* ]{}[**42**]{} (1978) 1654;\
A. Nelson, [*Phys. Lett.* ]{}[**136B**]{} (1984) 165;\
S. Barr, [*Phys. Rev. Lett.* ]{}[**53**]{} (1984) 329;\
S. Barr and D. Seckel, [*Nucl. Phys.* ]{}[**B233**]{} (1984) 116;\
S. Barr and A. Zee, [*Phys. Rev. Lett.* ]{}[**55**]{} (1985) 2253;\
A. Dannenberg, L. Hall and L. Randall, [*Nucl. Phys.* ]{}[**B271**]{} (1986) 157;\
S. Barr and A. Masiero, [*Phys. Rev.* ]{}[**D38**]{} (1988) 366;\
S. Barr and G. Segre, [*Phys. Rev.* ]{}[**D48**]{} (1993) 302;\
M. Dine, R. Leigh and A. Kagan, [*Phys. Rev.* ]{}[**D48**]{} (1993) 2214.
R. Peccei and H. Quinn, [*Phys. Rev. Lett.*]{}[**38**]{} (1977) 1440.
C. Bachas, C. Fabre and T. Yanagida, [*Phys. Lett.* ]{}[**B370**]{} (1996) 49.
|
---
abstract: 'We record $\binom{42}2+\binom{23}2+\binom{13}2=1192$ functional identities that, apart from being amazingly amusing by themselves, find applications in derivation of Ramanujan-type formulas for $1/\pi$ and in computation of mathematical constants.'
address:
- 'Institute of Natural and Mathematical Sciences, Massey University—Albany, Private Bag 102904, North Shore Mail Centre, Auckland 0745, New Zealand'
- 'Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany'
- 'School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan NSW 2308, Australia'
author:
- Shaun Cooper
- Wadim Zudilin
title: Hypergeometric modular equations
---
It is all about $\pi$.
Introduction and statement of results {#sec1}
=====================================
One of the best known results of Ramanujan is his collection of series for $1/\pi$ in [@ramanujan_pi]. One of memorable achievements of Jonathan and Peter Borwein was proving the entries in Ramanujan’s collection some 70 years later [@agm]. As a representative example, we quote the series [@ramanujan_pi Eq. (29)] $$\label{pi01}
\sum_{n=0}^\infty \frac{(\frac12)_n^3}{n!^3} \biggl(n+ \frac{5}{42}\biggr)
\biggl(\frac{1}{2}\biggr)^{6n}
= \frac{1}{\pi} \times \frac{8}{21}.$$ Here and below we use $$(s)_n=\frac{\Gamma(s+n)}{\Gamma(s)}=\begin{cases}
s(s+1)\dotsb(s+n-1) & \text{if $n=1,2,\ldots$}\,, \\
1 & \text{if $n=0$},
\end{cases}$$ for the Pochhammer symbol (or shifted factorial), as well as the related notation $${ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{m+1}F_{m}\biggl(\genfrac..{0pt}{}{a_0,a_1,\dots,a_m}{b_1,\dots,b_m};x\biggr) \endgroup
}
=\sum_{n=0}^\infty\frac{(a_0)_n(a_1)_n\dotsb(a_m)_n}{n!\,(b_1)_n\dotsb(b_m)_n}\,x^m
\label{pFq}$$ for the generalized hypergeometric function.
Ramanujan’s series for $1/\pi$ may all be expressed in the form $$\biggl(\alpha + x\,\frac{{{\mathrm d}}}{{{\mathrm d}}x}\biggr)
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,s,1-s}{1,1};x\biggr) \endgroup
}\bigg|_{x=x_0}=\frac{\beta}{\pi}$$ where $\alpha$, $\beta$ and $x_0$ are algebraic numbers and $s \in \{\frac16,\frac14,\frac13,\frac12\}$. In the example , we have $$\alpha=\frac{5}{42}, \quad \beta=\frac{8}{21}, \quad x=\frac{1}{64} \quad\text{and}\quad s=\frac12.$$ Behind such identities there is a beautiful machinery of modular functions, something that was hinted by Ramanujan a century ago and led to the earlier and later proofs [@agm; @domb; @Chud].
A more recent investigation on interrelationships among Ramanujan’s series in the works of several authors [@aycock; @mathematika; @translation; @rogers; @Zu; @Zu5] suggests considering transformation formulas of the type $$\label{generaltransformation}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,s,1-s}{1,1};x(p)\biggr) \endgroup
} = r(p)\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,t,1-t}{1,1};y(p)\biggr) \endgroup
},$$ where $x(p)$, $y(p)$ and $r(p)$ are algebraic functions of $p$, and $s,t \in \{\frac16,\frac14,\frac13,\frac12\}$. For example, Aycock [@aycock] was mainly interested in using instances of to derive formulas such as . The discussion of transformation formulas of the type in [@aycock pp. 15–16] contains just eight formulas that were obtained by searching the literature. Just to mention a few, we list the examples[^1] $$\begin{aligned}
\label{eg0}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,\frac12,\frac12}{1,1};x\biggr) \endgroup
}
&= \frac{2}{\sqrt{4-x}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac16,\frac12,\frac56}{1,1};\frac{27x^2}{(4-x)^3}\biggr) \endgroup
},
\\
\label{eg1}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};\frac{256x}{(1+27x)^4}\biggr) \endgroup
}
&= \frac{1+27x}{1+3x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};\frac{256x^3}{(1+3x)^4}\biggr) \endgroup
}
\\ \intertext{and}
\label{eg2}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};4x(1-x)\biggr) \endgroup
}
&= \frac{3}{\sqrt{9-8x}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac16,\frac12,\frac56}{1,1};\frac{64x^3(1-x)}{(9-8x)^3}\biggr) \endgroup
}.\end{aligned}$$
We should point out that algebraic transformations of hypergeometric functions, in particular, of modular origin, are related to the monodromy of the underlying linear differential equations. This is a reasonably popular topic, with Goursat’s original 140-page contribution [@goursat] as the starting point. See [@almkvistetal; @maier; @vidunas] for some recent developments.
The goal of this work is to systematically organize and classify identities of the type . In particular, we will show that the functions that occur in – are part of a single result that asserts that forty-two functions are equal. Our results also encapsulate identities such as $$\label{pb}
f_{6b}(x)=\frac{1}{1-x}f_{6c}\biggl(\frac{x}{1-x}\biggr)
\quad\text{and}\quad
f_{6c}(x)=\frac{1}{1+x}f_{6b}\biggl(\frac{x}{1+x}\biggr)$$ where $$f_{6b}(x)=\sum_{n=0}^\infty \biggl\{\sum_{j+k+\ell=n} \biggl(\frac{n!}{j!k!\ell!}\biggr)^2\biggr\}x^n$$ and $$f_{6c}(x)=\sum_{n=0}^\infty \biggl\{ \sum_{j+k=n}\biggl(\frac{n!}{j!k!}\biggr)^3\biggr\} x^n;$$ that is, $f_{6b}(x)$ is the generating function for sums of squares of trinomial coefficients, while $f_{6c}(x)$ is the generating function for sums of cubes of binomial coefficients. Results for Apéry, Domb and Almkvist–Zudilin numbers, as well as sums of the fourth powers of binomial coefficients, will also appear as special cases of our results.
Our results also include transformation formulas such as $$\begin{gathered}
\frac{1}{(1-4x)^{5/2}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,\frac12,\frac12}{1,1};-64x\biggl(\frac{1+x}{1-4x}\biggr)^5\biggr) \endgroup
}
\\
=\frac{1}{(1-4x)^{1/2}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,\frac12,\frac12}{1,1};-64x^5\biggl(\frac{1+x}{1-4x}\biggr)\biggr) \endgroup
}\end{gathered}$$ are part of a thirteen-function identity.
This work is organized as follows. In the next section, some sequences and their generating functions are defined.
The main results are stated in Sections \[sec3\], \[sec4\] and \[sec5\]. Each section consists of a single theorem that asserts that a large number of functions are equal.
Short proofs, using differential equations, are given in Section \[sec6\]. Alternative proofs using modular forms, that help put the results into context, are given in Section \[sec7\].
Several special cases are elucidated in Section \[sec8\]. An application to Ramanujan’s series for $1/\pi$, using some of Aycock’s ideas, is given in Section \[sec9\].
Definitions and background information {#sec2}
======================================
The series in that defines the hypergeometric functions ${}_2F_1$ and ${}_3F_2$ converges for $|x|<1$. Clausen’s identity [@aar p. 116] is $$\biggl\{{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{a+b+\frac12};x\biggr) \endgroup
}\biggr\}^2
={ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{2a,2b,a+b}{2a+2b,a+b+\frac12};x\biggr) \endgroup
}.$$ It may be combined with the quadratic transformation formula [@aar p. 125] $${ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{2a,2b}{a+b+\frac12};x\biggr) \endgroup
} = { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{a+b+\frac12};4x(1-x)\biggr) \endgroup
}$$ to give $$\biggl\{{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{2a,2b}{a+b+\frac12};x\biggr) \endgroup
}\biggr\}^2
= { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{2a,2b,a+b}{2a+2b,a+b+\frac12};4x(1-x)\biggr) \endgroup
}.$$ We will be interested in the special case $2a=s$, $2b=1-s$, that is $$\label{clauclau}
\biggl\{{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{s,1-s}{1};x\biggr) \endgroup
}\biggr\}^2
={ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,s,1-s}{1,1};4x(1-x)\biggr) \endgroup
}$$ where $s$ assumes one the values $1/6$, $1/4$, $1/3$ or $1/2$. For cosmetic reasons we introduce the following nicknames for these special instances of hypergeometric functions: $$f_\ell(x)={ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{s,1-s}{1};C_sx\biggr) \endgroup
}$$ and $$F_\ell(x)={ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,s,1-s}{1,1};4C_sx\biggr) \endgroup
},$$ where $\ell=\ell_s=1,2,3,4$ and $C_s=432,64,27,16$ for $s=1/6,1/4,1/3,1/2$, respectively. The arithmetic normalization constants $C_s$ are introduced in such a way that the series $f_\ell(x)$ and $F_\ell(x)$ all belong to the ring $\mathbb Z[[x]]$; for example, $$\begin{gathered}
F_1(x)=\sum_{n=0}^\infty\binom{6n}{3n}\binom{3n}{2n}\binom{2n}{n}x^n,
\quad
F_2(x)=\sum_{n=0}^\infty\binom{4n}{2n}{\binom{2n}{n}}^2x^n,
\\
F_3(x)=\sum_{n=0}^\infty\binom{3n}{n}{\binom{2n}{n}}^2x^n
\quad\text{and}\quad
F_4(x)=\sum_{n=0}^\infty{\binom{2n}{n}}^3x^n.
\end{gathered}
\label{bhf}$$ The convergence domains of the series for $f_\ell(x)$ and $F_\ell(x)$ are then $|x|<1/C_s$ and $|x|\le1/(4C_s)$, respectively.
Our further examples of the series from $\mathbb Z[[x]]$ are generating functions of so-called Apéry-like sequences. Let $\alpha$, $\beta$ and $\gamma$ be fixed, and consider the recurrence relations $$\label{recw1}
(n+1)^2t(n+1)=(\alpha n^2+\alpha n+\beta)t(n)+\gamma\,n^2t(n-1)$$ and $$(n+1)^3T(n+1) = -(2n+1)(\alpha n^2+\alpha n+\alpha-2\beta)T(n)-(\alpha^2+4\gamma) n^3T(n-1).$$ We assume that $n$ is a non-negative integer in each recurrence relation, and use the single initial condition $t(0)=T(0)=1$ to start each sequence.
------------------------- --------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------
$(\alpha,\beta,\gamma)$ $t(n)$ $T(n)$
$(11,3,1)$ $\displaystyle{\sum_{j} {\binom{n}{j}}^2\binom{n+j}{j}}$ ${\displaystyle{\sum_{j}} (-1)^{j+n}\binom{n}{j}^3 \binom{4n-5j}{3n}}$
$(-17,-6,-72)$ $\displaystyle{\sum_{j,\ell}(-8)^{n-j}\binom{n}{j}\binom{j}{\ell}^3}$ $\displaystyle{\sum_{j} \binom{n}{j}^2\binom{n+j}{j}^2}$
$(10,3,-9)$ $\displaystyle{\sum_{j} \binom{n}{j}^2\binom{2j}{j}} = \sum_{j+k+\ell=n}\biggl(\frac{n!}{j!k!\ell!}\biggr)^2$ $\displaystyle{(-1)^{n}\sum_{j} \binom{n}{j}^2\binom{2j}{j}\binom{2n-2j}{n-j}}$
$(7,2,8)$ $\displaystyle{\sum_{j} \binom{n}{j}^3}$ ${\displaystyle{\sum_{j}} (-3)^{n-3j}\binom{n+j}{j}\binom{n}{j}\binom{n-j}{j}\binom{n-2j}{j}}$
------------------------- --------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------
: Solutions to recurrence relations
\[table1\]
Define the generating functions $$f(x) = \sum_{n=0}^\infty t(n)x^n, \quad
F(x) = \sum_{n=0}^\infty \binom{2n}{n} t(n)x^n \quad\text{and}\quad
G(x) = \sum_{n=0}^\infty T(n)x^n.$$ It is known [@almkvistetal; @ctyz] that $$\begin{aligned}
\label{zs}
f(x)^2
&=\frac{1}{1+\gamma x^2}\,F\biggl(\frac{x(1-\alpha x-\gamma x^2)}{(1+\gamma x^2)^2}\biggr)
\\
&=\frac{1}{1-\alpha x-\gamma x^2}\,G\biggl(\frac{x}{1-\alpha x-\gamma x^2}\biggr).
\nonumber\end{aligned}$$ When $\gamma=0$, the first equality in reduces to .
Let $f_{6a}(x)$, $F_{6a}(x)$ and $G_{6a}(x)$ be the functions $f(x)$, $F(x)$ and $G(x)$ respectively, for the parameter values $$(\alpha,\beta,\gamma)=(-17,-6,-72).$$ Similarly, let $f_{6b}(x)$, $F_{6b}(x)$ and $G_{6b}(x)$ be the respective functions that correspond to the values $$(\alpha,\beta,\gamma)=(10,3,-9),$$ while $f_{6c}(x)$, $F_{6c}(x)$ and $G_{6c}(x)$ are the respective functions that correspond to the values $$(\alpha,\beta,\gamma)= (7,2,8)$$ and $f_5(x)$, $F_5(x)$ and $G_5(x)$ are the respective functions that correspond to the values $$(\alpha,\beta,\gamma)= (11,3,1).$$ Formulas for the coefficients $t(n)$ and $T(n)$ that involve sums of binomial coefficients are known in the four special cases defined above, and these are listed in Table \[table1\]. The entries for $t(n)$ come from a list that is originally due to D. Zagier [@zagier Section 4]. It may also be mentioned that the numbers $T(n)$ in the cases $(\alpha,\beta,\gamma)=(-17,-6,-72)$, $(10,3,-9)$ and $(7,2,8)$ are called the Apéry numbers, Domb numbers and Almkvist–Zudilin numbers, respectively.
Finally, let $H(x)$ be defined by $$H(x)=\sum_{n=0}^\infty \biggl\{\sum_{j=0}^n {\binom{n}{j}}^4\biggr\}x^n.$$
In our results below we use the labels “Level 1”, “Level 2” etc., to distinguish the appearance of different generating functions; the function $H(x)$ is labeled “Level 10” while the other labels can be extracted from the subscripts of the corresponding functions. Levels themselves, in particular, their origins and meaning, are discussed further in Section \[sec7\] in the context of modular forms.
Results: Part 1 {#sec3}
===============
Our first meta-identity is the subject of the following theorem.
\[t1\] The following forty-two functions are equal, in a neighborhood of $p=0$: $$\begin{aligned}
\intertext{Level $1$\textup:}
\label{11}
\lefteqn{
\frac{1}{(1 +4p -8p^2 ) ^{1/2}(1 +228p-408p^2 -128p^3 -192p^4+768p^5-512p^6) ^{1/2}}
} \\
&\quad\times
F_1\biggl(\frac{p( 1-p)^3 ( 1-4p) ^{12} ( 1-2p)( 1+2p)^3 }{ (1 +4p -8p^2 )^3 (1 +228p-408p^2 -128p^3 -192p^4+768p^5-512p^6)^3}\biggr)
\nonumber
\displaybreak[2]\\
\label{12}
&=\frac{1}{(1 -2p + 4p^2) ^{1/2}(1 -6p +240p^2-920p^3+960p^4-96p^5 +64p^6) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {p^2 ( 1-p )^6 ( 1-4p )^6( 1-2p )^2 ( 1+2p )^6} { (1 -2p + 4p^2)^3 (1 -6p +240p^2-920p^3+960p^4-96p^5 +64p^6)^3} \biggr)
\nonumber
\displaybreak[2]\\
\label{13}
&=\frac{1}{(1+4p -8p^2 ) ^{1/2} (1 -12p +72p^2-128p^3-192p^4+768p^5-512p^6 ) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {p^3 ( 1-p ) ( 1-4p )^4 ( 1-2p )^3 ( 1+2p ) } { (1+4p -8p^2 )^3 (1 -12p +72p^2-128p^3-192p^4+768p^5-512p^6 )^3}\biggr)
\nonumber
\displaybreak[2]\\
\label{14}
&=\frac{1}{(1-2p -2p^2 ) ^{1/2}(1-6p +6p^2+16p^3+204p^4-456p^5-8p^6) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {p^4 ( 1-p ) ^{12} ( 1-4p )^3 ( 1-2p ) ( 1+2p )^3}{ (1-2p -2p^2 )^3(1-6p +6p^2+16p^3+204p^4-456p^5-8p^6)^3} \biggr)
\nonumber
\displaybreak[2]\\
\label{16}
&=\frac{1}{(1-2p+ 4p^2 ) ^{1/2}( 1-6p+40p^3-96p^5 +64p^6) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {p^6 ( 1-p )^2 ( 1-4p )^2 ( 1-2p )^6 ( 1+2p )^2}{ (1-2p+ 4p^2 )^3( 1-6p+40p^3-96p^5 +64p^6)^3 } \biggr)
\nonumber
\displaybreak[2]\\
\label{112}
&=\frac{1}{(1-2p -2p^2 ) ^{1/2} (1-6p +6p^2+16p^3-36p^4+24p^5 -8p^6) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {p^{12} ( 1-p )^4 ( 1-4p ) ( 1-2p )^3 ( 1+2p ) } { (1-2p -2p^2 )^3 (1-6p +6p^2+16p^3-36p^4+24p^5 -8p^6)^3} \biggr)
\nonumber
\displaybreak[2]\\
\label{1m1}
&=\frac{1}{(1 -8p +4p^2 ) ^{1/2} (1 -240p +1932p^2 -5888p^3 +7728p^4-3840p^5+64p^6) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {-p ( 1-p )^3 ( 1-4p )^3 ( 1-2p )^4 ( 1+2p ) ^{12}} { (1 -8p +4p^2 )^3 (1 -240p +1932p^2 -5888p^3 +7728p^4-3840p^5+64p^6)^3}\biggr)
\nonumber
\displaybreak[2]\\
\label{1m3}
&=\frac{1}{(1 -8p +4p^2 ) ^{1/2} (1 +12p^2 -128p^3 +48p^4+64p^6) ^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {-p^3 ( 1-p ) ( 1-4p ) ( 1-2p )^{12} ( 1+2p )^4} { (1 -8p +4p^2 )^3 (1 +12p^2 -128p^3 +48p^4+64p^6)^3}\biggr)
\nonumber
\displaybreak[2]\\
\intertext{Level $2$\textup:}
\label{21}
&=\frac{1}{1+20p-48p^2+32p^3-32p^4}\,
F_2\biggl(\frac {p ( 1-p )^3 ( 1-4p )^6 ( 1-2p ) ( 1+2p )^3} { (1 +20p -48p^2 +32p^3-32p^4)^4}\biggr)
\\
\label{22}
&=\frac{1}{1-4p+24p^2-40p^3 -8p^4}\,
F_2\biggl(\frac {p^2 ( 1-p )^6 ( 1-4p )^3 ( 1-2p ) ( 1+2p )^3}{ ( 1-4p+24p^2 -40p^3 -8p^4)^4}\biggr)
\displaybreak[2]\\
\label{23}
&=\frac{1}{1-4p+32p^3-32p^4}\,
F_2\biggl(\frac {p^3 ( 1-p ) ( 1-4p )^2 ( 1-2p )^3 ( 1+2p ) } { (1 -4p +32p^3-32p^4 )^4}\biggr)
\displaybreak[2]\\
\label{26}
&=\frac{1}{1-4p+8p^3-8p^4}\,
F_2\biggl(\frac {p^6 ( 1-p )^2 ( 1-4p ) ( 1-2p )^3 ( 1+2p ) } { (1-4p +8p^3 -8p^4 )^4}\biggr)
\displaybreak[2]\\
\label{2m1}
&=\frac{1}{1-28p+96p^2-112p^3+16p^4}\,
F_2\biggl(-\frac {p ( 1-p )^3 ( 1-4p )^3 ( 1-2p )^2 ( 1+2p )^6} { (1 -28p +96p^2 -112p^3+16p^4)^4}\biggr)
\displaybreak[2]\\
\label{2m3}
&=\frac{1}{1-4p-16p^3+16p^4}\,
F_2\biggl(-\frac {p^3 ( 1-p ) ( 1-4p ) ( 1-2p )^6 ( 1+2p )^2} { (1 -4p -16p^3+16p^4)^4}\biggr)
\displaybreak[2]\\
\intertext{Level $3$\textup:}
\label{31}
&=\frac{1}{(1+4p-8p^2)^2}\,
F_3\biggl(\frac {p ( 1-p ) ( 1-4p )^4 ( 1-4p^2 ) }{ ( 1+4p-8p^2 )^6}\biggr)
\\
\label{32}
&=\frac{1}{(1-2p+4p^2)^2}\,
F_3\biggl(\frac {p^2 ( 1-p )^2 ( 1-4p )^2 ( 1-4p^2 )^2}{ ( 1-2p+4p^2 )^6}\biggr)
\displaybreak[2]\\
\label{34}
&=\frac{1}{(1-2p-2p^2)^2 }\,
F_3\biggl(\frac {p^4 ( 1-p )^4 ( 1-4p ) ( 1-4p^2 ) }{ ( 1-2p-2p^2 )^6}\biggr)
\displaybreak[2]\\
\label{3m}
&=\frac{1}{(1-8p+4p^2)^2 }\,
F_3\biggl(-\frac {p ( 1-p ) ( 1-4p ) ( 1-4p^2 )^4 }{ ( 1-8p+4p^2 )^6}\biggr)
\displaybreak[2]\\
\intertext{Level $4$\textup:}
\label{41}
&=\frac {1}{ ( 1-2p ) ( 1+2p )^3}\,
F_4\biggl(\frac {p ( 1-p )^3 ( 1-4p )^3}{ ( 1-2p )^2 ( 1+2p )^6}\biggr)
\\
\label{43}
&=\frac {1}{ ( 1-2p )^3 ( 1+2p ) }\,
F_4\biggl(\frac {p^3 ( 1-p ) ( 1-4p ) }{ ( 1-2p )^6 ( 1+2p )^2}\biggr)
\displaybreak[2]\\
\label{4m1}
&=\frac {1}{ ( 1-4p )^3 }\,
F_4\biggl(-\frac {p ( 1-2p )(1+2p)^3(1-p)^3 }{( 1-4p )^6}\biggr)
\displaybreak[2]\\
\label{4m3}
&=\frac {1}{ ( 1-4p ) }\,
F_4\biggl(-\frac {{p^3} ( 1-2p )^3(1+2p)(1-p) }{( 1-4p )^2}\biggr)
\displaybreak[2]\\
\label{4m2}
&=\frac {1}{(1-2p)^{1/2}(1+2p)^{3/2} ( 1-4p ) ^{3/2} }\,
F_4\biggl(-\frac {p^2 ( 1-p )^6}{(1-2p)(1+2p)^3 ( 1-4p )^3}\biggr)
\displaybreak[2]\\
\label{4m6}
&=\frac {1}{(1-2p)^{3/2}(1+2p)^{1/2} ( 1-4p ) ^{1/2} }\,
F_4\biggl(-\frac{p^6( 1-p )^2}{(1-2p)^3(1+2p) ( 1-4p )}\biggr)
\displaybreak[2]\\
\intertext{Level $6$, functions $F$\textup:}
\label{Faa1}
&=\frac{1}{1-16p+24p^2+32p^3-32p^4}\,F_{6a}\biggl(\frac{p(1-p)(1-4p)^2(1-2p)(1+2p)}{(1-16p+24p^2+32p^3-32p^4)^2}\biggr)
\\
\label{Faa2}
&=\frac{1}{1-4p-12p^2+32p^3-8p^4}\,F_{6a}\biggl(\frac{p^2(1-p)^2(1-4p)(1-2p)(1+2p)}{(1-4p-12p^2+32p^3-8p^4)^2}\biggr)
\\
\label{Faa3}
&=\frac{1}{1+8p-48p^2+32p^3+16p^4}\,F_{6a}\biggl(\frac{-p(1-p)(1-4p)(1-2p)^2(1+2p)^2}{(1+8p-48p^2+32p^3+16p^4)^2}\biggr)
\displaybreak[2]\\
\label{Fbb1}
&=\frac{1}{(1-2p+4p^2)(1+4p-8p^2)}\,F_{6b}\biggl(\frac{p(1-p)(1-4p)^2(1-2p)(1+2p)}{(1-2p+4p^2)^2(1+4p-8p^2)^2}\biggr)
\\
\label{Fbb2}
&=\frac{1}{(1-2p-2p^2)(1-2p+4p^2)}\,F_{6b}\biggl(\frac{p^2(1-p)^2(1-4p)(1-2p)(1+2p)}{(1-2p-2p^2)^2(1-2p+4p^2)^2}\biggr)
\\
\label{Fbb3}
&=\frac{1}{(1-2p+4p^2)(1-8p+4p^2)}\,F_{6b}\biggl(\frac{-p(1-p)(1-4p)(1-2p)^2(1+2p)^2}{(1-2p+4p^2)^2(1-8p+4p^2)^2}\biggr)
\displaybreak[2]\\
\label{Fcc1}
&=\frac{1}{1+8p^2-32p^3+32p^4}\,F_{6c}\biggl(\frac{p(1-p)(1-4p)^2(1-2p)(1+2p)}{(1+8p^2-32p^3+32p^4)^2}\biggr)
\\
\label{Fcc2}
&=\frac{1}{1-4p+4p^2+8p^4}\,F_{6c}\biggl(\frac{p^2(1-p)^2(1-4p)(1-2p)(1+2p)}{(1-4p+4p^2+8p^4)^2}\biggr)
\\
\label{Faa3}
&=\frac{1}{1-8p+32p^2-32p^3+16p^4}\,F_{6c}\biggl(\frac{-p(1-p)(1-4p)(1-2p)^2(1+2p)^2}{(1-8p+32p^2-32p^3+16p^4)^2}\biggr)
\displaybreak[2]\\
\intertext{Level $6$, functions $G$\textup:}
\label{aa1}
&=\frac{1}{(1+2p)(1-p)}\,G_{6a}\biggl(\frac{p(1-4p)^2(1-2p)}{(1+2p)(1-p)}\biggr)
\\
\label{aa2}
&=\frac{1}{(1-2p)(1-p)^2}\,G_{6a}\biggl(\frac{p^2(1+2p)(1-4p)}{(1-2p)(1-p)^2}\biggr)
\\
\label{aa3}
&=\frac{1}{(1-p)(1-4p)(1-2p)^2}\,G_{6a}\biggl(\frac{-p(1+2p)^2}{(1-p)(1-4p)(1-2p)^2}\biggr)
\displaybreak[2]\\
\label{bb1}
&=\frac{1}{(1-4p)^2}\,G_{6b}\biggl(\frac{p(1+2p)(1-2p)(1-p)}{(1-4p)^2}\biggr)
\\
\label{bb2}
&=\frac{1}{(1-2p)(1+2p)(1-4p)}\,G_{6b}\biggl(\frac{p^2(1-p)^2}{(1-2p)(1+2p)(1-4p)}\biggr)
\\
\label{bb3}
&=\frac{1}{(1-2p)^2(1+2p)^2}\,G_{6b}\biggl(\frac{-p(1-p)(1-4p)}{(1-2p)^2(1+2p)^2}\biggr)
\displaybreak[2]\\
\label{cc1}
&=\frac{1}{(1-p)(1+2p)(1-4p)^2}\,G_{6c}\biggl(\frac{p(1-2p)}{(1-p)(1+2p)(1-4p)^2}\biggr)
\\
\label{cc2}
&=\frac{1}{(1+2p)(1-p)^2(1-4p)}\,G_{6c}\biggl(\frac{p^2(1-2p)}{(1+2p)(1-p)^2(1-4p)}\biggr)
\\
\label{cc3}
&=\frac{1}{(1-p)(1-4p)(1+2p)^2}\,G_{6c}\biggl(\frac{-p(1-2p)^2}{(1-p)(1-4p)(1+2p)^2}\biggr).\end{aligned}$$
Results: Part 2 {#sec4}
===============
The results in this section are consequences of the results in previous section by taking square roots. For example, by we find that the ${}_3F_2$ hypergeometric function in is related to the ${}_2F_1$ function by $$\begin{gathered}
\label{claueg}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};256\,{\frac {p ( 1-p )^3 ( 1-4p )^6
( 1-2p ) ( 1+2p )^3}
{ (1 +20p -48p^2 +32p^3-32p^4)^4}}\biggr) \endgroup
}
\\
= \biggl\{{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac14,\frac34}{1};{\frac {64p ( 1-p )^3 ( 1-2p ) ( 1+2p )^3}
{ (1 +20p -48p^2 +32p^3-32p^4)^2}}\biggr) \endgroup
}\biggr\}^2.\end{gathered}$$ The argument in the ${}_2F_1$ function is obtained by noting that the solution of $$4x(1-x) = 256\,\frac{p(1-p)^3(1-4p)^6(1-2p )( 1+2p )^3}{ (1 +20p -48p^2 +32p^3-32p^4)^4}$$ that satisfies $x=0$ when $p=0$ is given by $$\label{qe}
x = \frac {64p ( 1-p )^3 ( 1-2p ) ( 1+2p )^3}{ (1 +20p -48p^2 +32p^3-32p^4)^2}.$$ By applying and taking the square root of the expression in we get an expression that involves the ${}_2F_1$ function. This can be done, in principle, for all of the hypergeometric expressions in –. In a similar way, the identity can be used to take square roots of the expressions in –. After taking square roots, the arguments of the resulting functions are not always rational functions of $p$ as they are for the example , above; sometimes they are algebraic funcions of $p$ involving square roots that arise from solving quadratic equations.
In Theorem \[t2\] below, we list all of the functions obtained by taking square roots of the expressions in – for which the arguments are rational functions of $p$. The identities have been numbered so that the formula $(4.x)$ in Theorem \[t2\] below is the square root of the corresponding expression $(3.x)$ in Theorem \[t1\] above.
\[t2\] The following twenty-three functions are equal, in a neighborhood of $p=0$: $$\begin{aligned}
\intertext{Level $2$\textup:}
\label{2F21}
\lefteqn{
\frac{1}{(1 +20p -48p^2 +32p^3-32p^4 )^{1/2}}\,
f_2\biggl(\frac{p ( 1-p )^3 ( 1-2p ) ( 1+2p )^3} { (1 +20p -48p^2 +32p^3-32p^4)^2}\biggr)
}
\\
\label{2F22}
&= \frac{1}{(1-4p+24p^2 -40p^3 -8p^4)^{1/2} }\,
f_2\biggl(\frac{p^2 ( 1-p )^6}{ ( 1-4p+24p^2 -40p^3 -8p^4)^2}\biggr)
\displaybreak[2]\\
\label{2F23}
&=\frac{1}{(1 -4p +32p^3-32p^4 )^{1/2} }\,
f_2\biggl(\frac{p^3 ( 1-p ) ( 1-2p )^3 ( 1+2p ) } { (1 -4p +32p^3-32p^4 )^2}\biggr)
\displaybreak[2]\\
\label{2F26}
&=\frac{1}{(1-4p +8p^3 -8p^4 )^{1/2} }\,
f_2\biggl(\frac {64p^6 ( 1-p )^2 } { (1-4p +8p^3 -8p^4 )^2}\biggr)
\displaybreak[2]\\
\label{2F2m1}
&=\frac{1}{(1 -28p +96p^2 - 112p^3+16p^4 )^{1/2}}\,
f_2\biggl(\frac {-p ( 1-p )^3 ( 1-4p )^3 } { (1 -28p +96p^2 -112p^3+16p^4)^2}\biggr)
\displaybreak[2]\\
\label{2F2m3}
&=\frac{1}{(1-4p-16p^3+16p^4)^{1/2}}\,
f_2\biggl(\frac{-p^3 ( 1-p ) ( 1-4p ) } { (1 -4p -16p^3+16p^4)^2}\biggr)
\displaybreak[2]\\
\intertext{Level $3$\textup:}
\label{2F31}
&=\frac{1}{(1+4p-8p^2)}\,f_3\biggl(\frac{p ( 1-2p )}{ ( 1+4p-8p^2 )^3}\biggr)
\\
\label{2F32}
&=\frac{1}{(1-2p+4p^2)}\,f_3\biggl(\frac{p^2 ( 1-2p )^2 }{( 1-2p+4p^2)^3}\biggr)
\displaybreak[2]\\
\label{2F34}
&=\frac{1}{(1-2p-2p^2)}\,f_3\biggl(\frac{p^4 ( 1-2p ) } { ( 1-2p-2p^2 )^3}\biggr)
\displaybreak[2]\\
\label{2F3m}
&=\frac{1}{(1-8p+4p^2)}\,f_3\biggl(\frac{-p( 1-2p )^4}{ ( 1-8p+4p^2 )^3}\biggr)
\displaybreak[2]\\
\intertext{Level $4$\textup:}
\label{2F41}
&=\frac{1}{(1-2p )^{1/2} ( 1+2p ) ^{3/2}}\,f_4\biggl(\frac{p ( 1-p )^3 }{ ( 1-2p ) ( 1+2p )^3}\biggr)
\\
\label{2F43}
&=\frac {1}{ ( 1-2p ) ^{3/2} ( 1+2p )^{1/2} }\,f_4\biggl(\frac{p^3 ( 1-p )}{ ( 1-2p )^3 ( 1+2p )}\biggr)
\displaybreak[2]\\
\label{2F4m1}
&=\frac {1}{ ( 1-4p )^{3/2} }\,f_4\biggl(\frac{-p (1-p)^3 }{( 1-4p )^3}\biggr)
\displaybreak[2]\\
\label{2F4m3}
&=\frac {1}{ ( 1-4p )^{1/2} }\,f_4\biggl(\frac {-p^3(1-p) }{( 1-4p )}\biggr)
\displaybreak[2]\\
\intertext{Level $6$\textup:}
\setcounter{equation}{24}
\label{a1}
&=\frac{1}{(1 -4p)^2}\,f_{6a}\biggl(\frac{p\,(1-2p)}{(1-4p)^2}\biggr)
\\
\label{a2}
&=\frac{1}{(1 -4p)(1+2p)}\,f_{6a}\biggl(\frac{p^2}{(1-4p)(1+2p)}\biggr)
\\
\label{a3}
&=\frac{1}{(1 + 2p)^2}\,f_{6a}\biggl(\frac{-p}{(1+2p)^2}\biggr)
\displaybreak[2]\\
\label{b1}
&=\frac{1}{(1 - p)(1+2p)}\,f_{6b}\biggl(\frac{p(1-2\, p)}{(1-p)(1+2p)}\biggr)
\\
\label{b2}
&=\frac{1}{(1 - p)^2}\,f_{6b}\biggl(\frac{p^2}{(1-p)^2}\biggr)
\\
\label{b3}
&=\frac{1}{(1 - p)(1-4p)}\,f_{6b}\biggl(\frac{-p}{(1-p)(1-4p)}\biggr)
\displaybreak[2]\\
\label{c1}
&=f_{6c}\bigl(p(1-2p)\bigr)
\\
\label{c2}
&=\frac{1}{1-2p}\,f_{6c}\biggl(\frac{p^2}{1-2p}\biggr)
\\
\label{c3}
&=\frac{1}{(1-2p)^2}\,f_{6c}\biggl( \frac{-p}{(1-2p)^2}\biggr).\end{aligned}$$
Results: Part 3 {#sec5}
===============
The results in this section involve transformations of degrees $2$, $5$ and $10$.
\[t3\] The following thirteen functions are equal, in a neighborhood of $p=0$: $$\begin{aligned}
\intertext{Level $1$\textup:}
\label{511}
\lefteqn{
\frac{1}{(1+236p +1440p^2+1920p^3+3840p^4+256p^5+256p^6)^{1/2}}
}
\\
&\quad\times
F_1\biggl(\frac {p ( 1-4p ) ^{10} ( 1+p )^5}{(1 +236p +1440p^2 +1920p^3 +3840p^4 +256p^5 + 256p^6 )^3}\biggr)
\nonumber
\displaybreak[2]\\
\label{512}
&=\frac{1}{(1-4p+240p^2-480p^3+1440p^4-944p^5+16p^6)^{1/2}}
\\
&\quad\times
F_1\biggl(\frac {p^2 ( 1-4p )^5 ( 1+p ) ^{10}}{ ( 1 -4p +240p^2 -480p^3 +1440p^4 -944p^5 + 16p^6)^3}\biggr)
\nonumber
\displaybreak[2]\\
\label{513}
&=\frac{1}{ (1 -4p +256p^5 + 256p^6) ^{1/2}}\,
F_1\biggl(\frac{p^5 ( 1-4p )^2 ( 1+p ) }{( 1 -4p +256p^5 + 256p^6 )^3}\biggr)
\displaybreak[2]\\
\label{514}
&=\frac{1}{ ( 1 -4p +16p^5 +16p^6 ) ^{1/2}}\,
F_1\biggl(\frac {p^{10} ( 1-4p ) ( 1+p )^2}{ ( 1 -4p +16p^5 +16p^6 )^3}\biggr)
\displaybreak[2]\\
\intertext{Level $2$\textup:}
\label{521}
&=\frac{1}{ ( 1+4p^2 ) ^{1/2} ( 1+22p-4p^2 ) }\,
F_2\biggl(\frac {p ( 1+p )^5 ( 1-4p )^5}{ ( 1+4p^2 )^2 ( 1+22p-4p^2 ) ^4}\biggr)
\\
\label{522}
&=\frac{1}{ ( 1+4p^2 )^{1/2} ( 1-2p-4p^2 ) }\,
F_2\biggl(\frac {p^5 ( 1+p ) ( 1-4p ) }{ ( 1+4p^2 )^2 ( 1-2p-4p^2 )^4}\biggr)
\displaybreak[2]\\
\intertext{Level $4$\textup:}
\label{541}
&=\frac{1}{ ( 1-4p ) ^{5/2} }\,
F_4\biggl(-p\biggl(\frac{ 1+p }{ 1-4p} \biggr)^5\biggr)
\\
\label{542}
&=\frac{1}{ ( 1-4p )^{1/2} }\,
F_4\biggl(-p^5\biggl(\frac{ 1+p }{ 1-4p} \biggr)\biggr)
\displaybreak[2]\\
\intertext{Level $5$, functions $F$\textup:}
\label{F5aa1}
&=\frac{1}{ ( 1 + 4p^2 )^{1/2} ( 1 + 4p + 8p^2 )}\,
F_5\biggl(\frac {p ( 1-4p )^2 ( 1+p ) }{ ( 1 + 4p^2 ) ( 1 + 4p + 8p^2 )^2 } \biggr)
\\
\label{F5aa2}
&=\frac{1}{( 1 + 4p^2 )^{1/2}( 1 -2p+2p^2 ) } \,
F_5\biggl( \frac {p^2 ( 1-4p ) ( 1+p )^2}{( 1 + 4p^2 )( 1 -2p+2p^2 )^2 } \biggr)
\displaybreak[2]\\
\intertext{Level $5$, functions $G$\textup:}
\label{5aa1}
&=\frac{1}{( 1+p ) ( 1-4p )^2} \,
G_5\biggl( \biggl(\frac p{ 1+p} \biggr) \biggl(\frac{1}{ 1-4p} \biggr)^2 \biggr)
\\
\label{5aa2}
&=\frac{1}{( 1+p )^2 ( 1-4p )} \,
G_5\biggl( \biggl(\frac p{ 1+p} \biggr)^2 \biggl(\frac{1}{ 1-4p} \biggr) \biggr)
\displaybreak[2]\\
\intertext{Level $10$, function $H$\textup:}
\label{10aa1}
&= \frac{1}{ ( 1+4p^2 ) ^{3/2}} \,
H \biggl( \frac {p ( 1+p ) ( 1-4p ) }{ ( 1+4p^2 )^2} \biggr).\end{aligned}$$
Proofs of Theorems \[t1\], \[t2\] and \[t3\] {#sec6}
============================================
In this section we will provide proofs of Theorems \[t1\], \[t2\] and \[t3\]. We begin with a proof of Theorem \[t2\] because it is the simplest and therefore the explicit details are the easiest to write down.
The hypergeometric function $${ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{s,1-s}{1};x\biggr) \endgroup
}$$ satisfies the second order linear differential equation $$\frac{{{\mathrm d}}}{{{\mathrm d}}x}\biggl(x(1-x)\,\frac{{{\mathrm d}}z}{{{\mathrm d}}x}\biggr)=s(1-s)z.$$ By changing variables, it can be shown that each function in – satisfies the differential equation $$\label{dzdp}
\frac{{{\mathrm d}}}{{{\mathrm d}}p}\biggl(p(1-p)(1-4p)(1-2p)(1+2p)\frac{{{\mathrm d}}y}{{{\mathrm d}}p}\biggr)
=2(1-4p)(1+4p-8p^2)y.$$ In a similar way, the recurrence relation implies that the each of the functions $f_{6a}$, $f_{6b}$ and $f_{6c}$ satisfies a second order linear differential equation of the form $$\label{zagde22}
\frac{{{\mathrm d}}}{{{\mathrm d}}x}\biggl(x(1-\alpha x-\gamma x^2)\frac{{{\mathrm d}}z}{{{\mathrm d}}x}\biggr)=(\beta+ \gamma x)z.$$ On specializing the parameters and changing variables, we may deduce that each function – also satisfies . By direct calculation, we find that the first two terms in the expansions about $p=0$ of each function in – or – are given by $$y=1+2p+O(p^2).
$$ It follows that the twenty-three functions – are all equal.
The proofs of Theorems \[t1\] and \[t3\] are similar and go as follows.
Each function in – (respectively, –) satisfies the same third order linear differential equation. Moreover, the first three terms in the expansion of each function in – (respectively, –) about $p=0$ are given by $$y=1 + 4p + 16p^2 +O(p^3) \qquad \text{(respectively, $y=1 + 2p + 6p^2 +O(p^3)$).}
$$ It follows that the forty-two functions in – (respectively, the thirteen functions in –) are equal.
It may be emphasized that the proofs outlined above are conceptually simple, establish the correctness of the results, and require very little mathematical knowledge. However, the proofs above are not illuminating: they give no insight into the structure of the identities, nor any reason for why the results exist or any clue as to how the identities were discovered. In the next section, the theory of modular forms will be used to give alternative proofs that also provide an explanation for how the identities were discovered.
In practice, the easiest way to show that each of the functions in – satisfies the differential equation is to expand each function as a power series about $p=0$ to a large number of terms, and then use computer algebra to determine the differential equation. For example using Maple, the differential equation satisfied by the function in can be determined from the first $50$ terms in the series expansion in powers of $p$, by entering the commands
> with(gfun):
> n := 64*p*(1-p)^3*(1-2*p)*(1+2*p)^3;
> d := (-32*p^4+32*p^3-48*p^2+20*p+1)^2;
> s := series(d^(-1/4)*hypergeom([1/4, 3/4], [1], n/d), p, 50):
> L := seriestolist(s):
> guesseqn(L, y(p));
This produces the output $$\begin{gathered}
\Big[\Big\{ \left( -16\,{p}^{5}+20\,{p}^{4}-5\,{p}^{2}+p \right) {
\frac {{\rm d}^{2}}{{\rm d}{p}^{2}}}y \left( p \right) + \left( -80\,{
p}^{4}+80\,{p}^{3}-10\,p+1 \right) {\frac {\rm d}{{\rm d}p}}y \left( p
\right)
\\
+ \left( -64\,{p}^{3}+48\,{p}^{2}-2 \right) y \left( p
\right) ,y \left( 0 \right) =1,\mbox {D} \left( y \right) \left( 0
\right) =2 \Big\}\Big] ,{\it ogf}\end{gathered}$$ which is equivalent to .
Modular origins {#sec7}
===============
We will now explain how the identities in Theorems \[t1\], \[t2\] and \[t3\] were found. The explanation also puts the formulas into context and reveals why they exist.
A modular explanation for Theorem \[t1\] requires the theory of modular forms for level $12$ as developed in [@cooperye]. Some of the details may be summarized as follows. Let $h$, $p$ and $z$ be defined by $$\begin{gathered}
h=q\prod_{j=1}^\infty \frac{(1-q^{12j-11})(1-q^{12j-1})}{(1-q^{12j-7})(1-q^{12j-5})},
\\
p=\frac{h}{1+h^2}
\quad\text{and}\quad
z=q\,\frac{{{\mathrm d}}}{{{\mathrm d}}q}\, \log h.\end{gathered}$$ Ramanujan’s Eisenstein series $P$ and $Q$ are defined by $$P(q)=1-24\sum_{j=1}^\infty \frac{jq^j}{1-q^j}
\quad\text{and}\quad
Q(q)=1+240\sum_{j=1}^\infty \frac{j^3q^j}{1-q^j}.$$ Dedekind’s eta-function is defined for ${\operatorname{Im}}\tau>0$ and $q=\exp(2\pi i\tau)$ by $$\eta(\tau)=q^{1/24}\prod_{j=1}^\infty (1-q^j).$$
We will require the following three lemmas, extracted from the literature.
\[7.1\] Suppose $k$ and $m$ are positive divisors of $12$ and $k>m$. Then there are rational functions $r_{k,m}(h)$ such that $$kP(q^k)-mP(q^m)=z \times r_{k,m}(h).$$ Furthermore, there are rational functions $s_m(h)$ and $t_m(h)$ such that $$Q(q^m)=z^2 \times s_m(h)
\quad\text{and}\quad
\eta^{24}(m\tau)= z^6 \times t_m(h).$$
The result for $P$ follows from [@cooperye Theorem 4.5]; the result for $Q$ is Theorem 4.6 in [@cooperye]; and the result for Dedekind’s eta-function is Theorem 4.2 in [@cooperye].
Explicit formulas for the rational functions $r_{k,m}(h)$, $s_m(h)$ and $t_m(h)$ can be determined from the information in [@cooperye]. Some examples will be given below, after Lemma \[7.3\].
\[7.2\] For $\ell\in\{1,2,3,4\}$, let $Z_\ell$ be defined by $$\label{zall4def2}
Z_\ell(q) = \begin{cases}
Q^{1/2}(q) & \text{if $\ell=1$,} \\[1.5mm]
\displaystyle{\frac{\ell P(q^\ell)-P(q)}{\ell-1}} & \text{if $\ell=2$, $3$ or $4$.}
\end{cases}$$ Then the following parameterizations of hypergeometric functions hold: $$\begin{gathered}
Z_1(q)=F_1\biggl(\biggl(\frac{\eta^4(\tau)}{Z_1(q)}\biggr)^6\biggr),
\quad
Z_2(q)=F_2\biggl(\biggl(\frac{\eta^2(\tau)\eta^2(2\tau)}{Z_2(q)}\biggr)^4\biggr),
\\
Z_3(q)=F_3\biggl(\biggl(\frac{\eta^2(\tau)\eta^2(3\tau)}{Z_3(q)}\biggr)^3\biggr)
\quad\text{and}\quad
Z_4(q)=F_4\biggl(\biggl(\frac{\eta^4(\tau)\eta^4(4\tau)}{\eta^4(2\tau)\,Z_4(q)}\biggr)^2\biggr),\end{gathered}$$ where, as usual, $q=\exp(2\pi i\tau)$ and the hypergeometric functions are as in .
These results may be found in [@bbg], [@domb] or [@cooperlms], or they can be proved by putting together identities in those references and applying the special case of Clausen’s identity given by .
The parameter $\ell$ in Lemma \[7.2\] is called the level. The next lemma gives the analogous results for level $6$.
\[7.3\] The following parameterizations hold: $$\begin{aligned}
\frac{\eta^6(\tau)\eta(6\tau)}{\eta^3(2\tau)\eta^2(3\tau)}
&=f_{6a}\biggl(\frac{\eta(2\tau)\eta^5(6\tau)}{\eta^5(\tau)\eta(3\tau)}\biggr),
\\
\frac{\eta^6(2\tau)\eta(3\tau)}{\eta^3(\tau)\eta^2(6\tau)}
&=f_{6b}\biggl(\frac{\eta^4(\tau)\eta^8(6\tau)}{\eta^8(2\tau)\eta^4(3\tau)}\biggr)
\intertext{and}
\frac{\eta(2\tau)\eta^6(3\tau)}{\eta^2(\tau)\eta^3(3\tau)}
&=f_{6c}\biggl(\frac{\eta^3(\tau)\eta^9(6\tau)}{\eta^3(2\tau)\eta^9(3\tau)}\biggr).\end{aligned}$$
This is explained in Zagier’s work [@zagier]. Our functions $f_{6b}$ and $f_{6c}$ correspond to the functions $f(z)$ in [@zagier] in the cases C and A, respectively. The function $f_{6a}$ corresponds to the function $f(z)$ in [@zagier] in case F but with $-q$ in place of $q$.
We are now ready to explain how the forty-two functions in Theorem \[t1\] arise, and why they are equal. We will only focus on the level 3 functions in – as an illustration; the other functions can be obtained by a similar procedure by working with the other levels.
By Lemma \[7.1\] and the explicit formulas in [@cooperye Theorem 4.2] we find that $$\eta^6(\tau)\eta^6(3\tau) = z^3 \times \frac{h(1+h^2)(1-h+h^2)}{(1-h^2)(1-h+h^2)^2}$$ and $$\frac12\bigl(3P(q^3)-P(q)\bigr)
=z \times \frac{(1+4h-6h^2+4h^3+h^4)^2}{(1+h^2)(1-h+h^2)(1-4h+h^2)(1-h^2)}.$$ Substituting these in the result for $Z_3(q)$ in Lemma \[7.2\] gives $$\begin{gathered}
z \times \frac{(1+4h-6h^2+4h^3+h^4)^2}{(1+h^2)(1-h+h^2)(1-4h+h^2)(1-h^2)} \\
= { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};108\frac{h(1+h^2)^4(1-h+h^2)(1-4h+h^2)^4(1-h^2)^2}{(1+4h-6h^2+4h^3+h^4)^6}\biggr) \endgroup
}.\end{gathered}$$ Under the change of variables $p=h/(1+h^2)$, this be written in the form $$\begin{gathered}
\label{nearly1}
\frac{z}{(1-p)(1-4p)\sqrt{1-4p^2}}\\
=\frac{1}{(1+4p-8p^2)^2}\;
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};108\,\frac {p ( 1-p ) ( 1-4p )^4 ( 1-4p^2) }
{(1+4p-8p^2)^6}\biggr) \endgroup
}.\end{gathered}$$ We will now obtain three further formulas akin to by replacing $q$ with $q^2$, $q^4$ or $-q$ in the identity for $Z_3(q)$ in Lemma \[7.2\]; that is, by replacing $\tau$ with $2\tau$, $4\tau$ and $\tau + \frac12$, respectively. First, replacing $\tau$ with $2\tau$ in the formula for $Z_3$ in Lemma \[7.2\] and using the parameterizations $$\eta^6(2\tau)\eta^6(6\tau) = z^3 \times \frac{h^2(1-h^2)}{(1+h^2)(1-h+h^2)(1-4h+h^2)}$$ and $$\frac12\bigl(3P(q^6)-P(q^2)\bigr)
=z \times \frac{(1-2h+6h^2-h^3+h^4)^2}{(1+h^2)(1-h+h^2)(1-4h+h^2)(1-h^2)}$$ leads to the identity $$\begin{gathered}
\label{nearly2}
\frac{z}{(1-p)(1-4p)\sqrt{1-4p^2}}\\
= \frac{1}{(1-2p+4p^2)^2}\,
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};108\,\frac {p^2 (1-p)^2 (1-4p)^2 (1-4p^2)^2}{(1-2p+4p^2)^6}\biggr) \endgroup
}.\end{gathered}$$ Similarly, replacing $\tau$ with $4\tau$ in the formula for $Z_3$ in Lemma \[7.2\] and using the parameterizations $$\eta^6(4\tau)\eta^6(12\tau) = z^3 \times \frac{h^4(1-h+h^2)}{(1+h^2)^2(1-h^2)(1-4h+h^2)^2}$$ and $$\frac12\bigl(3P(q^{12})-P(q^4)\bigr)
= z \times \frac{(1-2h-2h^3+h^4)^2}{(1+h^2)(1-h+h^2)(1-4h+h^2)(1-h^2)}$$ leads to the identity $$\begin{gathered}
\label{nearly3}
\frac{z}{(1-p)(1-4p)\sqrt{1-4p^2}}\\
= \frac{1}{(1-2p-2p^2)^2 }\,
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};108\,\frac {p^4 (1-p)^4 (1-4p) (1-4p^2)}{(1-2p-2p^2)^6}\biggr) \endgroup
}.\end{gathered}$$ In order to replace $q$ with $-q$, equivalently $\tau$ with $\tau+\frac12$, use the identity $$\prod_{j=1}^\infty (1-(-q)^j) = \prod_{j=1}^\infty \frac{(1-q^{2j})^3}{(1-q^j)(1-q^{4j})}$$ to get $$\label{minus1}
\eta^6(\tau)\eta^6(3\tau)\bigg|_{\tau \mapsto \tau+\frac12}
=\frac{-\eta^{18}(2\tau)\eta^{18}(2\tau)}{\eta^6(\tau)\eta^6(3\tau)\eta^6(4\tau)\eta^6(12\tau)},$$ and use the identity $$\label{pminus}
P(-q) = -P(q)+6P(q^2)-4P(q^4)$$ to deduce that $$\begin{gathered}
\label{minus2}
\lefteqn{ \frac12\bigl(3P(-q^3)-P(-q)\bigr)} \\
= -\frac12\bigl(3P(q^3)-P(q)\bigr) + 3\bigl(3P(q^6)-P(q^2)\bigr)-2\bigl(3P(q^{12})-P(q^4)\bigr).
\end{gathered}$$ Then, and can be used along with the parameterizations given above, to replace $q$ with $-q$ in the formula for $Z_3$ in Lemma \[7.2\] and produce the identity $$\begin{gathered}
\label{nearly4}
\frac{z}{(1-p)(1-4p)\sqrt{1-4p^2}}\\
= \frac{1}{(1-8p+4p^2)^2 }\,
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};-108\,\frac{p (1-p) (1-4p) (1-4p^2)^4}{(1-8p+4p^2)^6}\biggr) \endgroup
}.\end{gathered}$$ Finally, equating , , and shows the equality of – in Theorem \[t1\].
The identities –, – and – can be obtained in the same way by using the results for $Z_1$, $Z_2$ and $Z_4$, respectively, in Lemma \[7.2\]. The identities – can be obtained using the parameterizations in Lemma \[7.3\] together with the identity .
Suppose that $y$ is the solution of the differential equation $$\frac{{{\mathrm d}}}{{{\mathrm d}}p} \biggl(p(1-p)(1-4p)(1-2p)(1+2p) \frac{{{\mathrm d}}y}{{{\mathrm d}}p}\biggr)
=2(1-4p)(1+4p-8p^2)y$$ that satisfies the initial conditions $y(0)=1$, $y'(0)=2$, and suppose $$y=\sum_{n=0}^\infty t(n)p^n$$ in a neighborhood of $p=0$. Then $$\begin{gathered}
(n+1)^2 t(n+1) \\
= (5n^2+5n+2)t(n) -4(5n^2-5n+2)t(n-2) + 16(n-1)^2 t(n-3),\end{gathered}$$ the coefficient of the term $t(n-1)$ being zero. Furthermore, $y$ and $p$ may be parameterized by the modular forms $$y = \frac{\eta(2\tau)\eta^6(3\tau)}{\eta^2(\tau)\eta^3(6\tau)}
\quad\text{and}\quad
p = \frac{\eta(\tau)\eta^3(12\tau)}{\eta^3(3\tau)\eta(4\tau)}.$$
The recurrence relation for the coefficients can be deduced immediately by substituting the series expansion into the differential equations. This is a routine procedure so we omit the details.
Next, let $$Y=\frac{z}{(1-p)(1-4p)\sqrt{1-4p^2}}.$$ By the ‘proof of Theorem \[t1\] using modular forms’ detailed above, and especially , , and , all forty-two functions in Theorem \[t1\] are different expressions for $Y$. By the change of variable $p=h/(1+h^2)$ and the formulas in [@cooperye], we have $$Y = z \times \frac{(1+h^2)^3}{(1-h^2)(1-h+h^2)(1-4h+h^2)} = \frac{\eta^2(2\tau)\eta^{12}(3\tau)}{\eta^4(\tau)\eta^6(6\tau)}.$$ By Clausen’s formula, the functions $y$ in Theorem \[t2\] are related to the functions $Y$ in Theorem \[t1\] by $Y=y^2$, and it follows that $$y = \frac{\eta(2\tau)\eta^6(3\tau)}{\eta^2(\tau)\eta^3(6\tau)}.$$ Moreover, by the formulas in [@cooperye] we have $$p = \frac{h}{1+h^2} = \frac{\eta(\tau)\eta^3(12\tau)}{\eta^3(3\tau)\eta(4\tau)}.$$ Finally, $y$ satisfies the required differential equation with respect to $p$, by .
Theorem \[t3\] can be proved in a similar way using the level $10$ function $$k=q\prod_{j=1}^\infty \frac{(1-q^{10j-9})(1-q^{10j-8})(1-q^{10j-2})(1-q^{10j-1})}
{(1-q^{10j-7})(1-q^{10j-6})(1-q^{10j-4})(1-q^{10j-3})}$$ and letting $$\label{pkk}
p=\frac{k}{1-k^2},$$ and then using the properties developed in [@cooper10] and [@cooper10a], along with the results for level 5 modular forms that are summarized in [@cc]. We omit most of the details as they are similar to the proof of Theorem \[t1\] given above. It is worth recording the modular parameterization.
Let the common power series of each expression in – be denoted by $$\sum_{n=0}^\infty b(n)p^n.$$ Then $$\frac{\eta(2\tau)\eta^{10}(5\tau)}{\eta^2(\tau)\eta^5(10\tau)}
=\sum_{n=0}^\infty b(n) \biggl(\frac{\eta(\tau)\eta^5(10\tau)}{\eta(2\tau)\eta^5(5\tau)}\biggr)^n.$$
By and [@cooper10 Theorem 3.5] we have $$\label{pk2}
p=\frac{k}{1-k^2} = \frac{\eta(\tau)\eta^5(10\tau)}{\eta(2\tau)\eta^5(5\tau)}.$$ Next, starting with and using the formula for $Z_2$ in Lemma \[7.2\], we get $$\begin{aligned}
\sum_{n=0}^\infty b(n)p^n
&= \frac{1}{(1+4p^2) ^{1/2} (1+22p-4p^2)} \\
& \qquad \times { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};256\,\frac{p (1+p)^5 (1-4p)^5}{(1+4p^2)^2 (1+22p-4p^2)^4}\biggr) \endgroup
}
\displaybreak[2]\\
&= \biggl(\frac{1}{p(1+p)^5(1-4p)^5}\biggr)^{1/4} \times \biggl(\frac{\eta^2(\tau)\eta^2(2\tau)}{Z_2(q)}\biggr)\times Z_2(q)
\\
&= \biggl(\frac{1}{p(1+p)^5(1-4p)^5}\biggr)^{1/4} \times \eta^2(\tau)\eta^2(2\tau).\end{aligned}$$ Now use to write $p$ in terms of $k$, and then use [@cooper10 Theorem 3.5] to express the resulting rational function of $k$ in terms of eta-functions, to get $$\label{pk3}
\sum_{n=0}^\infty b(n)p^n = \frac{\eta^{10}(5\tau)}{\eta^4(\tau)\eta(2\tau)\eta^5(10\tau)} \times \eta^2(\tau)\eta^2(2\tau)
= \frac{\eta(2\tau)\eta^{10}(5\tau)}{\eta^2(\tau)\eta^5(10\tau)}.$$ The proof may be completed by substituting the result of into .
Special cases {#sec8}
=============
Many of the transformation formulas in Theorems \[t1\], \[t2\] and \[t3\] can be simplified, sometimes significantly, by changing variables. We give several examples.
This example is from a paper by N.D. Baruah and B.C. Berndt [@baruah] and the book by J.M. Borwein and P.B. Borwein [@agm pp. 180–181]. Let $X=4x(1-x)$. Then $$\begin{aligned}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,\frac12,\frac12}{1,1};X\biggr) \endgroup
}
&= \frac{1}{1-x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,\frac12,\frac12}{1,1};\frac{-4x}{(1-x)^2}\biggr) \endgroup
}
\\
&= \frac{1}{\sqrt{1-x}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac12,\frac12,\frac12}{1,1};\frac{-x^2}{4(1-x)}\biggr) \endgroup
}
\displaybreak[2]\\
&=\frac{1}{1+x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};\frac{16x(1-x)^2}{(1+x)^4}\biggr) \endgroup
}
\displaybreak[2]\\
&=\frac{1}{1-2x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};\frac{-16x(1-x)}{(1-2x)^4}\biggr) \endgroup
}
\displaybreak[2]\\
&= \frac{2}{\sqrt{4-X}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac16,\frac12,\frac56}{1,1};\frac{27X^2}{(4-X)^3}\biggr) \endgroup
}
\\
&= \frac{1}{\sqrt{1-4X}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac16,\frac12,\frac56}{1,1};\frac{-27X}{(1-4X)^3}\biggr) \endgroup
}.\end{aligned}$$
Let $$x=\frac {16p (1-p)^3}{(1-2p) (1+2p)^3}$$ in each of , , , , , and , respectively.
This example was considered by J. Guillera and W. Zudilin [@translation Eq. (20)]: $$\begin{aligned}
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};4x(1-x)\biggr) \endgroup
}
&= \frac{1}{\sqrt{1+8x}}\,
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac16,\frac12,\frac56}{1,1};\frac{64x(1-x)^3}{(1+8x)^3}\biggr) \endgroup
} \\
&= \frac{3}{\sqrt{9-8x}}\,
{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac16,\frac12,\frac56}{1,1};\frac{64x^3(1-x)}{(9-8x)^3}\biggr) \endgroup
}.\end{aligned}$$
Let $$x=\frac{27p^2 ( 1-2p )^2 }{( 1-2p+4p^2)^3}$$ in , and , respectively.
The identities in this example are from the unorganized pages of Ramanujan’s second notebook [@notebooks p. 258]: $$\sqrt{1+2x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac13,\frac23}{1};\frac{27x^2(1+x)^2}{4(1+x+x^2)^3}\biggr) \endgroup
}
= (1+x+x^2)\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac12,\frac12}{1};\frac{x^3(2+x)}{1+2x}\biggr) \endgroup
}$$ and $$\begin{gathered}
(2+2x-x^2)\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac13,\frac23}{1};\frac{27x(1+x)^4}{2(1+4x+x^2)^3}\biggr) \endgroup
}
\\
= 2(1+4x+x^2)\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac13,\frac23}{1};\frac{27x^4(1+x)}{2(2+2x-x^2)^3}\biggr) \endgroup
}.\end{gathered}$$ Proofs, by a different method, have been given by Berndt et al. [@bbg Theorems 5.6 and 6.4].
Take $p=-x/2$ in and to obtain $$\begin{gathered}
\frac{1}{(1+x+x^2)}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac13,\frac23}{1};\frac{27x^2 ( 1+x )^2 }{4( 1+x+x^2)^3}\biggr) \endgroup
}
\\
=\frac {1}{ ( 1+x ) ^{3/2} ( 1-x )^{1/2} }\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac12,\frac12}{1};\frac{-x^3(2+x)}{(1+x)^3(1-x)}\biggr) \endgroup
}.\end{gathered}$$ Now apply Pfaff’s transformation [@aar Theorem 2.2.5] $${ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,b}{c};z\biggr) \endgroup
} = (1-z)^{-a}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{a,c-b}{c};\frac{-z}{1-z}\biggr) \endgroup
}$$ to the right-hand side to obtain the first identity.
The second identity is obtained simply by putting $p=-x/2$ in and .
This example was studied by M. Rogers [@rogers Theorem 3.1] and by H.H. Chan and W. Zudilin [@mathematika Theorems 3.2 and 4.2]: $$\begin{aligned}
G_{6b}(x)
&= \frac{1}{1+16x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};\frac{108x}{(1+16x)^3}\biggr) \endgroup
} \\
&= \frac{1}{1+4x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac13,\frac12,\frac23}{1,1};\frac{108x^2}{(1+4x)^3}\biggr) \endgroup
}
\intertext{and}
G_{6c}(x)
&= \frac{1}{1+27x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};\frac{256x}{(1+27x)^4}\biggr) \endgroup
} \\
&= \frac{1}{1+3x}\, { \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{3}F_{2}\biggl(\genfrac..{0pt}{}{\frac14,\frac12,\frac34}{1,1};\frac{256x^3}{(1+3x)^4}\biggr) \endgroup
}.\end{aligned}$$
The first group of identities is obtained by taking $$x=\frac{p(1-p)(1+2p)(1-2p)}{(1-4p)^2}$$ in , and . The second group of identities may be proved by taking $$x=\frac{p(1-2p)}{(1-p)(1+2p)(1-4p)^2}$$ in , and .
\[cz6\] This example shows that the result of H.H. Chan and W. Zudilin [@mathematika Theorem 2.2] is subsumed by Theorem \[t1\]: $$\begin{aligned}
\frac{1}{1+x}\,G_{6a}\biggl(\frac{x(1-8x)}{1+x}\biggr)
&=\frac{1}{1-8x}\,G_{6b}\biggl(\frac{x(1+x)}{1-8x}\biggr),
\\
\frac{1}{1-x}\,G_{6a}\biggl(\frac{x(1-9x)}{1-x}\biggr)
&=\frac{1}{1-9x}\,G_{6c}\biggl(\frac{x(1-x)}{1-9x}\biggr)
\\ \intertext{and}
\frac{1}{1+8x}\,G_{6b}\biggl(\frac{x(1+9x)}{1+8x}\biggr)
&=\frac{1}{1+9x}\,G_{6c}\biggl(\frac{x(1+8x)}{1+9x}\biggr).\end{aligned}$$
The three identities may be obtained by setting $$x=p(1-2p), \quad x=\frac{p(1-2p)}{(1+2p)(1-p)}\quad\text{and}\quad x=\frac{p(1-2p)}{(1-4p)^2}$$ respectively, in , and .
Alternative proofs of the identities in Example \[cz6\] may be given by taking $$x=\frac{p^2}{1-2p},\quad x=\frac{p^2}{(1-p)^2}\quad\text{and}\quad x=\frac{p^2}{1-2p-8p^2}$$ in , and , or by taking $$x=\frac{-p}{(1-2p)^2},\quad x=\frac{-p}{(1-p)(1-4p)} \quad\text{and}\quad x=\frac{-p}{(1+2p)^2}$$ in , and .
\[6weight1\] Part of this example was mentioned in the identity as part of the introduction: $$\begin{aligned}
f_{6a}(x)
&=\frac{1}{1+9x}\,f_{6b}\biggl(\frac{x}{1+9x}\biggr)
=\frac{1}{1+8x}\,f_{6c}\biggl(\frac{x}{1+8x}\biggr),
\\
f_{6b}(x)
&=\frac{1}{1-9x}\,f_{6a}\biggl(\frac{x}{1-9x}\biggr)
=\frac{1}{1-x}\,f_{6c}\biggl(\frac{x}{1-x}\biggr),
\\
f_{6c}(x)
&=\frac{1}{1-8x}\,f_{6a}\biggl(\frac{x}{1-8x}\biggr)
=\frac{1}{1+x}\,f_{6b}\biggl(\frac{x}{1+x}\biggr).\end{aligned}$$
Each of the three sets of identities can be proved by taking $$x=\frac{p(1-2p)}{(1-4p)^2},\quad
x=\frac{p(1-2p)}{(1-p)(1+2p)}\quad\text{and}\quad
x=p(1-2p),$$ respectively, in , and .
Here we give representations for the functions in the previous example in terms of hypergeometric functions: $$\begin{aligned}
f_{6b}(x)
&=\frac{1}{\sqrt{1+18x-27x^2}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac14,\frac34}{1};\frac{64x}{(1+18x-27x^2)^2}\biggr) \endgroup
}
\\
&=\frac{1}{\sqrt{1-6x-3x^2}}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac14,\frac34}{1};\frac{64x^3}{(1-6x-3x^2)^2}\biggr) \endgroup
}
\intertext{and}
f_{6c}(x)
&=\frac{1}{1+4x}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac13,\frac23}{1};\frac{27x}{(1+4x)^3}\biggr) \endgroup
}
\\
&=\frac{1}{1-2x}\,{ \begingroup
\begingroup\lccode`~=`,
\lowercase{\endgroup\def~}{\pFcomma\mkern\pFqskip} \mathcode`,=\string"8000
{}_{2}F_{1}\biggl(\genfrac..{0pt}{}{\frac13,\frac23}{1};\frac{27x^2}{(1-2x)^3}\biggr) \endgroup
}.\end{aligned}$$
The first set of identities may be proved by taking $$x=\frac{p(1-2p)}{(1-p)(1+2p)}$$ in , and . To prove the second set of identities, take $x=\nobreak p(1-2p)$ in , and .
Just as for Example \[cz6\], the identities in Example \[6weight1\] can be given alternative proofs using , and , or by using , and .
Applications {#sec9}
============
In this section, we will show how the transformation formulas in Theorem \[t1\] can be used to establish the equivalence of several of Ramanujan’s series for $1/\pi$. In the remainder of this section, we will use the binomial representation of the related hypergeometric functions, so that the resulting formulas will be consistent with the data in [@cc Tables 3–6].
\[aycock1\] The following series identities are equivalent in the sense that any one can be obtained from the others by using the transformation formulas in Theorem : $$\label{pi1}
\sum_{n=0}^\infty \binom{6n}{3n}\binom{3n}{n}\binom{2n}{n}\biggl(n+\frac{3}{28}\biggr) \biggl(\frac{1}{20}\biggr)^{3n}
= \frac{5\sqrt{5}}{28} \times \frac{1}{\pi},$$ $$\label{pi2}
\sum_{n=0}^\infty \binom{4n}{2n}\binom{2n}{n}^2\biggl(n+\frac{3}{40}\biggr) \biggl(\frac{1}{28}\biggr)^{4n}
= \frac{49\,\sqrt{3}}{360} \times \frac{1}{\pi}$$ and $$\label{pi3}
\sum_{n=0}^\infty \binom{2n}{n}^3\biggl(n+\frac{1}{4}\biggr) \biggl(\frac{-1}{64}\biggr)^{n}
= \frac{1}{2\pi}.$$
In order to prove Theorem \[aycock1\], it will be convenient to make use of the following simple lemma.
\[translate\] Let $x$, $y$ and $r$ be analytic functions of a complex variable $p$, and suppose that $x(0)=y(0)=0$ and $r(0)=1$. Suppose that a transformation formula of the form $$\label{h1}
\sum_{n=0}^\infty a(n)\, x^n = r\, \sum_{n=0}^\infty b(n)\, y^n$$ holds in a neighborhood of $p=0$. Let $\lambda$ be an arbitrary complex number. Then $$\label{h3}
\sum_{n=0}^\infty a(n)\, (n+\lambda)\, x^n
= \sum_{n=0}^\infty b(n)\,\biggl(n\,\frac{xr}{y}\,\frac{{{\mathrm d}}y}{{{\mathrm d}}x} + \biggl(x\,\frac{{{\mathrm d}}r}{{{\mathrm d}}x}+\lambda r\biggr)\biggr)y^n.$$
Applying the differential operator $x\,\frac{{{\mathrm d}}}{{{\mathrm d}}x}$ to gives $$\label{h2}
\sum_{n=0}^\infty a(n)\, n\, x^n = \frac{xr}{y}\,\frac{{{\mathrm d}}y}{{{\mathrm d}}x}\,\sum_{n=0}^\infty b(n)\,n\,y^n
+x\,\frac{{{\mathrm d}}r}{{{\mathrm d}}x}\,\sum_{n=0}^\infty b(n)\,y^n.$$ Taking a linear combination of and gives the required result.
We are now ready for:
By and in Theorem \[t1\], the functions $$\begin{aligned}
x &= \frac {p^2 ( 1-p )^6 ( 1-4p ) ^{
6} ( 1-2p )^2 ( 1+2p )^6}
{ (1 -2p + 4p^2)^3
(1 -6p +240p^2-920p^3+960p^4-96p^5 +64p^6)^3}\\
y &={{\frac {p^3 ( 1-p ) ( 1-4p )^2
( 1-2p )^3 ( 1+2p ) }
{ (1 -4p +32p^3-32p^4 )^4}}
}
\intertext{and}
r &= \frac
{ (1 -2p + 4p^2) ^{1/2}
(1 -6p +240p^2-920p^3+960p^4-96p^5 +64p^6) ^{1/2}}
{(1 -4p +32p^3-32p^4 )}\end{aligned}$$ satisfy the hypotheses of Lemma \[translate\], where the coefficients are given by $$a(n) = \binom{6n}{3n}\binom{3n}{n}\binom{2n}{n}
\quad\text{and}\quad
b(n) = \binom{4n}{2n}{\binom{2n}n}^2.$$ Setting $$p=\frac14(1+3\sqrt{2}-3\sqrt{3})$$ gives $$x=\biggl(\frac{1}{20}\biggr)^3,\qquad y=\biggl(\frac{1}{28}\biggr)^4 \quad\text{and}\quad r=\frac{9}{28}\sqrt{10}.$$ The derivatives can be calculated by the chain rule and we find that $$\begin{aligned}
\frac{{{\mathrm d}}y}{{{\mathrm d}}x}\biggr|_{p=\frac14\left(1+3\sqrt{2}-3\sqrt{3}\right)}
&=\frac{5^5 }{7^6} \times \frac{\sqrt{6}}{3}
\\ \intertext{and}
\frac{{{\mathrm d}}r}{{{\mathrm d}}x}\bigg|_{p=\frac14\left(1+3\sqrt{2}-3\sqrt{3}\right)}
&=\frac{2^2 \times 3^2 \times 5^3}{7^3} \times \sqrt{5}\,(20\sqrt{3}-21\sqrt{2}).\end{aligned}$$ Substituting these values in and taking $\lambda=3/28$ gives $$\begin{gathered}
\sum_{n=0}^\infty \binom{6n}{3n}\binom{3n}{n}\binom{2n}{n}\biggl(n+\frac{3}{28}\biggr) \biggl(\frac{1}{20}\biggr)^{3n}
\\
= \frac{150}{343} \times \sqrt{15}\,
\sum_{n=0}^\infty \binom{4n}{2n}\binom{2n}{n}^2\biggl(n+\frac{3}{40}\biggr) \biggl(\frac{1}{28}\biggr)^{4n}.\end{gathered}$$ This shows that the series evaluations and are equivalent.
To show that is equivalent to , use and as motivation to define $$\begin{aligned}
x &=\frac {p^3 ( 1-p ) ( 1-4p )^2 ( 1-2p )^3 ( 1+2p ) } { (1 -4p +32p^3-32p^4 )^4},
\\
y &=\frac {-p( 1-2p )(1+2p)^3(1-p)^3 }{( 1-4p )^6}
\intertext{and}
r &= \frac{(1 -4p +32p^3-32p^4 )}{(1 -4p )^3}\end{aligned}$$ and take $$a(n) = \binom{4n}{2n}{\binom{2n}n}^2
\quad\text{and}\quad
b(n) = {\binom{2n}n}^3.$$ Then compute the required derivatives, let $p$ have the same value as above, and put $\lambda=3/40$ in . We omit the details as they are similar to the above.
In order to explain why the particular transformation formulas , and were used in the proof of Theorem \[aycock1\], we use the classification of series, such as –, by modular forms. In [@cc Theorem 2.1], it is shown that series such as – can be classified according to three parameters:
1. the level $\ell$;
2. the degree $N$; and
3. the nome $q$.
Tables of series for $1/\pi$ for various parameter values are given in [@cc Tables 3–6] and [@aldawoud]. The relevant parameters corresponding to the series , and are given by $$(\ell,N,q) = (1,2, e^{-2\pi\sqrt{2}}),\quad (2,9,e^{-3\pi\sqrt{2}})\quad \text{and}\quad
(4,2,-e^{-\pi\sqrt{2}}),$$ respectively. If the $q$-parameters are denoted by $q_1$, $q_2$ and $q_3$, respectively, then $$q_1=q_0^2,\quad q_2=q_0^3\quad\text{and}\quad q_3=-q_0$$ where $q_0=e^{-\pi\sqrt{2}}$. The relevant issue is that $q_1$, $q_2$ and $q_3$ are all integral powers of a common value $q_0$, and for $q_3$ there is also a sign change. The values of $q_1$ and $q_2$ suggest that quadratic and cubic transformation formulas be used, respectively, while the value of $q_3$ suggests a change of sign is involved. The corresponding hypergeometric functions which have these properties, for the relevant levels, are given by , and .
The entries in Tables 3–6 of [@cc] may be further analyzed by their $q$-values to obtain similar relations. This leads to the following equivalence classes of Ramanujan-type series for $1/\pi$ in Theorems \[th93\]–\[th97\].
\[th93\] Ramanujan’s series and in [@ramanujan_pi], namely, $$\label{irrational}
\sum_{n=0}^\infty {\binom{2n} n}^3 \biggl(n+\frac{31}{270+48\sqrt{5}}\biggr) \frac{(\sqrt{5}-1)^{8n}}{2^{20n}}
= \frac{16}{15+21\sqrt{5}} \times \frac{1}{\pi}$$ and $$\label{notirrational}
\sum_{n=0}^\infty \binom{3n}n{\binom{2n}n}^2 \biggl(n+\frac{4}{33}\biggr) \frac{1}{15^{3n}}
= \frac{5\,\sqrt{3}}{22} \times \frac{1}{\pi}$$ are equivalent in the sense that one may be deduced from the other by using transformation formulas in Theorem .
The clue is to observe that the series and correspond to the data $$(\ell,N,q)=(4,15,e^{-\pi\sqrt{15}})
\quad\text{and}\quad
(\ell,N,q)=(3,5,e^{-2\pi\sqrt{5/3}}),$$ respectively, in the classifications in [@aldawoud Table 3.9] and [@cc Table 5]. The values of $q$ are related by $$e^{-\pi\sqrt{15}} = q_0^3
\quad\text{and}\quad
e^{-2\pi\sqrt{5/3}}=q_0^2,
\qquad\text{where}\quad
q_0=e^{-\pi\sqrt{5/3}}.$$ Therefore, we seek a cubic transformation formula from the level $4$ theory and a quadratic transformation formula form the cubic theory. The relevant functions occur in and , respectively. The proof may be completed by applying the result of Lemma \[translate\] and copying the procedure in the proof of Theorem \[aycock1\]. We omit the details, as they are similar, except to say that the value of $p$ in this case is given by $$p=\frac12\bigl(8-4\sqrt{3}+3\sqrt{5}-2\sqrt{15}\bigr).
\qedhere$$
The series is notable for being the only one of Ramanujan’s 17 examples to contain an irrational value for the power series variable; the other 16 series all involve rational numbers.
In the remaining examples, we will be more brief.
The Ramanujan-type formulas for $1/\pi$ given by the following data in Tables of [@cc] are equivalent: $$(\ell,N,q) =
(1,3, e^{-2\pi\sqrt{3}}),\;
(1,27,-e^{-3\pi\sqrt{3}}),\;
(3,9,-e^{-\pi\sqrt{3}}),\;
(4,3,e^{-\pi\sqrt{3}})$$ and $(3,4,e^{-4\pi/\sqrt{3}})$.
For the first four sets of parameter values, use , , and , and let $p$ be the smallest positive root of $$\biggl(2p+\frac{1}{2p}\biggr)^3-120\biggl(2p+\frac{1}{2p}\biggr)^2+480\biggl(2p+\frac{1}{2p}\biggr)-496=0$$ so that $p\approx 0.00431456$.
The series corresponding to the last two sets of parameter values $(3,4,e^{-4\pi/\sqrt{3}})$ and $(4,3,e^{-\pi\sqrt{3}})$ can be shown to be equivalent using and and using the value $p=1-\sqrt{3}/2$.
The Ramanujan-type formulas for $1/\pi$ given by the following data in Tables of [@cc] are equivalent: $$(\ell,N,q) =
(1,4, e^{-4\pi}),\;
(2,2,e^{-2\pi}),\;
(2,9,-e^{-3\pi}),\;
(4,4,-e^{-2\pi}).$$
Use , , and , and take $$p=\frac14\bigl(7+3\sqrt{3}-{\textstyle\sqrt{72+42\sqrt{3}}}\bigr) \approx 0.0412759.
\qedhere$$
The Ramanujan-type formulas for $1/\pi$ given by the following data in Tables of [@cc] are equivalent: $$(\ell,N,q) =
(1,7, e^{-2\pi\sqrt{7}}),\;
(1,7,-e^{-\pi\sqrt{7}}),\;
(2,7,-e^{-\pi\sqrt{7}}),\;
(4,7,e^{-\pi\sqrt{7}}).$$
Use , , and , and let $p$ be the smallest positive root of $$\begin{gathered}
\biggl(2p+\frac{1}{2p}\biggr)^4 - 2044\biggl(2p+\frac{1}{2p}\biggr)^3
+15360\biggl(2p+\frac{1}{2p}\biggr)^2
\\
-38416\biggl(2p+\frac{1}{2p}\biggr)+31984=0\end{gathered}$$ so that $p \approx 0.000245523$.
\[th97\] The Ramanujan-type formulas for $1/\pi$ given by the following data in Tables of [@cc] are equivalent: $$(\ell,N,q) =
(2,3, e^{-2\pi\sqrt{3/2}}),\;
(3,2,e^{-2\pi\sqrt{2/3}}).$$
Use and and take $$p=\frac14(1+\sqrt{3}-\sqrt{6}).
\qedhere$$
Finally, we notice that the identities in Theorems \[t1\], \[t2\] and \[t3\] can be used in designing AGM-type algorithms [@agm] for the effective computation of $\pi$ and other mathematical constants. The details of such applications can be found in [@CGSZ; @guillera].
**Acknowledgments.** A part of the work was done during the second author’s stay at the Max-Planck-Institut für Mathematik, Bonn, in May–June 2016. He thanks the staff of the institute for the excellent conditions he experienced when conducting this research.
[99]{}
Aldawoud, A.M.: Ramanujan-type series for $1/\pi$ with quadratic irrationals. Master of Science Thesis, Massey University, Auckland (2012)
Almkvist, G., van Straten, D., Zudilin, W.: Generalizations of Clausen’s formula and algebraic transformations of Calabi–Yau differential equations. Proc. Edinb. Math. Soc. **54**, 273–295 (2011)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia Math. Appl., Vol. 71. Cambridge University Press, Cambridge (1999)
Aycock, A.: On proving some of Ramanujan’s formulas for $1/\pi$ with an elementary method. An unpublished manuscript; [arXiv:1309.1140v2 \[math.NT\]](http://arxiv.org/abs/1309.1140) (2013)
Baruah, N.D., Berndt, B.C.: Eisenstein series and Ramanujan-type series for $1/\pi$. Ramanujan J. **23**, 17–44 (2010)
Berndt, B.C., Bhargava, S., Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. **347**, 4163–4244 (1995)
Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Canad. Math. Soc. Series Monographs Advanced Texts. John Wiley, New York (1987)
Chan, H.H., Chan, S.H., Liu, Z.-G.: Domb’s numbers and Ramanujan–Sato type series for $1/\pi$. Adv. Math. **186**, 396–410 (2004)
Chan, H.H., Cooper, S.: Rational analogues of Ramanujan’s series for $1/\pi$. Math. Proc. Cambr. Phil. Soc. **153**, 361–383 (2012)
Chan, H.H., Tanigawa, Y., Yang, Y., Zudilin, W.: New analogues of Clausen’s identities arising from the theory of modular forms. Adv. Math. **228**, 1294–1314 (2011)
Chan, H.H., Zudilin, W.: New representations for Apéry-like sequences. Mathematika **56**, 107–117 (2010)
Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In: Ramanujan Revisited (Urbana-Champaign, Ill. 1987), pp. 375–472. Academic Press, Boston, MA (1988)
Cooper, S.: Inversion formulas for elliptic functions. Proc. Lond. Math. Soc. **99**, 461–483 (2009)
Cooper, S.: On Ramanujan’s function $k(q)=r(q)r^2(q^2)$. Ramanujan J. **20**, 311–328 (2009)
Cooper, S.: Level $10$ analogues of Ramanujan’s series for $1/\pi$. J. Ramanujan Math. Soc. **27**, 59–76 (2012)
Cooper, S., Guillera, J., Straub, A., Zudilin, W.: Crouching AGM, Hidden Modularity. Preprint; [arXiv:1604.01106 \[math.NT\]](http://arxiv.org/abs/1604.01106) (2016)
Cooper, S., Ye, D.: The level $12$ analogue of Ramanujan’s function $k$. J. Aust. Math. Soc. **101**, 29–53 (2016)
Goursat, É.: Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique. Ann. Sci. ENS Sér. 2 **10**, 3–142 (1881) Guillera, J.: New proofs of Borwein-type algorithms for Pi. Integral Transform Spec. Funct. **27**, 775–782 (2016)
Guillera, J., Zudilin, W.: Ramanujan-type formulae for $1/\pi$: The art of translation. In: The Legacy of Srinivasa Ramanujan, B. C. Berndt and D. Prasad (eds.). Ramanujan Math. Soc. Lecture Notes Series, vol. 20, 181–195 (2013)
Maier, R.S.: Algebraic hypergeometric transformations of modular origin. Trans. Am. Math. Soc. **359**, 3859–3885 (2007)
Ramanujan, S.: Notebooks, vol. 2. Tata Institute of Fundamental Research, Bombay (1957)
Ramanujan, S.: Modular equations and approximations to $\pi$, Quart. J. Math. **45**, 350–372 (1914); Reprinted in [@RCW pp. 23–39] (2000)
Ramanujan, S.: Collected Papers. 3rd printing. Am. Math. Soc. Chelsea, Providence, RI (2000)
Rogers, M.: New ${}_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\pi$. Ramanujan J. **18**, 327–340 (2009)
Vidūnas, R.: Algebraic transformations of Gauss hypergeometric functions. Funkcial. Ekvac. **52**, 139–180 (2009)
Zagier, D.: Integral solutions of Ap[é]{}ry-like recurrence equations. In: Groups and symmetries. CRM Proc. Lecture Notes, vol. 47, pp. 349–366. Am. Math. Soc., Providence, RI (2009)
Zudilin, W.: Ramanujan-type formulae for $1/\pi$: A second wind? In: Modular Forms and String Duality (Banff, June 3–8, 2006), N. Yui, H. Verrill and C.F. Doran (eds.). Fields Inst. Comm. Ser., vol. 54, 179–188. Am. Math. Soc., Providence, RI (2008)
Zudilin, W.: Lost in translation. In: Advances in Combinatorics, Waterloo Workshop in Computer Algebra, W80 (May 26–29, 2011), I. Kotsireas and E.V. Zima (eds.), pp. 287–293. Springer, New York (2013)
[^1]: Some misprints in [@aycock p. 15] have been corrected.
|
---
abstract: 'Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability greater than 25/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Li’s phase matching. In this paper we introduce a multi-phase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multi-phase operation with six iterations yields a success probability between 99.8% and 100% for a fraction of marked states of 1/10 or larger.'
author:
- 'F.M. Toyama'
- 'W. van Dijk'
- 'Y. Nogami'
- 'M. Tabuchi'
- 'Y. Kimura'
title: 'Multi-phase matching in the Grover algorithm'
---
Introduction {#sec:intro}
============
The quantum search algorithm introduced by Grover [@grover96; @grover97a; @grover97b; @grover01] constitutes a major advance in quantum computing. It enables us to find a marked state stored in a database state consisting of $N$ unordered basis states in only ${\cal O}(\sqrt{N})$ Grover operations. A number of modifications and generalizations of the original Grover search algorithm have been proposed [@grover98; @long99; @long01; @collins02; @tulsi06; @li07; @younes07]. In particular, phase matching methods in the Grover algorithm have been extensively examined [@long99; @long01; @li07]. The outcome of the search algorithm is characterized in terms of $P(\lambda)$, the probability of obtaining an equal-amplitude superposition of the marked states where $\lambda$ is the ratio of the marked states to all the states stored in the original database state.
Recently, Li and Li [@li07] proposed a new phase matching for the Grover algorithm and they obtained an improved success probability $P(\lambda)$ over a wide range of the ratio $\lambda$. They introduced the set of the Grover operators (details are described in Eqs. (\[eq:1\]) and (\[eq:2\])): $U =
I-(1-e^{i\alpha})\sum_{l=0}^{M-1}|t_l\rangle\langle t_l|$ and $V =
Ie^{i\beta}+(1-e^{i\beta})|0^{\otimes n}\rangle \langle 0^{\otimes
n}|$. The phase factor $e^{i\beta}$ in the first term of the operator $V$ was first introduced in Ref. [@li07]. In the new phase matching the number of phases is the same as the usual one but the form of the phase shift operator $V$ is different. Li and Li found the remarkable result that *a single Grover operation* of the new phase matching yields $P(\lambda)>25/27$ for $1/3\leq\lambda\leq 1$. This is significant in the sense that with *only one Grover operation* the efficiency of the Grover algorithm is substantially improved in the range of values of $\lambda$ where the efficiency of the original algorithm deteriorates.
This phase matching has another interesting aspect that was not explicitly pointed out by Li and Li [@li07]. For a given values of $\lambda$ in the range $1/4\leq\lambda \leq 1$, one Grover operation with the phases $\alpha=-\beta=\arccos(1-1/2\lambda)$ yields exactly $P=1$. \[See Eq. (\[eq:12a\]) in the following.\] This results was obtained earlier by Chi and Kim [@chi97] who considered a modified Grover operator of arbitrary phase. The special case of $\lambda=1/2$ yields $\alpha=-\beta=\pm\pi/2$, which are the phases found in Ref. [@li07]. This aspect of the phase matching is also significant because it implies that one can always find the equal-amplitude superposition of the marked states by only one Grover operation when $\lambda$ is greater than 1/4 by tuning the phases $\alpha$ and $\beta$ appropriately for the given $\lambda$. Conditions for a success probability of unity have been studied by previous authors. See, for example, Refs. [@hoyer00; @long01a].
It should be pointed out, however, that the so-called new phase matching of Ref. [@li07] is equivalent to the original phase matching of Long *et al.* [@long99]. When the second operator is defined as $V'=e^{-i\beta}V$, it becomes the phase-matching operator of Long *et al.* The only difference between the two is that the overall state is multiplied by a phase factor and so the amplitudes of the components are different, but the probabilities are the same. Thus the remarkable result of Li and Li can also be seen to follow from the operator of Long *et al.* Analytically the formulation by Li and Li is somewhat more transparent and hence we use it throughout this paper, except in the Appendix where we explicitly show the equivalence of the two formulations by calculating the probability profile.
Thus a number of aspects of the Grover algorithm with phase matching, already alluded to, are of particular interest and they form the objectives of this study. We focus on high success probabilities with as few iterations as possible in order to enhance the efficiency of the quantum search. We emphasize the following three objectives: (1) the elucidation of features of the phase-matched Grover operations with a small number of iterations that yield success probabilities $P(\lambda)$ close to one over a wide range of values of $\lambda$, (2) given a value of $\lambda$ the determination of the phase-matched Grover operator(s) that results in $P(\lambda)=1$ exactly, and (3) the elucidation of the features of the phase-matched Grover operators that allow us to obtain $P(\lambda)=1$ for very small values of $\lambda$.
In this paper we explore the search algorithm with these objectives in mind using the advantages of a few multiple Grover operations with phase matching. It is well known that a multiple application of the original Grover operation gives rise to intensive oscillations of $P$ as a function of $\lambda$ and such oscillations deteriorate the efficiency of the algorithm. This undesirable feature remains even in the new phase matching of Li and Li, as we will illustrate. We show that if we introduce a *multi-phase* matching subject to a certain matching rule, we can obtain a multiple Grover operation that yields a success probability almost constant and unity over a wide range of $\lambda$, e.g., $0.1 \leq \lambda\leq 1$. This is also significant in the sense that when $\lambda$ is greater than a small minimum value we can always find the superposition of the marked states with high degree of certainty without (re)tuning the phases.
In the next section we set up the algorithm of the multi-phase matching in the framework of the phase matching of Li and Li [@li07] and analyze the efficiency of the algorithm by considering a single matched phase and a two-stage multi-phase matching. We also obtain an exemplar of a good probability profile for a six-stage multi-phase matched operator. In Sec. \[k\_iterations\] we consider the success probability for small $\lambda$ by using the Grover operations with a phase other than $\pi$. We summarize our results in Sec. \[summary\].
Multi-phase matching in the framework of the new phase matching {#multiphase}
===============================================================
The new phase matching in the Grover algorithm proposed by Li and Li [@li07] is defined with the two operators, $$\label{eq:1}
U=I-(1-e^{i\alpha})\sum_{l=0}^{M-1}|t_l\rangle\langle t_l|$$ $$\label{eq:2}
V=Ie^{i\beta}+(1-e^{i\beta})|0^{\otimes n}\rangle\langle 0^{\otimes n}|.$$ where $|0^{\otimes n}\rangle$ is the $n$-qubits initial state, $M$ is the number of target (marked) states stored in an unstructured database state, and the $|t_l\rangle$ denote the target or marked states. The database state is given as $|\phi\rangle =
H^{\otimes n}|0^{\otimes n}\rangle$, where $H$ is the Walsh-Hadamard transformation. The state $|\phi\rangle$ is an equally-weighted superposition of the $N=2^n$ basis states, $|w_l\rangle, \ l =
0,\dots,N-1$. The fraction $\lambda$ of the target states is defined as $\lambda = M/N$. The $U$ and $V$ of Eqs. (\[eq:1\]) and (\[eq:2\]) are both unitary as was shown in Ref. [@li07]. With $\alpha=\beta=\pi$, $U$ and $V$ reduce to the Grover operators of the original algorithm. As we mentioned in Sec. \[sec:intro\], Li and Li showed explicitly that *a single Grover operation* of the new phase matching $(H^{\otimes n}VH^{\otimes n})UH^{\otimes n}|0^{\otimes
n}\rangle$ with $\alpha = -\beta=\pi/2$ yields a success probability $P(\lambda)>25/27$ for $1/3\leq\lambda\leq 1$.
We introduce a multi-phase matching within the framework of the new phase matching. We rewrite the database state $|\phi\rangle =
H^{\otimes n}|0^{\otimes n}\rangle = N^{-1/2}\sum_{l=0}^{N-1}
|\omega_l\rangle$ in terms of $\lambda$ as $$\begin{aligned}
\label{eq:3}
|\phi\rangle = \frac{1}{\sqrt{N}}\sum_{l=0}^{N-1}|\omega_l\rangle
& = & \sqrt{\frac{N-M}{N}}|R\rangle +
\sqrt{\frac{M}{N}}|T\rangle \nonumber \\
& = & \sqrt{1-\lambda}|R\rangle + \sqrt{\lambda} |T\rangle,\end{aligned}$$ where $$\label{eq:4}
|R\rangle = \frac{1}{\sqrt{N-M}}\sum_{l=0}^{N-M-1}|r_l\rangle,
\ \ |T\rangle = \frac{1}{\sqrt M}\sum_{l=0}^{M-1}|t_l\rangle.$$ The state $|T\rangle$ is the uniform superposition of the marked states and $|R\rangle$ is that of the remaining states $|r_l\rangle$. They are both normalized to unity and orthogonal to each other. In the following, for convenience, we work in the two-dimensional space defined by the basis $\{|R\rangle, |T\rangle\}$. The two-dimensional representations of $U$ and $H^{\otimes n}VH^{\otimes n} = Ie^{i\beta}
+ (1-e^{i\beta})|\phi\rangle\langle\phi|$ are
$$\label{eq:5}
U:\begin{pmatrix} 1 & 0 \\ 0 & e^{i\alpha} \end{pmatrix},
\ \ \
H^{\otimes n}VH^{\otimes n} :
\begin{pmatrix}
(1-e^{i\beta})(1-\lambda)+e^{i\beta} & (1-e^{i\beta})
\sqrt{\lambda(1-\lambda)} \\
(1-e^{i\beta})\sqrt{\lambda(1-\lambda)} & (1-e^{i\beta})\lambda
+ e^{i\beta} \end{pmatrix}.$$
We write the multiple Grover operation with the multiple phases $\alpha_j$ and $\beta_j$ $(j=1,\dots,k)$ as $$\label{eq:7}
\begin{pmatrix} u_k \\ d_k \end{pmatrix}
= G(\alpha_k,\beta_k)G(\alpha_{k-1},\beta_{k-1})\cdots G(\alpha_1,\beta_1)
\begin{pmatrix} \sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix},$$ where one Grover operation $G(\alpha_j,\beta_j)$ $(j=1,\dots,k)$ in this representation is $$\label{eq:8}
G(\alpha_j,\beta_j)=
\begin{pmatrix}
(1-e^{i\beta_j})(1-\lambda) + e^{i\beta_j} &
(e^{i\alpha_j} - e^{i(\alpha_j+\beta_j)})\sqrt{\lambda(1-\lambda)} \\
(1-e^{i\beta_j})\sqrt{\lambda(1-\lambda)} &
(e^{i\alpha_j} - e^{i(\alpha_j+\beta_j)})\lambda + e^{i(\alpha_j+\beta_j)}
\end{pmatrix}.$$
The success probability of finding the superposition of target states is given by $P_k(\lambda)\equiv|d_k|^2$.
We now consider the one- and two-pair-phase cases before increasing the phase-matching to six different pairs of phases in order to obtain $P(\lambda)$ nearly equal to unity over a large range of values of $\lambda$. In other words, we discuss the $k=1$ and the $k=2$ cases in detail first, and then proceed to the numerical results of the $k=6$ case.
Multi-phase matching with one pair of phases {#one}
--------------------------------------------
When $k=1$, Eq. (\[eq:7\]) reduces to $$\label{eq:10}
\begin{pmatrix}u_1 \\ d_1 \end{pmatrix} = G_1(\alpha,\beta)
\begin{pmatrix}\sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}.$$ Since we first focus on cases of complete success $P=|d_1|^2 = 1 -
|u_1|^2 = 1$, we can equivalently consider the condition $u_1 = 0$. In general $$\label{eq:11}
u_1 = \sqrt{1-\lambda} \, [1-\lambda+e^{i\beta}\lambda + (e^{i\alpha}
-e^{i(\alpha+\beta)})\lambda \, ].$$ The condition that $u_1$ be zero leads to $$\begin{aligned}
\label{eq:12}
\frac{1}{\lambda} & = & 1 -\cos\alpha -\cos\beta
- \cos(\alpha+\beta) \nonumber \\
&& \ \ + i[\sin{(\alpha+\beta)}-\sin\alpha-\sin\beta].\end{aligned}$$ The fact that $\lambda$ must be real implies that (1) $\beta =
-\alpha$, (2) either $\alpha$ or $\beta$ are zero, or (3) both $\alpha$ and $\beta$ are zero. When $\beta =0$ in the operator $V$ of Eq. (\[eq:2\]), the operator is the identity and the overall effect of operator $U$ of Eq. (\[eq:1\]) by itself would cause the phase of the marked states to be changed, but the probabilities of marked and unmarked states would remain the same. When $\alpha=0$, then $U=I$ and $G=H^{\otimes n} VH^{\otimes n}$. The initial state $|\phi\rangle$ is an eigenvector of $G$ with eigenvalue 1. Thus $G$ does not cause any evolution in $|\phi\rangle$. The success probability is $P=\lambda$, which is the success probability of the classical algorithm. As no quantum improvement to the search algorithm is achieved, we eliminate the case of $\alpha=0$ and any nonzero $\beta$ from the solutions of Eq. (\[eq:12\]). Thus only the solution $\beta=-\alpha$ is meaningful, and yields, as mentioned in Sec. \[sec:intro\] and in Ref. [@chi97], $P=1$ when $$\label{eq:12a}
\alpha = -\beta = \arccos(1-1/2\lambda).$$ Since $\lambda$ lies between zero and one, the range of $\alpha$ is $\pi/3 \leq \alpha \leq \pi$. The boundary point of this range $\alpha=\pi/3$ occurs when $P(\lambda=1)=1$, and similarly $\alpha=\pi$ when $P(\lambda=1/4)=1$.
We can express $P(\lambda)$ as a function of $\lambda$ depending on the parameter $\alpha$, $$\begin{aligned}
\label{eq:13}
P(\lambda) & = & 1 - |u_1|^2 \nonumber \\
& = & \lambda \, [5-4 \, \cos\alpha
- 4 \, (1-\cos\alpha) \, (2-\cos\alpha) \, \lambda \nonumber \\
&& \ \ \ \ +4 \, (1-\cos\alpha)^2\lambda^2].\end{aligned}$$ For $\alpha = \pi/2$ the equation reduces to Eq. (14) of Li and Li [@li07] . Since Eq. (\[eq:13\]) is cubic in $\lambda$ we expect a local maximum and a local minimum in the range $0<\lambda\leq
1$ at $\lambda_\mathrm{max}$ and $\lambda_\mathrm{min}$ respectively, where $$\label{eq:14}
\lambda_\mathrm{max}=\frac{1}{2 \, (1-\cos\alpha)}, \ \
\lambda_\mathrm{min} = \frac{5-4\cos\alpha}{6 \, (1-\cos\alpha)}.$$ Furthermore the extrema are $$\label{eq:15}
P(\lambda_\mathrm{max}) = 1, \ \ P(\lambda_\mathrm{min}) =
\frac{(1+\cos\alpha)(5-4\cos\alpha)^2}{27 \, (1-\cos\alpha)}.$$ We illustrate different cases in Fig. \[fig:1\].
It is evident that the $\alpha=\pi/2$ case, which is the one used by Li and Li [@li07], gives the optimal profile for the success probability. Optimal here could be defined as the largest average $P$ over the range of $\lambda$, or the largest range of $\lambda$ over which $P\geq 25/27$.
Multi-phase matching with two pairs of phases {#two}
---------------------------------------------
We now consider Eq. (\[eq:7\]) for $k=2$ and we again concentrate on the upper component of the vector $(u_2,d_2)^T$. The general expression for it is too lengthy to give here, but again we demand that for an arbitrary value of $\lambda$ the imaginary part is zero to obtain the matching relationship for the phases. Apart from a factor of $\sqrt{1-\lambda}$ the expression of $\mathrm{Im} \, u_2$ contains a term in $\lambda$ and another in $\lambda^2$. Demanding that the coefficients of each power of $\lambda$ vanishes gives us two equations involving $\alpha_1$, $\alpha_2$, $\beta_1$, and $\beta_2$. Solving for $\beta_1$ and $\beta_2$ in terms of $\alpha_1$ and $\alpha_2$ we obtain the following four solutions: $$\label{eq:16}
\left.
\begin{array}{l}
\{\beta_1=-\alpha_1, \ \beta_2 = 0\} \nonumber \\
\{\beta_1=-\alpha_2, \ \beta_2 = -\alpha_1\} \nonumber \\
\{\beta_1 = \beta_2 = 0\} \nonumber \\
\{\beta_1 = 0, \ \beta_2 = -(\alpha_1 + \alpha_2)\}.
\end{array}
\right.$$ Since one of $\beta_1$ and $\beta_2$ is zero for the first and last solution, the operation is then reduced to one iteration, and for the third solution the two iterations would not change the probabilities of the marked and unmarked states. Thus the only solution that gives new information is the one where $\beta_1=-\alpha_2$ and $\beta_2=-\alpha_1$. (The fact that $\mathrm{Im} \, u_2 = 0$ is a necessary, but not a sufficient, condition for this solution.) After obtaining the matched phases for which $\mathrm{Im} \, u_2 =0$, we set $\mathrm{Re} \, u_2 = 0$ to solve for the values of $\lambda$ which gives $P=1$.
The expression for $u_2$ is then real and can be written as $$\begin{aligned}
\label{eq:17}
u_2 & = & \{1+ 2\, [(1-\cos\alpha_1)(-2+\cos\alpha_2)
-\sin\alpha_1\sin\alpha_2]\, \lambda \nonumber \\
& & + 4\, (1-\cos\alpha_1) (1-\cos\alpha_2) \, \lambda^2 \}
\sqrt{1-\lambda}.\end{aligned}$$ The factor multiplying $\sqrt{1-\lambda}$ is quadratic in $\lambda$ and hence it can vanish for two values of $\lambda$. Thus we can ask ourselves the questions, suppose two values of $\lambda$ between zero and one are given at which $P(\lambda)=1$, what are the corresponding values of $\alpha$ and what limits are there on the possible values of $\lambda$ that satisfy $P(\lambda)=1$? If $\lambda_1$ and $\lambda_2$ are the roots of the equation $$\label{eq:17a}
u_2(\lambda)/\sqrt{1-\lambda}=0,$$ then $\cos\alpha_1$ and $\cos\alpha_2$ satisfy the equations
$$\begin{aligned}
\label{eq:18}
&& 8\lambda_1\lambda_2\cos^3\alpha_2 + [4(\lambda_1+\lambda_2)
(1-\lambda_1-\lambda_2) - 8\lambda_1\lambda_2]\cos^2\alpha_2
+ [8(\lambda_1+\lambda_2)^2 - 12(\lambda_1+\lambda_2)-8\lambda_1\lambda_2
+ 4]\cos\alpha_2 \nonumber \\
&& \hspace{1.7in} - 4(\lambda_1+\lambda_2)^2 + 8(\lambda_1+\lambda_2)
-5 + 8\lambda_1\lambda_2 = 0,\end{aligned}$$
$$\label{eq:19}
\cos\alpha_1 = 1 - \frac{1}{4(1-\cos\alpha_2)\lambda_1\lambda_2}.$$
In order to have a sense of the values of $\alpha_1$ and $\alpha_2$ that are valid, we have minimally the condition that the discriminant of Eq. (\[eq:17a\]) (quadratic in $\lambda$) should be nonnegative to avoid complex values of $\lambda$. In Fig. [\[fig:3a\]]{} we plot the discriminant as a surface $z= D(\alpha_1,\alpha_2)$; the intersection of the surface with the $xy$ plane gives the boundary of the non-allowed $\alpha_1$ and $\alpha_2$ values.
Given $\lambda_1$ and $\lambda_2$ one can solve Eq. (\[eq:18\]) for $\cos\alpha_2$ and using it we obtain $\cos\alpha_1$ from the second equation. Only those solutions that yield real angles $\alpha_1$ and $\alpha_2$ are meaningful for the unitary operators. The minimum value of $\lambda$ for which $P=1$ occurs when $\alpha_1=\alpha_2 =
\pi$. In that case $\lambda= (3-\sqrt{5})/8 =0.09549$. It can be shown that varying $\alpha_1$ or $\alpha_2$ by a small amount away from $\pi$ always leads to an increase in the $\lambda$ which corresponds to the smaller of the two values of $\lambda$. When we let $\alpha_{1,2}=\pi+\epsilon_{1,2}$ we obtain a change in the smaller $\lambda$ of $$\label{eq:20}
\Delta\lambda = \frac{1}{160}\left[\left(2\sqrt{5}\epsilon_1
+\frac{5-3\sqrt{5}}{\sqrt{2\sqrt{5}}}\epsilon_2\right)^2+(22-8\sqrt{5})
\epsilon_2^2\right],$$ which is positive regardless of the signs of $\epsilon_{1,2}$. The larger $\lambda$ can increase or decrease with changes in the phases(s).
We obtain a particular example using the procedure described above. We search through combinations of $\lambda_{1,2}$ and find that $\lambda_1 = 2/5$ and $\lambda_2=4/5$ give good results. In this case $\alpha_1=1.00889485$ and $\alpha_2=2.30794928$. We find local minima of $P(\lambda)$ at $\lambda = 0.5767$ and $\lambda = 0.9433$ at which $P = 0.9936$ and 0.9966, respectively. The corresponding graph of the success probability as a function of $\lambda$ obtained with the two-stage multi-phase operator is shown in Fig. \[fig:3\] and compared with double iterations of the Grover operation and that of Li and Li [@li07].
It would be interesting to examine a classical counterpart of $P(\lambda)$. The probability of failing to find one of $M$ marked objects out of $N$ objects is $(N-M)/N = 1-\lambda$. The probability of failing twice in a row is $$(1-\lambda)\left(\frac{N-1-M}{N-1}\right)=(1-\lambda)
\left(1-\frac{\lambda}{1-1/N}\right).$$ The probability of failing $k$ times in a row is $$\begin{aligned}
&& (1-\lambda)\left(1-\frac{\lambda}{1-1/N}\right)\cdots
\left(1-\frac{\lambda}{1-(k-1)/N}\right) \nonumber \\
&&=\prod_{n=1}^k
\left[1-\lambda\left(1-\frac{n-1}{N}\right)^{-1}\right].
\nonumber\end{aligned}$$ Thus the probability of finding at least one of the $M$ items in $k$ successive attempts is $$\label{eq:class}
P_\mathrm{classical}(\lambda) = 1 -
\prod_{n=1}^k\left[1-\lambda\left(1-\frac{n-1}{N}\right)^{-1}\right].$$ If $k\ll N$, this probability is approximately $P_\mathrm{classical}(\lambda)\approx 1-(1-\lambda)^k$, which we interpret as the classical counterpart of $P(\lambda)$. This probability with $k=2$ is also plotted in Fig. \[fig:3\].
Multi-phase matching with six pairs of phases {#six}
---------------------------------------------
We show that if we match the multi-phase $\alpha_j$ and $\beta_j$ $(j=1,\dots,k)$ with $k=6$ in accordance with a certain matching rule (best fit), we can obtain a multiple Grover operation that yields $P(\lambda)\approx 1$ in a wide range of $\lambda$. We found this best solution for six Grover iterations by a nonlinear fitting to the ideal probability curve $P(\lambda)=1$ for $0<\lambda\leq 1$. The phases $\alpha_j$ and $\beta_j$ found in this way are given in the left side of Table \[table:1\].
$~~j~~$ $\beta_j$ $\lambda_j^{(P(\lambda_j)=1)}$ $P(\lambda_j^{(\mathrm{local~min})})$
--------- ------------- --------------- -------------------------------- ---------------------------------------
1 1.20560132 $-\alpha_6$ 0.10777 0.9980
2 1.29806396 $-\alpha_5$ 0.23793 0.9993
3 1.31701508 $-\alpha_4$ 0.41889 0.9996
4 1.33356767 $-\alpha_3$ 0.62393 0.9997
5 0.47289426 $-\alpha_2$ 0.81366 0.9997
6 1.66668634 $-\alpha_1$ 0.94483 0.9995
: Phase parameters for the six-parameter multi-phase matching, and results for local maxima ($P(\lambda)=1$) and local minima of $P(\lambda)$.[]{data-label="table:1"}
It is remarkable that $\alpha_j$ and $\beta_j$ are matched to each other such that $\alpha_j=-\beta_{6-j+1}$. The signs of $\alpha_j$ and $\beta_j$ are opposite to each other, which is consistent with the case of the new phase matching of Ref. [@li07], i.e., the $k=1$ case with $\alpha_1=-\beta_1 = \pi/2$. The matching rule $\alpha_j =
-\beta_{k-j+1}$ between the multi-phases $\alpha_j$ and $\beta_j$ holds for any $k$ in the best solution obtained by the nonlinear fitting to the ideal probability curve $P(\lambda)=1$ for $0 < \lambda
\leq 1$, although we omit to show cases other than those for which $k=1$, 2, and 6.
Fig. \[fig:4\] shows the success probabilities obtained by six Grover operations with the multi-phase matching of Eq. (\[eq:7\]). The inset of Fig. \[fig:4\] shows that there are six values of $\lambda_i$ at which $P(\lambda_i)=1$ exactly. They are given in the right side of Table \[table:1\] along with local minimum values of the function $P(\lambda)$ which occur between 0.1 and 1.
We studied the $k=5$ case in the same way and obtained a graph similar to Fig. \[fig:4\] with $P(\lambda)=1$ for *five* values of $\lambda$ other than unity. The local minima of $P(\lambda)$ are lower and the minimum value of $\lambda$ for which $P(\lambda)=1$ is slightly larger than in the $k=6$ case. The matching rule $\alpha_j=-\beta_{k-j+1}$ is also satisfied for the $k=5$ case as it was for $k=1$, 2, and 6 cases. We are confident that for any $k>1$ this matching rule for the best fit holds so that in general one finds $k$ values of $\lambda$ for which $P(\lambda)=1$ and the smallest $\lambda$ for which $P(\lambda)=1$ decreases as $k$ increases.
Returning to the six-stage multiple phase operation, we define $P_j(\lambda)$ $(j=1,\dots,6)$ as the success probability curves after $j$ steps of the six-stage multi-phase operation. As seen in the Figs. \[fig:4\] and \[fig:5\], $P_6(\lambda)\approx 1$ is achieved for $0.1\leq\lambda\leq 1$ in the sixth Grover operation. This is significant in the sense that if $\lambda$ is greater than 0.1, we can always find the superposition of marked states by just six Grover operations. In contrast to the shape of the curve for $P_6(\lambda)$, Fig. \[fig:5\] shows that each $P_j(\lambda)$ for $j=1,\dots,5$ depends strongly on $\lambda$ and is far from the desired success probability $P_6(\lambda)$. The curves do not monotonically approach the desired success probability $P_6(\lambda)$ when $\lambda > 0.05$. In particular, $P_5(\lambda)$ is quite different from the desired probability $P_6(\lambda)$. However, in the final (sixth) step the desired probability $P_6(\lambda)$ is obtained. This is in contrast to the fixed-point iteration schemes studied in Refs. [@grover05; @tulsi06; @younes07].
Figure \[fig:6\] shows the success probabilities obtained by six Grover operations with the single phase matching with $\alpha_j=-\beta_j=\pi/2$ $(j=1,\dots,6)$ , where we showed only $P_1(\lambda)$, $P_3(\lambda)$ and $P_6(\lambda)$. The $P_1(\lambda)$ is the success probability of the new phase matching obtained by Li and Li [@li07]. As stressed in Ref. [@li07], the success probability is substantially improved in $\lambda>1/3$ by a single Grover operation, compared with that of the original Grover algorithm indicated by the yellow line, where the probability is plotted for optimal iteration times indicated by $k$. However, $P_3(\lambda)$ and $P_6(\lambda)$ obtained by multiple Grover operations with the single phase matching show intensive oscillations with $\lambda$. As we have shown, such undesirable oscillations can be eliminated by the multi-phase matching subject to the matching rule $\alpha_j=-\beta_{6-j+1}$.
Here we should note that the nonlinear fitting is not unique. The phases $\alpha_j$ and $\beta_j$ given in Table \[table:1\] were obtained by minimizing the function $\sum_i\chi_i^2$ where the $\chi_i$ are the differences at $\lambda=\lambda_i$ of the ideal probability $P(\lambda)=1$ and the probability function $P_6(\lambda)$. If we take, for example, a function such as $\sum_i|\chi_i|$ we obtain another solution. Although this solution gives almost the same $P_6(\lambda)$, the probability curve is shifted slightly toward larger values of $\lambda$, so that the local extrema are also slightly moved to the right. Since we emphasize obtaining $P(\lambda)\approx 1$ over as wide a range of $\lambda$ as possible we adopted the solution that uses the $\chi^2_i$ for the fitting.
Iteration of Grover’s operation with phase other than $\pi$ {#k_iterations}
===========================================================
In this section we consider the repeated application of Grover’s original operation generalized to have a phase other than $\pi$. We focus in particular on cases with small $\lambda$ for which the success probability with the multi-phase matching is small, and determine the conditions that yield success probabilities close to unity.
Consider a single Grover operation with matched phase, Eq. (\[eq:8\]), but with $\beta=-\alpha$, $$\label{eq:a1}
G_1 = \begin{pmatrix} (1-e^{-i\alpha})(1-\lambda)+ e^{-i\alpha} &
(e^{i\alpha}-1)\sqrt{\lambda(1-\lambda)} \\
(1-e^{-i\alpha})\sqrt{\lambda(1-\lambda)} &
(e^{i\alpha}-1)\lambda+1
\end{pmatrix}.$$ Note that $\det G_1 = 1$. We obtain eigenvalues $\sigma$ of the matrix $G_1$ by solving $$\label{eq:a2}
f(\sigma)=\det(G_1-\sigma I) = 0.$$ The characteristic function $f(\sigma)$ is $$\label{eq:a3}
f(\sigma) = \sigma^2 + 2[-1+(1-\cos\alpha)\lambda]\sigma + 1.$$ The equation $f(\sigma)=0$ yields solutions $$\label{eq:a4}
\sigma = 1 - (1-\cos\alpha)\lambda \pm i\sqrt{(1-\cos\alpha)\lambda
[2-(1-\cos\alpha)\lambda]}$$ We define $x$ as $$\label{eq:a4a}
x =(1-\cos\alpha)\lambda,$$ so that the eigenvalues can be written as $$\label{eq:a5}
\sigma = e^{\pm i\phi}, \ \ \phi =
\arctan\left(\frac{\sqrt{x(2-x)}}{1-x}\right).$$ We choose the definition of the arc tangent so that as $x$ varies from 0 to 2, $\phi$ goes from 0 to $\pi$. We can rewrite the function $f(\sigma)$ as $$\label{eq:a6}
f(\sigma) = \sigma^2 - 2\sigma\cos\phi +1.$$ By the Cayley-Hamilton theorem [@pipes58 page 91] $f(G_1) =0$, so that we obtain the identity $$\label{eq:a7}
G_1^2 = 2G_1\cos\phi -1.$$ This means that $G_1$ iterated any number of times can be written as a linear expression of $G_1$. In fact for $k$ iterations it can be shown by induction [@sprung93] that $$\label{eq:a8}
G_1^k = \frac{1}{\sin\phi}\left[G_1\sin(k\phi) - \sin((k-1)\phi)\right].$$ Consider now the $k$ iterations of the Grover operation, so that $$\label{eq:a9}
\begin{pmatrix} u_k \\ d_k \end{pmatrix} = G_1^k
\begin{pmatrix} \sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}.$$ This yields
$$\label{eq:a10}
\begin{pmatrix} u_k \\ d_k \end{pmatrix} = \frac{1}{\sin\phi} \left[
\sin{(k\phi)} \begin{pmatrix} (1-e^{-i\alpha})(1-\lambda)+ e^{-i\alpha} &
(e^{i\alpha}-1)\sqrt{\lambda(1-\lambda)} \\
(1-e^{-i\alpha})\sqrt{\lambda(1-\lambda)} &
(e^{i\alpha}-1)\lambda+1
\end{pmatrix} - \sin{((k-1)\phi)} \ I \right]
\begin{pmatrix} \sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}.$$
Thus the expression for $u_k$ is $$\label{eq:a11}
u_k = \frac{\sqrt{1-\lambda}}{\sin\phi}\left\{\sin{(k\phi)}(1-2x)
-\sin{((k-1)\phi)}
\right\}.$$ We require $u_k=0$ so that $P=1$. A trivial solution is $\lambda=1$. We also note that $\phi=0$ yields $x=-1/(2k)$. Since $x$ must be positive $\sin{\phi}\neq 0$. Thus we need to solve only $$\label{eq:a12}
\sin{(k\phi)}(1-2x)-\sin{((k-1)\phi)} = 0.$$ The solutions are values of $x=(1-\cos{\alpha})\lambda$ for which $P=1$. Thus we have $P=1$ for combinations of $\alpha$ and $\lambda$. For instance, when $\alpha=\pi$, then $\lambda = x/2$. In Fig. \[fig:ap1\], we display the $P(\lambda)$ curves for six ($k=6$) iterations when $\alpha$ has different values.
For large $k$ we can estimate the smallest value of $\lambda$ for which $P$ is unity. We rewrite Eq. (\[eq:a12\]) so that $$\label{eq:a13}
\tan(k\phi) = \frac{\sin\phi}{\cos\phi-1+2x}.$$ The value of $x$ which is the solution occurs for the $x$ coordinate of the point of intersection of the curves represented by the left side and the right side of Eq. (\[eq:a13\]). The curve on the right is a smoothly decreasing positive function starting at infinity when $x=0$ and asymptotically approaching the positive $x$ axis. The curve on the left starts at zero and increases to positive infinity when $k\phi= \pi/2$. When $k$ is large this occurs for small values of $\phi$ or small values of $x$. Thus using the condition $\phi\approx
\pi/(2k)$, we obtain $$\label{eq:a14}
\arctan\frac{\sqrt{x(2-x)}}{1-x} {\mathrm{\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$}}}\frac{\pi}{2k}
\ \ \mathrm{or} \ \ \frac{\sqrt{x(2-x)}}{1-x} {\mathrm{\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$}}}\frac{\pi}{2k}.$$ This leads to the approximation of the smallest value of $x$ for which $P$ is one as $x_\mathrm{min} {\mathrm{\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$}}}\pi^2/(8k^2)$; for $\alpha =
\pi$ (the Grover case) $\lambda_\mathrm{min}= x_\mathrm{min}/2$. This approximation leads to $\lambda_\mathrm{min} = 0.017$ for $k=6$ and $\alpha=\pi$, whereas the exact solution of Eq. (\[eq:a13\]) gives $0.014$. As $k$ gets larger the approximation improves further.
The Grover algorithm as a special case
--------------------------------------
We recover the Grover algorithm starting with Eq. (\[eq:a11\]) and setting $\alpha=\pi$ or $x=2\lambda$. Then it follows from Eq. (\[eq:a5\]) that $\sin\phi = \sqrt{4\lambda(1-\lambda)}$ and $\cos\phi = (1-2\lambda)$. The $u_k$ of Eq. (\[eq:a11\]) can be reduced to $$\label{eq:a15}
u_k=-\sqrt{\lambda}\sin(k\phi) + \sqrt{1-\lambda}\cos(k\phi).$$ Define $\sin\theta = \sqrt{\lambda}$. Then $$\label{eq:a16}
u_k=\cos(k\phi+\theta).$$ We can show that $\phi = 2\theta$, so that $$\label{eq:a17}
u_k = \cos[(2k+1)\theta] = \cos[(2k+1)\arcsin(\sqrt{\lambda})].$$ This is Eq. (6) of Ref. [@li07]. Furthermore $$\label{eq:a18}
P=1-u_k^2 = \sin^2[(2k+1)\arcsin(\sqrt{\lambda})].$$ Thus each iteration effectively rotates the state through an angle of $\theta/2 = \arcsin(\sqrt{\lambda})/2$. We can use this to estimate the number of iterations that are required to obtain $P(\lambda) =1$. That occurs when the argument of the sine function in Eq. (\[eq:a18\]) is $\pi/2$, i.e., $$\label{eq:a19}
k=\frac{1}{2}\left(\frac{\pi}{2\theta}-1\right).$$ For small $\theta$ (or large $k$) $$\label{eq:a20}
k \approx\frac{\pi}{4\theta} \approx \left[\frac{\pi}{4\theta}\right]
= \mathrm{~integer~value~of~} \frac{\pi}{4\theta} \approx \left[
\frac{\pi}{4}\frac{1}{\sqrt{\lambda}}\right].$$ Thus after approximately $\pi/(4\sqrt{\lambda})$ iterations one has certainty of having found the superposition of marked states. Classically the number of search operations to have this certainty is on the average approximately $1/(2\lambda)=N/(2M)$ for $N$ much larger than $M$. By the same reasoning we find $P=0$ with twice as many quantum iterations. Thus by continuing to iterate indefinitely we can end with any probability of success. However, if we iterate close to the number that gives 100% probability of success we have a good approximation to a successful search.
Effect of phase $\mathbf{\alpha}$ {#general_phase}
---------------------------------
For a general value of $x=(1-\cos\alpha)\lambda$, Eq. (\[eq:a11\]) can be written as $$\label{eq:a21}
u_k = A\cos(k\phi+\theta),$$ where $$\label{eq:a22}
A=\sqrt{\frac{2(1-\lambda)}{2-x}}, \ \ \ \theta =
\arctan\sqrt{\frac{x}{2-x}}.$$ Since $P(\lambda) = 1 - u_k^2 = 1 - A^2\cos^2(k\phi+\theta)$, the minimum of $P(\lambda)$ is $$\label{eq:a23}
P_\mathrm{min}(\lambda) = 1 - A^2 = \frac{\lambda(1+\cos\alpha)}
{2-(1-\cos\alpha)\lambda}.$$ In Fig. \[fig:ap2\] it is seen that $P_\mathrm{min}(\lambda)$ has at most a linear rise as $\lambda$ increases from zero to one.
Summary
=======
We have proposed a multi-phase matching for the Grover search algorithm, which is an extension of the new phase matching proposed in Ref. [@li07]. The multi-phase matching is characterized by multiple Grover operations with two kinds of multi-phases $\alpha_j$ and $\beta_j$ $(j=1,\dots,k)$. We showed that if we match $\alpha_j$ and $\beta_j$ in accordance with the rule $\alpha_j = -\beta_{k-j+1}$ for a given $k$ we can obtain an optimal solution for $\alpha_j$, $\beta_j$ that gives a success probability curve such that it is almost constant and unity in a wide range of the fraction of marked states. As an example we presented an optimal solution obtained for $k=6$. The solution yields the desired success probability $P = 1$ to within 0.2% for the fraction of the marked states greater than 0.1. This is significant in the sense that when the fraction of marked states is greater than 0.1, we can always with a high degree of confidence find a uniform superposition of the marked states by repeating the Grover operation just six times.
To clarify the mechanism of the multi-phase matching we studied in detail the one- and two-iteration cases. We showed that it is possible to obtain $P=1$ exactly for a particular fraction $\lambda$ by tuning the phases. This can be generalized to having $k$ values of $\lambda$ for which $P(\lambda)=1$ when we go to a $k$-iteration scheme.
One can obtain $P=1$ for a given very small $\lambda$ by using the original Grover algorithm or the phase-matched version of it. In this case usually a specified large number of iterations is required. Further study is needed to obtain an efficient algorithm for extremely small $\lambda$.
This work was supported by the Japan Society for the Promotion of Sciences and the Natural Sciences and Engineering Research Council of Canada.
Equivalence of two phase-matching schemes
=========================================
Li and Li [@li07] claim to have generalized the Long phase-matching algorithm in order to produce a higher success probability. In actual fact the phase matching of Li and Li and that of Long *et al.* [@long99] result in the same success probability. We show that in the following.
Instead of Eqs. (\[eq:1\]) and (\[eq:2\]), Long *et al.* work with the operators $$\label{eq:aa1}
U=I-(1-e^{i\theta})\sum_{l=0}^{M-1}|t_l\rangle\langle t_l|$$ $$\label{eq:aa2}
V=I-(1-e^{i\phi})|0^{\otimes n}\rangle\langle 0^{\otimes n}|.$$ These unitary transformations lead to the Grover operator (in the notation of this paper) $G(\theta,\phi)$, where $$\label{eq:aa3}
G =
\begin{pmatrix}
1-(1-e^{i\phi})(1-\lambda) &
-(1 - e^{i\phi})e^{i\theta}\sqrt{\lambda(1-\lambda)} \\
-(1-e^{i\phi})\sqrt{\lambda(1-\lambda)} &
[1-(1 - e^{i\phi})\lambda]e^{i\theta}
\end{pmatrix}.$$ For one operation we calculate the final state $$\label{eq:aa4}
\begin{pmatrix}u \\ d \end{pmatrix} = G(\theta,\phi)
\begin{pmatrix}\sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}$$ with $$\label{eq:aa5}
u=\sqrt{1-\lambda}[1-(1-e^{i\phi})(1-\lambda) - (1-e^{i\phi})
e^{i\theta}\lambda].$$ Setting $u=0$ we obtain (in addition to $\lambda=1$) the solution $$\label{eq:aa6}
\phi=\theta, \ \ \ \lambda= \frac{1}{2}\frac{\cos\theta +1}{\sin^2\theta}.$$ Note that the signs of $\phi$ and $\theta$ are the same, unlike the opposite signs of the matched phases of Li and Li, i.e., $\beta=-\alpha$. In order that $0 < \lambda\leq 1$ with this phase matching, $\theta$ varies from $\pi/3$ to $\pi$. For $P(\lambda) =
1-|u|^2$, we obtain the expression of Eq. (\[eq:13\]) with $\alpha$ replaced by $\theta$. Thus the impressive result by Li and Li of a single phase-matched Grover operation can also be obtained with the earlier-proposed operation of Long *et al.* However, the formulation of Li and Li results in $\mathrm{Im} \, u = 0$ when $\beta = -\alpha$, whereas $\mathrm{Im} \, u \neq 0$ when $\phi=\theta$ for the operator of Long *et al.* It should be noted however that the remarkable single-operation result was first reported by Li and Li [@li07]. Although the probabilities are the same the amplitudes are not, and Li and Li’s formulation gives a more straightforward derivation of the probabilities. (See Sec. IIA.) One can relate the two formulations by suggesting that instead of the operator acting on $(\sqrt{1-\lambda},\sqrt{\lambda})^T$ initially, in the case of Long *et al.* it operates on this state multiplied by a phase factor.
[17]{}
natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, in ** (, , ), pp. .
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , , , , , , , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ().
, ().
, ****, ().
, ****, ().
, ****, ().
, ** (, , ), ed.
, , , ****, ().
|
---
abstract: 'Tomograms, a generalization of the Radon transform to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signal and are robust in the presence of noise. We provide an explicit construction of tomogram transforms for many pairs of noncommuting operators in one and two dimensions and illustrations of their use for denoising, detection of small signals and component separation.'
author:
- 'F. Briolle[^1], V. I. Man’ko[^2], B. Ricaudand R. Vilela Mendes[^3] [^4]'
date:
title: 'Non-commutative tomography: A tool for data analysis and signal processing'
---
Introduction
============
Integral transforms [@handbook] [@BWolf79] are very useful for signal processing in communications, engineering, medicine, physics, etc. Linear and bilinear transforms have been used. Among the linear transforms, Fourier [@Fourier1888] and wavelets [@Combes90] [[@Daubechies90] [@Chui92] are the most popular. Among the bilinear ones, the Wigner–Ville quasidistribution [@Wigner32] [@Ville48] provides information in the joint time–frequency domain with good energy resolution. A joint time–frequency description of signals is important, because in many applications (biomedical, seismic, radar, etc.) the signals are of finite (sometimes very short) duration. However, the oscillating cross-terms in the Wigner–Ville quasidistribution make the interpretation of this transform a difficult matter. Even if the average of the cross-terms is small, their amplitude may be greater than the signal in time–frequency regions that carry no physical information. To profit from the time–frequency energy resolution of the bilinear transforms while controlling the cross-terms problem, modifications to the Wigner–Ville transform have been proposed. Transforms in the Cohen class [@Cohen1] [@Cohen2] make a two-dimensional filtering of the Wigner–Ville quasidistribution and the Gabor spectrogram [@Gabor] is a truncated version of this quasidistribution. The difficulties with the physical interpretation of quasidistributions arise from the fact that time and frequency correspond to two noncommutative operators. Hence a joint probability density cannot be defined.]{} Even in the case of positive quasiprobabilities like the Husimi–Kano function [@Husimi] [@Kano], an interpretation as a joint probability distribution is also not possible because the two arguments of the function are not simultaneously measurable random variables.
Recently, a new type of strictly positive bilinear transforms has been proposed [@MendesPLA] [@MankoJPA], called *tomograms*, which are a generalization of the Radon transform [@Radon] to noncommutative pairs of operators. The Radon–Wigner transform [@Radon1] [@Radon2] is a particular case of such noncommutative tomography technique. The tomograms are strictly positive probability densities, provide a full characterization of the signal and are robust in the presence of noise.
A unified framework to characterize linear transforms, quasidistributions and tomograms was developed in Ref.[@MankoJPA]. This is briefly summarized in Section 2. Then Sections 3,4,6 and 7 contains an explicit construction of tomogram transforms for many pairs of noncommuting operators in one and two dimensions. Some of these transforms have been used in the past [@reflecto1] [@reflecto2], others are completely new.
It is in the time-frequency plane that most signal processing experts have developed their intuition, not in the eigenspaces associated to the new tomograms. Therefore, to provide a qualitative intuition on the way the tomograms explore the time-frequency plane, we have provided graphical spectrograms of the eigenstates on which the signal is projected by the tomograms.
In Section 5, an interpretation of the tomograms is given as operator symbols of the set of projection operators in the space of signals. This provides a very general framework to deal with all kinds of custom-designed integral transforms both for deterministic and random signals. It also provides an alternative framework for an algebraic formulation of signal processing.
Finally, an illustration of how such transforms may be used to analyze signals is contained in Section 8. A brief review of denoising, detection of small signals and component separation, done in the past, is included as well as an application of one of the new transforms.
Linear transforms, quasi-distributions and tomograms
====================================================
Consider signals $f(t)$ as vectors $\mid f\rangle $ in a dense nuclear subspace $\mathcal{N}$ of a Hilbert space $\mathcal{H}$ with dual space $%
\mathcal{N}^{*}$ (with the canonical identification $\mathcal{N\subset N}%
^{*} $) and a family of operators $\left\{ U(\alpha ):\alpha \in I, I \subset \mathbb{R}^n\right\} $ defined on $\mathcal{N}^{*}$ . In most cases of interest $U\left( \alpha
\right) $ generates a unitary group $U\left( \alpha \right) =e^{iB\left(
\alpha \right) }$. In this setting three types of integral transforms are constructed.
Let $h\in \mathcal{N}^{*}$ be a reference vector and let $U$ be such that the linear span of $\left\{ U(\alpha )h\in \mathcal{N}^{*}:\alpha \in I\right\} $ is dense in $\mathcal{N}^{*}$ . In the set $\left\{ U(\alpha )h\right\} $, a complete set of vectors can be chosen to serve as a basis.
**1 - Linear transforms** $$W_{f}^{(h)}(\alpha )=\langle U\left( \alpha \right) h\mid f\rangle
\label{2.1}$$
**2 - Quasi-distributions** $$Q_{f}(\alpha )=\langle U\left( \alpha \right) f\mid f\rangle \label{2.2}$$
**3 - Tomograms**
If $U\left( \alpha \right) $ is a unitary operator there is a self-adjoint operator $B\left( \alpha \right) $ such that $U\left( \alpha \right)
=e^{iB\left( \alpha \right) }$. The tomogram is $$M_{f}^{(B)}(X)=\langle f\mid \delta \left( B\left( \alpha \right) -X\right)
\mid f\rangle \label{2.3}$$
$X$ takes values on the spectrum of $B(\alpha )$. Considering a set of generalized eigenstates (in $\mathcal{N}^{\ast }$) of $B(\alpha )$, one obtains for the kernel $$\langle Y\mid \delta \left( B(\alpha )-X\right) \mid Y^{\prime }\rangle
=\delta (Y^{\prime }-X)\,\delta (Y-Y^{\prime })=\langle Y\mid X\rangle
\langle X\mid Y^{\prime }\rangle$$Therefore, we may identify $\delta \left( B(\alpha )-X\right) $ with the projector $\mid X\rangle \langle X\mid $ $$\delta \left( B(\alpha )-X\right) =\mid X\rangle \langle X\mid =P_{X}$$From this, it follows $$M_{f}^{(B)}=\langle f\mid \delta \left( B(\alpha )-X\right) \mid f\rangle
=\langle f\mid X\rangle \langle X\mid f\rangle =|\langle X\mid f\rangle |^{2}
\label{2.4}$$showing the positivity of the tomogram and its nature as the squared amplitude of the projection on generalized eigenvectors of $B(\alpha )$. Let, by a unitary transformation $S$, $B(\alpha )$ be transformed to$$SB(\alpha )S^{\dagger }=B^{^{\prime }}(\alpha )$$If $\left\{ \mid Z\rangle \right\} $ is the set of (generalized) eigenvectors of $B^{^{\prime }}(\alpha )$, $\left\{ S^{\dagger }\mid
Z\rangle \right\} $ is a set of eigenvectors for $B$. Therefore, $$M_{f}^{(B)}(Z)=\langle f\mid \delta \left( B(\alpha )-Z\right) \mid f\rangle
=|\langle Z\mid S\mid f\rangle |^{2}=\langle f\mid S^{\dagger }\mid Z\rangle
\langle Z\mid S\mid f\rangle$$
For normalized $\mid f\rangle $, $$\langle f\mid f\rangle =1$$ the tomogram is normalized $$\int M_{f}^{(B)}\left( X\right) \,dX=1 \label{2.5}$$ It is a probability distribution for the random variable $X$ corresponding to the observable defined by the operator $B\left( \alpha \right) $. The tomogram is a homogeneous function $$M_{f}^{(B/p)}(X)=|p|M_{f}^{(B)}(pX) \label{2.6}$$
**Examples:**
If $U\left( \alpha \right) $ is unitary generated by $B_{F}\left(
\overrightarrow{\alpha }\right) =\alpha _{1}t+i\alpha _{2}\frac{d}{dt}$ and $%
h$ is a (generalized) eigenvector of the time-translation operator the linear transform $W_{f}^{(h)}(\alpha )$ is the Fourier transform. For the same $B_{F}\left( \overrightarrow{\alpha }\right) $**,** the quasi-distribution** **$Q_{f}(\alpha )$ is the ambiguity function.
The Wigner–Ville transform [[@Wigner32] [@Ville48]]{} is the quasi-distribution $Q_{f}(\alpha )$ for the following $B-$operator $$B^{(WV)}(\alpha _{1},\alpha _{2})=-i2\alpha _{1}\frac{d}{dt}-2\alpha _{2}t+%
\frac{\pi \left( t^{2}-\frac{d^{2}}{dt^{2}}-1\right) }{2}\, \label{2.7}$$
The wavelet transform is $W_{f}^{(h)}(\alpha )$ for $B_{W}\left(
\overrightarrow{\alpha }\right) =\alpha _{1}D+i\alpha _{2}\frac{d}{dt}$, $D$ being the dilation operator $D=-\frac{1}{2}\left( it\frac{d}{dt}+i\frac{d}{dt%
}t\right) $. The wavelets $h_{s,\,\tau }(t)$ are kernel functions generated from a basic wavelet $h(\tau )$ by means of a translation and a rescaling $%
(-\infty <\tau <\infty ,$ $s>0)$: $$h_{s,\,\tau }(t)=\frac{1}{\sqrt{s}}\,h\left( \frac{t-\tau }{s}\right)
\label{2.8}$$using the operator $$U^{(A)}(\tau ,s)=\exp (i\tau \hat{\omega})\exp (i\log \,sD), \label{2.9}$$$$h_{s,\tau }(t)=U^{(A)\dagger }(\tau ,s)h(t). \label{2.10}$$For normalized $h(t)$ the wavelets $h_{s,\,\tau }(t)$ satisfy the normalization condition $$\int |h_{s,\,\tau }(t)|^{2}\,dt=1.$$The basic wavelet (reference vector) may have different forms, for example, $$h(t)=\frac{1}{\sqrt{\pi }}\,e^{i\omega _{0}t}\,e^{-t^{2}/2}, \label{2.11a}$$or $$h(t)=(1-t^{2})\,e^{-t^{2}/2} \label{2.11b}$$called the Mexican hat wavelet.
The Bertrand transform [@BerBerJMP] [@Baran] is $Q_{f}(\alpha )$ for $B_{W}\smallskip $.
Linear, bilinear and tomogram transforms are related to one another by
$$M_{f}^{(B)}(X)=\frac{1}{2\pi }\int Q_{f}^{(kB)}(\alpha )\,e^{-ikX}\,dk$$
$$Q_{f}^{(B)}(\alpha )=\int M_{f}^{(B/p)}(X)\,e^{ipX}\,dX$$
$$Q_{f}^{(B)}(\alpha )=W_{f}^{(f)}(\alpha )$$
$$W_{f}^{(h)}(\alpha )=\frac{1}{4}\int e^{iX}\left[
\begin{array}{c}
M_{f_{1}}^{(B)}(X)-iM_{f_{2}}^{(B)}(X) \\
-M_{f_{3}}^{(B)}(X)+iM_{f_{4}}^{(B)}(X)%
\end{array}%
\right] \,dX$$
with $$\begin{aligned}
&\mid &f_{1}\rangle =\mid h\rangle +\mid f\rangle ;\qquad \mid f_{3}\rangle
=\mid h\rangle -\mid f\rangle \\
&\mid &f_{2}\rangle =\mid h\rangle +i\mid f\rangle ;\qquad \mid f_{4}\rangle
=\mid h\rangle -i\mid f\rangle\end{aligned}$$
One-dimensional tomograms
=========================
As shown in (\[2.4\]) a tomogram corresponds to projections on the eigenstates of the $B$ operators. These operators are linear combinations of different (commuting or noncommuting) operators, $$B=\mu O_{1}+\nu O_{2}$$Therefore the tomogram explores the signal along lines in the plane $\left(
O_{1},O_{2}\right) $. For example the tomogram$$M_{f}^{(S)}\left( X,\mu ,\nu \right) =\langle f\mid \delta \left( \mu t+\nu
\omega -X\right) \mid f\rangle \label{3.1}$$with $\omega =i\frac{d}{dt}$, is the expectation value of an operator delta-function in the state $\mid f\rangle $, the support of the delta-function being a line in the time–frequency plane $$X=\mu t+\nu \omega \label{3.2}$$Therefore, $M_{f}^{(S)}\left( X,\mu ,\nu \right) $ is the marginal distribution of the variable $X$ along this line in the time–frequency plane. The line is rotated and rescaled when one changes the parameters $\mu
$ and $\nu $. In this way, the whole time–frequency plane is sampled and the tomographic transform contains all the information on the signal.
It is clear that, instead of marginals collected along straight lines on the time–frequency plane, one may use other curves to sample this space. It has been shown in [@MankoJPA] that the tomograms associated to the affine group, for example $$M_{f}^{(A_{t})}\left( X,\mu ,\nu \right) =\langle f\mid \delta \left( \mu
t+\nu \frac{t\omega +\omega t}{2}-X\right) \mid f\rangle \label{3.3}$$correspond to hyperbolas in the time-frequency plane. This point of view has been further explored in [@Mankonew] defining tomograms in terms of marginals over surfaces generated by deformations of families of hyperplanes or quadrics. However not all tomograms may be defined as marginals on lines in the time-frequency plane.
Here we construct the tomograms corresponding to a large set of operators defined in terms of (one-dimensional) time. Of particular interest are the tomograms associated to finite-dimensional Lie algebras.
1D conformal group tomograms
----------------------------
The generators of the one-dimensional conformal group are, $$\begin{array}{l}
\omega =i\frac{d}{dt} \\
D=i\left( t\frac{d}{dt}+\frac{1}{2}\right) \\
K=i\left( t^{2}\frac{d}{dt}+t\right)%
\end{array}
\label{3.4}$$ One may construct t[omograms using the following operators:]{}
*Time-frequency* $$B_{1}=\mu t+i\nu \frac{d}{dt} \label{3.5}$$
*Time-scale* $$B_{2}=\mu t+i\nu \left( t\frac{d}{dt}+\frac{1}{2}\right) \label{3.6}$$
*Frequency-scale* $$B_{3}=i\mu \frac{d}{dt}+i\nu \left( t\frac{d}{dt}+\frac{1}{2}\right)
\label{3.7}$$
*Time-conformal* $$B_{4}=\mu t+i\nu \left( t^{2}\frac{d}{dt}+t\right) \label{3.8}$$
The construction of the tomograms reduces to the calculation of the generalized eigenvectors of each one of the $B_{i}$ operators
$B_{1}\psi _{1}\left( \mu ,\nu ,t,X\right) =X\psi _{1}\left( \mu ,\nu
,t,X\right) $$$\psi _{1}\left( \mu ,\nu ,t,X\right) =\exp i\left( \frac{\mu t^{2}}{2\nu }-%
\frac{tX}{\nu }\right) \label{3.9}$$ with normalization $$\int dt\psi _{1}^{*}\left( \mu ,\nu ,t,X\right) \psi _{1}\left( \mu ,\nu
,t,X^{\prime }\right) =2\pi \nu \delta \left( X-X^{\prime }\right)
\label{3.10}$$
$B_{2}\psi _{2}\left( \mu ,\nu ,t,X\right) =X\psi _{2}\left( \mu ,\nu
,t,X\right) $$$\psi _{2}\left( \mu ,\nu ,t,X\right) =\frac{1}{\sqrt{\left| t\right| }}\exp
i\left( \frac{\mu t}{\nu }-\frac{X}{\nu }\log \left| t\right| \right)
\label{3.11}$$ $$\int dt\psi _{2}^{*}\left( \mu ,\nu ,t,X\right) \psi _{2}\left( \mu ,\nu
,t,X^{\prime }\right) =4\pi \nu \delta \left( X-X^{\prime }\right)
\label{3.12}$$
$B_{3}\psi _{3}\left( \mu ,\nu ,\omega ,X\right) =X\psi _{3}\left( \mu ,\nu
,\omega ,X\right) $$$\psi _{3}\left( \mu ,\nu ,t,X\right) =\exp \left( -i\right) \left( \frac{\mu
}{\nu }\omega -\frac{X}{\nu }\log |\omega |\right) \label{3.13}$$ $$\int d\omega \psi _{1}^{*}\left( \mu ,\nu ,\omega ,X\right) \psi _{1}\left(
\mu ,\nu ,\omega ,X^{\prime }\right) =2\pi \nu \delta \left( X-X^{\prime
}\right) \label{3.14}$$
$B_{4}\psi _{4}\left( \mu ,\nu ,t,X\right) =X\psi _{4}\left( \mu ,\nu
,t,X\right) $$$\psi _{4}\left( \mu ,\nu ,t,X\right) =\frac{1}{\left| t\right| }\exp i\left(
\frac{X}{\nu t}+\frac{\mu }{\nu }\log \left| t\right| \right) \label{3.15}$$ $$\int dt\psi _{4}^{*}\left( \mu ,\nu ,t,s\right) \psi _{4}\left( \mu ,\nu
,t,s^{\prime }\right) =2\pi \nu \delta \left( s-s^{\prime }\right)
\label{3.16}$$
Then the tomograms are:
*Time-frequency tomogram* $$M_{1}\left( \mu ,\nu ,X\right) =\frac{1}{2\,\pi |\nu |}\left| \int \exp %
\left[ \frac{i\mu t^{2}}{2\,\nu }-\frac{itX}{\nu }\right] f(t)\,dt\right|
^{2} \label{3.17}$$
*Time-scale tomogram* $$M_{2}(\mu ,\nu ,X)=\frac{1}{2\pi |\nu |}\left| \int dt\,\frac{f(t)}{\sqrt{|t|%
}}e^{\left[ i\left( \frac{\mu }{\nu }t-\frac{X}{\nu }\log |t|\right) \right]
}\right| ^{2} \label{3.18}$$
*Frequency-scale tomogram* $$M_{3}(\mu ,\nu ,X)=\frac{1}{2\pi |\nu |}\left| \int d\omega \,\frac{f(\omega
)}{\sqrt{|\omega |}}e^{\left[ -i\left( \frac{\mu }{\nu }\omega -\frac{X}{\nu
}\log |\omega |\right) \right] }\right| ^{2} \label{3.19}$$ $f(\omega )$ being the Fourier transform of $f(t)$
*Time-conformal tomogram* $$M_{4}(\mu ,\nu ,X)=\frac{1}{2\pi |\nu |}\left\vert \int dt\,\frac{f(t)}{|t|}%
e^{\left[ i\left( \frac{X}{\nu t}+\frac{\mu }{\nu }\log |t|\right) \right]
}\right\vert ^{2} \label{3.20}$$The tomograms $M_{1},M_{2}$ and $M_{4}$ interpolate between the (squared) time signal and its projection on the $\psi _{i}\left( \mu ,\nu ,t,X\right) $ functions for $\mu =0$. Fig.\[realpart\] shows the typical behavior of the real part of these functions.
\[htb\] ![Typical behaviour of the real part of the functions $\protect\psi _{1}$,$\protect\psi _{2}$ and $\protect%
\psi _{4}$ at $\protect\mu =0$[]{data-label="realpart"}](fig1.eps "fig:"){width="12cm"}
Figs.\[timefreq\],\[timescale\] and \[conformal\] illustrate how the tomograms $M_{1},M_{2}$ and $M_{4}$ explore the time-frequency space by plotting the spectrograms of typical vectors $\psi _{1},\psi _{2}$ and $\psi _{4}$.
![Modulus of the Short-time Fourier transform of 4 vectors of the time-frequency tomogram for some fixed $\protect\theta$, $\protect\mu=\cos%
\protect\theta$, $\protect\nu=\sin\protect\theta$. A vector is a linear chirp, hence a line in the time-frequency plane. Moreover, each vector is a frequency-translated version of the one which starts at the origin. Since it forms an orthogonal basis, the sum of all the vectors cover the entire time-frequency plane. The parameter $\protect\theta$ allows to change the slope of the line in the time-frequency plane.[]{data-label="timefreq"}](lintomo1.eps){width="12cm"}
![Modulus of the Short-time Fourier transform of 4 vectors of the time-scale tomogram for $\protect\mu=0$, $\protect\nu=1$ (left) and $\protect%
\mu=\protect\sqrt(2)/2$, $\protect\nu=\protect\sqrt(2)/2$ (right). Each vector is an hyperbolic chirp. Two of them correspond to positive $X$ and two of them to negative $X$. Due to the sampling used in the numerical computation, some aliasing phenomenum occurs at times close to zero. There is a axis of symetry: the line of zero frequency on the left graph. This axis is shifted in frequency when $\protect\mu$ and $\protect\nu$ are changed.[]{data-label="timescale"}](hyptomo1.eps){width="7cm"}
![Modulus of the Short-time Fourier transform of 4 vectors of the time-scale tomogram for $\protect\mu=0$, $\protect\nu=1$ (left) and $\protect%
\mu=\protect\sqrt(2)/2$, $\protect\nu=\protect\sqrt(2)/2$ (right). Each vector is an hyperbolic chirp. Two of them correspond to positive $X$ and two of them to negative $X$. Due to the sampling used in the numerical computation, some aliasing phenomenum occurs at times close to zero. There is a axis of symetry: the line of zero frequency on the left graph. This axis is shifted in frequency when $\protect\mu$ and $\protect\nu$ are changed.[]{data-label="timescale"}](hyptomo2.eps){width="7cm"}
![Modulus of the Short-time Fourier transform of 4 vectors of the time-conformal tomogram for $\protect\mu=0$, $\protect\nu=1$. Due to the sampling used in the numerical computation, some aliasing phenomenum occurs at times close to zero. Some interferences between the vectors occur for large time. Two vectors correspond to positive $X$ and two to negative $X$.[]{data-label="conformal"}](conftomo1.eps){width="12cm"}
In a similar way, tomograms may be constructed for any operator of the general type $$B_{4}=\mu t+i\nu \left( g\left( t\right) \frac{d}{dt}+\frac{1}{2}\frac{%
dg\left( t\right) }{dt}\right)$$the generalized eigenvectors being $$\psi _{g}\left( \mu ,\nu ,t,X\right) =\left\vert g\left( t\right)
\right\vert ^{-1/2}\exp i\left( -\frac{X}{\nu }\int^{t}\frac{ds}{g\left(
s\right) }+\frac{\mu }{\nu }\int^{t}\frac{sds}{g\left( s\right) }\right)$$
Another finite-dimensional algebra
----------------------------------
Another finite-dimensional Lie algebra which may be used to construct tomograms, exploring other features of the signals, is generated by $\mathbf{%
1}$, $t$ and
$$\begin{array}{l}
\omega =i\frac{d}{dt} \\
D=i\left( t\frac{d}{dt}+\frac{1}{2}\right) \\
F=-\frac{1}{2}\left( \frac{d^{2}}{dt^{2}}-t^{2}+1\right) \\
\sigma =\frac{1}{2}\left( \frac{d^{2}}{dt^{2}}+t^{2}+1\right)%
\end{array}%$$
Of special interest are the tomograms related to the operators $$B_{F}=\mu t+\nu F$$ and $$B_{\sigma }=\mu t+\nu \sigma$$
As before, the construction of the tomograms relies on finding a complete set of generalized eigenvectors for the operators $B_{F}$ and $B_{\sigma }$. With $y=t+\frac{\mu }{\nu }$ one defines creation and annihilation operators $$\begin{aligned}
a &=&\frac{1}{\sqrt{2}}\left( y+\frac{d}{dy}\right) \\
a^{\dagger } &=&\frac{1}{\sqrt{2}}\left( y-\frac{d}{dy}\right)\end{aligned}$$obtaining $$B_{F}=\nu \left( a^{\dagger }a-\frac{\mu ^{2}}{2\nu ^{2}}\right)$$$$B_{\sigma }=\nu \left( aa-\frac{\mu ^{2}}{2\nu ^{2}}\right)$$Therefore for $B_{F}$ one has an orthonormalized complete set of eigenvectors $$\psi _{n}^{(F)}\left( t\right) =u_{n}\left( t+\frac{\mu }{\nu }\right)$$with a discrete set of eigenvalues $X_{n}=\nu \left( n+\frac{1}{2}\right) -%
\frac{\mu ^{2}}{2\nu }$$$B_{F}\psi _{n}^{(F)}\left( t\right) =X_{n}\psi _{n}^{(F)}\left( t\right)$$the function $u_{n}$ being $$u_{n}\left( y\right) =\left( \pi ^{1/2}2^{n}n!\right) ^{-1/2}\left( y-\frac{d%
}{dy}\right) ^{n}e^{-\frac{y^{2}}{2}}$$The tomogram $M_{f}^{(F)}\left( \mu ,\nu ,X_{n}\right) $ is $$M_{f}^{(F)}\left( \mu ,\nu ,X_{n}\right) =\left\vert \int \psi _{n}^{(F)\ast
}\left( t\right) f\left( t\right) dt\right\vert ^{2}$$
For $B_{\sigma }$ one uses a basis of coherent states $$\begin{aligned}
\phi _{\lambda }\left( y\right) &=&e^{\lambda a^{\dagger }-\lambda ^{\ast
}a}u_{0}\left( y\right) \\
&=&e^{\frac{\left\vert \lambda \right\vert ^{2}}{2}}\sum_{n=0}\frac{\lambda
^{n}}{\sqrt{n!}}u_{n}\left( y\right)\end{aligned}$$with decomposition of identity $$\frac{1}{\pi }\int \phi _{\lambda }\left( y\right) \phi _{\lambda }^{\ast
}\left( y\right) d^{2}\lambda =1$$Then, a set of generalized eigenstates of $B_{\sigma }$ is $$\psi _{\lambda }^{(\sigma )}\left( \mu ,\nu ,t\right) =\phi _{\lambda
}\left( t+\frac{\mu }{\nu }\right)$$with eigenvalues $$B_{\sigma }\psi _{\lambda }^{(\sigma )}\left( \mu ,\nu ,t\right) =X_{\lambda
}\psi _{\lambda }^{(\sigma )}\left( \mu ,\nu ,t\right)$$$$X_{\lambda }=\nu \left( \lambda ^{2}-\frac{\mu ^{2}}{2\nu ^{2}}\right)$$the tomogram being $$M_{f}^{\left( \sigma \right) }\left( \mu ,\nu ,X_{\lambda }\right)
=\left\vert \int \psi _{\lambda }^{(\sigma )\ast }\left( \mu ,\nu ,t\right)
f\left( t\right) dt\right\vert ^{2}$$This tomogram is closely related to the Sudarshan-Glauber P-representation [@Sudarshan] [@Glauber63].
Multidimensional tomograms
==========================
Several types of multidimensional tomograms may be obtained from generalizations of the one-dimensional ones. Consider a signal $%
f(t_{1},t_{2})$. The tomogram will depend on a vector variable $\vec{X}%
=\left( X_{1},X_{2}\right) $ and four real parameters $\mu _{1}$, $\mu _{2}$,$\nu _{1}$, and $\nu _{2}$. For example, the two-dimensional time-frequency tomogram will be $$M(\vec{X},\vec{\mu},\vec{\nu})=\frac{1}{2\pi |\nu _{1}|}\,\frac{1}{2\pi |\nu
_{2}|}\left\vert \int f(t_{1},t_{2})\exp \left( \frac{i\mu _{1}}{2\nu _{1}}%
\,t_{1}^{2}-\frac{iX_{1}}{\nu _{1}}\,t_{1}+\frac{i\mu _{2}}{2\nu _{2}}%
\,t_{2}^{2}-\frac{iX_{2}}{\nu _{2}}\,t_{2}\right)
\,dt_{1}\,dt_{2}\right\vert ^{2} \label{Mw}$$From this one may also construct a *center of mass tomogram* $$\begin{aligned}
&&M_{\mathrm{cm}}(Y,\vec{\mu},\vec{\nu})=\int M(\vec{X},\vec{\mu},\vec{\nu}%
)\,\delta (Y-X_{1}-X_{2})\,dX_{1}\,dX_{2}=\int \delta (Y-X_{1}-X_{2})\,\frac{%
1}{2\pi |\nu _{1}|}\,\frac{1}{2\pi |\nu _{2}|} \\
&&\times \left\vert \int f(t_{1},t_{2})dt_{1}\,dt_{2}\,\exp \left( \frac{%
i\mu _{1}}{2\nu _{1}}\,t_{1}^{2}-\frac{iz_{1}X_{1}}{\nu _{1}}+\frac{i\mu _{2}%
}{2\nu _{2}}\,t_{2}^{2}-\frac{iz_{2}X_{2}}{\nu _{2}}\right) \right\vert
^{2}\,dX_{1}\,dX_{2}\end{aligned}$$the center of mass tomogram being normalized $$\int M_{\mathrm{cm}}(X,\vec{\mu},\vec{\nu})\,dX=1$$and a homogeneous function $$M_{\mathrm{cm}}(\lambda X,\lambda \vec{\mu},\lambda \vec{\nu})=\frac{1}{%
|\lambda |}\,M_{\mathrm{cm}}(X,\vec{\mu},\vec{\nu}).$$The generalization to $N$ channels is straightforward.
As in the one-dimensional case, useful tomograms may be constructed from the operators of Lie algebras. For example, given the generators of the conformal algebra in $\mathbb{R}^{d}$, $d\geq 2$, $$\begin{array}{l}
\omega _{k}=i\frac{\partial }{\partial t_{k}} \\
D=i\left( t\bullet \nabla +\frac{d}{2}\right) \\
R_{j,k}=i\left( t_{j}\frac{\partial }{\partial t_{k}}-t_{k}\frac{\partial }{%
\partial t_{j}}\right) \\
K_{j}=i\left( t_{j}^{2}\frac{\partial }{\partial t_{j}}+t_{j}\right)%
\end{array}%$$Let, in two dimensions, $t_{1}=t$ and $t_{2}=x$. The tomograms corresponding to the operators $$\begin{aligned}
B_{\omega } &=&\mu _{1}t+\mu _{2}x+\nu _{1}\omega _{1}+\nu _{2}\omega _{2} \\
B_{D} &=&\mu _{1}t+\mu _{2}x+\nu D \\
B_{\omega } &=&\mu _{1}t+\mu _{2}x+\nu _{1}K_{1}+\nu _{2}K_{2}\end{aligned}$$are, as in (\[Mw\]), straightforward generalizations of the corresponding one-dimensional ones. For the operator $$B_{R}=\mu _{1}t+\mu _{2}x+\nu R_{1,2}$$the eigenstates are $$\psi ^{(R)}\left( \overset{\rightarrow }{\mu },\nu ,x,t,X\right) =\exp \frac{%
i}{\nu }\left( \mu _{1}x-\mu _{2}t+X\tan ^{-1}\frac{t}{x}\right)$$and the tomogram $$M_{f}\left( \overset{\rightarrow }{\mu },\nu ,X\right) =\left\vert \int \psi
^{(R)\ast }\left( \overset{\rightarrow }{\mu },\nu ,x,t,X\right) f\left(
x,t\right) dxdt\right\vert ^{2}$$
The tomograms as operator symbols
=================================
Tomograms may be described not only as amplitudes of projections on a complete basis of eigenvectors of a family of operators, but also as operator symbols. That is, as a map of operators to a space of functions where the operators non-commutativity is replaced by a modification of the usual product to a star-product.
Let $\hat{A}$ be an operator in Hilbert space $\mathcal{H}$ and $\hat{U}(%
\vec{x})$, $\hat{D}(\vec{x})$ two families of operators called *dequantizers* and *quantizers*, respectively, such that $$\text{Tr}\left\{ \,\hat{U}(\vec{x})\hat{D}(\vec{x}^{\prime })\right\}
=\delta (\vec{x}-\vec{x}^{\prime }) \label{5.2}$$The labels $\vec{x}$ (with components $x_{1},x_{2},\ldots x_{n}$) are coordinates in a linear space $V$ where the functions (operator symbols) are defined. Some of the coordinates may take discrete values, then the delta function in (\[5.2\]) should be understood as a Kronecker delta. Provided the property (\[5.2\]) is satisfied, one defines the *symbol of the operator* $\hat{A}$ by the formula $$f_{A}(\vec{x})=\text{Tr}\left\{ \hat{U}(\vec{x})\hat{A}\right\} ,
\label{5.3}$$assuming the trace to exist. In view of (\[5.2\]), one has the reconstruction formula $$\hat{A}=\int f_{A}(x)\hat{D}(\vec{x})\,d\vec{x} \label{5.4}$$The role of quantizers and dequantizers may be exchanged. Then $$f_{A}^{d}(\vec{x})=\text{Tr}\left\{ \hat{D}(\vec{x})\,\hat{A}\right\}
\label{5.6}$$is called the dual symbol of $f_{A}(\vec{x})$ and the reconstruction formula is $$\hat{A}=\int f_{A}^{d}(x)\hat{U}(\vec{x})\,d\vec{x} \label{5.7}$$Symbols of operators can be multiplied using the star-product kernel as follows $$f_{A}(\vec{x})\star f_{B}(\vec{x})=\int f_{A}(\vec{y})f_{B}(\vec{z})K(\vec{y}%
,\vec{z},\vec{x})\,d\vec{y}\,d\vec{z} \label{5.9}$$the kernel being$$K(\vec{y},\vec{z},\vec{x})=\text{Tr}\left\{ \hat{D}(\vec{y})\hat{D}(\vec{z})%
\hat{U}(\vec{x})\right\} \label{5.10}$$The star-product is associative, $$\left( f_{A}(\vec{x})\star f_{B}(\vec{x})\right) \star f_{C}(\vec{x})=f_{A}(%
\vec{x})\star \left( f_{B}(\vec{x})\star f_{C}(\vec{x})\right) \label{5.11}$$this property corresponding to the associativity of the product of operators in Hilbert space.
With the dual symbols the trace of an operator may be written in integral form $$\text{Tr}\left\{ \,\hat{A}\hat{B}\right\} =\int f_{A}^{d}(\vec{x})f_{B}(\vec{%
x})\,d\vec{x}=\int f_{B}^{d}(\vec{x})f_{A}(\vec{x})\,d\vec{x}. \label{5.13}$$
For two different symbols $f_{A}(\vec{x})$ and $f_{A}(\vec{y})$ corresponding, respectively, to the pairs ($\hat{U}(\vec{x})$,$\hat{D}(\vec{x%
})$) and ($\hat{U}_{1}(\vec{y})$,$\hat{D}_{1}(\vec{y})$), one has the relation $$f_{A}(\vec{x})=\int f_{A}(\vec{y})K(\vec{x},\vec{y})\,d\vec{y}, \label{5.14}$$with intertwining kernel $$K(\vec{x},\vec{y})=\text{Tr}\left\{ \hat{D}_{1}(\vec{y})\hat{U}(\vec{x}%
)\right\} \label{5.15}$$
Let now each signal $f\left( t\right) $ be identified with the projection operator $\Pi _{f}$ on the function $f\left( t\right) $, denoted by$$\Pi _{f}=\left\vert f\right\rangle \left\langle f\right\vert \label{5.15a}$$Then the tomograms and also other transforms are symbols of the projection operators for several choices of quantizers and dequantizers.
Some examples:
\# The *Wigner-Ville function*: is the symbol of $\mid f\rangle
\langle f\mid $ corresponding to the dequantizer $$\hat{U}(\vec{x})=2\hat{\mathcal{D}}(2\alpha )\hat{P},\qquad \alpha =\frac{%
t+i\omega }{\sqrt{2}}\, \label{5.16}$$where $\hat{P}$ is the inversion operator $$\hat{P}f(t)=f(-t) \label{5.18}$$and $\hat{\mathcal{D}}(\gamma )$ is a displacement operator $$\hat{\mathcal{D}}(\gamma )=\exp \left[ \frac{1}{\sqrt{2}}\gamma \left( t-%
\frac{\partial }{\partial t}\right) -\frac{1}{\sqrt{2}}\gamma ^{\ast }\left(
t+\frac{\partial }{\partial t}\right) \right] \label{5.19}$$The quantizer operator is $$\hat{D}(\vec{x}):=\hat{D}(t,\omega )=\frac{1}{2\pi }\hat{U}(t,\omega ),
\label{5.20}$$$t$ and $\omega $ being time and frequency.
The Wigner–Ville function is $$W(t,\omega )=2\text{Tr}\left\{ \mid f\rangle \langle f\mid \hat{D}(2\alpha )%
\hat{D}\right\} \label{5.21}$$or, in integral form $$W(t,\omega )=2\int f^{\ast }(t)\hat{\mathcal{D}}(2\alpha )f(-t)\,dt
\label{5.22}$$
\# The *symplectic tomogram* or time-frequency tomogram of $\mid
f\rangle \langle f\mid $ corresponds to the dequantizer $$\hat{U}(\vec{x}):=\hat{U}(X,\mu ,\nu )=\delta \left( X\hat{1}-\mu \hat{t}%
-\nu \hat{\omega}\right) , \label{5.23}$$with $$\hat{t}f(t)=tf(t),\qquad \hat{\omega}f(t)=-i\frac{\partial }{\partial t}%
\,f(t) \label{5.24}$$and $X,\mu ,\nu \in R$. The quantizer of the symplectic tomogram is $$\hat{D}(\vec{x}):=\hat{D}(X,\mu ,\nu )=\frac{1}{2\pi }\,\exp \left[ i\left( X%
\hat{1}-\mu \hat{t}-\nu \hat{\omega}\right) \right] \label{5.25}$$
\# The *optical tomogram* is the same as above for the case $$\mu =\cos \theta ,\qquad \nu =\sin \theta . \label{5.26}$$Thus the optical tomogram is $$\begin{aligned}
M(X,\theta ) &=&\text{Tr}\left\{ \mid f\rangle \langle f\mid \delta \left( X%
\hat{1}-\mu \hat{t}-\nu \hat{\omega}\right) \right\} \notag \\
&=&\frac{1}{2\pi }\int f^{\ast }(t)e^{ikX}\exp \left[ ik\left( X-t\cos
\theta +i\,\frac{\partial }{\partial t}\sin \theta \right) \right]
f(t)\,dt\,dk \notag \\
&=&\frac{1}{2\pi |\sin \theta |}\left\vert \int f(t)\exp \left[ i\left(
\frac{\cot \theta }{2}t^{2}-\frac{Xt}{\sin \theta }\right) \right]
dt\right\vert ^{2}. \label{5.28}\end{aligned}$$
One important feature of the formulation of tomograms as operator symbols is that one may work with deterministic signals $f\left( t\right) $ as easily as with probabilistic ones. In this latter case the projector in (\[5.15a\]) would be replaced by$$\Pi _{p}=\int p_{\mu }\left\vert f_{\mu }\right\rangle \left\langle f_{\mu
}\right\vert d\mu \label{5.29}$$with $\int p_{\mu }d\mu =1$, the tomogram being the symbol of this new operator.
This also provides a framework for an algebraic formulation of signal processing, perhaps more general than the one described in [@Moura1] [@Moura2]. There, a signal model is a triple $\left( \mathcal{A},%
\mathcal{M},\Phi \right) $ $\mathcal{A}\ $being an algebra of linear filters, $\mathcal{M}$ a $\mathcal{A}$-module and $\Phi $ a map from the vector space of signals to the module. With the operator symbol interpretation both (deterministic or random) signals and (linear or nonlinear) transformations on signals are operators. By the application of the dequantizer (Eq. \[5.3\]) they are mapped onto functions, the filter operations becoming star-products.
Rotated-time tomography
=======================
Now we consider a version of tomography where a discrete random variable is used as an argument of the probability distribution function. We call this tomography *rotated time tomography*. It is a variant of the spin-tomographic approach for the description of discrete spin states in quantum mechanics. For a finite duration signal $f(t)$, with $0\leq t\leq T$,we consider discrete values of time $f(t_{m})\equiv f_{m}$, where with the labeling $m=-j,-j+1,-j+2,\ldots ,0,1,\ldots ,j-1,j$ they are like the components of a spinor $\mid f\rangle $. This means that we split the interval $[0,T]$ onto $N$ parts at time values $t_{-j},t_{-j+1},\ldots
,t_{j} $ and replace the signal $f(t)$, a function of continuous time, by a discrete set of values organized as a spinor. By dividing by a factor we normalize the spinor, i.e., $$\langle f\mid f\rangle =\sum_{m=-j}^{j}|f_{m}|^{2}=1 \label{S1}$$Without loss of generality, we consider the “spin” values to be integers, i.e., $j=0,1,2,\ldots $ and use an odd number $N=2j+1$ of values.
In this setting, $\mid f\rangle $ being a column vector, we construct the $N$$\times $$N$ matrix $$\rho =\mid f\rangle \langle f\mid \label{S2}$$with matrix elements $$\rho _{mm^{\prime }}=f_{m}f_{m^{\prime }}^{\ast }. \label{S3}$$The tomogram is defined as the probability-distribution function $$\mathcal{M}(m,u)=|\langle m\mid u\mid f\rangle |^{2},\qquad m=-j,\ldots
,j-1,j \label{S4}$$where $u$ is the unitary $N$$\times $$N$ matrix $$uu^{\dagger }=1_{N} \label{S5}$$For this matrix we use an unitary irreducible representation of the rotation group (or $SU(2)$) with matrix elements $$\begin{aligned}
u_{mm^{\prime }}(\theta ) &=&\frac{(-1)^{j-m^{\prime }}}{(m+m^{\prime })!}%
\left[ \frac{(j+m)!(j+m^{\prime })!}{(j-m)!(j-m^{\prime })!}\right]
^{1/2}\left( \sin \frac{\theta }{2}\right) ^{m-m^{\prime }}\left( \cos \frac{%
\theta }{2}\right) ^{m+m^{\prime }} \notag \\
&&\times \mathcal{F}_{j-m}\left( 2m+1,m+m^{\prime 2}\frac{\theta }{2}\right)
\label{S61}\end{aligned}$$$\mathcal{F}_{j-m}$ being a function with Jacobi polynomial structure expressed in terms of hypergeometric function as $$\begin{aligned}
\mathcal{F}_{n}(a,b,t) &=&F(-n,a+n,b;t)=\frac{(b-1)!}{(b+n-1)!}%
\,t^{1-b}(1-t)^{b-a}\left( \frac{d}{dt}\right) ^{n}\left[
t^{b+n-1}(1-t)^{a-b+1}\right] \notag \\
&& \label{S71}\end{aligned}$$The dequantizer in the rotated-time tomography is $$\hat{U}(\vec{x})\equiv U(m,\vec{n})=\delta (m1-u^{\dagger }J_{z}u)=\delta
\left( m1-\vec{n}\vec{J}\right) \label{S8}$$where $J_{z}$ is the matrix with diagonal matrix elements $$(J_{z})_{mm^{\prime }}=m\delta _{mm^{\prime }} \label{S9}$$The vector $\vec{n}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi
,\cos \theta )$ determines a direction in 3D space. The matrix (\[S61\]) was written for $\varphi =0$ but, if this angle is nonzero, the matrix element has to be multiplied by the phase factor $e^{im\varphi }$.
The quantizer can take several forms:
In integral form, it reads $$\hat{D}(m,\vec{n})=\frac{2j+1}{\pi }\int_{0}^{2\pi }\sin ^{2}\frac{\gamma }{2%
}\exp (-i\vec{J}\vec{n})\gamma \,d\gamma (\cdots ) \label{S10}$$The tomogram $\mathcal{M}(m,u)$ is a nonnegative normalized probability distribution depending on the direction $\vec{n}$, i.e., $\mathcal{M}%
(m,u)\geq 0$ and $$\sum_{m=-j}^{j}\mathcal{M}(m,u)=1 \label{S11}$$To compute the tomogram for a given direction with angles $\varphi =0$ and $%
\theta $, one has to estimate $$\mathcal{M}(m,\theta )=\sum_{m^{\prime \prime },m^{\prime
}=-j}^{j}u_{mm^{\prime }}^{\ast }(\theta )f_{m}f_{m^{\prime \prime }}^{\ast
}u_{m^{\prime \prime }m}(\theta ) \label{S12}$$where the matrix $u_{m^{\prime \prime }m}(\theta )$ is given by (\[S61\]). The following form for the matrix $u_{m^{\prime }m}(\theta )$ is more convenient for numerical calculations: $$u_{m^{\prime }m}(\theta )=\left[ \frac{(j+m^{\prime })!(j-m^{\prime })!}{%
(j+m)!(j-m)!}\right] ^{1/2}\left( \cos \frac{\theta }{2}\right) ^{m^{\prime
}+m}\left( \sin \frac{\theta }{2}\right) ^{m^{\prime }-m}P_{j-m^{\prime
}}^{m^{\prime }-m,m^{\prime }+m}(\cos \theta ) \label{LL}$$where $P_{n}^{a,b}$ are Jacobi polynomials.
In principle, one could use not only the unitary matrix in (\[S61\]) but arbitrary unitary matrices. They contain a larger number of parameters (equal to $N^{2}-1)$ and can provide additional information on the signal structure.
How the time-rotated tomogram explores the time-frequency plane is, as before, illustrated by spectrograms of the eigenstates (Figs.\[trot1\] and \[trot2\]). For $m=0$, formula (70) reduces to the set of normalized associated Legendre functions $L^{m'}_j$: $$u_{m',0}(\theta)=\sqrt{\frac{2}{2j+1}}L_j^{m'}(\cos(\theta)).$$ The normalized associated Legendre functions are related to the unmormalized ones $P^{m'}_j$ through: $$L_j^{m'}(\cos(\theta))=\sqrt{\frac{2j+1}{2}\frac{(j-m')}{(j+m')}}P^{m'}_j(\cos\theta).$$ In the tomogram, $\theta$ is the parameter labelling the vectors of the basis associated to $m=0,m'$. The index $j$ is the variable. In order to illustrate the effect of this tomogram, we computed numerically some vectors in the time-frequency plane (Figs. \[rotatomo1\] and \[rotatomo2\]). In the discrete setting, If we choose $m'=N$, where $N$ is the number of points, the $\{L_j^{N}\}_j$ form an orthonormal basis of the discrete time-frequency plane. Hence the projection on the eigenvectors of the rotated tomogram with $m=0,m'=N$ can be seen as the projection on the bended lines in the time-frequency plane. This tomogram should be adapted for the study of functions which possess certain symetry in the time-frequency plane.
![Modulus of the short-time Fourier transform of the sum of 4 vectors of the rotated-time tomogram. Each $u_{m',0}(\theta,j)$ is a bended line in the time-frequency plane where $\theta$ fix the size. Here $N=1571$ and $\theta=\pi/8,2\pi/8,3\pi/8,\pi/2$. As $\theta$ increases, the line is stretched to the right until it breaks in two parts for $\pi/2$.[]{data-label="rotatomo1"}](rotatomo1.eps){width="12cm"}
\[trot1\]
![Modulus of the short-time Fourier transform of the sum of 4 vectors of the rotated-time tomogram. Here $N=1571$ and $\theta=5\pi/8,6\pi/8,7\pi/8$ and $\pi-\pi/16$. Each $u_{m',0}(\theta,j)$ is made of two bended lines in the time-frequency plane, one in the upper-half plane and one in the lower-half plane.[]{data-label="rotatomo2"}](rotatomo2.eps){width="12cm"}
\[trot2\]
Hermite basis tomography
========================
Here we consider a dequantizer $$\hat{U}(n,\alpha )=\hat{\mathcal{D}}(\alpha )\mid n\rangle \langle n\mid
\hat{\mathcal{D}}^{\dagger }(\alpha ),\qquad \alpha =|\alpha |e^{i\theta
_{\alpha }} \label{PN1}$$and a quantizer $$\hat{D}(n,\alpha )=\frac{4}{\pi (1-\lambda ^{2})}\left( \frac{\lambda +1}{%
\lambda -1}\right) ^{n}\hat{\mathcal{D}}(\alpha )\left( \frac{\lambda -1}{%
\lambda +1}\right) ^{n}\hat{\mathcal{D}}(-\alpha ) \label{PN2}$$where $-1<\lambda <1$ is an arbitrary parameter and $n$ is related to the order of an Hermite polynomial. This is analogous to the use of a photon number basis in quantum optics.
For any signal $f(t)$, one has the probability distribution (tomogram) $$\mathcal{M}_{f}(n,\alpha )=\mbox{Tr}\,\mid f\rangle \langle f\mid \hat{U}%
(n,\alpha ) \label{PN3}$$and, from the tomogram, the signal is reconstructed by $$\mid f\rangle \langle f\mid =\sum_{n=0}^{\infty }\int d^{2}\alpha \mathcal{M}%
(n,\alpha )\hat{D}(n,\lambda ) \label{PN4}$$One has $\mathcal{M}(n,\alpha )\geq 0$ and $$\sum_{n=0}^{\infty }\mathcal{M}_{f}(n,\alpha )=1 \label{PN5}$$for any complex $\alpha $. For an arbitrary operator $\hat{A}$, one has $$\hat{I}\hat{A}=\sum_{n=0}^{\infty }\int d^{2}\alpha \hat{D}(n,\alpha )%
\mbox{Tr}\left( \hat{U}(n,\alpha )\hat{A}\right) , \label{PN6}$$where $\hat{I}$ is the identity operator.
The explicit form of the tomogram for a signal function $f(t)$ is $$\mathcal{M}_{f}(n,\lambda )=\left\vert \langle f\mid \hat{\mathcal{D}}%
(\alpha )\mid n\rangle \right\vert ^{2}=\left\vert \int f^{\ast
}(t)f_{n,\alpha }(t)\,dt\right\vert ^{2} \label{PN7}$$where $$f_{n,\alpha }(t)=\hat{\mathcal{D}}(\alpha )\left[ \pi
^{-1/4}(2^{n}n!)^{-1/2}e^{-t^{2}/2}H_{n}(t)\right] \label{PN8}$$$H_{n}(t)$ being an Hermite polynomial.
Thus, one has $$f_{n,\alpha }(t)=\pi ^{-1/4}(2^{n}n!)^{-1/2}e^{-(\alpha ^{2}-\alpha ^{\ast
2})/4}e^{[(\alpha -\alpha ^{\ast })t]/\sqrt{2}}e^{-\tilde{t}^{2}/2}H_{n}(%
\tilde{t}) \label{PN9}$$and $$\tilde{t}=t-\frac{\alpha +\alpha ^{\ast }}{\sqrt{2}}\,. \label{PN10}$$For fixed $|\alpha |$ the tomogram is a function of the discrete set $%
n=0,1,\ldots $ and the phase factor $\theta _{\alpha }$.
How the Hermite basis tomogram explores the time-frequency plane is, as before, illustrated by spectrograms of the eigenstates (Fig.\[hermite\]). In the particular case where $\alpha=0$, the functions $f_{n,0}$ are the Hermite functions. Their time-frequency representation has been calculated on Figure \[Hermite0\]. It shows that the tomogram at $\alpha=0$ is suited for rotation invariant functions in the time-frequency plane. One can see from (79) that: for real $\alpha$ this pattern is shifted in time and for purely imaginary $\alpha$ the pattern is shifted in frequency. The pattern can be shifted in both time and frequency by choosing the appropriate complex value for $\alpha$.
![Modulus of the short-time Fourier transform of the sum of 4 Hermite functions. Each ring is a Hermite function. Here, the number of points is $N=2000$. The picture has been centered, the origin has been set to Time $t=1000$, Frequency $f=0$. That is to say, $t=-N/2+l\Delta t$ for $l\in[0,N)$, $\Delta t=1$. The smallest circle is for $n=5$ and in increasing size order $n=500$, $n=1000$, $n=1500$, respectively.[]{data-label="Hermite0"}](hermtomo1.eps){width="12cm"}
\[hermite\]
Some applications
=================
The tomograms are squared amplitudes of the signal projections on families of unitarily equivalent basis (labelled by the $\mu ,\nu $ parameters). By inspecting the unfolding of these (probability) amplitudes as the parameters change, several features of the signals are put into evidence. Here we review briefly three such applications, namely denoising, detection of small signals and component decomposition, which use the time-frequency tomogram. Then the time-scale tomogram will be used to analyse a turbulent velocity fluctuations signal.
For the finite-time signals, instead of (\[3.17\]), we consider the finite-time tomogram $$M_{1}(\theta ,X)=\left| \int_{t_{0}}^{t_{0}+T}\ f^{*}(t)\psi _{\theta
,X}^{(1)}\left( t\right) \,dt\right| ^{2}=\left| <f,\psi ^{(1)}>\right| ^{2}
\label{4.1}$$ with $$\psi _{\theta ,X}^{(1)}\left( t\right) =\frac{1}{\sqrt{T}}\exp \left( \frac{%
i\cos \theta }{2\sin \theta }\,t^{2}-\frac{iX}{\sin \theta }\,t\right)
\label{4.2}$$ and $\mu =\cos \theta ,\nu =\sin \theta $.
$\theta $ is a parameter that interpolates between the time and the frequency operators, running from $0$ to $\pi /2$ whereas $X$ is allowed to be any real number.
Detection of small signals
--------------------------
As an example [@MendesPLA] consider a signal generated as a superposition of a normally distributed random amplitude - random phase noise (with total duration $T=1$) with a sinusoidal signal of same average amplitude, operating only during the time $0.45-0.55$. The signal to noise power ratio is $1/10$. The true nature of the signal is not revealed neither from its time development nor from its Fourier spectrum. However computing the tomogram (see the contour plot in Fig.\[noise\])
\[htb\] ![Detection of small signals in noise[]{data-label="noise"}](fig2.eps "fig:"){width="12cm"}
one sees clearly a sequence of small peaks connecting a time around $0.5$ to a frequency around $200$. The signature that the signal leaves on the tomogram is a manifestation of the fact that, despite its low signal to noise ratio, there is a certain number of directions in the $(t,\omega )$ plane along which detection happens to be more favorable. For different trials the coherent peaks appear at different locations, but the overall geometry of the ridge is the same. On the other hand, a ridge of small peaks is reliable because the rigorous probability interpretation of $M(\theta ,X)$ renders the method immune to spurious effects.
Denoising and component decomposition
-------------------------------------
Most natural and man-made signals are nonstationary and have a multicomponent structure. Therefore separation of its components is an issue of great technological relevance. However, the concept of signal component is not uniquely defined. The notion of *component* depends as much on the observer as on the observed object. When we speak about a component of a signal we are in fact referring to a particular feature of the signal that we want to emphasize. For signals that have distinct features both in the time and the frequency domain, the time-frequency tomogram is an appropriate tool.
Here again consider finite-time tomograms as in (\[4.1\]). For all different $\theta $’s the $U({\theta })$, of which $B\left( \theta \right) $ is the self-adjoint generator, are unitarily equivalent operators, hence all the tomograms share the same information.
First we select a subset $X_{n}$ in such a way that the corresponding family $\left\{ \psi _{\theta ,X_{n}}^{(1)}\left( t\right) \right\} $ is orthogonal and normalized, $$<\psi _{\theta ,X_{n}}^{(1)}\psi _{\theta ,X_{m}}^{(1)}>=\delta _{m,n}
\label{4.3}$$ This is possible by taking the sequence $$X_{n}=X_{0}+\frac{2n\pi }{T}\sin \theta \hspace{2cm}n\in \mathbb{Z}
\label{4.4}$$ where $X_{0}$ is freely chosen (in general we take $X_{0}=0$). We then consider the projections of the signal $f(t)$ $$c_{X_{n}}^{\theta }(f)=<f,\psi _{\theta ,X_{n}}^{(1)}> \label{4.5}$$
*Denoising* consists in eliminating the $c_{X_{n}}^{\theta }(f)$ such that $$\left| c_{X_{n}}^{\theta }(f)\right| ^{2}\leq \epsilon \label{4.6}$$ for some threshold $\epsilon $. This power selective denoising is more robust than, for example, frequency filtering which may also eliminate important signal information.
The *component separation technique* is based on the search for an intermediate value of ${\theta }$ where a good compromise might be found between time localization and frequency information. This is achieved by selecting subsets $\mathcal{F}_{k}$ of the $X_{n}$ and reconstructing partial signals ($k$-components) by restricting the sum to $$f_{k}(t)=\sum_{n\in \mathcal{F}_{k}}c_{X_{n}}^{\theta }(f)\psi _{\theta
,X_{n}}(t) \label{4.7}$$ for each $k$.
As an example consider the following signal
$$y(t)=y_{1}(t)+y_{2}(t)+y_{3}(t)+b(t) \label{4.8}$$
$$\begin{aligned}
y_{1}\left( t\right) &=&\exp \left( i25t\right) ,t\in \left[ 0,20\right]
\notag \\
y_{2}\left( t\right) &=&\exp \left( i75t\right) ,t\in \left[ 0,5\right]
\notag \\
y_{3}\left( t\right) &=&\exp \left( i75t\right) ,t\in \left[ 10,20\right]
\label{4.9}\end{aligned}$$
Separation is impossible both at the time ($\theta =0$) and the frequency ($%
\theta =\frac{\pi }{2}$) axis. However, at some intermediate $\theta $ value one obtains distinct probability peaks (Fig.\[pi/5\]), which after the projections (\[4.7\]) allows an accurate separation of the signal components (Figs.\[y2component\] and \[y3component\])
\[htb\] ![The tomogram at $\protect\theta %
=\protect\pi /5$ used for the component separation[]{data-label="pi/5"}](fig3.eps "fig:"){width="12cm"}
\[htb\] ![The $y_{2}$ component[]{data-label="y2component"}](fig4.eps "fig:"){width="12cm"}
\[htb\] ![The $y_{3}$ component[]{data-label="y3component"}](fig5.eps "fig:"){width="12cm"}
Component decomposition of more complex signals (nonlinear chirps overlapping in both the time and the frequency domains and experimental reflectometry signals) has been successfully carried out by this technique [@reflecto1] [@reflecto2].
Tomograms and turbulent velocity fluctuations
---------------------------------------------
Here we report briefly on an analysis by the tomographic technique of a velocity fluctuation signal of a turbulent flow in a wind tunnel. It illustrates the fact that the choice of the pair of non-commuting operators in tomogram, should be adapted to the signal under study. As before we use finite-time tomograms in the interval $\left( t_{0},t_{0}+T\right) $. For the finite-time (time-frequency) tomogram $M_{1}$, the normalization and a set of $X_{n}$’s leading to an orthonormalized set of eigenstates has already been written in (\[4.1\])-(\[4.2\]).
For future reference we include here the corresponding sets of orthonormalized eigenstates for the finite-time time-scale tomogram $%
M_{2}(\mu ,\nu ,X)$ (Eq.\[3.18\]) and for the finite-time time-conformal tomogram $M_{4}(\mu ,\nu ,X)$ (Eq.\[3.20\]): $$M_{2}\left( \theta ,X\right) =\left\vert \int_{t_{0}}^{t_{0}+T}\ f^{\ast
}(t)\psi _{\theta ,X}^{(2)}\left( t\right) \,dt\right\vert ^{2}=\left\vert
<f,\psi ^{(2)}>\right\vert ^{2} \label{4.10}$$$$\psi _{\theta ,X}^{(2)}\left( t\right) =\frac{1}{\sqrt{\log \left\vert
t_{0}+T\right\vert -\log \left\vert t_{0}\right\vert }}\frac{1}{\sqrt{%
\left\vert t\right\vert }}\exp i\left( \frac{\cos \theta }{\sin \theta }\,t-%
\frac{X}{\sin \theta }\,\log \left\vert t\right\vert \right) \label{4.11}$$$$X_{n}=X_{0}+\frac{2n\pi }{\log \left\vert t_{0}+T\right\vert -\log
\left\vert t_{0}\right\vert }\sin \theta \hspace{2cm}n\in \mathbb{Z}
\label{4.12}$$and $$M_{4}(\theta ,X)=\left\vert \int_{t_{0}}^{t_{0}+T}\ f^{\ast }(t)\psi
_{\theta ,X}^{(4)}\left( t\right) \,dt\right\vert ^{2}=\left\vert <f,\psi
^{(4)}>\right\vert ^{2} \label{4.13}$$$$\psi _{\theta ,X}^{(4)}\left( t\right) =\sqrt{\frac{t_{0}\left(
t_{0}+T\right) }{T}}\frac{1}{\left\vert t\right\vert }\exp i\left( \frac{%
\cos \theta }{\sin \theta }\,\,\log \left\vert t\right\vert +\frac{X}{t\sin
\theta }\right) \label{4.14}$$$$X_{n}=X_{0}+\frac{t_{0}\left( t_{0}+T\right) }{T}2\pi n\sin \theta \hspace{%
2cm}n\in \mathbb{Z} \label{4.15}$$
\[htb\] ![Contour plot of the tomogram for a velocity fluctuations signal[]{data-label="velocity1"}](fig8_3_3.eps "fig:"){width="12cm"}
\[htb\] ![The tomogram $M_{2}\left( X_{n},%
\protect\theta \right) $ at $\protect\theta =1.26$[]{data-label="velocity2"}](fig8_3_2.eps "fig:"){width="12cm"}
Analyzing the turbulent velocity fluctuations signal with these tomograms, one notices that except for some features on the frequency axis corresponding to some dominating frequencies, no interesting structures are put into evidence when one use the time-frequency tomogram. The situation is more interesting for the time-scale tomogram $M_{2}\left( \theta ,X\right) $. In Fig.\[velocity1\] we show a contour plot for $M_{2}\left( \theta ,X\right) $ corresponding to a section of 1000 data points. For intermediate regions of $%
\theta $ one notices, a strong concentration of energy in a few regions. This is put into evidence by a cut at $\theta =1.26$ (Fig.\[velocity2\]). Projecting out the signal corresponding to these regions with the corresponding $\psi
_{\theta ,X}^{(2)}\left( t\right) $’s at this $\theta $, one sees that although the signal has many complex features most of the energy is concentrated in fairly regular structures. Fig.\[structure\] shows the structure $\eta
\left( t\right) $ corresponding to the second peak in Fig.\[velocity2\].
\[htb\] ![The structure $\protect\eta %
\left( t\right) $ corresponding to the second peak in Fig.7[]{data-label="structure"}](fig8_3_4.eps "fig:"){width="12cm"}
Conclusions
===========
Tomograms provide a two-variable characterization of signals which, due to its rigorous probabilistic interpretation, is robust and free of artifacts and ambiguities. For each particular signal that one wants to analyse the choice of the appropriate tomogram depends not only on the signal but also on the features that we might want to identity or emphasize. So far we have explored component separation, denoising and identification of small signal in noise, but other features may also benefit from the robust probabilistic of the tomographic analysis. This was our main motivation to include here a long list of many different operator choice leading to different classes of tomograms.
The description of the tomograms as operator symbols, with the corresponding quantizers and dequantizers, not only provides an alternative formulation but may also be used to extend the algebraic signal processing formalism to a wider nonlinear context.
[99]{} A. D. Poularikas (ed.); *The Transforms and Applications Handbook*, CRC Press & IEEE Press, Boca Raton, Florida (1996).
K.-B. Wolf; *Integral Transforms in Science and Engineering*, Plenum Press, New York (1979).
J. B. J. Fourier; *Théorie Analytique de la Chaleur*, in: G. Darbous (ed.), *Oeuvres de Fourier*, Gauthiers-Villars, Paris (1888), Tome premier.
J. M. Combes, A. Grossmann, and Ph. Tchamitchian (eds.); *Wavelets*, Springer, Berlin (1990), 2nd edition.
I. Daubechies; *The wavelet transform: time–frequency localization and signal analysis*, IEEE Trans. Inform. Theory, **36**, No. 5 (1990) 961–1005.
C. K. Chui (ed.); *Wavelets: A Tutorial. Theory and Applications*, Academic, Boston (1992), Vol. 2.
E. Wigner; *On the quantum correction for thermodynamic equilibrium*, Phys. Rev., 40 (1932) 749–759.
J. Ville; *Théorie et applications de la notion de signal analytique*, Cables et Transmission, 2 A (1948) 61–74.
L. Cohen; *Generalized phase-space distribution functions*, J. Math. Phys. 7 (1966) 781–806.
L. Cohen; *Time–frequency distributions. A review*, Proc. IEEE 77 (1989) 941–981.
S. Qian and D. Chen; *Joint time–frequency analysis*, Prentice-Hall, Englewood Cliffs, N. J. (1995).
K. Husimi; *Some formal properties of the density matrix*, Proc. Phys. Mat. Soc. Jpn, 22 (1940) 264–314.
Y. Kano; *A new phase-space distribution function in the statistical theory of the electromagnetic field*, J. Math. Phys. 6 (1965) 1913–1915.
V. I. Man’ko and R. Vilela Mendes; *Noncommutative time–frequency tomography*, Phys. Lett. A, 263 (1999) 53–59.
M. A. Man’ko, V. I. Man’ko and R. Vilela Mendes; *Tomograms and other transforms: A unified view*, J. Phys. A: Math. and Gen.34 (2001) 8321-8332.
S. R. Deans; *The Radon Transform and Some of Its Applications*, John Wiley & Sons, New York 1983.
J. C. Woods and D. T. Barry; *Linear signal synthesis using the Radon–Wigner transform*, IEEE Trans. Signal Process. 42 (1994) 2105–2111.
S. Granieri, W. D. Furlan, G. Saavedra, and P. Andrés; *Radon–Wigner display: a compact optical implementation with a single varifocal lens*, Appl. Opt. 36 (1997) 8363–8369.
J. Bertrand and P. Bertrand; *A class of affine Wigner functions with extended covariance properties*, J. Math. Phys., 33 (1992) 2515–2527.
P. Goncalvés and R. G. Baraniuk; *A pseudo-Bertrand distribution for time–scale analysis*, IEEE Signal Process. Lett. 3 (1996) 82–84.
M. Asorey, P. Facchi, V.I. Manko, G. Marmo, S. Pascazio and E.C.G. Sudarshan; *Generalized tomographic maps*, Physical Review A 77 (2008) 042115.
E. C. G. Sudarshan, *Equivalence of semiclassical and quantum-mechanical descriptions of statistical light beams*, Phys. Rev. Lett. 10 (1963) 277–279.
R. J. Glauber; *Coherent and incoherent states of the radiation fields*, Phys. Rev. 131 (1963) 2766–2788; *Photon correlations*, Phys. Rev. Lett. 10 (1963) 84–86.
M. Püschel and J. M. F. Moura; *Algebraic signal processing theory: Foundation and 1-D time*, IEEE Trans.on Signal Process. 56 (2008) 3572-3585.
M. Püschel and J. M. F. Moura; *Algebraic signal processing theory: 1-D space*, IEEE Trans.on Signal Process. 56 (2008) 3586-3599.
F. Briolle, R. Lima, V. I. Man’ko and R. Vilela Mendes; *A tomographic analysis of reflectometry data I: Component factorization*, Meas. Sci. Technol. 20 (2009) 105501.
F. Briolle, R. Lima and R. Vilela Mendes; *A tomographic analysis of reflectometry data II: The phase derivative*, Meas. Sci. Technol. 20 (2009) 105502.
[^1]: Centre de Physique Théorique, CNRS Luminy, case 907, F-13288 Marseille Cedex 9, France, Francoise.Briolle@univmed.fr
[^2]: P.N. Lebedev Physical Institute, Moscow, Russia, manko@sci.lebedev.ru
[^3]: CMAF, Complexo Interdisciplinar, Universidade de Lisboa, Av. Gama Pinto, 2 - P1699 Lisboa Codex, Portugal, e-mail: vilela@cii.fc.ul.pt, http://label2.ist.utl.pt/vilela/
[^4]: IPFN, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, Portugal
|
---
abstract: |
A non-orthogonal multiple access (NOMA) approach that always outperforms orthogonal multiple access (OMA) called Fair-NOMA is introduced. In Fair-NOMA, each mobile user is allocated its share of the transmit power such that its capacity is always greater than or equal to the capacity that can be achieved using OMA. For any slow-fading channel gains of the two users, the set of possible power allocation coefficients are derived. For the infimum and supremum of this set, the individual capacity gains and the sum-rate capacity gain are derived. It is shown that the ergodic sum-rate capacity gain approaches 1 b/s/Hz when the transmit power increases for the case when pairing two random users with i.i.d. channel gains. The outage probability of this approach is derived and shown to be better than OMA.
The Fair-NOMA approach is applied to the case of pairing a near base-station user and a cell-edge user and the ergodic capacity gap is derived as a function of total number of users in the cell at high SNR. This is then compared to the conventional case of fixed-power NOMA with user-pairing. Finally, Fair-NOMA is extended to $K$ users and it is proven that the capacity can always be improved for each user, while using less than the total transmit power required to achieve OMA capacities per user.
author:
- 'José Armando Oviedo, and Hamid R. Sadjadpour, [^1][^2]'
title: 'A Fair Power Allocation Approach to NOMA in Multi-user SISO Systems '
---
\[section\]
Introduction
============
*Orthogonal multiple access* (OMA) is defined as a system that schedules multiple users in non-overlapping time slots or frequency bands during transmission. Therefore, if the signals for $k$ users, $k=1,\ldots, K$ are scheduled for transmission over a time period $T$, where $T$ is less than the coherence time of the channel, each user transmits only $T/K$ amount of the total transmission period (or fraction of the total bandwidth) with the entire transmit power $\xi$ allocated to that user.
*Non-orthogonal multiple access* (NOMA) schedules transmission of the $K$ users’ signals simultaneously over the entire transmission period and bandwidth. Since the total transmit power $\xi $ must be shared between the $K$ users, a fraction $a_k\in(0,1)$ of the transmit power is allocated to user $k$, and $\sum_{k=1}^K a_k \leq 1$. In NOMA, each user employs successive interference cancellation (SIC) at the receiver to remove the interference of the signals from users with lesser channel gains [@InfTh:CT].
An approach called Fair-NOMA is proposed for $K$ users in future wireless cellular networks. The underlying fundamental property of Fair-NOMA is that users will always be guaranteed to achieve a capacity at least as good as OMA. The average capacities of two random users with i.i.d. channel SNR gains are derived along with the expected increase in capacity between OMA and NOMA.
Another unique feature of our approach compared to the previous work is the fact that prior studies on NOMA have focused on demonstrating that NOMA has advantages for increasing the capacity of the network when users are scheduled and paired based on their channel conditions (i.e. their location in the cell). Fair-NOMA does not rely on this condition since users’ channel conditions are i.i.d. (i.e. location in the cell is not considered). Hence, all users will have equal opportunity to be scheduled, and thus is also completely “fair” from a time-sharing perspective. However, Fair-NOMA can be applied to any system with any scheduling and user-pairing approach.
The paper is organized as follows. Previous contributions on NOMA are discussed in section \[sec:previous\]. The assumptions and system model are outlined in section \[sec:system\]. Section \[sec:fairNOMA\] defines the Fair-NOMA power allocation region $\mathcal{A}_\text{FN}$, and develops its basic properties. The analysis of the effects of Fair-NOMA on the capacity of each user together with simulation results are provided in section \[sec:analysis\]. The improvement in outage probability is derived and demonstrated in section \[sec:outage\]. The application of Fair-NOMA to opportunistic user-pairing with near and cell-edge users is discussed and analyzed in section \[sec:fairNOMAMUD\]. Section \[sec:MU-NOMA\] defines Fair-NOMA for multi-user SISO systems. Finally, section \[sec:conclusion\] concludes the paper and discusses future work.
Previous Work on NOMA {#sec:previous}
=====================
The concept of NOMA is based on using superposition coding (SC) at the transmitter and successive interference cancellation (SIC) at the receivers. This was shown to achieve the capacity of the channel by Cover and Thomas [@InfTh:CT]. The existence of a set of power allocation coefficients that allow all of the participating users to achieve capacity at least as good as OMA was suggested in [@FundWiCom:Tse].
Non-orthogonal access approaches using SC for future wireless cellular networks were mentioned in [@CompOMANOMA:WXP] as a way to increase single user rates when compared to CDMA. Schaepperle and Ruegg [@4GNOSig:SR] evaluated the performance of non-orthogonal signaling using SC and SIC in single antenna OFDMA systems using very little modifications to the existing standards, as well as how user pairing impacts the throughput of the system when the channel gains become increasingly disparate. This was then applied [@WCSCMA:Schaep] to OFDMA wireless systems to evaluate the performance of cell edge user rates, proposing an algorithm that attempts to increase the average throughput and maintain fairness. These works do not assume to have the exact channel state information at the transmitter.
The concept of NOMA is evaluated through simulation for full channel state information at the transmitter (CSIT) in the uplink [@ULNOMA:TakedaHiguchi] and downlink [@DLNOMA:TomidaHiguchi], where the throughput of the system is shown to be on average always better than OMA when considering a fully defined cellular system evaluation, with both users occupying all of the bandwidth and time, and was compared to FDMA with each user being assigned an orthogonal channel. In [@SLDLNOMA:Saitoetal], the downlink system performance throughput gains are evaluated by incorporating a complete simulation of an LTE cellular system (3GPP). Kim et. al. [@NOMABF:KKSK] developed an optimization problem that finds the power allocation coefficients for a broadcast MIMO NOMA system with $N$ base-station antennas serving $2N$ simultaneous users, where user-pairing is based on clustering two users with similar channel vector directions. Choi [@BFMIMONOMA:Choi] extends this work for a base-station with $L$ antennas and $K$ user-pairs ($L\geq2K$) by creating a two-stage beamforming approach with NOMA, such that closed-form solutions to the power allocation coefficients are found.
More recently, Sun et. al. explored the MIMO NOMA system ergodic capacity [@ErgMIMONOMA:SHIP] when the base-station has only statistical CSIT. Given that a near user has a larger expected channel gain than the cell-edge user, properties of the power allocation coefficients are derived, and a suboptimal algorithm to solving for the power allocation coefficients is proposed which maximizes the ergodic capacity.
Fairness in NOMA systems is addressed in some works. The uplink case in OFDMA systems is addressed in [@5GNOMAUp:AXIT] by using an algorithm that attempts to maximize the sum throughput, with respect to OFDMA and power constraints. The fairness is not directly addressed in the problem formulation, but is evaluated using Jain’s fairness index. In [@PropFairNOMA:LMP], a proportional fair scheduler and user pair power allocation scheme is used to achieve fairness in time and rate. In [@FairnessNOMA5G:TK], fairness is achieved in the max-min sense, where users are paired such that their channel conditions are not too disparate, while the power allocation maximizes the rates for the paired users.
Ding et. al. [@5GNOMA:DFP] provide an analysis for fixed-power NOMA (F-NOMA) and cognative radio NOMA (CR-NOMA). In F-NOMA, with a cell that has $N$ total users, it is shown that the probability that NOMA outperforms OMA asymptotically approaches $1$. In CR-NOMA, a primary user is allowed all of the time and bandwidth, unless an opportunistic secondary user exists with a stronger channel condition relative to the primary user, such that transmitting both of their signals will not reduce the primary user’s SINR below some given threshold. It is shown that the diversity order of the $n$-th user is equal to the order of the weaker $m$-th user, leading to the conclusion that this approach benefits from pairing the two users with the strongest channels.
The main contribution of this work is to demonstrate that NOMA capacity can fundamentally always outperform OMA capacity for each user, regardless of the channel conditions of the users, and to derive exactly what the power allocation should be for each user to achieve this, based on their channel gains. Furthermore, the expected sum-rate capacity gain made when the users are paired with i.i.d. random channels is 1 bps/Hz at high SNR, even in the extreme case when all of the transmit power is allocated to the stronger user. Furthermore, the outage probabilities are also derived for this case, and shown to decrease for each user, but significantly for the weaker user. The approximate sum-rate capacity gain for the case of pairing the strongest and weakest users in the cell is derived for the high SNR regime, and compared to user-pairing with fixed-power approaches. Lastly, the more general case of $K$ user SISO NOMA is considered, and it is fundamentally proven that a NOMA power allocation strategy always exists that achieves equal or greater capacity per user when compared to OMA.
System Model and Capacity {#sec:system}
=========================
Let a mobile user $i$ have a signal $x_i$ transmitted from a single antenna base-station (BS). The channel gain is $h_i\in\mathbb{C}$ with SNR gain p.d.f. $f_{|h|^2}(w) = \frac{1}{\beta}e^{-\frac{w}{\beta}}$, and receiver noise is complex-normal distributed $z_i\sim\mathcal{CN}(0,1)$. In the two-user case, if user-1 and user-2 transmit their signals half of period $T$ utilizing the entire transmit power $\xi$, then the received signal for each user is $y_i = h_i \sqrt{\xi} x_i + z_i, i=1,2$. If $\mathbb{E}[ |x_i|^2 ] = 1$, the information capacity of each user is $C_i^\text{O} = \frac{1}{2}\log_2\left(1 + \xi |h_i|^2\right).$ The sum-rate capacity for OMA is therefore $S_\text{O} = C_1^\text{O}+C_2^\text{O}$. For the case of NOMA, it is assumed that user-2 channel gain is the larger one ($|h_2|^2 > |h_1|^2$), then user-2 can perform SIC at the receiver by treating its own signal as noise and decoding user-1’s signal first. If the power allocation coefficient for user-2 is $a\in(0,1/2)$, then user-1’s signal is allocated $1-a$ transmit power, and the received signals for both users are $$\begin{aligned}
y_i &= \sqrt{(1-a)\xi }h_i x_1 + \sqrt{a\xi }h_i x_2 + z_i, i=1,2.$$ Since $|h_2|^2>|h_1|^2$, it follows that $\frac{(1-a)\xi |h_2|^2}{a\xi |h_2|^2+1} > \frac{(1-a)\xi |h_1|^2}{a\xi |h_1|^2+1},$ which allows user-2’s receiver to perform SIC and remove the interference from user-1’s signal. Hence, the capacity for each user is $$\begin{aligned}
&C_1^\text{N}(a) = \log_2\left( 1 + \dfrac{(1-a)\xi |h_1|^2}{a\xi |h_1|^2 + 1}\right), \\
&C_2^\text{N}(a) = \log_2\left( 1 + a\xi |h_2|^2\right).\end{aligned}$$ The sum-rate capacity for NOMA is therefore $S_\text{N}(a) = C_1^\text{N}(a)+C_2^\text{N}(a)$. These capacity expressions are used in each case of OMA and NOMA to find the values of $a$ that make NOMA “fair.”
Fair-NOMA Power Allocation Region {#sec:fairNOMA}
=================================
In order for user-1 NOMA capacity to be greater than or equal to OMA capacity, it must be true that $C_1^\text{N}(a)\geq C_1^\text{O}$. Solving this inequality for $a$ gives $a \leq \frac{\sqrt{1 + \xi |h_1|^2} - 1}{\xi |h_1|^2}$. Similarly, for user-2 when $C_2^\text{N}(a)\geq C_2^\text{O}$ results in $a \geq \frac{\sqrt{1 + \xi |h_2|^2} - 1}{\xi |h_2|^2}$. Both the upper and lower bounds on the transmit power fraction $a$ to achieve better sum and individual capacities have the form $a(x) = (\sqrt{1 + \xi x}-1)/(\xi x)$.
Define $$\begin{array}{ccc} a_{\inf} = \dfrac{\sqrt{1 + \xi |h_2|^2} - 1}{ \xi |h_2|^2}$ and $a_{\sup} = \dfrac{\sqrt{1 + \xi |h_1|^2} - 1}{\xi |h_1|^2}. \end{array}$$ Then by Property 1 in [@FairNOMAInfocom2016], it is clear that if $|h_2|^2>|h_1|^2 \Rightarrow a_\text{sup} > a_\text{inf}$. The Fair-NOMA power allocation region is therefore defined as $\mathcal{A}_\text{FN}=[a_\text{inf}, a_\text{sup}]$, and selecting any $a\in\mathcal{A}_\text{FN}$ gives $ C_1^\text{N}(a)\geq C_1^\text{O},$ $C_2^\text{N}(a)\geq C_2^\text{O},$ and $S_\text{N}(a) > S_\text{O}$. Since the sum-rate capacity $S_\text{N}(a)$ is a monotonically increasing function of $a$, then $a_\text{sup} = \arg{\displaystyle\max_{a\in\mathcal{A}_\text{FN}}}(C_2^\text{N}(a))$ also maximizes $S_\text{N}(a)$. The sum-rate capacity of NOMA is strictly larger than the sum-rate capacity of OMA because at the least one of the user’s capacities always increases.
\[thm:thm1\] For a two-user NOMA system that allocates power fraction $1-a$ to user-1 and $a$ to user-2, such that $a\in\mathcal{A}_\text{FN}$, the sum-rate $S_\text{N}(a)$ is a monotonically increasing function of both $|h_1|^2$ and $|h_2|^2$.
See appendix \[proof:thm1\].
This result implies that as the channel gain for the weaker user increases, the total capacity increases while the power allocation to the stronger user decreases. This means that, as the channel gain $|h_1|^2$ increases towards the value of $|h_2|^2$, then the capacity gain by user-1 is greater than the capacity loss by user-2. In the extreme case where $ |h_1|^2 \rightarrow |h_2|^2$, then $a_\text{sup}\rightarrow a_\text{inf}$, and both $C_1^\text{N}(a)$ and $C_2^\text{N}(a)\rightarrow C_2^\text{O}$. In other words, the Fair-NOMA capacity is upper bounded by the capacity obtained by allocating all of the transmit power to the stronger user. This is somewhat related to the multiuser diversity concept result in [@knopp95] for OMA systems, which suggests allocating all the transmit power to the stronger users will increase the overall capacity of the network.
In contrast, with the increase in $|h_2|^2$, $C_2^\text{N}(a_\text{sup})$ increases and hence the capacity gains from Fair-NOMA increase. Therefore with Fair-NOMA, as is the same with the previously obtained result for fixed-power allocation NOMA, when $|h_2|^2 - |h_1|^2$ increases, $S_\text{N}(a)-S_\text{O}$ also increases [@InfTh:CT; @5GNOMA:DFP]. This will be further exemplified in Section \[sec:fairNOMAMUD\], Theorem \[thm:NOMAminmax\].
Analysis of Fair-NOMA Capacity {#sec:analysis}
==============================
Expected Value of Fair-NOMA Capacity
------------------------------------
The expected value of the Fair-NOMA capacities of the two users depend on the power allocation coefficient $a$. In order to determine the bounds of this region, the expected value of capacity of each user is derived for the cases of $a_\text{inf}$ and $a_\text{sup}$ and compared with that of OMA.
Since the channels of the two users are i.i.d. random variables, let the two users selected have channel SNR gains of $|h_i|^2$ and $|h_j|^2$, where $f_{|h|^2}(x)=\frac{1}{\beta}e^{-\frac{x}{\beta}}$. Since we call the user with weaker (stronger) channel gain user-$1$ (user-$2$), then $|h_1|=\min\{|h_i|^2, |h_j|^2\}$ and $|h_2|^2=\max\{|h_i|^2, |h_j|^2\}$. Therefore, the joint pdf of $|h_1|^2$ and $|h_2|^2$ is $$f_{|h_1|^2, |h_2|^2}(x_1,x_2) = \frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}.$$ It is shown [@FairNOMAInfocom2016] that the ergodic capacities and the sum-rate of users in OMA are $$\begin{aligned}
&\mathbb{E}[ C_1^\text{O} ] = \frac{e^{\frac{2}{\beta\xi}}}{\ln(4)} E_1\left(\frac{2}{\beta\xi}\right),\\
&\mathbb{E}[ C_2^\text{O} ] = \frac{e^{\frac{1}{\beta\xi}}}{\ln(2)}E_1\left(\frac{1}{\beta\xi}\right) - \frac{e^{\frac{2}{\beta\xi}}}{\ln(4)} E_1\left(\frac{2}{\beta\xi}\right),\\
&\mathbb{E}[S_{\text{O}} ] = \frac{e^{\frac{1}{\beta\xi}}}{\ln(2)}E_1\left(\frac{1}{\beta\xi}\right)\end{aligned}$$ where $E_1(x) =\int_x^\infty u^{-1}e^{-u} du$ is the well-known exponential integral. Note that $\mathbb{E}[ C_1^\text{O} ]=\mathbb{E}[ C_1^\text{N}(a_\text{sup}) ]$ and $\mathbb{E}[ C_2^\text{O} ] = \mathbb{E}[ C_2^\text{N}(a_\text{inf}) ]$.
It is also shown [@FairNOMAInfocom2016] that $$\begin{aligned}
\label{eq:C1ainf} &\mathbb{E}\left[ C_1^\text{N}(a_\text{inf}) \right] = \frac{3e^{\frac{2}{\beta\xi}}}{\ln(4)}E_1\left(\frac{2}{\beta\xi}\right) \\
&- \int_0^\infty \frac{2}{\beta\ln(2)} \cdot \exp\left(-\frac{x}{\beta}\left( \frac{\sqrt{1+\xi x}-2}{\sqrt{1+\xi x}-1}\right)\right)\nonumber\\
& \times \left( E_1\left( \frac{x}{\beta(\sqrt{1+\xi x}-1)} \right) - E_1\left(\frac{x\sqrt{1+\xi x}}{\beta(\sqrt{1+\xi x}-1)} \right) \right) dx, \nonumber\end{aligned}$$ and $$\begin{aligned}
\label{eq:C2asup}
&\mathbb{E}[ C_2^\text{N}(a_\text{sup}) ] = \frac{e^{\frac{2}{\beta\xi}}}{\ln(4)} E_1\left(\frac{2}{\beta\xi}\right) + \int_0^\infty \frac{2}{\beta\ln(2)} \\
&\times \exp\left(-\frac{x}{\beta}\left(\frac{\sqrt{1+\xi x} -2}{\sqrt{1+\xi x} -1}\right)\right) E_1\left( \frac{x\sqrt{1+\xi x}}{\beta(\sqrt{1+\xi x} - 1)} \right)dx.\nonumber\end{aligned}$$
At high SNR ($\xi\gg 1$), the approximate capacities are $$\begin{aligned}
& C_i^\text{O} \approx \frac{1}{2}\log_2(\xi|h_i|^2), \\
& C_1^\text{N}(a_\text{inf}) \approx \frac{1}{2}\log_2(\xi|h_2|^2), \\
& C_2^\text{N}(a_\text{sup})\approx \log_2\left(\sqrt{\frac{\xi}{|h_1|^2}}|h_2|^2\right).\end{aligned}$$ This implies that when $\xi\gg 1$, $C_1^\text{N}(a_\text{sup}) \approx C_2^\text{O}$. The high SNR approximations lead to following result for the difference in the expected capacity gains, i.e., $\Delta S(a) = S_\text{N}(a)-S_\text{O}$.
\[thm:DiffCap1\] In a two-user SISO system with $|h_i|^2\sim\mathrm{Exponential}(\frac{1}{\beta})$ and at high SNR regime, the increase in sum capacity is $\mathbb{E}[\Delta S(a)]\approx 1$ bps/Hz, $\forall a\in\mathcal{A}_\text{FN}$.
See appendix \[proof:DiffCap1\].
This interesting result means that when the transmit power approaches infinity, the average increase in sum capacity is the same for both $a_\text{inf}$ and $a_\text{sup}$ and is equal to $1\text{ bps/Hz}$. Equivalently, it means that $\forall a\in\mathcal{A}_\text{FN}$, both users experience an expected increase in capacity over OMA of $c$ and $1-c$ where $c\in[0,1]$.
Comparison of Theoretical and Simulations Results
-------------------------------------------------
Fair-NOMA theoretical results are compared with simulation when $\beta=1$. In figure \[fig:oma\], the capacity of NOMA is compared with that of OMA for both users, including the high SNR approximations. As can be seen, the theoretical derivations match the simulation results. The performances of $C_1^\text{N}(a_\text{inf})$ and $C_2^\text{N}(a_\text{sup})$ are plotted. The simulation of $\mathbb{E}[C_1^\text{N}(a_\text{inf})]$ matches the theoretical result in equation (\[eq:C1ainf\]), and the simulation of $\mathbb{E}[C_2^\text{N}(a_\text{sup})]$ matches the theoretical result in equation (\[eq:C2asup\]). The high SNR approximations show to be very close for values of $\xi>25$ dB. Since $C_2^\text{O}=C_2^\text{N}(a_\text{inf})$ and $C_1^\text{O}=C_1^\text{N}(a_\text{sup})$, it is apparent from the plots that the gain in performance is always approximately $1\text{ bps/Hz}$ for one of the users and also the sum capacity when using Fair-NOMA [@FairNOMAInfocom2016].
![\[fig:oma\]Comparing the capacity of NOMA and OMA](NOMA_2user_nomaVSoma_theory_compareALL2-eps-converted-to.pdf){width="\columnwidth"}
Outage Probability of Fair-NOMA {#sec:outage}
===============================
Suppose that the minimum rate that is allowed by the system to transmit a signal is $R_0$. The probability that a user cannot achieve this rate with any coding scheme is given by $\mathrm{Pr}\{C < R_0\}$. As with the average capacity analysis, the outage performance of NOMA is analyzed by looking at $a_\text{inf}$ and $a_\text{sup}$, and then draw logical conclusions from that.
The outage probability of user-1 using OMA is given by $$\begin{aligned}
p_{\text{O},1}^\text{out}&=\mathrm{Pr}\left\{\log_2(1+\xi|h_1|^2)^{1/2} < R_0\right\}\\ &= \int_0^{\frac{4^{R_0}-1}{\xi}}\int_{x_1}^\infty \frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}dx_2dx_1 \\
&= 1-\exp\left(-\tfrac{2(4^{R_0}-1)}{\beta\xi}\right)\nonumber.\end{aligned}$$ For user-2 using OMA, the outage probability is given by $$\begin{aligned}
p_{\text{O},2}^\text{out}=&\mathrm{Pr}\left\{\log_2(1+\xi|h_2|^2)^{1/2} < R_0\right\}\\
=& \int_0^{\frac{4^{R_0}-1}{\xi}}\int_{x_1}^{\frac{4^{R_0}-1}{\xi}} \frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}dx_2dx_1, \\
=& 1 + \exp\left(-\tfrac{2(4^{R_0}-1)}{\beta\xi}\right) - 2\exp\left(-\tfrac{4^{R_0}-1}{\beta\xi}\right).\nonumber\end{aligned}$$
Denote the NOMA outage probability for user-$i$ as $p_{\text{N},i}^\text{out}(a)$ such that $p_{\text{N},i}^\text{out}(a) = \mathrm{Pr}\{C_i^\text{N}(a) < R_0 \}$ for $i=1,2$. It should be obvious that $p_{\text{N},1}^\text{out}(a_\text{sup})=p_{\text{O},1}^\text{out}$ and $p_{\text{N},2}^\text{out}(a_\text{inf})=p_{\text{O},2}^\text{out}$. The outage probabilities $p_{\text{N},1}^\text{out}(a_\text{inf})$ and $p_{\text{N},2}^\text{out}(a_\text{sup})$ are provided in the following property.
\[prop:outage\] Outage Probabilities $p_{\text{N},1}^\text{out}(a_\text{inf})$ and $p_{\text{N},2}^\text{out}(a_\text{sup})$:
(a) \[item:outage1\] The outage probability for user-1 at $a=a_\text{inf}$ is given by $$\begin{aligned}
\label{eq:NOMAoutMU1} p_{\text{N},1}^\text{out}(a_\text{inf}) = 1 + e^{-\frac{\alpha_2}{\beta}} - \frac{2}{\beta}\int_{\alpha_2}^\infty e^{-\frac{x(\alpha_1+1)}{\beta}}dx,
\end{aligned}$$ where $\alpha_1$ and $\alpha_2$ are defined as $$\begin{aligned}
&\alpha_1= \tfrac{2^{R_0}-1}{\xi x + 2^{R_0}(1-\sqrt{1+\xi x}) }, \\
&\alpha_2= \tfrac{4^{R_0}-2}{2\xi} + \sqrt{ \tfrac{4^{R_0}-1}{\xi^2}+\tfrac{(4^{R_0}-2)^2}{4\xi^2} }.
\end{aligned}$$
(b) \[item:outage2\] The outage probability for user-2 at $a=a_\text{sup}$ is given by
$$\begin{aligned}
\hspace{-0.2in} p_{\text{N},2}^\text{out}(a_\text{sup}) = 1 + e^{-\frac{2(4^{R_0}-1)}{\beta\xi}} - 2e^{-\frac{2(2^{R_0}-1)}{\beta\xi}} + (2^{R_0}-1)\nonumber\\
e^{\frac{(2^{R_0}-3)^2}{4\beta\xi} } \cdot\sqrt{\tfrac{\pi}{\beta\xi}}\left[\mathrm{erfc}\left(\tfrac{2^{R_0}+1}{2\sqrt{\beta\xi}}\right) - \mathrm{erfc}\left(\tfrac{3(2^{R_0})-1}{2\sqrt{\beta\xi}}\right)\right].
\label{eq:outage2}
\end{aligned}$$
See appendix \[proof:Outage\_Probabilities\].
There is no closed form solution for the integral in $p_{\text{N},1}^\text{out}(a_\text{inf})$, however it can be easily computed by a computer.
Figure \[fig:nomaout\] plots the outage probabilites of OMA and NOMA for different values of $a$ and for $R_0=2$ bps/Hz. The probability of user-1 experiencing an outage is clearly greater than for user-2. However, the reduction of the outage probability for user-1 using $a=a_\text{inf}$ becomes significant as $\xi$ increases, to the effect of nearly 1 order of magnitude drop-off when $\xi$ is really large. The outage probability reduction for user-2 is not as significant as the improvement made by user-1. However, when $a=a_\text{sup}$, the same outage probability can be obtained using NOMA with $\xi$ approximately 2 dB less than is required when using OMA. Thus, even when the power allocation coefficient $a$ is restricted to being in $\mathcal{A}_\text{FN}$, the probability of users to be able to achieve their minimum service requirement rates $R_0$ is improved, and especially improved for the weaker channel gain. Even when the power allocation coefficient $a=(a_\text{inf}+a_\text{sup})/2$, the outage probabilities of both users improves significantly when using NOMA.
![\[fig:nomaout\]Outage probabilities of NOMA and OMA as functions of $\xi$.](NOMA_outage-eps-converted-to.pdf){width="\columnwidth"}
Fair-NOMA in Opportunistic User-Pairing {#sec:fairNOMAMUD}
=======================================
It has been suggested in [@5GNOMA:DFP] that the best NOMA performance is obtained when user channel conditions are most disparate, i.e. pairing the user with the weakest channel condition and the user with the strongest channel condition together. However, it is not known what the expected capacity gap is in this case, particularly for the case when both users are allocated power such that they both always outperform their OMA performance. Since the power allocation scheme where NOMA outperforms OMA with probability of 1 has been defined, regardless of number of users, this approach can also be applied here.
Suppose there exists a set of $K$ mobile users in a cell, and two of these users can be scheduled during the same transmission period. It is of particular interest to select the users that have the largest difference in channel SNR gain. If the channel SNR gains of the users are i.i.d. $\mathrm{Exponential}(\frac{1}{\beta})$, and the two selected users have the minimum and maximum channel SNR gains, how much of an improvement in the sum-rate capacity will be observed by using NOMA versus OMA?
Analysis of Fair-NOMA with Opportunistic User-Pairing
-----------------------------------------------------
Let $|h_0|^2 = \min(|h_1|^2, \ldots,|h_K|^2)$ and $|h_M|^2 = \max(|h_1|^2, \ldots,|h_K|^2)$. In order to compute the expected sum-rate capacity, the joint CDF $F_{|h_0|^2,|h_M|^2}(x_0, x_M)$ and PDF of $f_{|h_0|^2,|h_M|^2}(x_0,x_M)$ are needed. It is easily shown that $$\begin{aligned}
&\mathrm{Pr}\{|h_M|^2<x_M\} = \mathrm{Pr}\{|h_0|^2<x_0, |h_M|^2<x_M \} \nonumber\\
& + \mathrm{Pr}\{|h_0|^2>x_0, |h_M|^2<x_M \},\\
\Rightarrow &F_{|h_0|^2,|h_M|^2}(x_0, x_M) = \mathrm{Pr}\{|h_0|^2<x_0, |h_M|^2<x_M \} \\
\label{eq:JointProb2}&= \mathrm{Pr}\{|h_M|^2<x_M\} - \mathrm{Pr}\{|h_0|^2>x_0, |h_M|^2<x_M \}.\end{aligned}$$ The first term on the right in equation (\[eq:JointProb2\]) is the CDF of the maximum of $K$ i.i.d. exponential random variables, which is given by $$\mathrm{Pr}\{|h_M|^2<x_M\} = (1-e^{-\frac{x_M}{\beta}})^K.$$ The second term can be easily computed. $$\begin{aligned}
\mathrm{Pr}\{|h_0|^2>x_0, |h_M|^2<x_M \} =& \int_{x_0}^{x_M}\hspace{-2mm}\cdots\int_{x_0}^{x_M} \prod_{k=1}^{K}\frac{e^{-\frac{x_k}{\beta}}}{\beta}dx_k \\
=& (e^{-\frac{x_0}{\beta}} - e^{-\frac{x_M}{\beta}})^K\end{aligned}$$ Therefore, the joint CDF is given by $$\begin{aligned}
F_{|h_0|^2,|h_M|^2}(x_0, x_M) =& (1 - e^{-\frac{x_M}{\beta}})^K-(e^{-\frac{x_0}{\beta}} - e^{-\frac{x_M}{\beta}})^K,\end{aligned}$$ and the joint PDF is $$\begin{aligned}
f_{|h_0|^2,|h_M|^2}&(x_0, x_M) \\
=&\frac{K(K-1)}{\beta^2}e^{-\frac{x_0+x_M}{\beta}}(e^{-\frac{x_0}{\beta}}-e^{-\frac{x_M}{\beta}})^{K-2}.\nonumber\end{aligned}$$ The following theorem provides the sum-rate capacity increase of NOMA when $\xi|h_0|^2\gg 1$.
\[thm:NOMAminmax\] Let $\{|h_1|^2, \ldots,|h_K|^2\}$ be the i.i.d. SISO channel SNR gains of $K$ users, such that the two users selected for transmission together have the minimum and maximum channel SNR gains. When $\xi\ |h_0|^2 \gg 1$, the sum-rate capacity increase from OMA to NOMA for $a = a_\text{sup}$ is $$\label{eq:NOMAMUDDiff}
\mathbb{E}[\Delta S(a_\text{sup})] \approx \frac{1}{2}\log_2(K) + \frac{1}{2}\sum_{m=2}^K\binom{K}{m}(-1)^{m} \log_2 (m) .$$
See appendix \[proof:NOMAminmax\].
This result is similar to the result obtained for the 2-by-2 MIMO case in Lemma 2, equation 33 in [@MIMONOMA:DAP], except a fixed-power allocation approach was used there, whereas the result above uses a Fair-NOMA power allocation approach. Although the expected capacity gap $\mathbb{E}[\Delta S(a)]$ increases when selecting $a=a_\text{sup}$, caution should be used when utilizing the fixed-power approach to not set $a$ too close to the value of $a_\text{inf}$. An approximation of the capacity gap using $\mathbb{E}[\Delta S(a_\text{inf})]$ for large $\xi$ and $K$ is given as $$\begin{aligned}
&\mathbb{E}[\Delta S(a_\text{inf})]\approx \frac{e^{\frac{K}{\beta\xi}}}{\ln(4)}E_1\left(\frac{K}{\beta\xi}\right)\\
&- \log_2\left(1+\frac{\sqrt{1+\xi(\psi(K+1)+\gamma)}-1)}{K(\psi(K+1)+\gamma)}\right).
\end{aligned}$$ where $\psi(w)=\Gamma'(w)/\Gamma(w)$ is the digamma function, $\Gamma(w)=\int_0^\infty u^{w-1}e^{-u}du$ is the gamma function, and $\gamma=-\int_0^\infty e^{-u}\ln(u)du$ is the Euler-Mascheroni constant. It can be seen in figure \[fig:NOMAmud\_deltaS\] that as the number of users increases, the expected capacity gap actually decreases. Therefore, even for fixed-power allocation approaches to NOMA, $a$ should be selected to be greater than $a_\text{inf}$ for the case of pairing minimum and maximum channel gain users.
This result shows that the sum-rate capacity difference increases as a function of $K$. However, this increase is slow. Nonetheless, there is a fundamental limit to the amount the capacity can increase when using Fair-NOMA, while maintaining the capacity of the weaker user equal to the capacity using OMA.
It is important to note that as the number of mobile users becomes very large, while pairing the strongest and weakest users together will give us the greatest increase in sum-rate capacity, it does not maximize sum-rate capacity itself. This can be seen from theorem \[thm:thm1\], which states that the sum-rate capacity actually increases as the channel gain of the weaker user monotonically increases. A practical way of viewing this issue is that, as the number of users $K$ increases, the weakest user has channel gain that in probability is too weak to achieve the quality of service threshold rate $R_0$.Should no outage rate be specified, the weaker user achieves such a low capacity, that the stronger user contributes most of the capacity, while using nearly half the transmit power, according to Property 1 from [@FairNOMAInfocom2016]. Hence, a little more than half of the transmit power is nearly wasted.
Comparing Simulation Results with Analysis
------------------------------------------
For the simulation results, the performance of Fair-NOMA combined with opportunistic user-pairing is compared to the performance of OMA and fixed-power NOMA. The simulations are run for different values of $\xi$ and $K$. For fixed-power NOMA, the power allocation coefficient is a constant value of $a=\frac{1}{5}$, such that the weaker user is allocated $\frac{4}{5}$ of the transmit power.
Figure \[fig:NOMAmud\_capacity\] shows the average capacities of both the weakest and strongest users versus $K$ and for each case of $a=a_\text{inf}$ and $a_\text{sup}$. The capacity of the stronger user is shown to exhibit the effects of multiuser diversity, since not only does its channel gain grow as $K$ increases, but also the power allocated also increases when $a=a_\text{sup}$, thus providing the increase in capacity predicted in equation (\[eq:NOMAMUDDiff\]). In the case of $a=a_\text{inf}$ the capacity is initially shown to increase as $K$ increases, due to $a_\text{inf}$ decreasing with $|h_M|^2$ according to Property 1 from [@FairNOMAInfocom2016]. However, as $K$ continues to increase, the weakest users capacity eventually begins to decrease due to its channel gain being the minimum of a large number of users, and thus this term begins to dominate the capacity behavior.
![\[fig:NOMAmud\_capacity\] Ergodic capacity with opportunistic user-pairing, $\xi=50$ dB](NOMA_minmax-eps-converted-to.pdf){width="\columnwidth"}
The sum-rate capacity for Fair-NOMA with $a=a_\text{sup}$, fixed-power NOMA with $a=\frac{1}{5}$, and OMA are shown in Fig. \[fig:NOMAmud\_sumcompare\]. As expected, the sum-rate capacity for each user at lower values of $\xi$ performs best when applying Fair-NOMA when compared to fixed-power NOMA. This is because Fair-NOMA always guarantees a capacity increase, i.e. with probability 1, while fixed-power NOMA only achieves higher capacity with probability as given in [@5GNOMA:DFP]. However, as $\xi$ increases, both capacities of Fair-NOMA and fixed-power NOMA approach the same value asymptotically. This agrees with the result obtained that at high SNR, the capacity gain should reach a limit when $\xi\rightarrow\infty$, no matter how much extra power is allocated to the stronger user.
![\[fig:NOMAmud\_sumcompare\] Comparison of Fair-NOMA, fixed-power NOMA, and OMA](NOMAmud_fix_compare-eps-converted-to.pdf){width="\columnwidth"}
Equation (\[eq:NOMAMUDDiff\]) shows that the capacity gain made by pairing the nearest and furthest cell-edge users is slow in $K$, and is due to the combined gain in capacity achieved by the strongest user and loss in capacity by the weakest user. This makes sense from multiple points of view. The expected value of power allocation coefficient $a_\text{sup}\rightarrow\frac{1}{2}$ when $K$ is large, due to the selection of the user with the weakest channel gain. In other words, as $K$ increases, the weakest user needs less power in NOMA to achieve the same capacity as it can using OMA. Hence, more power goes to the stronger user. Figure \[fig:NOMAmud\_deltaS\] plots the simulation of $\mathbb{E}[\Delta S(a_\text{sup})]$ for $\xi=50$ dB, and the approximation given by (\[eq:NOMAMUDDiff\]). Notice that the simulation and approximation seem to slightly diverge as the number of users increases. This is because the approximation in (\[eq:NOMAMUDDiff\]) needs a sufficiently large value of $\xi$ as the number of users increases for the simulation and approximation to become tighter. However, $\xi=50$ dB was used because it is a large but still realistic value of $\xi$. Since the number of users $K$ cannot become arbitrarily large, the approximation remains tight for realistic values of $\xi$ and $K$.
![\[fig:NOMAmud\_deltaS\] Difference in ergodic capacity with opportunistic user-pairing; $\xi=50$ dB](NOMA_minmax_sumrateDiff3-eps-converted-to.pdf){width="\columnwidth"}
Multi-user NOMA in SISO Systems {#sec:MU-NOMA}
===============================
So far, the treatment of Fair-NOMA has focused on the two-user case. Consider an OMA system, where $K$ users have their information transmitted over $K$ orthogonal time slots (or frequency bands) during a total time of $T$ (and bandwidth $B$). For each user $k$, the capacity of user $k$ is given by $$C_k^\text{O} = \frac{1}{K}\log_2(1 + \xi|h_k|^2), \forall k=1,\ldots, K.$$ When applying NOMA to this system, the information of each user occupies the entire time $T$ (bandwidth $B$) simultaneously. Hence, a superposition coding strategy must be used, in which all $K$ users must share the total transmit power $\xi$. User $k$ must perform SIC of each message that is intended for the other users $l$ that have weaker channel conditions than user $k$. The channel gains are ordered as $|h_1|^2<|h_2|^2<\cdots<|h_K|^2$. Lets define the power allocation coefficients $\{b_1, \ldots, b_K\}$, where $b_k$ is the power allocation coefficient for user $k$ and $$\label{eq:munomapower}
\sum_{k=1}^K b_k \leq 1.$$ Therefore, the capacity of user $k$ for $1 \leq k \leq K$ is given by $$C_k^\text{N}(b_1, \ldots, b_K) = \log_2\left(1 + \frac{b_k\xi|h_k|^2}{1 + \xi|h_k|^2\sum_{l=k+1}^K b_l } \right).$$ In order for $C_k^\text{N}(b_1, \ldots, b_K) > C_k^\text{O}$, the inequality must be solved for $b_k$ assuming that equation (\[eq:munomapower\]) is true. Since user $K$ does not receive any interference power after decoding all of the other users’ messages, solving for $b_K$ is straight forward. $$\begin{aligned}
C_K^\text{O} \leq C_K^\text{N}(b_1, \ldots, b_K) \Rightarrow b_K \geq \frac{(1+\xi|h_K|^2)^\frac{1}{K}-1}{\xi|h_K|^2}.\end{aligned}$$ For users $k=1,\ldots, K-1$, the power allocation for each user is conditioned on $C_k^\text{O} \leq C_k^\text{N}(b_1, \ldots, b_K)$ which results in $$\begin{aligned}
b_k \geq \dfrac{[(1+\xi|h_k|^2)^\frac{1}{K}-1]\left(1+ \xi|h_k|^2\sum_{l=k+1}^K b_l \right)}{\xi|h_k|^2}.\end{aligned}$$ As expected, the power allocation of the users with weaker channel gains depend on the power allocation of the users with stronger channel gains.
Notice that in the above derivation, the total power allocation was not necessarily used. Consider the case where $\sum_{k=1}^K a_k = 1$ and the case where user 1 capacity in OMA and NOMA are equal. Therefore, $C_1^\text{O} = C_1^\text{N}\Rightarrow$ $$\begin{aligned}
&\Rightarrow\log_2(1+\xi|h_1|^2)^\frac{1}{K} = \log_2\left(1+\frac{a_1\xi|h_1|^2}{1 + \xi|h_1|^2\sum_{l=2}^K a_l }\right) \nonumber \\
&\Rightarrow (1+\xi|h_1|^2)^\frac{1}{K} = \frac{1+\xi|h_1|^2}{1 + \xi|h_1|^2(1-a_1) }.
\label{eq:powerint1}\end{aligned}$$ Solving for $a_1$ gives $$\label{eq:powera1}
a_1 = \frac{1+\xi|h_1|^2 - (1+\xi|h_1|^2)^\frac{K-1}{K}}{\xi|h_1|^2}.$$ Note that both sides of equation (\[eq:powerint1\]) are greater than 1 which means $0<a_1<1, \forall \xi>0$. Define $A_1 = 1-a_1$ as the sum of the interference coefficients to user 1. Therefore, $$A_1 = \frac{(1+\xi|h_1|^2)^\frac{K-1}{K} -1}{\xi|h_1|^2},$$ and $0<A_1<1$. In general, the power allocation coefficient required for the NOMA capacity of user $k$ to equal the OMA capacity of user $k$ can be derived by solving the equation $$\begin{aligned}
&C_k^\text{O} = C_k^\text{N}(a_1, \ldots, a_K)\nonumber\\
\label{eq:powerboundK}\Rightarrow& (1+\xi|h_k|^2)^\frac{1}{K} = \frac{1 + A_{k-1}\xi|h_k|^2}{1 + (A_{k-1}-a_k)\xi|h_k|^2}, \end{aligned}$$ $\forall k\in\{2,\ldots,K\}, \xi>0$, where $A_{k-1} = 1-\sum_{l=1}^{k-1}a_l$. The following property for the set of power allocation coefficients $\{a_1,\ldots,a_K\}$ arises from solving equation (\[eq:powerboundK\]).
\[prop:multipower\] If the set of power allocation coefficients $\{a_1,\ldots,a_K\}$ are derived from equations (\[eq:powerint1\]) and (\[eq:powerboundK\]), then $$a_k\in(0, 1), \hspace{5mm}\text{ and }\hspace{5mm}\sum_{k=1}^K a_k \leq 1.$$
See appendix \[proof:multipower\].
This is an important property, because it sets the precedent for the existence of a set of NOMA power allocation coefficients that (i) achieves at least OMA capacity for every user in the current transmission time period, and (ii) allows for at least one user to have a capacity greater than OMA capacity.
The power allocation coefficient $a_k$ considers interference received from users with higher channel gains to be at a maximum. However, the coefficients $b_k$ consider the minimum power allocation. Note that in power allocation for multiuser Fair-NOMA using $a_k$ coefficients, the allocation process begins with user having weakest channel (first user) and allocate enough power to have a capacity of at least equal to OMA capacity for the first user. Then, the process continues with the next user until all the power is allocated amongst all users, i.e., the last user with strongest channel receives the remaining power allocation that results in higher capacity than OMA for that user. When $b_k$ coefficients are used for power allocations, the power allocation process begins with the user with strongest channel, $K^{th}$ user and assign enough power to achieve the same capacity as OMA for that user. The process then continues with the next user until the process reaches the first user. Therefore, it is clear that $$\begin{aligned}
&b_k < a_k\\
\text{and } & C_k^\text{N}(b_1,\ldots, b_K) < C_k^\text{N}(a_1,\ldots, a_K), \forall k.\end{aligned}$$ Hence, property \[prop:multipower\] highlights that there always exists a power allocation scheme in the general multiuser NOMA case that always achieves higher capacity than OMA, while keeping the total transmit power to $\xi$. This minimum power allocation requirement is demonstrated in figure \[fig:totalpower\]. The most interesting aspect of this result is that the same capacity of OMA can be achieved using Fair-NOMA with potentially much less total transmit power by using $b_k$ coefficients. This can be useful if the purpose of NOMA is to minimize the total transmit power in the network.
![\[fig:totalpower\] Minimum total power allocation in NOMA required to achieve capacity equal to OMA per user, $K=5$](totalpowernomamu-eps-converted-to.pdf){width="\columnwidth"}
Conclusion and Future Work {#sec:conclusion}
==========================
Fair-NOMA approach is introduced which allows two paired users to achieve capacity greater than or equal to the capacity with OMA. Given the power allocation set $\mathcal{A}_\text{FN}$ for this scheme, the ergodic capacity for the infimum and supremum of this set is derived for each user, and the expected asymptotic capacity gain is found to be 1 bps/Hz. The outage probability was also derived and it is shown that when $a=a_\text{inf}$, the outage performance of the weaker user significantly improves over OMA, where as the outage performance of the stronger user improves by at most roughly 2dB.
Fair-NOMA is applied to opportunistic user-pairing and the exact capacity gain is computed. The performance of Fair-NOMA is compared with a fixed-power NOMA approach to show that even when the power allocation coefficient $a=a_\text{sup}$ becomes less than the fixed-power allocation coefficient, the capacity gain is the same at high SNR, while Fair-NOMA clearly outperforms the fixed-power approach at low SNR.
The concept of Fair-NOMA can be extended to MIMO systems. In [@MIMONOMA:DAP], a similar result is found for the approximate expected capacity gap of a 2-user 2-by-2 MIMO NOMA system. In order to eliminate the existing possibility that NOMA does not outperform OMA in capacity for any user, the Fair-NOMA approach can be applied to users that are utilizing the same degree of freedom from the base-station. By ordering the composite channel gains, which include the transmit and receive beamforming applied to the channel, $K$ users on the same transmit beam can have their signals superpositioned, and then SIC can be done at their receivers to obtain their own signal with minimum interference. Receive beamforming is used to eliminate the interference from the transmit beams’ signals. The power allocation region can then be derived in the same manner as in Section \[sec:MU-NOMA\], and NOMA can then be used to either increase the capacity gap as is done in [@MIMONOMA:DAP], or to minimize the transmit power required to achieve the same capacity as in OMA, similar to what was done in Section \[sec:MU-NOMA\].
Finally, it is necessary to demonstrate a full system analysis and simulation of the impact of NOMA on bit-error rate (BER). It has been shown that the BER is very tightly approximated by the outage probability in [@MUDDIV:ZT], and therefore a tight approximation of the BER performance is given in this work. However, the transmission of signals using superposition coding and different information rates for the users are factors that impact the analysis of BER for the signal of the weaker user, especially because constellation sizes will most likely be different for the two superpositioned signals.
Proof of Theorem \[thm:thm1\] {#proof:thm1}
=============================
For the case when $a = a_\text{inf}$, the proof is trivial. Proving for the case when $a=a_\text{sup}$ then suffices to show it is true for all $a\in\mathcal{A}_\text{FN}$, because the $C_1^\text{N}(a)$ performance is lower-bounded by the case when $a=a_\text{sup}$, while for $C_2^\text{N}(a)$ the performance will only improve for $a>a_\text{inf}$. In order for $S$ to be monotonically increasing function of $|h_i|^2$, it must be shown that $dS/d|h_i|^2 > 0$, $\forall |h_i|^2$. In the case of $|h_2|^2$, $C_1^\text{N}(a_\text{sup})$ does not factor in, so $\frac{dS}{d|h_2|^2} = \frac{dC_2^\text{N}(a_\text{sup}) }{d|h_2|^2} = \frac{a_\text{sup}\xi}{a_\text{sup}\xi|h_2|^2} > 0,$ $\forall |h_2|^2$. The case of $|h_1|^2$ goes as follows. $$\begin{aligned}
&\frac{dS}{d|h_1|^2}
=\frac{1}{\ln 2}\left[\frac{\xi}{2(1+\xi|h_1|^2)}\right.\\
& + \left. \frac{\frac{\xi|h_2|^2}{2|h_1|^2\sqrt{1+\xi|h_1|^2}} + \frac{|h_2|^2}{|h_1|^4}(1-\sqrt{1+\xi|h_1|^2} )}{1+\frac{|h_2|^2}{|h_1|^2}(\sqrt{1+\xi|h_1|^2} - 1)}\right]\\
=& \frac{\left\{\begin{array}{l}\xi(|h_1|^4+|h_1|^2|h_2|^2[(1+\xi|h_1|^2)^{\frac{1}{2}}-1]\\ + 2|h_2|^2(1+\xi|h_1|^2) + \xi|h_1|^2|h_2|^2(1+\xi|h_1|^2)^{\frac{1}{2}}\\ - 2|h_2|^2(1+\xi|h_1|^2)^{\frac{3}{2}}\end{array} \right\} }{ 2|h_1|^2\ln(2)(1+\xi|h_1|^2)(|h_1|^2+|h_2|^2(\sqrt{1+\xi|h_1|^2}-1) ) }
\end{aligned}$$ The numerator above can be simplified as $$\begin{aligned}
& \xi|h_1|^4 + |h_2|^2 \\
&\times[ 2\xi|h_1|^2\sqrt{1+\xi|h_1|^2} + \xi|h_1|^2 + 2 - 2(1+\xi|h_1|^2)^\frac{3}{2}].
\end{aligned}$$ The value inside the square brackets can be simplified to $$\begin{aligned}
=& 2+\xi|h_1|^2 - 2\sqrt{1+\xi|h_1|^2}.
\end{aligned}$$ Since $2+\xi|h_1|^2 - 2\sqrt{1+\xi|h_1|^2} \geq 0$ because $\xi^2|h_1|^4 \geq 0$, then $$\begin{aligned}
\Rightarrow & \xi|h_1|^4+|h_2|^2(2+\xi|h_1|^2 - 2\sqrt{1+\xi|h_1|^2}) > 0\\
\Rightarrow&\frac{dS}{d|h_1|^2}> 0.
\end{aligned}$$ Since $|h_1|^2<|h_2|^2$, $\frac{dS}{d|h_1|^2}>0$, and $\frac{dS}{d|h_2|^2}>0$, then $S$ is a monotonically increasing function with respect to $|h_1|^2$ and $|h_2|^2$.
Proof of Theorem \[thm:DiffCap1\] {#proof:DiffCap1}
=================================
For $\xi\gg 1$, $$\begin{aligned}
\Delta C_1(a_\text{inf})\approx \Delta C_2(a_\text{sup}) \approx \tfrac{1}{2}\log_2(|h_2|^2)-\tfrac{1}{2}\log_2(|h_1|^2).
\end{aligned}$$The expected value of $\Delta C_1(a_\text{inf})$ and $\Delta C_2(a_\text{sup})$ is then $$\begin{aligned}
&\mathbb{E}\left[\frac{1}{2}\log_2(\frac{|h_2|^2}{|h_1|^2})\right] \approx \int_0^\infty \hspace{-2mm}\int_{0}^{x_2}\frac{1}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}\log_2(x_2)dx_1dx_2 \\
&- \int_0^\infty\hspace{-2mm}\int_{x_1}^\infty \frac{1}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}\log_2(x_1)dx_2dx_1\\
=& \int_0^\infty\frac{1}{\beta}e^{-\frac{x_2}{\beta}}\log_2(x_2)dx_2 - \int_0^\infty\frac{1}{\beta}e^{-\frac{2x_2}{\beta}}\log_2(x_2)dx_2 \\
&- \int_0^\infty\frac{1}{\beta}e^{-\frac{2x_1}{\beta}}\log_2(x_1)dx_1\\
\stackrel{(a)}{=}& \int_0^\infty\frac{1}{\beta}e^{-\frac{x_2}{\beta}}\log_2(x_2)dx_2
- 2\int_0^\infty\frac{1}{\beta}e^{-\frac{2x_2}{\beta}}\log_2(x_2)dx_2 \\
\stackrel{(b)}{=}& \int_0^\infty\frac{1}{\beta}e^{-\frac{x_2}{\beta}}\log_2(x_2)dx_2 - \int_0^\infty\frac{1}{\beta}e^{-\frac{x}{\beta}}\log_2(x)dx\\
& + \int_0^\infty \frac{1}{\beta}e^{-\frac{x}{\beta}}\log_2(2)dx
\stackrel{(c)}{=} 1,
\end{aligned}$$ where $(a)$ is true because the second two integrals are actually the same integral adding together, $(b)$ is true by making the substitution $x= 2x_2$. and $(c)$ is true because the first two integrals are the same integral subtracting each other. Since $S_\text{N}(a)$ is a monotonically increasing function of $a$, and $S_\text{N}(a_\text{inf})-S_\text{O} = \Delta C_1(a_\text{inf})$ and $S_\text{N}(a_\text{sup})-S_\text{O}=\Delta C_2(a_\text{sup})$, then when $\xi\gg 1$, $\Delta S = S_\text{N}(a)-S_\text{O}\approx 1, \forall a\in\mathcal{A}_\text{FN}$.
Proof of Outage Probability Results {#proof:Outage_Probabilities}
===================================
Proof of property \[prop:outage\](\[item:outage1\])
---------------------------------------------------
$p_{\text{N},1}^\text{out}(a_\text{inf}) $ $$\begin{aligned}
= & \mathrm{Pr}\left\{ \log_2\left(\tfrac{1+\xi|h_1|^2}{1+a_\text{inf}\xi|h_1|^2}\right) < R_0\right\}\\
\label{eq:p1ainf1} = & \mathrm{Pr}\left\{ |h_1|^2 < \tfrac{|h_2|^2(2^{R_0}-1)}{\xi|h_2|^2 + 2^{R_0}(1-\sqrt{1+\xi|h_2|^2})} \right\}
\end{aligned}$$ Since $|h_1|^2<|h_2|^2$, then there are two cases as $$\begin{aligned}
&|h_2|^2 \lessgtr \tfrac{|h_2|^2(2^{R_0}-1)}{\xi|h_2|^2 + 2^{R_0}(1-\sqrt{1+\xi|h_2|^2})} = \alpha_1.
\end{aligned}$$ Solving above for $|h_2|^2$ gives $$\begin{aligned}
\Longrightarrow& |h_2|^2 \lessgtr \tfrac{4^{R_0}-2}{2\xi}+\sqrt{\tfrac{4^{R_0}-1}{\xi^2} + \tfrac{(4^{R_0}-2)^2}{4\xi^2}}=\alpha_2.
\end{aligned}$$ This allows the event in equation (\[eq:p1ainf1\]) to be written as the two mutually exclusive events given in $$\begin{aligned}
&\left\{ |h_1|^2 < \alpha_1\right\} \\
&=\left\{ |h_1|^2<|h_2|^2, |h_2|^2< \alpha_2 \right\} \bigcup \left\{ |h_1|^2<\alpha_1, |h_2|^2 > \alpha_2 \right\}. \nonumber
\end{aligned}$$ The probability in equation (\[eq:p1ainf1\]) can then be written as $\mathrm{Pr}\{|h_1|^2<\alpha_1\}=\mathrm{Pr}\{ |h_1|^2<|h_2|^2, |h_2|^2< \alpha_2 \} + \mathrm{Pr}\{|h_1|^2<\alpha_1, |h_2|^2 > \alpha_2\}$. The first probability is equal to $$\begin{aligned}
&\mathrm{Pr}\{ |h_1|^2<|h_2|^2, |h_2|^2< \alpha_2 \}=\int_0^{\alpha_2}\int_0^{x_2}\frac{2}{\beta}e^{-\frac{x_1+x_2}{\beta}}dx_1dx_2 \nonumber\\
\label{eq:p1ainf2}&=1+e^{-\frac{2\alpha_2}{\beta}}-2e^{-\frac{\alpha_2}{\beta}}.
\end{aligned}$$ The second probability is found to be $$\begin{aligned}
& \mathrm{Pr}\{|h_1|^2<\alpha_1, |h_2|^2 > \alpha_2\}=\int_{\alpha_2}^\infty\int_0^{\alpha_1}\frac{2}{\beta}e^{-\frac{x_1+x_2}{\beta}}dx_1dx_2 \nonumber\\
\label{eq:p1ainf3}&= 2e^{-\frac{\alpha_2}{\beta}} -\frac{2}{\beta}\int_{\alpha_2}^\infty e^{-\frac{x_2+\alpha_1}{\beta}}dx_2,
\end{aligned}$$ where the integral in equation (\[eq:p1ainf3\]) has no known closed-form solution. Combining equations (\[eq:p1ainf2\]) and (\[eq:p1ainf3\]), gives $$\begin{aligned}
p_{\text{N},1}^\text{out}(a_\text{inf})=1+e^{-\frac{2\alpha_2}{\beta}}-\frac{2}{\beta}\int_{\alpha_2}^\infty e^{-\frac{x_2+\alpha_1}{\beta}}dx_2
\end{aligned}$$
Proof of property \[prop:outage\](\[item:outage2\])
---------------------------------------------------
$p_{\text{N},2}^\text{out}(a_\text{sup})$ $$\begin{aligned}
=& \mathrm{Pr}\left\{ \log_2(1+\tfrac{|h_2|^2}{|h_1|^2}(\sqrt{1+\xi|h_1|^2} -1)) < R_0 \right\}\\
\label{eq:pout2noma1}=& \mathrm{Pr}\left\{ |h_1|^2 > \tfrac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \tfrac{2|h_2|^2}{2^{R_0}-1} \right\}.
\end{aligned}$$ Since $0<|h_1|^2<|h_2|^2$ is always true, then the domain of $|h_2|^2$ that makes the statement $\frac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \frac{2|h_2|^2}{2^{R_0}-1}>0$ true or false must be found, and thus gives us two intervals for $|h_2|^2$. $$\begin{aligned}
&\tfrac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \tfrac{2|h_2|^2}{2^{R_0}-1} \lessgtr 0\\
\Longrightarrow &|h_2|^2 \lessgtr \tfrac{2(2^{R_0}-1)}{\xi}.
\end{aligned}$$ For the case of $|h_2|^2 < \frac{2(2^{R_0}-1)}{\xi}$, which gives $\frac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \frac{2|h_2|^2}{2^{R_0}-1} < 0$, the event is explicitly written as $$\begin{aligned}
\label{eq:outage1asup}\mathcal{A}_1^\text{out}=\left\{0<|h_1|^2<|h_2|^2, 0<|h_2|^2<\tfrac{2(2^{R_0}-1)}{\xi}\right\}.
\end{aligned}$$ For the case of $|h_2|^2 > \frac{2(2^{R_0}-1)}{\xi}$, the interval for $|h_1|^2$ is $\frac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \frac{2|h_2|^2}{2^{R_0}-1}< |h_1|^2<|h_2|^2$, so it must also be true that $|h_2|^2 > \frac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \frac{2|h_2|^2}{2^{R_0}-1}$. This gives $|h_2|^2 < \frac{(2^{R_0}-1)^2 + 2(2^{R_0}-1)}{\xi} = \frac{4^{R_0}-1}{\xi}$, and therefore the interval for this event is explicitly written as $$\begin{aligned}
\label{eq:outage2asup} \mathcal{A}_2^\text{out}=\left\{ \phantom{\frac{A}{B}}\right. \hspace{-2mm}& \tfrac{\xi|h_2|^4-2|h_2|^2(2^{R_0}-1)}{(2^{R_0}-1)^2}< |h_1|^2<|h_2|^2, &\\
&\left.\tfrac{2(2^{R_0}-1)}{\xi}<|h_2|^2< \frac{4^{R_0}-1}{\xi} \right\}&\nonumber
\end{aligned}$$ Now the probability above can be derived by computing the probabilities of the two disjoint regions as $$\begin{aligned}
&\mathrm{Pr}\left\{ |h_1|^2 > \tfrac{\xi|h_2|^4}{(2^{R_0}-1)^2} - \tfrac{2|h_2|^2}{2^{R_0}-1} \right\}= \mathrm{Pr}\{\mathcal{A}_1^\text{out}\} + \mathrm{Pr}\{\mathcal{A}_2^\text{out}\}.
\end{aligned}$$ The first probability is computed by $$\begin{aligned}
&\mathrm{Pr}\{\mathcal{A}_1^\text{out}\} {\color{black} = \int_0^{\frac{2(2^{R_0}-1)}{\xi}}\int_0^{x_2}\frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}dx_1dx_2\nonumber}\\
\label{eq:p2asup1}&=1+e^{-\frac{4(2^{R_0}-1)}{\beta\xi}} - 2e^{-\frac{2(2^{R_0}-1)}{\beta\xi}}.
\end{aligned}$$ Let $K_1=\frac{2(2^{R_0}-1)}{\xi}$ and $K_2=\frac{4^{R_0}-1}{\xi}$. Then the second probability is given by $$\begin{aligned}
&\mathrm{Pr}\{\mathcal{A}_2^\text{out}\} = \int_{K_1}^{K_2}\int_{\frac{\xi x_2^2-2x_2^{\phantom{2}}(2^{R_0}-1)}{(2^{R_0}-1)^2}}^{x_2}\frac{2}{\beta^2}e^{-\frac{x_1+x_2}{\beta}}dx_1dx_2\\
\label{eq:p2asup2p1}&= \int_{K_1}^{K_2} \frac{2}{\beta}\left( e^{-\left(\frac{x_2}{\beta}+\frac{\xi x_2^2-2x_2^{\phantom{2}}(2^{R_0}-1)}{\beta(2^{R_0}-1)^2}\right) } - e^{-\frac{2x_2}{\beta}} \right)dx_2
\end{aligned}$$ The second term in the integral in equation (\[eq:p2asup2p1\]) can be easily computed to be $$\begin{aligned}
&\label{eq:p2asup2p1-1}\int_{K_1}^{K_2} \frac{2}{\beta}e^{-\frac{2x_2}{\beta}}dx_2=-e^{- \frac{2(4^{R_0}-1)}{\beta\xi}} + e^{-\frac{4(2^{R_0}-1)}{\beta\xi}}.
\end{aligned}$$ The first integral in equation (\[eq:p2asup2p1\]) is computed by completing the square in the exponent as $$\begin{aligned}
& \int_{K_1}^{K_2} \frac{2}{\beta} e^{-\left(\frac{x_2}{\beta}+\frac{\xi x_2^2-2x_2^{\phantom{2}}(2^{R_0}-1)}{\beta(2^{R_0}-1)^2}\right) } dx_2\\
\label{eq:p2asup2p2}&= {\color{black}\int_{K_1}^{K_2} \frac{2}{\beta} e^{-\frac{\xi}{\beta(2^{R_0}-1)^2}\left(x_2^2 + \frac{x_2^{\phantom{2}}(2^{R_0}-1)(2^{R_0}-3)}{\xi}\right) } dx_2.}
\end{aligned}$$ Since $$\begin{aligned}
&x_2^2 + \tfrac{x_2(2^{R_0}-1)(2^{R_0}-3)}{\xi}\\
=&\left(x_2 + \tfrac{(2^{R_0}-1)(2^{R_0}-3)}{2\xi}\right)^2 - \left(\tfrac{(2^{R_0}-1)(2^{R_0}-3)}{2\xi}\right)^2\hspace{-1mm},
\end{aligned}$$ then equation (\[eq:p2asup2p2\]) equals $$\begin{aligned}
\label{eq:p2asup2p3}&= \frac{2}{\beta} e^{\frac{(2^{R_0}-3)^2}{4\beta\xi}}\int_{K_1}^{K_2} e^{-\frac{\xi}{\beta(2^{R_0}-1)^2}\left(x_2^{\phantom{2}} + \frac{(2^{R_0}-1)(2^{R_0}-3)}{2\xi}\right)^2} dx_2 .
\end{aligned}$$ By using the substitution $$\begin{aligned}
\label{eq:p2outsub}u(x_2) = \tfrac{1}{2^{R_0}-1}\sqrt{\tfrac{\xi}{\beta}}\left(x_2 + \tfrac{(2^{R_0}-1)(2^{R_0}-3)}{2\xi}\right),
\end{aligned}$$ the integral in equation (\[eq:p2asup2p3\]) equals $$\begin{aligned}
\label{eq:p2asup2p4}&=\frac{2}{\beta} e^{\frac{(2^{R_0}-3)^2}{4\beta\xi}}\int_{u(K_1)}^{u(K_2)} e^{-u^2}\cdot(2^{R_0}-1)\sqrt{\frac{\beta}{\xi}}du.\\
&=\label{eq:p2asup2p5}(2^{R_0}-1)e^{\frac{(2^{R_0}-3)^2}{4\beta\xi}}\sqrt{\frac{\pi}{\beta\xi}}\cdot[\mathrm{erfc}\left(u(K_1)\right) - \mathrm{erfc}\left(u(K_2)\right)],
\end{aligned}$$ where $u(x)$ is obtained by equation (\[eq:p2outsub\]), and thus $u(K_1) = \frac{2^{R_0}+1}{2\sqrt{\beta\xi}}$, $u(K_2) = \frac{3(2^{R_0})-1}{2\sqrt{\beta\xi}}$, and $\mathrm{erfc}(z)=\frac{2}{\sqrt{\pi}}\int_z^\infty e^{-u^2} du$ is the complementary error function. Thus, combining equations (\[eq:p2asup1\], \[eq:p2asup2p1-1\], \[eq:p2asup2p5\]) results in equation (\[eq:outage2\]).
Proof of Theorem \[thm:NOMAminmax\] {#proof:NOMAminmax}
===================================
When $\xi |h_0|^2 \gg 1$, $\Delta S(a_\text{sup}) \approx \frac{1}{2}(\log_2(|h_M|^2)-\log_2(|h_0|^2))$. Therefore by equation (\[eq:expminmaxlong\]),
$$\begin{aligned}
\mathbb{E}\left[ \Delta S(a_\text{sup}) \right] \approx &\int_0^\infty\int_0^{x_M}\tfrac{1}{2}\log_2 (x_M) \tfrac{K(K-1)}{\beta^2}e^{-\frac{x_0+x_M}{\beta}}(e^{-\frac{x_0}{\beta}}-e^{-\frac{x_M}{\beta}})^{K-2}dx_0dx_M \nonumber\\
\label{eq:expminmaxlong}&- \int_0^\infty\int_{x_0}^\infty\tfrac{1}{2}\log_2 (x_0) \tfrac{K(K-1)}{\beta^2}e^{-\frac{x_0+x_M}{\beta}}(e^{-\frac{x_0}{\beta}}-e^{-\frac{x_M}{\beta}})^{K-2}dx_Mdx_0
\end{aligned}$$
------------------------------------------------------------------------
$$\begin{aligned}
&\mathbb{E}\left[ \Delta S(a_\text{sup}) \right] \\
\approx& \int_0^\infty\log_2 (x_M) \frac{K}{2\beta}e^{-\frac{x_M}{\beta}}(1-e^{-\frac{x_M}{\beta}})^{K-1}dx_M \\
&- \int_0^\infty\log_2 (x_0) \frac{K}{2\beta}e^{-\frac{Kx_0}{\beta}}dx_0 \\
=& \int_0^\infty\log_2 (x_M) \frac{K}{2\beta}e^{-\frac{x_M}{\beta}} \sum_{n=0}^{K-1} \tbinom{K-1}{n}(-1)^{n}e^{-\frac{nx_M}{\beta}}dx_M \\
&- \int_0^\infty\log_2 (x_0) \frac{K}{2\beta}e^{-\frac{Kx_0}{\beta}}dx_0 \\
=& \sum_{n=0}^{K-1} \int_0^\infty\log_2 \left(\frac{x}{n+1}\right) \frac{K}{2(n+1)\beta} \tbinom{K-1}{n}(-1)^{n}e^{-\frac{x}{\beta}}dx \\
&- \int_0^\infty\log_2\left(\frac{x}{K}\right) \frac{1}{2\beta}e^{-\frac{x}{\beta}}dx \\
=& \int_0^\infty \log_2 \left(\frac{K}{x}\cdot \prod_{n=0}^{K-1} \left(\frac{x}{n+1}\right)^{\binom{K}{n+1}(-1)^{n}} \right) \frac{1}{2\beta}e^{-\frac{x}{\beta}}dx \\
=& \frac{1}{2}\log_2(K) + \frac{1}{2}\sum_{m=1}^K\tbinom{K}{m}(-1)^{m} \log_2 (m) .\end{aligned}$$
Proof of Property \[prop:multipower\] {#proof:multipower}
=====================================
It is already established that $a_1, A_1\in(0,1)$. The power allocation coefficient for user 2 is found by the equation $$\begin{aligned}
\label{eq:poweru2derive}
(1+\xi|h_2|^2)^\frac{1}{K} =& \frac{1+A_1\xi|h_2|^2}{1+(A_1-a_2)\xi|h_2|^2}\nonumber\\
\Longrightarrow a_2 =& \frac{(1+A_1\xi|h_2|^2)[(1+\xi|h_2|^2)^\frac{1}{K}-1]}{\xi|h_2|^2(1+\xi|h_2|^2)^\frac{1}{K}}.\end{aligned}$$ If the following is true $$\label{ineq:poweru2}
1< \frac{1+A_1\xi|h_2|^2}{1+(A_1-a_2)\xi|h_2|^2} < 1+A_1\xi|h_2|^2,$$ then clearly $a_2\in(0, A_1)$. However, for equation (\[eq:poweru2derive\]) and inequality (\[ineq:poweru2\]) to be true, it must be true that $$1< (1+\xi|h_2|^2)^\frac{1}{K} < 1+A_1\xi|h_2|^2.$$ It is trivial to show that $1<(1+\xi|h_2|^2)^\frac{1}{K}, \forall \xi, |h_2|^2>0$. To show that $(1+\xi|h_2|^2)^\frac{1}{K}<1+A_1\xi|h_2|^2, \forall \xi>0$, the inequality is rearranged so that $$\gamma_2 < A_1,$$ where $$\gamma_k = \frac{(1+\xi|h_k|^2)^\frac{1}{K}-1}{\xi|h_k|^2}.$$ The inequality $\gamma_2 < A_1$ is clearly true because $\gamma_2 < \gamma_1$, and $\gamma_1 < A_1$ because $(1+\xi|h_1|^2)^\frac{m}{K} < (1+\xi|h_1|^2)^\frac{K-1}{K},\forall m<K-1$. Therefore, equation (\[eq:poweru2derive\]) and inequality (\[ineq:poweru2\]) are true. In a similar manner, in order for the power allocation coefficient $a_k$ for user $k$ to be less than total interference $A_{k-1}$ received by user $k-1$, the following must be true: $$\begin{aligned}
\label{eq:munomapowerk1}&a_k = \frac{(1+A_{k-1}\xi|h_k|^2)[(1+\xi|h_k|^2)^\frac{1}{K}-1]}{\xi|h_k|^2(1+\xi|h_k|^2)^\frac{1}{K}},\\
\label{eq:munomapowerk2}&1< \frac{1+A_{k-1}\xi|h_k|^2}{1+(A_{k-1}-a_k)\xi|h_k|^2} < 1+A_{k-1}\xi|h_k|^2,\\
\label{eq:munomapowerk3}&1< (1+\xi|h_k|^2)^\frac{1}{K} < 1+A_{k-1}\xi|h_k|^2.\end{aligned}$$ Equation (\[eq:munomapowerk1\]) is true by solving eq. (\[eq:powerboundK\]), while (\[eq:munomapowerk2\]) states that $a_k\in(0,A_{k-1})$ and (\[eq:munomapowerk3\]) requires that user $k$’s OMA capacity is feasible within $a_k\in(0,A_{k-1})$, given the channel condition of user $k$. Therefore, (\[eq:munomapowerk3\]) leads to $$\begin{aligned}
&\gamma_k < A_{k-1} = A_{k-2}-a_{k-1} = \frac{A_{k-2}-\gamma_{k-1}}{(1+\xi|h_{k-1}|^2)^\frac{1}{K}} \nonumber\\
\label{eq:lbfrac0} &\Longrightarrow A_{k-2}> \nonumber\\
&\frac{(1+\xi|h_k|^2)^\frac{1}{K}-1}{\xi|h_k|^2}(1+\xi|h_{k-1}|^2)^\frac{1}{K} +\frac{(1+\xi|h_{k-1}|^2)^\frac{1}{K}-1}{\xi|h_{k-1}|^2}\end{aligned}$$ Since the function $$\label{eq:monotonicfrac}
f(x) = \frac{(1+x)^\frac{m}{K}-1}{x}, \forall m<K, m\text{ and }K\in\mathbb{N}$$ is a monotonically decreasing function of $x$, then [ $$\begin{aligned}
\label{eq:lbfrac1} &\frac{(1+\xi|h_k|^2)^\frac{1}{K}-1}{\xi|h_k|^2}(1+\xi|h_{k-1}|^2)^\frac{1}{K}
+ \frac{(1+\xi|h_{k-1}|^2)^\frac{1}{K}-1}{\xi|h_{k-1}|^2} \\
&< \frac{(1+\xi|h_{k-1}|^2)^\frac{1}{K}-1}{\xi|h_{k-1}|^2}(1+\xi|h_{k-1}|^2)^\frac{1}{K}
+\frac{(1+\xi|h_{k-1}|^2)^\frac{1}{K}-1}{\xi|h_{k-1}|^2} \\
\label{eq:lbfrac2} &= \frac{(1+\xi|h_{k-1}|^2)^\frac{2}{K}-1}{\xi|h_{k-1}|^2} < A_{k-2}
= A_{k-3}-a_{k-2} \nonumber\\
&= \frac{ A_{k-3}-\gamma_{k-2}}{(1+\xi|h_{k-2}|^2)^\frac{1}{K}} \\
\label{eq:lbfrac3} & \Longrightarrow A_{k-3}> \frac{(1+\xi|h_{k-1}|^2)^\frac{2}{K}-1}{\xi|h_{k-1}|^2} (1+\xi|h_{k-2}|^2)^\frac{1}{K} \\
& +\frac{(1+\xi|h_{k-2}|^2)^\frac{1}{K}-1}{\xi|h_{k-2}|^2}.\end{aligned}$$]{} The inequality in (\[eq:lbfrac3\]) has the same form as the inequality in (\[eq:lbfrac0\]), so the same steps taken in inequalities (\[eq:lbfrac1\]) and (\[eq:lbfrac2\]) can be used repeatedly, until the following is obtained $$\frac{(1+\xi|h_{1}|^2)^\frac{k-1}{K}-1}{\xi|h_{1}|^2} \leq A_{1},$$ which is true $\forall k\leq K$. Hence, this series of inequalities shows that the transmit power allocation coefficient $a_k$ required for user $k$ to achieve OMA capacity is always less than the total interference coefficient received by user $k-1$, which equals the total fraction of power available for users $k,\ldots,K$.
[2]{} T.M. Cover and J.A. Thomas, [*Elements of Information Theory*]{}, John Wiley & Sons, New York, U.S.A., 1991.
D. Tse and P. Viswanath, [*Fundamentals of Wireless Communication*]{}, Cambridge University Press, 2005
P. Wang, J. Xiao, L. Ping, “Comparison of Orthogonal and Non-Orthogonal Approaches to Future Wireless Cellular Systems,” [*IEEE Vehicular Technology Magazine*]{}, Volume 1, Issue 3, pp. 4-11, Sept. 2006.
J. Schaepperle and A. Ruegg, “Enhancement of Throughput Fairness in 4G Wireless Access Systems by Non-Orthogonal Signaling,” [*Bell Labs Tecnical Journal*]{}, Vol 13, Issue 4, pp 59-77, 2009.
J. Schaepperle, “Throughput of a Wireless Cell Using Superposition Based Multiple-Acess with Optimized Scheduling,” [*Personal Indoor and Mobile Radio Communications, 21st International Symposium on*]{}, pp 212-217, 2010.
T. Takeda, K. Higuchi, “Enhanced User Fairness Using Non-orthogonal Access and SIC in Cellular Uplink,” [*IEEE Vehicular Technology Conference*]{}, Fall 2011.
S. Tomida, K. Higuchi, “Non-orthogonal Access with SIC in Cellular Downlink for User Fairness Enhancement,” [*ISPACS*]{}, Dec 2011.
Y. Saito, A. Benjebbour, Y. Kishiyama, and T. Nakamura, “System-Level Performance Evaluation of Downlink Non-orthogonal Multiple Access (NOMA),” [*PIMRC 2013*]{}, pp. 611-615.
B. Kim, S. Lim, H. Kim, S. Suh, J. Kwun , S. Choi, C. Lee, S. Lee, and Daesik Hong, “Non-Orthogonal Multile Access in a Downlink Multiuser Beamforming System,” [*IEEE MILCOM 2013*]{}
J. Choi, “Minimum Power Multicast Beamforming with Superposition Coding for Multiresolution Broadcastand Application to NOMA Systems,” [*IEEE Transactions on Communications*]{}, Vol 63, Issue 3, pp. 791-800, 2014.
Q. Sun, S. Han, C.L. I, and Z. Pan, “On the Ergodic Capacity of MIMO NOMA Systems,” [*IEEE Wireless Communications Letters*]{}, Vol 4, No 4, August 2015.
M. Al-Imari, P. Xiao, M.A. Imran, R. Tafazolli, “Uplink Non-Orthogonal Multiple Access for 5G Wireless Networks,” [*Wireless Communications Systems, 11th International Symposium on*]{}, pp 781-785, August 2014.
F. Liu, P. Mähönen, M. Petrova, “Proportional Fairness-Based User Pairing and Power Allocation for Non-Orthogonal Multiple Access,” [*PIMRC, 26th Annual*]{}, pp. 1306-1310, 2015.
S. Timotheou and I. Krikidis, “Fairness for Non-Orthogonal Multiple Access in 5G Systems,” [*IEEE Signal Processing Letters*]{}, Vol 22, No 10, October 2015.
Z. Ding, P. Fan, H.V. Poor, “Impact of User Pairing on 5G Non-Orthogonal Multiple Access Downlink Transmissions,” [*IEEE Transactions on Vehicular Technology*]{}, Future Issue, 22 September 2015.
J.A. Oviedo, H.R. Sadjadpour, “A New NOMA Approach for Fair Power Allocation,” [*IEEE INFOCOM 2016 Workshop on 5G & Beyond - Enabling Technologies and Applications*]{}.
R. Knopp and P. Humblet, “Information Capacity and Power Control in Single-Cell Multiuser Communications,” [*Proc. of IEEE ICC*]{}, Seattle, Washington, USA, June 18-22 1995.
Z. Ding, F. Adachi, H.V. Poor, “The Application of MIMO to Non-Orthogonal Multiple Access,” [*IEEE Transactions on Wireless Communication*]{}, Vol 15, No 1, Jan 2016
L. Zheng and D.N.C. Tse, “Diversity and Multiplexing: A Fundamental Tradeoff in Multiple-Antenna Channels,” [*IEEE Transactions on Information Theory*]{}, Vol 49, No 5, May 2003
[José Armando Oviedo]{} José Armando Oviedo received the B.S. in Electrical Engineering in 2009 from the California State Polytechnic University, Pomona, and the M.S. in Electrical Engineering in 2010 from the University of California, Riverside. He is currently pursuing the Ph.D. in Electrical Engineering at the University of California, Santa Cruz.
His main areas of interest are non-orthogonal multiple access and multi-user diversity in multi-user wireless communication systems.
[Hamid R. Sadjadpour]{} Hamid R. Sadjadpour (S’94–M’95–SM’00) received the B.S. and M.S. degrees from the Sharif University of Technology, and the Ph.D. degree from the University of Southern California at Los Angeles, Los Angeles, CA. In 1995, he joined the AT&T Research Laboratory, Florham Park, NJ, USA, as a Technical Staff Member and later as a Principal Member of Technical Staff. In 2001, he joined the University of California at Santa Cruz, Santa Cruz, where he is currently a Professor. He has authored over 170 publications. He holds 17 patents. His research interests are in the general areas of wireless communications and networks. He has served as a Technical Program Committee Member and the Chair in numerous conferences. He is a co-recipient of the best paper awards at the 2007 International Symposium on Performance Evaluation of Computer and Telecommunication Systems and the 2008 Military Communications conference, and the 2010 European Wireless Conference Best Student Paper Award. He has been a Guest Editor of EURASIP in 2003 and 2006. He was a member of the Editorial Board of Wireless Communications and Mobile Computing Journal (Wiley), and the Journal OF Communications and Networks.
[^1]: Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
[^2]: The authors are with the Department of Electrical Engineering, University of California, Santa Cruz, CA 95064 USA. (e-mail: {xmando, hamid}@soe.ucsc.edu.
|
---
abstract: 'We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time algorithm for finding an optimal solution. For the latter problem, we give a linear-time algorithm for finding a feasible solution whose size is equal to the obvious lower bound or more than that by one.'
author:
- 'Tatsuya Matsuoka[^1]'
- 'Shun Sato[^2]'
date: 'September, 2017'
title: Making Bidirected Graphs Strongly Connected
---
Introduction {#intro}
============
Problems to make a given graph (strongly) connected are well-investigated. The minimum number of additional edges to make a given undirected graph connected and that of additional arcs to make a given directed graph strongly connected [@ET1976] are well-known.
; (v1) at (-1.4,2) ; (v2) at (-2,0.5) ; (v3) at (-2,-1) ; (v4) at (1,1.5) ; (v5) at (0,0) ; (v6) at (-0.5,-1.4) ; (v7) at (1.8,0.6) ; (v8) at (1.4,-1.5) ; (v9) at (2.9,-1) ; (v10) at (3,1.6) ; (v11) at (3.9,-0.4) ; / ǐn [v2/v4,v2/v5,v3/v6,v4/v7,v5/v7,v5/v6,v5/v8,v6/v8,v7/v8,v10/v11]{} () to (); (v3) to \[out=130,in=230,looseness=10\] (v3); (v9) to \[out=220,in=320,looseness=10\] (v9); (v10) to \[out=40,in=140,looseness=10\] (v10); ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
The concept of bidirected graphs (Figure \[FigBidirectedGraph\]; the precise definition will be given later in Section 2) was introduced by Edmonds and Johnson [@EJ1970]. It is a common generalization of undirected graphs and directed graphs. For bidirected graphs, Ando, Fujishige and Nemoto [@AFN1996] defined the notion of strong connectivity and gave a linear-time algorithm for the strongly connected component decomposition. However, problems to make a given bidirected graph strongly connected have not been formulated.
In this paper, we consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs.
Related Works
-------------
It is obvious that the minimum number of additional edges to make a given undirected graph connected is fewer than the number of connected components of a given graph by one. Eswaran and Tarjan [@ET1976] gave the minimum number of additional arcs to make a given directed graph strongly connected and that of additional edges to make a given undirected graph *bridge-connected* (*2-edge-connected*) or *biconnected* (*2-vertex-connected*). Linear-time algorithms for finding an optimal solution of these problems are also given in [@ET1976]. Note that they defined an operation called “condensation” which transforms a general directed graph to an acyclic directed graph. We can focus on the acyclic case since for this problem we can obtain a solution of the original problem by solving the problem on the condensed graph. For a directed graph $G=(V, A)$, $v\in V$ is a *source* if $\delta (v)\geq 1,\ \rho (v)=0$, a *sink* if $\rho (v)\geq 1,\ \delta (v)=0$ and an *isolated vertex* if $\rho (v)=\delta (v)=0$ (in directed graphs, $\delta$ and $\rho$ denote the out-degree and in-degree functions, respectively).
Let $G=(V, A)$ be an acyclic directed graph with the set $S\subseteq V$ of sources, the set $T\subseteq V$ of sinks and the set $Q\subseteq V$ of isolated vertices $(|S|+|T|+|Q|>1)$. Then the minimum number of additional arcs to make the given graph strongly connected is $\max\{ |S|, |T|\} +|Q|$.
For an undirected graph $G=(V, E)$, $v\in V$ is called a *pendant* if $\delta (v)=1$ and $V'\subseteq V$ is called a *pendant block* if it is a 2-vertex-connected component and it contains exactly one *cutnode* (for undirected graphs, $\delta$ denotes the degree function). Note that $v\in V$ is a cutnode if the original graph is connected and the graph induced by $V\setminus \{ v\} $ is disconnected. Similarly, $V'\subseteq V$ is called an *isolated block* if it is a 2-vertex-connected component and it contains no cutnode.
Let $G=(V, E)$ be an undirected graph with the set $P\subseteq V$ of pendants and the set $Q\subseteq V$ of isolated vertices $(|P|+|Q|>1)$. Then the minimum number of additional edges to make the given graph 2-edge-connected is $\lceil |P|/2\rceil +|Q|$.
Let $G=(V, E)$ be an undirected graph with the set $\mathcal{P}\subseteq 2^V$ of pendant blocks and the set $\mathcal{Q}\subseteq 2^V$ of isolated blocks $(|\mathcal{P}|+|\mathcal{Q}|>1)$. Then the minimum number of additional edges to make the given graph 2-vertex-connected is $\max \left\{ d-1, \lceil |\mathcal{P}|/2\rceil +|\mathcal{Q}|\right\}$. Here, $$\begin{aligned}
d:=&\max \{ \# (\mbox{2-vertex-connected components containing }v)\mid v\in V \} \\
&+\# (\mbox{connected components})-1.\end{aligned}$$
On the other hand, problems on bidirected graphs also have been considered in the literature. Ando, Fujishige and Nemoto [@AFN1996] gave a linear-time algorithm for strongly connected component decomposition of bidirected graphs. This algorithm is made use of for the block triangularization of skew-symmetric matrices [@Iwata1998]. Bidirected graphs are also used in the field of computational biology [@MB2009; @MGMB2007; @Yasuda2015].
The strongly connected component decomposition of a bidirected graph [@AFN1996] is obtained by the ordinary strongly connected component decomposition of the associated directed graph, *skew-symmetric graph*, which will be used in Section 3. As pointed out in [@AFN1996], the same graph is used by Zaslavsky [@Zaslavsky1991] for the study of *signed graphs* [@Harary19531954]. The notion of skew-symmetric graphs is defined first by Tutte [@Tutte1967] with the name “antisymmetrical digraphs” independent from bidirected graphs. There are also various problems on skew-symmetric graphs, and they have been intensively studied [@GK1995; @GK1996; @GK2004]. Study on bisubmodular polyhedra also made use of this skew-symmetric graph [@AF1996] with the name “exchangeability graph.”
Our Contribution
----------------
In this paper, we formulate problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. Since self-loops have influence on the strong connectivity on bidirected graphs, these two problems arise depending on how to treat self-loops.
We first define the procedure called “condensation” on bidirected graphs. We can reduce general cases to acyclic cases by this operation for the above two problem settings. This can be done by using the strongly connected component decomposition algorithm for bidirected graphs devised by Ando, Fujishige and Nemoto [@AFN1996]. This is similar to the fact that the condensation on directed graphs is done by using strongly connected component decomposition of directed graphs [@ET1976 Lemma 1]. However, since there are signs on each arc in bidirected graphs, we must define the appropriate signs for each arc on the condensed bidirected graph.
We discuss the two versions of the problems on bidirected graphs. For the problem on signs, the obvious lower bound can be obtained from the necessity for connectivity of the underlying graph and a condition on signs around each vertex. We show that this lower bound can be achieved for any acyclic bidirected graph and give a linear-time algorithm for finding an optimal solution. For the problem on arcs, we give a linear-time algorithm for finding a feasible solution whose size is equal to the obvious lower bound or more than that by one.
Organization
------------
The organization of the rest of this paper is as follows. We give definitions and notation in Section 2. In Section 3, we give two problem settings dealt with in this paper and devise the condensation operation on bidirected graphs, which reduces a general case to an acyclic case. These two problems are discussed in Sections 4 and 5, respectively. Section 6 is devoted to concluding remarks involving other problem settings.
Preliminaries
=============
In this section, we introduce definitions and notation used in this paper. Definitions in this section mainly refer Ando and Fujishige [@AF1994] and Ando, Fujishige and Nemoto [@AFN1996].
A *bidirected graph* is a triplet of a vertex set $V$, an arc set $A$ and a boundary operator $\partial : A\to 3^V (:=\{ (X, Y)\mid X, Y\subseteq V, X\cap Y=\emptyset \} )$ such that $\partial a=(X_a, Y_a)$ satisfies $1\leq |X_a|+|Y_a|\leq 2$ for each $a\in A$. We use the notation $|\partial a|:=|X_a|+|Y_a|$. Let $\partial^+: A\to 2^V$ and $\partial^-: A\to 2^V$ denote the operators with $\partial^+a=X_a$ and $\partial^-a=Y_a$. This can be regarded that the signs are put on endpoints of *links* or on *self-loops* by $\partial^+$ and $\partial^-$ (here we call an arc a link if it connects two distinct vertices). In other words, $\partial^+a$ and $\partial^-a$ are the sets of endpoints of $a$ with the signs “$+$” and “$-$”, respectively. We call an arc $a$ with $\partial a=(\{ v\} , \emptyset )$ a plus-loop at $v$ and $a$ with $\partial a=(\emptyset , \{ v\} )$ a minus-loop at $v$.
For simplicity, we define some other notation. Let $\bar{\partial}: A\to 2^V$ denote the operator with $a\mapsto \partial^+a\cup \partial^-a$ for each $a\in A$. For a bidirected graph $G=(V, A; \partial )$, let $\bar{G}=(V, A)$ be the undirected graph omitting the signs of $G$ (the *underlying graph* of $G$). We write as $a=(u, v)$ if $\bar{\partial}a=\{ u, v\}$. Let us define a sign operator $\pi : \left\{ (a, u)\mid a\in A, u\in \bar{\partial}a\right\} \to \{ +, -\}$ as $\pi (a, u)=+$ if $u\in \partial^+a$ and $\pi (a, u)=-$ if $u\in \partial^-a$. Let “$(u, v)$ with $(\pi_1, \pi_2 )$” $(\pi _1, \pi _2\in \{ +, -\} )$ denote an arc $a=(u, v)$ with $\pi (a, u)=\pi_1 $ and $\pi (a, v)=\pi_2 $.
An arc $a\in A$ is said to be *positively (resp. negatively) incident to* $v$ if $v\in \partial^+a$ $(\mbox{resp. }v\in \partial^-a)$. Arcs $a\in A$ and $a'\in A$ are said to be *oppositely incident to* $v$ if $a$ is positively (resp. negatively) incident to $v$ and $a'$ is negatively (resp. positively) incident to $v$.
An alternating sequence of vertices and arcs $(v_0, a_1, v_1, a_2, \ldots , a_l, v_l)\ (l\geq 1)$ is called a *path* if $a_i$ and $a_{i+1}$ are oppositely incident to $v_i$ $(i=1, 2, \ldots , l-1)$, $a_1$ is incident to $v_0$ and $a_l$ is incident to $v_l$. This is called ($\pi (a_1, v_0)$, $\pi (a_l, v_l)$)-path from $v_0$ to $v_l$. A path with $v_0=v_l$ is called a *cycle* with a *root* $v_0(=v_l)$. If $a_l$ and $a_1$ are oppositely incident to $v_0$ and it includes distinct vertices, we call it a *proper* cycle. A cycle which is not proper is called an *improper* cycle. An improper cycle with $\pi (a_1, v_0)=\pi (a_l, v_l(=v_0))=+ \ (\mathrm{resp.} -)$ is called a *positive (resp. negative) improper cycle*. If a graph does not contain a proper cycle, we call it an *acyclic* graph (Note that this definition is different from that of “strongly acyclic” or “weakly (node- or edge-) acyclic” in [@Babenko2006]).
For a bidirected graph $G=(V, A; \partial )$, two vertices $v, v'\in V$ are called *strongly connected* if $G$ contains two paths $(v, a_1^1, v_1^1, a_2^1, \ldots , a_{l_1}^1, v')$ and $(v, a_1^2, v_1^2, a_2^2, \ldots , a_{l_2}^2, v')$ such that $a_1^1$ and $a_1^2$ are oppositely connected to $v$ and that $a_{l_1}^1$ and $a_{l_2}^2$ are oppositely connected to $v'$. Note that these two paths need not to be vertex-disjoint. A binary relation on $V$ can be defined by this strong connectivity: $v\sim v'$ if $v$ and $v'$ are strongly connected. By assuming that $v\sim v$ for all $v\in V$, we obtain the equivalence relation $\sim$ on $V$. Each equivalence class of $V$ on $\sim$ is called *strongly connected component* and $G$ is called *strongly connected* if $G$ has only one strongly connected component.
A vertex $v\in V$ is called *inconsistent* if there exist improper cycles with root $v$ $C_1=(v, a_1^1, v_1^1, a_2^1, \ldots , a_{l_1}^1, v)$ and $C_2=(v, a_1^2, v_1^2, a_2^2, \ldots , a_{l_2}^2, v)$ such that $a_{l_2}^2$ and $a_1^1$ are oppositely incident to $v$. It is stated in [@AFN1996] that if $u$ and $v$ are strongly connected and $u$ is inconsistent, then $v$ is also inconsistent. Thus, the notion of inconsistency can also be naturally defined on strongly connected components.
Settings and the Condensation Operation
=======================================
In this section, we introduce the problem settings we tackle in this paper and explain the operation called condensation.
Problem Settings
----------------
We deal with the problems of the following type.
Let $G=(V, A; \partial )$ be a bidirected graph. Find additional arcs $A'$ and a boundary operator $\partial' : A\cup A'\to 3^V (\partial' a=\partial a\ (\forall a\in A))$ minimizing $F(A', \partial'):=\sum_{a'\in A'}f(\partial' a')$ $(f: \{ (X, Y)\mid X, Y\subseteq V, X\cap Y=\emptyset , 1\leq |X|+|Y|\leq 2\} \to \mathbb{R})$ such that $G':=(V, A\cup A'; \partial' )$ is a strongly connected bidirected graph. \[ProbGeneral\]
Note that Problem \[ProbGeneral\] is NP-hard in general. This can easily be shown by following the argument in the proof of [@ET1976 Theorem 1] as follows: we show this by reducing the following directed Hamiltonian cycle problem to Problem \[ProbGeneral\] with a certain function $f$.
Let $D=(V, A)$ be a directed graph. Find a directed Hamiltonian cycle in $D$. \[ProbDHC\]
Set $f(\partial' a')=1$ if $a'=(v_1, v_2)$ with $(+, -)$ and there exists $a=(v_1, v_2)$ in $D$ and set $f(\partial' a')=2$ for any other possible arc $a'$. There exists a solution of Problem \[ProbGeneral\] with respect to $ G = (V, \emptyset; \partial) $ satisfying $F(A', \partial' )=|V|$ if and only if there exists a solution of Problem \[ProbDHC\]. Since Problem \[ProbDHC\] is NP-complete [@Karp1972], Problem \[ProbGeneral\] is NP-hard.
For the problem on undirected graphs or directed graphs similar to Problem \[ProbGeneral\], it is natural to define $F(A', \partial' ):=|A'|$, i.e., minimization of the cardinality of additional edge (or arc) set. For bidirected graphs, however, there are two reasonable candidates of $F(A', \partial' )$, i.e., $\sum_{a'\in A'}|\partial' a'|$ and $|A'|$. In other words, $f(\partial' a'):=|\partial' a'|$ in the former setting and $f(\partial' a'):=1$ in the latter setting. The former means the minimization of the number of the additional signs on arcs and the latter means that of arcs themselves. In other words, the cost of a link is twice higher than that of a self-loop for the former problem and is the same for the latter problem. These natural two problems arise because self-loops have influence on strong connectivity in bidirected graphs (see, e.g., Figure \[FigEx\]). Note that self-loops do not have any influence on the structure of (strong) connectivity for undirected graphs or directed graphs.
Reduction to Acyclic Case
-------------------------
We present a technique for reducing general cases to acyclic cases for Problem \[ProbGeneral\] with respect to $F(A', \partial' )=\sum_{a'\in A'}|\partial' a'|$ or $F(A', \partial' )=|A'|$.
For directed graphs, Eswaran and Tarjan \[6\] first condense the given directed graph to focus on acyclic cases. There, the condensed graph $\tilde{G} = (\tilde{V}, \tilde{A} )$ is obtained from the strongly connected component decomposition $C_1,C_2,\dots ,C_k$ of the original graph $G = (V,A)$, where $$\begin{aligned}
\tilde{V} &:=\{ v_{C_1}, v_{C_2}, \ldots , v_{C_k}\},\\
\tilde{A} &:=\left\{ (v_{C_j}, v_{C_k})\mid \exists v\in V(C_j), \exists v'\in V(C_k)\mbox{ s.t. }(v, v')\in A\right\}.\end{aligned}$$
For bidirected graphs, we can utilize the linear-time algorithm for strongly connected component decomposition devised by Ando, Fujishige and Nemoto [@AFN1996]. Precisely speaking, in order to appropriately define signs in the condensed graph, we use the strongly connected component decomposition of the associated skew-symmetric graph, which corresponds to the strongly connected component decomposition of the original bidirected graph $ G = (V, A; \partial) $ [@AFN1996 Corollary 5.4].
In Algorithm CONDENSE($G$) described below, Steps 1–3 based on the steps of the strongly connected component decomposition algorithm of [@AFN1996].
**Algorithm** CONDENSE($G$)
Step 1
: Construct the associated skew-symmetric graph $G^{\pm}=(V^+\cup V^-, A^{\pm}) $, where $V^+$ and $V^-$ are copies of $V$ ($v^+\in V^+$ and $v^-\in V^-$ denote the copy of $v\in V$) and $A^{\pm}$ are defined by $$A^{\pm}=\{ (v^{\pi (a,v)}, w^{-\pi (a,w)})\mid a\in A, \bar{\partial}a=\{ v,w\} \}.$$ Note that $v$ can be equal to $w$. Here, $-\pi$ is equal to $-$ (resp. $+$) if $\pi =+$ (resp. $-$).
Step 2
: Decompose $G^{\pm}$ into strongly connected components $G^{\pm}_j=(U^{\pm}_j, B^{\pm}_j)$ $( j\in J)$.
Step 3
: For each $j\in J$, define $$U_j=\{ v\in V\mid v^+\in U^{\pm}_j\} \cup \{ v\in V\mid v^-\in U^{\pm}_j\}.$$ Then, define $W_i$ $(i\in I)$ be the distinct members of $U_j$ $(j\in J)$ and partition $I$ into $I_1$ and $I_2$ so that $W_i$ appears twice (resp. once) in the family $\{ U_j\mid j\in J\}$ for each $i\in I_1$ (resp. $I_2$).
Step 4
: For each $i\in I_1$, let $U_j^{\pm}$ be one of two strongly connected components corresponding to $W_i$. If $U_j^{\pm}$ includes both an element in $V^+$ and an element in $V^-$, then for each $v^-\in V^-\cap U_j^{\pm}$ swap this for the counterpart.
Step 5
: Make a skew-symmetric graph $\hat{G}^{\pm}=(\hat{V}^+\cup \hat{V}^-, \hat{A})$ from $G^{\pm}$ as follows. Let $\hat{v}_i^+$ (resp. $\hat{v}_i^-$) be a representative of $W_i$. Let $\hat{V}^+:=\{ \hat{v}_i^+\mid i\in I\}$ and $\hat{V}^-:=\{ \hat{v}_i^-\mid i\in I\}$. By using the map $\alpha : V^+\cup V^-\to \hat{V}^+\cup \hat{V}^-$ defined by $$\alpha (v^{\pi})=\hat{v}_i^{\pi}\quad (v\in W_i, \pi \in \{ +, -\} ),$$ the arc set $\hat{A}$ is defined by $$\hat{A}=\{ (\alpha (v), \alpha (w))\mid (v, w)\in A^{\pm}\ (\alpha (v)\neq \alpha (w))\} .$$
Step 6
: Return the bidirected graph $\tilde{G}$ corresponding to the skew-symmetric graph $\hat{G}^{\pm}$.
Note that the strongly connected component $W_i$ is inconsistent if and only if $i\in I_2$ (see, Corollary 5.4 of [@AFN1996]).
From a feasible solution of the problem of minimization on signs or arcs for $\tilde{G}$, we can obtain a feasible solution for $G$ with the same value for the function $F$. Conversely, from any feasible arc set for the original problem on $G$ we can obtain a solution for the problem on $\tilde{G}$ whose cost is less than or equal to the original cost. These hold since a condensed graph of any strongly connected bidirected graph is strongly connected and each link in the obtained solution graph corresponds with links in the original solution graph. Thus validity of the above condensation holds.
Minimization on Signs
=====================
In this section, we deal with Problem \[ProbGeneral\] with $F(A', \partial' )=\sum_{a'\in A'}|\partial' a'|$.
We first give some definitions on bidirected graphs. Let $\gamma (=\gamma (G))$ denote the number of connected components in the underlying graph $\bar{G}$ of $G$. A vertex $v\in V$ is called a *source* (resp. a *sink*) if $v$ is included in a connected component in $\bar{G}$ which has more than one vertices and any $a\in A$ connected to $v$ in $G$ is positively (resp. negatively) incident to $v$. The set of sources is denoted by $S(=S(G))$ and that of sinks is denoted by $T(=T(G))$. A vertex $v\in V$ is called an *isolated vertex* if there exists no arc connected to $v$. The set of isolated vertices is denoted by $Q(=Q(G))$. A vertex $v\in V$ is called a *pseudo-isolated vertex* if $\{ v\}$ is the connected component with only one vertex in $\bar{G}$ and there exists a self-loop at $v$. The set of pseudo-isolated vertices is denoted by $Q'(=Q'(G))$.
When adding an arc $a=(u, v)$ to a bidirected graph $G$, we write “with proper signs” if signs on $a$ are as follows: $\pi (a, u)$ is equal to $+$ if $\{ a\in A\mid u\in \partial^+a\}$ is empty for the current bidirected graph and $\pi (a, u)$ is equal to $-$ otherwise. The sign $\pi (a, v)$ is determined in the same way.
Now, let us consider Problem \[ProbGeneral\] with respect to the number of additional signs on an acyclic bidirected graph $G=(V, A; \partial )$. Since a bidirected graph is strongly connected only if its underlying graph is connected, the value of the objective function for a feasible solution must be greater than or equal to $2(\gamma -1)$. On the other hand, a bidirected graph with $|V|>1$ is strongly connected only if there are no sources, sinks, isolated vertices and pseudo-isolated vertices. Therefore, the number of additional signs to make a bidirected graph strongly connected is greater than or equal to $|S|+|T|+|Q'|+2|Q|$. Summing up, we obtain the lower bound $\max \{ 2(\gamma -1), |S|+|T|+|Q'|+2|Q|\}$. Actually, this lower bound can be achieved.
Let $G=(V, A; \partial )$ be an acyclic bidirected graph with $|V|>1$. Then the minimum number of $\sum_{a'\in A'}|\partial' a'|$ such that $G'=(V, A\cup A'; \partial' )$ is a strongly connected bidirected graph is $\max \{ 2(\gamma -1), |S|+|T|+|Q'|+2|Q|\}$. \[ThmSign\]
We now describe an algorithm for constructing an optimal solution (whose size is equal to the lower bound). Let $C_1^1, C_2^1, \ldots , C_{k_1}^1, C_1^2, C_2^2, \ldots , C_{k_2}^2, \ldots , C_1^K, C_2^K, \ldots , C_{k_K}^K$ be the distinct vertex sets of connected components of $\bar{G}$ such that each $C_i^j$ contains just $j$ elements of $S\cup T\cup Q'\cup Q$. Note that $\sum_{i=1}^Kk_i=\gamma$ and $\sum_{i=1}^Kik_i=|S|+|T|+|Q'|+|Q|$.
**Algorithm** ADDITIONAL SIGNS($G$)
Step 1
: Let $A':=\emptyset$.
Step 2
: Let $u_1, u_2, \ldots , u_{L_1} \ (L_1:=k_1-|Q|)$ be the elements of $\left( \bigcup_{i=1}^{k_1}C_i^1 \right) \cap (S\cup T\cup Q')$. If $L_1=\gamma =1$, add a self-loop at $u_1$ to $A'$ with a proper sign and go to Step 6. If $L_1=\gamma >1$, add $\{ (u_1, u_i)\mid 2\leq i \leq L_1\}$ to $A'$ with proper signs and go to Step 6.
Step 3
: Let $\mathcal{C}=\left\{ C_1^2, C_2^2, \ldots , C_{k_2}^2, \ldots , C_1^K, C_2^K, \ldots , C_{k_K}^K \right\}$. For each $C\in \mathcal{C}$, pick up two distinct elements of $C\cap (S\cup T)$ and label them as $l_i, r_i \ (i=1, 2, \ldots , |\mathcal{C}|)$. Label the rest of the elements of $\bigcup_{C\in \mathcal{C}}C\cap (S\cup T)$ as $w_1, w_2, \ldots , w_{L_2}$ with $L_2 :=\sum_{i=3}^{K}(i-2)k_{i}$. Add $\left\{ (u_i, w_i)\mid 1\leq i\leq \min\{ L_1, L_2 \} \right\}$ to $A'$ with proper signs.
Step 4
: Let $q_1, \ldots , q_{|Q|}$ be the elements of $Q$ and define $l_{|\mathcal{C}|+i}=r_{|\mathcal{C}|+i}=q_i$ for $i = 1, \ldots , |Q|$. Add $\left\{ (r_i, l_{i+1})\mid 1\leq i<|\mathcal{C}|+|Q|\right\}$ to $A'$ with proper signs.
Step 5
: Compare $L_1$ with $L_2$.
Step 5-1
: If $L_2 \leq L_1-2$, add $(u_{L_2+1}, l_1)$ and $\left\{ (u_i, r_{|\mathcal{C}|+|Q|})\mid L_2+1<i\leq L_1\right\}$ to $A'$ with proper signs.
Step 5-2
: If $L_2 =L_1-1$, add $(u_{L_1}, l_1)$ and a self-loop at $r_{|\mathcal{C}|+ |Q|}$ to $A'$ with proper signs.
Step 5-3
: If $L_2 \geq L_1$, add self-loops at $l_1$, $r_{|\mathcal{C}|+|Q|}$ and $w_i$ for $i=L_1+1, L_1+2, \ldots , L_2$ to $A'$ with proper signs.
Step 6
: Return $G'=(V, A\cup A'; \partial' )$.
Steps 3 and 4 are like as in Figure \[FigAlgo1\].
; (l1) at (-5.5,0.5) ; ; (r1) at (-4.5,-0.5) ; ; (-5,0) circle \[x radius=1.2, y radius=1\]; at (-5,1) [$C^2_1$]{}; (l2) at (-2.5,0.5) ; ; (r2) at (-1.5,-0.5) ; ; (w1) at (-1.8,0.3) ; ; (-2,0) circle \[x radius=1.2, y radius=1\]; at (-2,1) [$C^3_1$]{}; (l3) at (0.5,0.5) ; ; (r3) at (1.5,-0.5) ; ; (w2) at (0.7,-0.3) ; ; (1,0) circle \[x radius=1.2, y radius=1\]; at (1,1) [$C^3_2$]{}; (vi1) at (3,0) ; ; ; (vi2) at (4,0) ; ; ; (vp1) at (-1,2) ; ; (-1,2) circle \[x radius=0.7, y radius=0.5\]; at (-1,2.5) [$C^1_1$]{}; (vp2) at (0,-2) ; ; (0,-2) circle \[x radius=0.7, y radius=0.5\]; at (0,-2.5) [$C^1_2$]{}; / ǐn [r1/l2,r2/l3,r3/vi1,vi1/vi2]{} () to (); / ǐn [vp1/w1,vp2/w2]{} () to ();
The above algorithm returns an optimal solution in linear time. This is confirmed by the following two lemmas.
The output of the above algorithm is strongly connected.
This can be confirmed by the following claims when $\gamma >L_1$. (It can be shown more easily when $\gamma =L_1$.)
The vertex set $\{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup Q$ is strongly connected.
For each $C\in \mathcal{C}$, there exists a path between $l$ and $r$ (vertices chosen as $l_i$ and $r_i$). This can be shown by the facts that $l$ and $r$ are connected in the underlying graph and that every vertex in $C$ has both plus and minus signs around it. Since $l_1$ and $r_{|\mathcal{C}|+|Q|}$ have a self-loop or an improper cycle, the claim holds.
Each vertex in $\{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup Q$ is inconsistent.
Suppose there is a $(+, +)$-path between $l_1$ and $r_{|\mathcal{C}|+|Q|}$. (The other case can be treated in the similar way.) By the algorithm, there are a negative improper cycle rooted at $l_1$ and that rooted at $r_{|\mathcal{C}|+|Q|}$. Thus due to the above $(+, +)$-path, there is a positive improper cycle rooted at $l_1$. Therefore $l_1$ is inconsistent and hence the claim holds.
For each vertex $v\in V\setminus (\{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup Q)$, there exist $v_1^*, v_2^*\in \{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup Q$ (not necessarily distinct), a path $P_1$ between $v$ and $v_1^*$ and a path $P_2$ between $v$ and $v_2^*$ such that end arcs of $P_1$ and $P_2$ connected to $v$ are oppositely incident.
By the algorithm ADDITIONAL SIGNS, each vertex in the resultant graph has both plus and minus signs around it. Fix a vertex $v\in V\setminus (\{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup Q)$. By the above two claims, there is an inconsistent strongly connected component including $\{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup Q$. Since the underlying graph is connected and each vertex has both signs around it, we can reach this component from $v$ regardless of the initial sign. Thus we can obtain $v_1^*, v_2^*, P_1$ and $P_2$.
If a vertex set $V'$ contains inconsistent vertices $v_1$ and $v_2$, and for each $v\in V'\setminus \{ v_1, v_2\}$ there are $(v, v_1)$-path and $(v, v_2)$-path with the opposite starting sign around $v$, then the whole $V'$ is strongly connected.
It holds since there exists a proper cycle including the above paths which passes $v_1$ and $v_2$ twice and $v$.
Next, we check the number of additional signs.
The number of additional signs is equal to $\max \{ 2(\gamma -1), |S|+|T|+|Q'|+2|Q|\}$.
If $L_1=\gamma =1$, add only one self-loop thus $1=\max \{ 0, 1\}$. If $L_1=\gamma >1$, add $\gamma -1$ links thus $2(\gamma -1)=\max \{ 2(\gamma -1), \gamma \}$.
Otherwise, we go to Step 3 and add $\min \{ L_1, L_2\}$ links. Next, we add $|\mathcal{C}|+|Q|-1$ links at Step 4.
At Step 5, we consider three cases. It should be noted that $L_2 \leq L_1 - 2$ holds if and only if $2(\gamma -1)\geq |S|+|T|+|Q'|+2|Q|$ holds due to the following relation: $$\begin{aligned}
L_2-(L_1-2)&=\sum_{i=2}^{K}(i-2)k_i-(k_1-|Q|)+2\\
&=\sum_{i=1}^K ik_i-2\gamma +|Q|+2\\
&=|S|+|T|+|Q'|+2|Q|-2(\gamma -1).\end{aligned}$$ If $L_2\leq L_1-2$, we add $L_1-L_2$ links, thus the number of additional signs is $$\begin{aligned}
&2\min \{ L_1, L_2\} +2(|\mathcal{C}|+|Q|-1)+2(L_1-L_2)\\
={}&2(|\mathcal{C}|+|Q|+L_1-1)\\
={}&2(|\mathcal{C}|+k_1-1)\\
={}&2(\gamma -1)\\
={}&\max \{ 2(\gamma -1), |S|+|T|+|Q'|+2|Q|\} .\end{aligned}$$ If $L_2=L_1-1$, we add a link and a self-loop, thus the number of additional signs is $$\begin{aligned}
&2\min \{ L_1, L_2\} +2(|\mathcal{C}|+|Q|-1)+3\\
={}&2\sum_{i=2}^K(i-2)k_i+2\sum_{i=2}^Kk_i+2|Q|+1\\
={}&\sum_{i=2}^K(i-2)k_i+\sum_{i=1}^Kk_i+\sum_{i=2}^Kk_i+|Q|\\
={}&\sum_{i=1}^Kik_i+|Q|\\
={}&|S|+|T|+|Q'|+2|Q|\\
={}&\max \{ 2(\gamma -1), |S|+|T|+|Q'|+2|Q|\} .\end{aligned}$$ Otherwise, the number of additional signs is $$\begin{aligned}
&2\min \{ L_1, L_2\} +2(|\mathcal{C}|+|Q|-1)+(L_2-L_1+2)\\
={}&2(|\mathcal{C}|+|Q|)+L_2+L_1\\
={}&2\sum_{i=2}^Kk_i+2|Q|+\sum_{i=2}^K(i-2)k_i+L_1\\
={}&\sum_{i=1}^Kik_i+2|Q|-k_1+L_1\\
={}&|S|+|T|+|Q'|+2|Q|\\
={}&\max \{ 2(\gamma -1), |S|+|T|+|Q'|+2|Q|\} .\end{aligned}$$ Therefore, the number of additional signs is equal to the obvious lower bound.
Both the above algorithm and the condensation algorithm run in linear time, thus one can obtain an optimal solution in linear time for a general input bidirected graph.
Problem \[ProbGeneral\] with $F(A', \partial' )=\sum_{a'\in A'}|\partial' a'|$ can be solved in linear time.
Minimization on Arcs
====================
In this section, we deal with Problem \[ProbGeneral\] with $F(A', \partial' )=|A'|$.
Let $\lambda (G)$ be defined by $\lambda (G):=\max \left\{ \gamma -1, \lceil (|S|+|T|+|Q'|)/2\rceil +|Q|\right\} $. Clearly, $\lambda (G)$ is the lower bound of Problem \[ProbGeneral\] with $F(A', \partial' )=|A'|$ (which can be derived as well as that for the problem on signs). Unfortunately, however, there is a small example which cannot be made strongly connected by $\lambda (G)$ additional arcs (see Figure \[FigEx\]), whereas we can always achieve the lower bound when we deal with the number of additional signs as shown in the previous section. For the original graph $G$ in Figure \[FigEx\] (a), we have $$\lambda (G)=\max \left\{ \bigg\lceil \frac{1+1+0}{2}\bigg\rceil +0\right\} =\max \left\{ 1, 0\right\} =1.$$ Since there exist a source $s$ and a sink $t$ in $G$, we must add an arc $a=(s, t)$ with proper signs in order to extinguish both source and sink with one arc (see Figure \[FigEx\] (b)). However, it is not strongly connected. Actually, the minimum number of additional arcs to make $G$ strongly connected is two and one of the optimal solutions is shown in Figure \[FigEx\] (c). On the other hand, there is also an graph $G$ which can be made strongly connected with $\lambda (G)$ additional arcs.
; (v1) at (2,2) ; (v2) at (0,1.5) ; (v3) at (-1,-0.5) ; (v4) at (1,-0.5) ; / ǐn [v1/v2,v2/v3,v3/v4,v4/v2]{} () to (); ; ; ; ; ; ; ; ; ; ;
\(a) Original graph.
; (v1) at (2,2) ; (v2) at (0,1.5) ; (v3) at (-1,-0.5) ; (v4) at (1,-0.5) ; / ǐn [v1/v2,v2/v3,v3/v4,v4/v2]{} () to (); ; ; ; ; ; ; ; ; (v2) to (v1); ; ; ; ;
\(b) One arc added.
; (v1) at (2,2) ; (v2) at (0,1.5) ; (v3) at (-1,-0.5) ; (v4) at (1,-0.5) ; / ǐn [v1/v2,v2/v3,v3/v4,v4/v2]{} () to (); ; ; ; ; ; ; ; ; (v2) to (v1); ; ; (v2) to \[out=45,in=135,looseness=10\] (v2); ; ; ;
\(c) Optimal solution.
Now we aim at obtaining the upper bound of the size of optimal solutions. Actually, we can show the next theorem.
Let $G=(V, A; \partial )$ be an acyclic bidirected graph. Then the minimum number of $|A'|$ such that $G'=(V, A\cup A'; \partial' )$ is a strongly connected bidirected graph is $\lambda (G)$ or $\lambda (G)+1$. \[ThmArc\]
Note that if the output of ADDITIONAL SIGNS($G$) contains at most one self-loop, then it is also an optimal solution of the problem of minimizing the number of additional arcs. If the output of ADDITIONAL SIGNS($G$) contains more than 1 self-loops, however, we cannot guarantee the optimality for the problem on arcs in general. We can construct a feasible solution of size $\lambda (G)$ or $\lambda (G)+1$ by the following algorithm.
**Algorithm** ADDITIONAL ARCS($G$)
Step 1
: Let $A':=\emptyset$.
Step 2
: Let $u_1, u_2, \ldots , u_{L_1} \ (L_1:=k_1-|Q|)$ be the elements of $\left( \bigcup_{i=1}^{k_1}C_i^1 \right) \cap (S\cup T\cup Q')$. If $L_1=\gamma =1$, add a self-loop at $u_1$ to $A'$ with a proper sign and go to Step 14. If $L_1=\gamma >1$, add $\{ (u_1, u_i)\mid 2 \leq i \leq L_1\}$ to $A'$ with proper signs and go to Step 14.
Step 3
: Let $\mathcal{C}=\left\{ C_1^2, C_2^2, \ldots , C_{k_2}^2, \ldots , C_1^K, C_2^K, \ldots , C_{k_K}^K \right\}$. For each $C\in \mathcal{C}$, pick up two distinct elements of $C\cap (S\cup T)$ and label them as $l_i, r_i\ (i=1, 2, \ldots , |\mathcal{C}|)$. Label the rest of the elements of $\bigcup_{C\in \mathcal{C}}C\cap (S\cup T)$ as $w_1, w_2, \ldots , w_{L_2}$ with $L_2 :=\sum_{i=3}^{K}(i-2)k_{i}$. Add $\left\{ (u_i, w_i)\mid 1\leq i\leq \min\{ L_1, L_2 \} \right\}$ to $A'$ with proper signs.
Step 4
: Let $q_1, \ldots , q_{|Q|}$ be the elements of $Q$ and define $l_{| \mathcal{C} |+i}=r_{| \mathcal{C} |+i}=q_i$ for $i=1, 2, \ldots , |Q|$.
Step 5
: If $L_2\leq L_1-2$, add $\left\{ (u_i, r_{|\mathcal{C}|+|Q|})\mid L_2 +1<i\leq L_1\right\}$, $\left\{ (r_i, l_{i+1}) \mid 1 \leq i<|\mathcal{C}|+|Q|\right\}$ and $(u_{L_2+1}, l_1)$ to $A'$ with proper signs and go to Step 14.
Step 6
: If $L_2=L_1-1$, add $\{ (r_i, l_{i+1})\mid 1\leq i<|\mathcal{C}|+|Q|\}$, $(u_{L_1}, l_1)$ and a self-loop at $r_{|\mathcal{C}|+|Q|}$ with proper signs and go to Step 14.
Step 7
: If $|Q|=1$, add a new vertex $q_2$ to $V$ and add $(q_1, q_2)$ with $(+, -)$ to $A'$. Otherwise, add $(q_i, q_{i+1})$ to $A'$ with $(+, -)$ for $i=1, 2, \ldots , |Q|-1$.
Step 8
: Define a new bidirected graph $\hat{G}=(\hat{V}, \hat{A}; \hat{\partial})$ from the bidirected graph $G'=(V, A\cup A'; \partial' )$ as follows: $$\hat{V}:=\{ l_i, r_i\mid 1\leq i\leq |\mathcal{C}|\} \cup \{ w_j\mid L_1<j\leq L_2\} \cup \{ q_1, q_{\max \{ 2, |Q|\} }\} ,$$ $$\hat{A}:=\left\{ a=(u, v)\mbox{ with }(\pi _u, \pi _v)\mid \exists (\pi _u, \pi _v)\mbox{-path from }u\mbox{ to }v\mbox{ in }G', \{ u, v\} \subseteq \hat{V}\right\} .$$
Step 9
: Construct a maximal matching $M=\{ m_1, m_2, \ldots , m_{|M|}\}$ ($m_i=(v_i^l, v_i^r)$) in the underlying graph of $\hat{G}$. Add $B:=\{ (v_i^r, v_{i+1}^l)\mid 1\leq i\leq |M|\ (v_{|M|+1}^l:=v_1^l)\} $ to $A'$ with proper signs.
Step 10
: Let $p_1, p_2, \ldots , p_l$ be the vertices in $\hat{V}$ which are not the endpoints of any element of $M$. Add $P:=\left\{ (p_{2i-1}, p_{2i})\mid 1\leq i\leq \lfloor l/2\rfloor \right\}$ with proper signs to $A'$. If $l$ is odd, add a self-loop at $p_l$ to $A'$ with a proper sign.
Step 11
: Let $\tilde{G}$ be the output and $\alpha$ be the map in Step 5 of CONDENSE($G=(V, A\cup A'; \partial' )$).
Step 12
: Let $\tilde{v}$ be the only one element of $S(\tilde{G})\cup T(\tilde{G})\cup Q'(\tilde{G})\cup Q(\tilde{G})$. If $\tilde{v}\in S(\tilde{G})$ (resp. $\tilde{v}\in T(\tilde{G})$), add a minus-loop (resp. a plus-loop) at an arbitrary element in $\alpha^{-1}(\tilde{v})$.
Step 13
: If $|Q|=1$ holds for the original input graph $G$, then remove the arc $(q_1, q_2)$ from $A'$.
Step 14
: Return $G'=(V, A\cup A'; \partial' )$.
Note that the above algorithm finds a feasible solution of size $\lambda (G)$ or $\lambda (G)+1$ in linear time (Table \[TableApprox\]).
Since the algorithm is the same as ADDITIONAL SIGNS($G$) when $L_2\leq L_1-1$, let us give a brief explanation on the case of $L_2\geq L_1-1$. It is sufficient to show that $|S(\hat{G})\cup T(\hat{G})\cup Q'(\hat{G})\cup Q(\hat{G})|=1$ holds after Step 11 is executed. By Step 10 of the algorithm, there is a proper cycle $C$ consisting of alternating sequence of $M$ and $P$ in $\hat{G}$. The other elements in $S(\hat{G})\cup T(\hat{G})\cup Q'(\hat{G})\cup Q(\hat{G})$ are included in some improper cycle the root of which is included in $C$. Therefore, there exists only one element of $S(\hat{G})\cup T(\hat{G})\cup Q'(\hat{G})\cup Q(\hat{G})$ after Step 11. If the graph is not strongly connected, the all vertices have become strongly connected by adding a self-loop with the proper sign.
The cardinality of the solution is $\lambda (G)+1$ only if $\tilde{v}\in S(\tilde{G})\cup T(\tilde{G})$. The above algorithm runs in linear time, thus the total algorithm runs in linear time for a general input bidirected graph.
For Problem \[ProbGeneral\] with $F(A', \partial' )=|A'|$, a feasible solution with $|A'|=\lambda (G)$ or $\lambda (G)+1$ can be found in linear time.
Optimum$\backslash$Output $\lambda$ $\lambda +1$
--------------------------- --------------- --------------
$\lambda$ optimal approximate
$\lambda +1$ $\not\exists$ optimal
: The relation between the output of the algorithm and the optimal solution.[]{data-label="TableApprox"}
Actually, there exists an example such that our algorithm returns the approximate solution (Figure \[FigApproxEx\]). The original graph has five vertices and six $(+, +)$-arcs (Figure \[FigApproxEx\] (a)). If one applies our algorithm to this graph, a solution of four arcs is obtained (Figure \[FigApproxEx\] (b)). However, there exists a solution of three arcs (Figure \[FigApproxEx\] (c)).
; (v1) at (-0.5,1.5) ; (v2) at (0.5,1.5) ; (w1) at (-1,-0.5) ; (w2) at (0,-0.5) ; (w3) at (1,-0.5) ; / ǐn [v1/w1,v1/w2,v1/w3,v2/w1,v2/w2,v2/w3]{} () to (); (w3) to \[out=-45,in=45,looseness=10\] (w3); (v2) to \[out=-45,in=45,looseness=10\] (v2);
\(a) Original graph.
; (v1) at (-0.5,1.5) ; (v2) at (0.5,1.5) ; (w1) at (-1,-0.5) ; (w2) at (0,-0.5) ; (w3) at (1,-0.5) ; / ǐn [v1/w1,v1/w2,v1/w3,v2/w1,v2/w2,v2/w3]{} () to (); (v1) to (w2); (v2) to (w1); (w3) to \[out=-45,in=45,looseness=10\] (w3); (v2) to \[out=-45,in=45,looseness=10\] (v2);
\(b) Output.
; (v1) at (-0.5,1.5) ; (v2) at (0.5,1.5) ; (w1) at (-1,-0.5) ; (w2) at (0,-0.5) ; (w3) at (1,-0.5) ; / ǐn [v1/w1,v1/w2,v1/w3,v2/w1,v2/w2,v2/w3]{} () to (); (v1) to (v2); (w1) to (w2); (w2) to (w3); (w3) to \[out=-45,in=45,looseness=10\] (w3); (v2) to \[out=-45,in=45,looseness=10\] (v2);
\(c) Optimal solution.
Concluding Remarks
==================
In this paper, we propose two types of problems to make a given bidirected graph strongly connected. The first one aims at minimizing the number of additional signs on arcs and the second one aims at minimizing the number of additional arcs. We give a linear-time algorithm to find an optimal solution for the former problem and a linear-time algorithm to find a feasible solution which can have one more arc than an optimal solution for the latter problem.
As future works, the following problem on minimization on arcs can be considered.
Let $G=(V, A; \partial )$ be a bidirected graph. Decide whether the minimum number of additional arcs to make $G$ strongly connected is $\lambda (G)$ or $\lambda (G)+1$.
Connectivity augmentation problems on bidirected graphs can also be considered, e.g., arc-connectivity augmentation. Let $G$ be $k$-arc-connected if $G'=(V, A\setminus A^{\circ}; \partial |(A\setminus A^{\circ}))$ is strongly connected for all $A^{\circ}\subseteq A$ with $|A^{\circ}|=k-1$.
Let $G=(V, A; \partial )$ be a bidirected graph and $k$ be a positive integer. Find additional arcs $A'$ and a boundary operator $\partial' : A\cup A'\to 3^V\ (\partial' a=\partial a\ (\forall a\in A))$ minimizing $F(A', \partial' )$ such that $G$ is $k$-arc-connected.
In a similar way, the definition of $k$-vertex-connectivity and the connectivity augmentation problem on it can be introduced.
Also, the generalization of the problem to make a given undirected graph connected or that to make a given directed graph strongly connected can be considered. Although bidirected graphs can be seen as the common generalization of undirected graphs and directed graphs, the problems in this paper are not the generalization of these classical problems because there is no restriction on additional arcs. For the case of directed graphs, the problem can be formulated as follows:
Let $G=(V, A; \partial )$ be a bidirected graph. Find additional arcs $A'$ and a boundary operator $\partial' : A\cup A'\to 3^V\ (\partial' a=\partial a\ (\forall a\in A))$ minimizing $|A'|$ such that $G':=(V, A\cup A'; \partial' )$ is a strongly connected bidirected graph and $\left| \partial'^+a'\right| =\left| \partial'^-a'\right| =1$ for all $a'\in A'$.
Acknowledgments {#acknowledgments .unnumbered}
===============
Both authors are supported by JSPS Research Fellowship for Young Scientists. The research of the first author was supported by Grant-in-Aid for JSPS Research Fellow Grant Number 16J06879.
[99]{} K. Ando, S. Fujishige: $\sqcup$,$\sqcap$-closed families and signed posets. *Discussion Paper Series*, **567**, Institute of Socio-Economic Planning, University of Tsukuba, 1994. K. Ando, S. Fujishige: On structures of bisubmodular polyhedra. *Mathematical Programming* **74**:293–317, 1996. K. Ando, S. Fujishige, T. Nemoto: Decomposition of a bidirected graph into strongly connected components and its signed poset structure. *Discrete Applied Mathematics*, **68**:237–248, 1996. M. A. Babenko: Acyclic bidirected and skew-symmetric graphs: algorithms and structure. *Computer Science - Theory and Applications, Proceedings of the First International Computer Science Symposium in Russia (LNCS 3967)*, 23–34, 2006. J. Edmonds, E. L. Johnson: Matching: a well-solved class of linear programs. In: R. Guy, H. Hanani, N. Sauer, J. Schönheim (Eds.): *Combinatorial Structures and Their Applications*, 88–92, 1970. K. P. Eswaran, R. E. Tarjan: Augmentation problems. *SIAM Journal on Computing*, **5**:653–665, 1976. A. V. Goldberg, A. V. Karzanov: Maximum skew-symmetric flows. *Proceedings of the Third Annual European Symposium on Algorithms*, 155–170, 1995. A. V. Goldberg, A. V. Karzanov: Path problems in skew-symmetric graphs. *Combinatorica*, **16**:353–382, 1996. A. V. Goldberg, A. V. Karzanov: Maximum skew-symmetric flows and matchings. *Mathematical Programming*, **100**:537–568, 2004. F. Harary: On the notion of balance of a signed graph. *Michigan Mathematical Journal*, **2**:143–146, 1953–1954. S. Iwata: Block triangularization of skew-symmetric matrices. *Linear Algebra and its Applications*, **273**:215–226, 1998. R. M. Karp: Reducibility among combinatorial problems. In: R. E. Miller, J. W. Thatcher (Eds.): *Complexity of Computer Computations*, 85–103, 1972. P. Medvedev, M. Brudno: Maximum likelihood genome assembly. *Journal of Computational Biology*, **16**:1101–1116, 2009. P. Medvedev, K. Georgiou, G. Myers, M. Brudno: Computability of models for sequence assembly. *Algorithms in Bioinformatics*, 289–301, 2007. W. T. Tutte: Antisymmetrical digraphs. *Canadian Journal of Mathematics*, **19**:1101–1117, 1967. T. Yasuda: Inferring chromosome structures with bidirected graphs constructed from genomic structural variations. Ph.D. Thesis, The University of Tokyo, 2015. T. Zaslavsky: Orientation of signed graphs. *European Journal of Combinatorics*, **12**:361–375, 1991.
[^1]: The University of Tokyo, Japan ([tatsuya\_matsuoka@mist.i.u-tokyo.ac.jp]{}).
[^2]: The University of Tokyo, Japan ([shun\_sato@mist.i.u-tokyo.ac.jp]{}).
|
---
abstract: 'A new type of self-similar potential is used to study a multibarrier system made of graphene. Such potential is based on the traditional middle third Cantor set rule combined with a scaling of the barriers height. The resulting transmission coefficient for charge carriers, obtained using the quantum relativistic Dirac equation, shows a surprising self-similar structure. The same potential does not lead to a self-similar transmission when applied to the typical semiconductors described by the non-relativistic Schrödinger equation. The proposed system is one of the few examples in which a self-similar structure produces the same pattern in a physical property. The resulting scaling properties are investigated as a function of three parameters: the height of the main barrier, the total length of the system and the generation number of the potential. These scaling properties are first identified individually and then combined to find general analytic scaling expressions. diazd@uaem.mx'
author:
- 'Díaz-Guerrero D. S.^1^\*, Gaggero-Sager L. M.^1^, Rodríguez Vargas I.^2^, & Naumis G.G.^3^'
title: 'Self-similar Charge Transport in Gapped Graphene'
---
Graphene[@novoselov2004] is considered as one of the most promising new materials. This first truly two-dimensional crystal [@novoselov2005] has impressive physical properties [@novoselov2004; @balandin2008]. As a consequence, graphene is not only important from a technological point of view, but also is considered as an inflexion point in quantum physics. For instance, charge carriers in graphene follow an effective quantum relativistic (Dirac) equation instead of the usual Schrödinger one. This leads to new effects [@Geim2007; @Novoselov2011; @Stone2012]. Specially, the scattering produced by external potentials or impurities is different from what is observed in ordinary materials. Charge carriers can be transmitted perfectly due to the Klein effect and random sustitutional impurities can produce multifractal states [@Naumis1994; @Naumis2002]. Thus, a whole new world is open to investigate the possible effects of external potential geometries, which can be imposed to graphene by different means, as for example, using certain type of substrates, strain, electrostatic gates, impurities, electromagnetic fields, etc. [@Giovannetti2007; @CastroNetoReview2009].
For applications, an important issue is how to engineer substrate-induced bandgap opening in epitaxial graphene [@Giovannetti2007]. This leads to investigate how different multibarrier geometries affect the properties of graphene. Among these possible geometries, self-similarity has a paramount importance, since scale invariance is a fundamental property of many natural phenomena, as can be corroborated in reports that ranges from the distribution and abundance of species, temporal occurrence of earthquakes, and even in the growth of complex networks and trees [@JHarte1999Science; @ACorral2004PRL; @CSong2006NaturePhysics; @CEloy2011PRL].
From the technological standpoint, self-similarity can also be exploited to produce useful devices [@MSun2006PRB; @FMiyamaru2009APL; @HXDing2012APL] . Particularly, photoconductive fractal antennas show an efficient multiband emission of terahertz radiation owing to the self-similarity of the fractal structure [@FMiyamaru2009APL].
Despite that self-similar structures are getting plenty of attention [@EMacia2006Review], only a handful of experiments corroborate a self-similar behavior in its physical properties [@ALavrinenko2002PRE; @BHou2004APL; @JEstevez2012JAP]. Likewise, many theoretical works claim that physical properties, such as tranmission or reflection probabilities, display self-similarity. Unfortunately, most of them are just a matter of visual perception, since the scaling properties are never reported. Even in the case of quasicrystals, which are considered as one of the archetypical examples of self-similar structures, it has been elusive to find the scaling properties of the corresponding physical properties [@Naumis-Aragon1996; @Nava-Taguena-delRio2009].
Here we propose a novel self-similar multibarrier structure in graphene in which the main physical property, namely charge transport, presents a surprising self-similar pattern. This self-similar structure does not produce a similar behavior in a usual semiconducting material described by the Schrödinger equation. Thus, two essential ingredients are shown to be into play: a self-similar structure and quantum relativistic mechanics.
{width="4in" height="4in"}
To build our multibarrrier structure in one direction (the $x$ axis), we implement a variant of the middle third Cantor case [@cantorSet] (see supplementary material) using square barriers as indicated in Fig. \[potential\]. We begin with a line-segment of length $Lt$, we divide it in thirds and put a rectangular barrier of energy $V_0$ in the middle third (its width is $Lt/3$). We call this the generation one ($G1$). The second step consists in taking the remaining line-segments, divide them into thirds and place within scaled copies of the existing barrier. We scale the height and width of the barrier by a scaling factor of $1/3$. Additionally the existing barrier is also scaled in its width. Since scaling is a contraction transformation, each scaled copy will have a unique fixed point, *see the fixed point theorem in metric spaces* [@ApostolBook]. By fixed point we meant an element of the function’s domain that is mapped to itself by the function. In this case, this point is a fixed side of the barriers, which will be chosen as the right-hand side. So, every successive iteration (generation) ($Gj$) of the structure is obtained by repeating the second step of the construction for the remaining line-segmentes, using as the “existing barrier” those added in the last iteration, as shown in Fig. \[potential\].
This potential can be described as follows. The $j$-th generation of the potential will consists of $N_{Gj}$ barriers; the corresponding barriers (those added in the $j$-th iteration) has a width $W_G(j)$ and energy (height) $V_G{j}$. Such barrier is obtained as a scaled replica of barriers from generation $G(j-1)$, $$\begin{aligned}
N_{Gj}=2^{Gj}-1, \nonumber \\
W_{Gj}=\alpha=\frac{Lt}{3^{Gj}}, \nonumber \\
V_{Gj}=\frac{V_0}{3^{Gj-1}}. \nonumber\end{aligned}$$
This bidimensional potential does not depend on the perpendicular direction ($y$ axis), i.e., $V(x,y)=V(x)$ and can be adapted to graphene [@novoselov2004; @KNovoselov2005Nature; @YZhang2005Nature; @Beenakker2009] by means of symmetry breaking substrates [@SZhou2007NMaterials; @JVGomes2008JPCM; @VHNguyen2011SST], by gated graphene or in other systems, like optical analogies, with a linear spectrum [@cheianov2007; @Hartmann2010; @Hartmann2014].
Let us now study the transmission coefficient of charge carriers in this multibarrier system. For graphene, we use the Viana-Gomes et al. formalism [@JVGomes2008JPCM] to compute the transmission coefficient of electrons in gapped graphene. The Hamiltonian for the Dirac electrons under a potential $V(x)$ is given by $$H=v_F{\bf \sigma}\cdotp {\bf p} + \sigma_z V(x),$$ where $V(x)=m(x)v_F^2$ if $x\in[x_1,x_2]$ and zero in other case. $V(x)$ defines the region where there is a mass term ($q$-region) and where carriers are massles Dirac fermions ($k$-region). The first term of the Hamiltonian is just the Dirac equation used to describe electrons in graphene, where $v_F$ is the Fermi velocity, ${\bf \sigma}$ the set of Pauli matrices and ${\bf p}$ the moment. For normal incidence on the $x$ direction the corresponding wavefunctions are [@JVGomes2008JPCM] $$\psi_k^{\pm}(x)=\frac{1}{\sqrt{2}}\left(
\begin{array}{c}
1 \\
u_\pm
\end{array}\right)
e^{\pm ik_xx},$$ and $$\psi_q^{\pm}(x)=\frac{1}{\sqrt{2}}\left(
\begin{array}{c}
1 \\
v_\pm
\end{array}\right)
e^{\pm iq_xx},$$ where $u_\pm =\pm \mbox{sign}(E)$ and $v_\pm =\frac{E-V(x)}{\hbar v_F(\pm q_x-ik_y)}$. Using these equations, one establishes the continuity conditions when the wave goes from the $k$-region to the $q$-region.
{width="4.5in" height="4.5in"}
With the potential described in Fig. \[potential\] at hand, we compute the transmission coefficient, defined as $$T_G(E)=\frac{|\psi_k^{\pm}(0)|^2}{|\psi_k^{\pm}(Lt)|^2}$$
using the transfer matrix formalism [@SokoulisBook; @JVGomes2008JPCM; @IRVargas2012JAP]. The transfer matrix can be obtained by imposing the continuity conditions of the wave functions in each region of the self-similar structure. From this we can extract the transmission coefficient for a given energy, see for instance [@JVGomes2008JPCM; @IRVargas2012JAP].
Since the construction of the multibarrier structure is based on the length-scaling of the previous generation barriers, and the addition of energy-scaled new barriers, we search for relations between transmission curves corresponding to different generations of the multibarrier structure. So, we fix the total length of the structure, denoted by $Lt$ (in this case we use the value of $Lt=3500 \ \mathring{A}$) and the height of the main barrier, denoted by $V_0$ (using the value of $V_0=1$ eV).
In figure \[gen-comparison\]a we show the transmission curves corresponding to generations 6 and 7 of the potential, denoted by $T_6(E)$ and $T_7(E)$ respectively. As we can see, it is qualitatively clear that they are related by some kind of transformation. This transformation turns out to be $T_6(E)=\left[T_7(E)\right]^9$, where the subindex stands for the generation number, as seen in figure \[gen-comparison\]b. Thus, $T_6(E)$ are $T_7(E)$ scaled by a single exponent. As may be expected, other bigger generations also follow the same rule. More formaly $$T_G(E)\approx \left[ T_{G+1}(E)\right]^9,
\label{g-scaling}$$ where $G\geq 6$ and $G$ stands for the generation number. This means that any two transmission curves that differ only in the generation of their correspondig potential, and are related by some power of the form $9^m$, where $m$ is the difference between the generations. This suggests that the transmission curves follows an scaling rule which is a signature of the self-similar phenomena. In fact, here we are in presence of the first true self-similar physical property of graphene. Morever, even for perfect fractals it is known that the physical properties usually follow a multifractal behavior [@Nava-Taguena-delRio2009; @NaumisMultiFrac2007] but here only one single fractal exponent is obtained.
It is important to remark here that there are always finite size effects due to the break in the scaling of the potential, which affects the first generations. In this particular case, one cannot extend self-similarity beyond the length of the system. This effect is explained in the supplementary section where we present the transmittance from generation $G1$ to $G9$. For low generations, the transmittance is basically dominated by the biggest energy barrier. Between $G5$ and $G6$, there is a dramatic change in the behavior since now the system has transmittance for energies lower than the main barrier height. As generations increase, scaling appears. Generation $6$ is the crossover since as shown in the supplementary section, geometrical finite size effects in the potential die out after it. Also, is clear that in real systems there is also another finite size effect at the bottom of the hierarchy, where the Dirac approach can not be made due to the rapid variation of the potential at atomic distances.
To highlight the difference with a typical Schrödinger semiconductor, in figure \[gen-comparison\], panels c) and d), we present the transmission calculated for the same potential generations using Al$_{x}$Ga$_{1-x}$As/GaAs. In such system, the height of the quantum wells are controlled by the Aluminium concentration ($x$).
So far, we have found a scaling between transmission curves that correspond to different generations of the potential, but it seems possible to extend the search for scaling features to other parameters. We have two options at hand, the energy of the starting barrier and the total length of the structure.
{width="4.5in" height="4.5in"}
Let us explore now the first option, that is, we take a fixed generation of the potential, say generation 6, and make the calculation for two starting energies. Lets take $V_0=0.5$ eV and $V_0=1.0$ eV. In this case, the appearance of the curves seen in figure \[H0Lt-comparison\]a, resembles those corresponding to a change in the generation of the potential. In the case of the nonrelativistic transmission coefficient (e. g. GaAs), for the same geometric set up, one observes in general a shift of the first transmission resonance. Nevertheless, in this case the general behavior of the transmission curve is qualitatively the same. Due to the resemblance of these curves with the ones of changing generations, we try the same kind of transformation but with a different power. In this case, one has to raise the curve corresponding to $V_0=0.5$ eV to the fourth power to get the one corresponding to $V_0=1.0$ eV, see figure \[H0Lt-comparison\]b. Thus $T_{1.0}(E)\approx T_{0.5}(E)^4$, where the subindex stands for the $V_0$ of the respective structure. In fact, this scaling behavior is more general, i. e., it applies for every multiple of two, resulting in the expression $$T_{\frac{1}{k}V_0}(E)^{k^2}\approx T_{V_0}(E).
\label{v-scaling}$$
Finally we compare curves corresponding to different lengths of the whole structure, denoted by $Lt$. Although nothing suggests scaling in this variant at all, see figure \[H0Lt-comparison\]c, our previous results lead us to a different conclusion. From the results for the case of changing $V_0$, we try to find a similar scaling law for $Lt$, but in this case, the scaling in the energy gives the relation, $$T_{\frac{1}{\alpha}Lt}(\frac{1}{\alpha}E)^{\alpha^2}\approx T_{Lt}(E).
\label{Lt-scaling}$$ With this scaling rule applied to the curve corresponding to $500$ $\mathring{A}$, a very good approximation is obtained, see figure \[H0Lt-comparison\]d.
Combining expressions (\[g-scaling\]), (\[v-scaling\]) and (\[Lt-scaling\]), it is possible to get a good approximation to almost every curve that results from the combination of parameters. To do this, all the parameters and the variable $E$ will be arguments of $T$, $$T(E,G,V_0,Lt)\approx T(\alpha E,G-m,\frac{1}{k}V_0,\alpha Lt)^{k^2/\alpha^2 9^m},
\label{unified-scaling}$$ for $G-m \geq 6$. All these approximations are quite useful to predict the transmission as function of the parameters, but it is necesary to test the correctness. So we calculate the root mean square of the difference between the target curve and the scaled one, we call it $drms$. For parameter $G$ (scaling between generations of the potential, see figure \[gen-comparison\]b) the $drms$ is $5.50\times 10^{-5}$; for $V_0$ (scaling in the energy of the main barrier, see figure \[H0Lt-comparison\]b) the $drms$ is $2.05\times 10^{-4}$; at last for $Lt$ (scaling in the total length of the multibarrier structure, see figure \[H0Lt-comparison\]d) the $drms$ is $1.43\times 10^{-3}$. From this it is quite clear that the better approximation is for the scaling between generations.
This lead us to examinate a wider range of energy and check our scaling-laws values (see supplementary material). An analysis of the results, shows that the fit is excellent for small energies but tends to decrease in quality as the energy increases, as one should expect in the limiting case of very high energies. Also, the error tends to grow for small transmittance energies. However, as seen in the supplementary material, if the error is computed for bigger generations, for example, for $G=7$ against scaled $G=8$, then the error decreases in a dramatic way, as expected for a good scaling relation. The drms for the Schroedinger case does not reduce as the generations grow, suggesting the absence of a fixed point.
The numerical results presented above suggests that this type of potential indeed shows scaling properties, although this scaling is not found in a single curve, but between curves corresponding to different parameters. This is a very interesting result because in fact the transmission coefficient is an intricate combination of the wave functions, and even more, for a multibarrier potential it is the result of the product of several transfer matrices. This suggests that the wave function have self-similar properties. That is the affirmation in the Gaggero-Pujals theorem [@Gaggero-Sager-chap] for nonrelativistic electrons. Thus, it seems plausible that an extension of that theorem could be established for relativistic equations. Another important possibility is to study the nonperpedicular incidence, since Klein tunneling is ruled out in this case [@katsnelson].
In conclusion, the self-similar potential proposed here show various kinds of scaling properties in the transmission curves. This scaling properties gave the possibility to obtain, given a transmission curve and its set of parameters, an almost perfect approximation of the curve correspondig to certain transformation of the original parameters, see (\[unified-scaling\]). It is also suggested that the scaling properties of the transmission curves are drastically affected by the mix of two geometric symmetries, for example, reflection and self-similarity, see [@Diaz-Guerrero2008; @Gaggero-Sager-chap]. We believe that the proposed potential is ideal to study scaling properties in multibarrier systems, although our preliminary results considering electrostatic self-similar barriers in grafene indicate also the presence of a self-similar behavior.
Finally, we found that two ingredients are needed in order to obtain a pure monofractal scaling of the transmittance, a [*self-similar barrier and quantum relativistc equations*]{}, since the non-relativistic version does not display scaling. A possible heuristic explanation for such phenomena lies in the linear energy-momentum dispersion relation of the Dirac equation.
[10]{}
Novoselov, K. S. et al. Electric Field Effect in Atomically Thin Carbon Films. *Science*, **(**306), 666–669 (2004).
Novoselov, K.S. et al. Two-dimensional atomic crystals *Proc. Natl. Acad. Sci. USA*, **102** 10451–10453 (2005).
Balandin, A.A. et al. Superior thermal conductivity of single-layer graphene. *Nano Lett.*, **8**, 902–907 (2008).
Geim, A. K., & Novoselov, K. S. The rise of graphene. *Nat. Mater.* **6**, 183–191 (2007).
Novoselov, K. S. Nobel lecture: Graphene: Materials in the flatland. *Rev. Mod. Phys.*, **83** 837–849 (2011).
Stone, D. A., Downing, C. A., & Portnoi, M. E. Searching for Confined Modes in Graphene Channels: The Variable Phase Method *Phys. Rev. B*, **86** 075464 (2012).
Naumis, G. G., Barrio, R. A., & Wang, C. Effects of frustration and localization of states in the Penrose lattice. *Phys. Rev. B*, **50** 9834–9842 (1994).
Naumis, G. G., Wang, C., & Barrio, R. A. Frustration effects on the electronic density of states of a random binary alloy *Phys. Rev. B*, **65** 134203 (2002).
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S., & Geim, A. K. The electronic properties of graphene. *Rev. Mod. Phys.*, **81** 109–162 (2009).
Giovannetti, G., Khomyakov, P. A., Brocks, G., Kelly, P. J., & van den Brink, J. Substrate-induced band gap in graphene on hexagonal boron nitride: Ab initio density functional calculations. *Phys. Rev. B*, **76** 073103 (2007).
Corral, A. Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes. *Phys. Rev. Lett.* **92** 108501 (2004).
Eloy, C. Leonardo’s Rule, Self-Similarity, and Wind-Induced Stresses in Trees *Phys. Rev. Lett.*, **107**, 258101 (2011).
Harte, J., Kinzig, A., & Green, J. Self-similarity in the distribution and abundance of species *Science*, **284** 334–336 (1999).
Song, C., Havlin, S., & Makse, H. A. Origins of fractality in the growth of complex networks *Nat. Phys.*, **2** 275–281 (2006).
Ding, H. X., Shen, Z. H., Ni, X. W., & Zhu, X. F. Multi-splitting and self-similarity of band gap structures in quasi-periodic plates of Cantor series. *Appl. Phys. Lett.*, **100**, 083501 (2012).
Miyamaru, F. et al. Emission of terahertz radiations from fractal antennas *Appl. Phys. Lett.*, **95** 221111 (2009).
Sun, M. et al. Transmission properties of dual-band cross-dipole fractal slit arrays for near- and mid-infrared wavelengths *Phys. Rev. B*, **74** 193404 (2006).
Maciá, E. The role of aperiodic order in science and technology. *Rep. Prog. Phys.*, **69** 397 (2006).
Hou, B., Xu, G., Wen, W., & Wong, G. K. L. Diffraction by an optical fractal grating *Appl. Phys. Lett.* **85** 6125 (2004).
Estevez, J. O., Arriaga, J., Méndez-Blas, A., Robles-Cháirez, M. G., & Contreras-Solorio, D. A. Experimental realization of the porous silicon optical multilayers based on the 1-s sequence *J. Appl. Phys.*, **111** 013103 (2012).
Lavrinenko, A. V., Zhukovsky, S. V., Sandomirski, K. S., & Gaponenko, S. V. Propagation of classical waves in nonperiodic media: Scaling properties of an optical Cantor filter *Phys. Rev. E*, **65** 036621 (2002).
Naumis, G.G., & Aragon, J. L. Substitutional disorder in a Fibonacci chain: Resonant eigenstates and instability of the spectrum *Phys. Rev. B*, **54**, 15079 (1996).
Nava, R., Tagueña-Martínez, J., del Rio, J. A., & Naumis, G. G. Perfect light transmission in Fibonacci arrays of dielectric multilayers. *J. Phys.: Condens. Matter*, **21**, 155901 (2009).
Cantor, Georg. Ueber unendliche, lineare punktmannichfaltigkeiten. *Mathematische Annalen*, **21**, 545–591 (1883).
Apostol, Tom M. *Mathematical Analysis*. \[92\] (Addison-Wesley Publishing Company), 1983.
Novoselov, K. S., et al. Two-dimensional gas of massless Dirac fermions in graphene. *Nature*, **438** 197–200 (2005).
Zhang, Y., Tan, Y. W., Stormer, H. L., & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. *Nature*, **438** 201–204 (2005).
Beenakker, C. W. J., Sepkhanov, R. A., Akhmerov, A. R. & Tworzydlo, J. Quantum Goos-Hänchen Effect in Graphene *Phys. Rev. Lett.*, **102** 146804 (2009).
Nguyen, V. H., Bournel, A., & Dollfus, P. Resonant tunneling structures based on epitaxial graphene on SiC *Semicond. Sci. Technol.*, **26** 125012 (2011).
Gomes, J. Viana, & Peres, N M R. Tunneling of Dirac electrons through spatial regions of finite mass. *J. Phys.: Condens. Matter*, **20** 325221 (2008).
Zhou, S. Y. et al. Substrate-induced bandgap opening in epitaxial graphene. *Nature Mater.*, **6** 770–775 (2007).
Cheianov, V. V, Fal’ko, V., & Altshuler, B. L. The Focusing of Electron Flow and a Veselago Lens in Graphene *p-n* Junctions *Science*, **315** 1252 (2007).
Hartmann, R. R., Robinson, N. J., & Portnoi, M. E. Smooth Electron Waveguides in Graphene *Phys. Rev. B*, **81** 245431 (2010).
Hartmann, R. R., & Portnoi, M. E. Quasi-exact Solution to the Dirac Equation for the Hyperbolic-secant Potential *Phys. Rev. A*, **89** 012101 (2014).
Rodríguez-Vargas, I., Madrigal-Melchor, J., & Oubram, O. Resonant tunneling through double barrier graphene systems: A comparative study of Klein and non-Klein tunneling structures. *J. Appl. Phys.*, **112** 073711 (2012).
Markos, P. and Soukoulis, C. M. *Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials*. \[3-10\] (Princeton University Press), 2008.
Naumis, Gerardo G. Minimal multifractality in the spectrum of a quasiperiodic Hamiltonian *Phys. Lett. A*, **365**, 171–174 (2007).
Gaggero-Sager, L. M., Pujals, E., Diaz-Guerrero, D. S., & Escorcia-Garcia, J. \[Self-Similarity in Semiconductors: Electronic and Optical Properties\]. *Optoelectronics - Materials and Techniques*. \[Prof. P. Predeep (Ed.)\] \[435–458\] (InTech) 2011. Available at:http://www.intechopen.com/books/optoelectronics-materials-and-techniques/self-similarity-in-semiconductors-electronic-and-optical-properties
Katsnelson, M. I., Novoselov, K. S., & Geim, A. K. Chiral tunnelling and the Klein paradox in graphene. *Nat. Phys.*, **2**, 620–625 (2006).
Diaz-Guerrero, D. S., Montoya, F., Gaggero-Sager, L.M., & Perez-Alvarez, R. Transmittance and fractality in a Cantor-like multibarrier system. *Progress In Electromagnetics Research Letters*, **2**, 149–155 (2008).
|
---
abstract: 'This paper presents a mutual information (MI) based algorithm for the estimation of full 6-degree-of-freedom (DOF) rigid body transformation between two overlapping point clouds. We first divide the scene into a 3D voxel grid and define simple to compute features for each voxel in the scan. The two scans that need to be aligned are considered as a collection of these features and the MI between these voxelized features is maximized to obtain the correct alignment of scans. We have implemented our method with various simple point cloud features (such as number of points in voxel, variance of z-height in voxel) and compared the performance of the proposed method with existing point-to-point and point-to-distribution registration methods. We show that our approach has an efficient and fast parallel implementation on GPU, and evaluate the robustness and speed of the proposed algorithm on two real-world datasets which have variety of dynamic scenes from different environments.'
author:
- 'Nikhil Mehta$^{1}$, James R. McBride$^{2}$ and Gaurav Pandey$^{2}$[^1][^2] [^3]'
bibliography:
- 'ref.bib'
title: '**Robust and Fast 3D Scan Alignment using Mutual Information** '
---
INTRODUCTION
============
A 3D alignment algorithm to determine the relative rigid body transformation between two partially overlapping scans is an underpinning tool for many applications in mobile robotics including localization, mapping and navigation systems. In this work, we consider a robot which obtains two 3D scans ($A$ and $B$) from two poses $P_0$ and $P_1$ via a 3D laser scanner. Provided that some part of the environment is common to both scans, it is generally possible to find a rigid-body transformation $T$ that can project the points in $P_1$ so that they align with $P_0$. The solution to the process of scan alignment ($T$) is parameterized by six values: three translation components ($t_x$, $t_y$ and $t_z$) and three rotation components ($\theta_x$, $\theta_y$ and $\theta_z$). The reason that scan alignment problem is at the center of most navigation, mapping and localization systems, is simply because the rigid body transformation $T$ derived from alignment is of higher quality than odometry estimate (due to wheel slippage and surface irregularities).
The primary challenge in the problem is to minimize the runtime complexity while maximizing the robustness of the solution. Most existing methods are either designed around computationally-efficient local searches which are not robust to initialization or global search methods which are computationally intense. At vehicle’s dead-reckoning error, the initial estimate can be far from global maximum resulting in large errors when using local-search methods. To counter this problem, most implementations currently involve running the registration algorithm at high frequency rate resulting in a constant load on computation resources.
![An overview of mutual information (MI) based scan alignment framework. MI maximization is applied over the voxelized-features computed from two partially overlapping scans. []{data-label="fig:algo_flow"}](images/flow_chart.png){width="0.95\linewidth"}
Our approach poses the alignment problem in a MI (mutual information) maximization framework: it finds the rigid body transformation that maximizes MI between the features of two scans. An overview of our approach is illustrated in Figure \[fig:algo\_flow\]. We show that even primitive features like variance and number of points give results better than the other state of the art point-to-point and point-to-distribution methods for large prior errors from odometry.
The central contributions of this paper are:
- We present a robust and fast framework for scan alignment by posing the problem in a MI maximization framework with simple voxelized features.
- Unlike other 3D scan alignment methods, our approach allows us to consider the no-feature voxels (unoccupied region in scene).
- We present a detailed empirical evaluation of our method in different environments: both urban and rural scenes. We compare our approach with a point-to-point and a point-to-distribution alignment method.
- We show how the proposed approach has an efficient fast and robust GPU implementation, freeing the CPU for other important tasks.
The quality and robustness of our method along with it’s ability to work in real-time, makes it ideal for mobile robotic systems in which accuracy is of high importance.
In the following section we present a brief overview of the prior work (Section \[related\_work\]). In section \[approach\], we describe the proposed framework. Empirical evaluation and comparison with other methods along with runtime analysis is shown in section \[evaluation\].
RELATED WORK {#related_work}
============
Iterative Closest Point (ICP) [@Besl:1992:MRS:132013.132022] is one of the most popular scan registration algorithms used to estimate the optimal transformation between two overlapping scans. In ICP, closest points between scan $A$ and reference scan $B$ are used to obtain a closed-form solution by optimizing the sum of squared distances (usually Euclidean distance). Performing the nearest neighbor search in ICP becomes a bottleneck due to its high computational cost, however, using a K-D Tree [@1240280] does mitigate the problem to an extent. Iterative Dual Correspondence (IDC) [@Lu1997] generates corresponding points for both rotation and translation separately, with optimization done in an alternate fashion. This improves the alignment accuracy when the initial estimate has large rotational error. Iterative Closest Line (ICL) [@55667789-MIT], [@297537764-MIT], [@Censi08anicp] is a variant of ICP in which instead of matching the points in both the scans, the query points in scan $B$ are matched to lines extracted from points in reference scan $A$. Generalized ICP (GICP) [@Segal-RSS-09], attaches a probabilistic model in estimating the correct corresponding points by taking the covariance structure derived from the local neighborhood in the environment.
Another widely used algorithm is Normal Distribution Transform (NDT) which was initially proposed for 2D data [@Biber2003TheND] and was later extended to three-dimensions. 3D-NDT [@Magnusson053dmodelling] is a point-to-distribution method in which the maximum likelihood estimate (MLE) of points in $B$ is maximized over the distribution of points in a gaussian mixture model derived from $A$. Supervoxel based NDT (SV-NDT) [@kim2016] is an extension to 3D-NDT in which segmentation on the basis of local-spatial structure is done over $A$ before creating a GMM model.
The problem of scan alignment with fused sensor data as input has also been extensively researched in the past two decades. Some of the work includes incorporating color information in the ICP framework ([@Johnson:1997:RIT:523428.825385], [@Godin2001AMF]) by augmenting RGB color channels to 3D coordinates and by exploiting the co-registration of 3D data with the available camera imagery to associate scale invariant feature transform (SIFT) [@Lowe:2004:DIF:993451.996342] or speeded up robust features (SURF) [@Lowe:2004:DIF:993451.996342] features to the 3D points as in [@f8a4caebc7cf412ca3ed2113bf6c94df].
Mutual information based alignment methods was first proposed in [@pmid8331222] for multi-modality medical images. Since then literature has been filled with work inspired by mutual information, these include minimization of distribution of joint histograms [@Hill1993], data alignment from multiple modalities (different sensors) as in [@Viola1997], [@pmid9101328], [@pmid10709702], and [@6386053]. In [@1216223], a comprehensive survey of mutual information based techniques being used in medical images is presented.
The proposed method is closely related to FPFH [@6386053] mainly because we use mutual information to compute the registration parameters. However, the feature selection and histogram creation method is different. In [@6386053] high dimensional features are computed using FPFH which are then quantized into one of the precomputed codewords. There is no denying that FPFH are better descriptors of a scan than simple voxelized features like z-variance, but in a MI based framework, we don’t need to find correspondences between patches, therefore computing FPFH only increases the run-time complexity due to nearest neighbor search and normal computation at each point. Moreover, unlike voxelized features, FPFH fails to take into account the empty region in the scan.
[.48]{} ![This figure shows the cropped 3D textured point cloud of a pair of scan transformed to the same frame of reference. (a) Transformation used is from the initial estimate. (b) Transformation applied is first improved/aligned using our approach.[]{data-label="fig:qualitative_analysis"}](images/trans_base_new.png "fig:"){width="\linewidth" height="\linewidth"}
[.48]{} ![This figure shows the cropped 3D textured point cloud of a pair of scan transformed to the same frame of reference. (a) Transformation used is from the initial estimate. (b) Transformation applied is first improved/aligned using our approach.[]{data-label="fig:qualitative_analysis"}](images/trans_mi_image.png "fig:"){width="\linewidth" height="\linewidth"}
METHODOLOGY {#approach}
===========
Approach
--------
A qualitative result of the proposed MI based approach using voxelized features is shown in Figure \[fig:qualitative\_analysis\]. In order to estimate the correct rigid body transformation $T$, represented by a $[4 \times 4]$ transformation matrix, we first initialize our scene with a large 3D grid with resolution $R_v$ ($1\times1\times1$) that encompasses both the scans $A$ and $B$. We create two voxel-point mapping tables by finding the corresponding voxel (the voxel which contains the point) for all points in scan A and B. Now we calculate simple features like (1) Variance of z-height, or (2) number of points in voxel. It is important to note that unoccupied voxels are also included in our voxel-point mapping table with $\phi$ (no-feature) value; this results in a voxel-feature mapping that contains large number of voxels with feature $\phi$. Intuition behind this approach is that in a perfectly aligned state not only the occupied voxels should be aligned, but unoccupied voxels should also be aligned in the overlapping region.
To obtain the statistical dependence using mutual information of two partially overlapping scans, we define an overlapping region which is a subgrid parameterized by six variables $[x_{min}^o, x_{max}^o, y_{min}^o, y_{max}^o, z_{min}^o, z_{max}^o]$ (where ’o’ is the overlapping region). We consider the feature maps of this overlapping subgrid as two random variables representing the structure of the environment. These random variables are used to calculate the mutual information of both the scans. The mutual information between these two random variables is the amount of information obtained about one variable, through the information about the other variable. Therefore, under the correct rigid body transformation, we can expect the information from one of the scans to give maximum amount of information about the other scan, thus rendering the MI to be maximum between the feature maps.
![This figure shows 4 joint histograms: 2 before alignment (Row 1) and 2 after alignment using the proposed method (Row 2). Each column represents two different test cases: in Column 1 the scan pair had the initial heading error of 4$^{\circ}$ and in Column 2 the scan pair had the initial heading error of 12${^\circ}$. X-axis and Y-axis of all the 4 joint histograms represent the variance of Z in a voxel for scan A and scan B respectively. Note how the dispersion in the joint histogram decreases (forming a straight line) after alignment in both the test cases. It can also be seen that the dispersion in joint histogram increases as we increase the heading error from Column 1 - Column 2 (4$^{\circ}$ to 12${^\circ}$). Best viewed in color.[]{data-label="fig:joint_histos"}](images/joint_histos_2_new.png){width="50.00000%"}
$$\label{eq:cost}
\eta = \operatorname*{arg\,max}_\eta MI(X, Y; \eta)$$
where $ \eta = [t_x, t_y, t_z, \theta_x, \theta_y, \theta_z] $ are the required transformation parameters, $ [X, Y] $ are the respective random variables representing the features of scan A and B in the corresponding overlapping region and $MI(X, Y; \eta)$ is the mutual information of the overlapping voxels under the transformation $\eta$.
$$\label{eq:mi}
MI(X,Y; \eta) = H(X;\eta) + H(Y;\eta) - H(X,Y;\eta)$$
where $ H(X;\eta) $ and $ H(Y;\eta) $ are the entropies of the random variables X and Y respectively, and $ H(X,Y;\eta) $ is the joint entropy of the two variables at transformation $\eta$.
$$\label{eq:marginal_A}
H(X;\eta) = -\sum_{x \in X} {\mathsf{P}}_X(x;\eta)\log {{\mathsf{P}}_X(x;\eta)}$$
$$\label{eq:marginal_B}
H(Y;\eta) = -\sum_{y \in Y} {\mathsf{P}}_Y(y;\eta)\log {{\mathsf{P}}_Y(y;\eta)}$$
$$\label{eq:joint_AB}
H(X,Y;\eta) = -\sum_{x \in X}\sum_{y \in Y} {\mathsf{P}}_{XY}(x,y;\eta)\log {{\mathsf{P}}_{XY}(x,y;\eta)}$$
Here ${\mathsf{P}}_X(x;\eta)$ is the marginal probability for $X=x$, ${\mathsf{P}}_Y(y;\eta)$ is the marginal probability for $Y=y$ and ${\mathsf{P}}_{XY}(x,y;\eta)$ is the joint probability for $X=x$ and $Y=y$ when transformation parameters are $\eta$. For simplicity, we have taken x and y to be either the variance in z or the number of points in a voxel.
In each iteration, the MI value changes and is maximized when all voxels (occupied and unoccupied) are aligned. It is important to note that with every iteration, the overlapping region will change and the correct subgrid will be found when the MI is maximum. Figure \[fig:joint\_histos\] represents the joint-distribution before (row 1) and after (row 2) MI maximization for two pairs of scan with different initial error. It can be seen that the joint distribution converges to a line passing through origin and becomes relatively uniform after the correct alignment.
Two scans A and B with some partial overlap. Initial guess of the rigid-body transformation $T_o$. Estimated registration parameter T\
*Initialization* : Extract the associated voxel ids for all points in scan A Create scan A feature map for all voxels\
*Alignment* : $T_c = T_o$ Transform scan B using current rigid body transformation $T_c$ Extract associated voxel ids for points in scan B Create scan B feature map for all voxels Define an overlapping bounding region: \[alignment:line:8\]
$x_{min}^o = max(x_{min}^A, x_{min}^B)$
$y_{min}^o = max(y_{min}^A, y_{min}^B)$
$z_{min}^o = max(z_{min}^A, z_{min}^B)$
$x_{max}^o = min(x_{max}^A, x_{max}^B)$
$y_{max}^o = min(y_{max}^A, y_{max}^B)$
$z_{max}^o = min(z_{max}^A, z_{max}^B)$ Consider random variable X and Y as the voxelized feature map for all voxels lying inside the overlap of scan A and scan B respectively Compute the marginal and joint entropy using the equations \[eq:marginal\_A\], \[eq:marginal\_B\] and \[eq:joint\_AB\] Calculate MI using \[eq:mi\]
Update $T_c$ using Nelder-Mead optimization
$T_c$
--------------------------------------------------------------------------------------- ---------------------------------------------------------------- ------------- ------------------ ------------------ ------------------ ----------------------- ----------------------- ----------------------- --
Dataset Environment Total Scans deviation($t_x$) deviation($t_y$) deviation($t_z$) deviation($\theta_x$) deviation($\theta_y$) deviation($\theta_z$)
\[0.2ex\] $\begin{matrix} \text{Ford Campus} \\ \text{Vision and Lidar} \end{matrix}$ Urban 3538 0.65 2.87 0.05 0.02 0.02 0.23
$\begin{matrix} \text{Kitti Vision} \\ \text{Benchmark Suite} \end{matrix}$ $\begin{matrix} \text{Rural /} \\ \text{Highway} \end{matrix}$ 4919 2.67 0.31 0.04 0.01 0.01 0.08
--------------------------------------------------------------------------------------- ---------------------------------------------------------------- ------------- ------------------ ------------------ ------------------ ----------------------- ----------------------- ----------------------- --
Optimization
------------
The cost function (\[eq:cost\]) is maximized at the correct value of rigid body transformation. Therefore, any optimization technique that iteratively converges to the local optimum can be used here. Some of the commonly used optimization techniques compute the gradient or hessian of the cost function ([@whittaker1967], [@doi:10.1137/0111030] and [@10.2307/43633451]). There are also gradient free direct optimization methods like *pattern search* [@doi:10.1137/S1052623493250780] or *simulated annealing* [@Kirkpatrick671]. In this work we use one such direct optimization method called *Nelder-Mead* optimization [@doi:10.1093/comjnl/7.4.308].
Nelder-Mead method initializes a simplex with (N+1) points on the cost surface, where N is the number of DOF (six in our case) in the optimization phase. In each iteration it tries to improve these set of points by series of steps (reflecting, expanding, shrinking or contracting) to obtain the point which minimizes the cost function. NM-Simplex is sensitive to initial simplex size \[$s_x$, $s_y$, $s_z$, $s_{roll}$, $s_{pitch}$, $s_{yaw}$\]. Choosing an initial simplex large can cause unnecessary steps in areas of little interest, while a small simplex can lead to a narrow search on the cost surface increasing the computation. This problem can be alleviated by making a simple assumption that most of the relative traversal will be in $t_x$, $t_y$ and $\theta_z$ for wheeled robots. The intuition being that for any wheeled-robotic platform, motion is constrained in roll, pitch and z(height). Similar observation was also made in the localization method proposed in [@wolcott2015fast]. Making this assumption allows us to explore large range for $t_x$, $t_y$ and $\theta_z$ on cost surface, while at the same time a constrained initial search in $t_z$, $\theta_x$ and $\theta_y$. We verify this assumption, by calculating the deviation of all 6-DOF in Table \[table:gt\_analysis\].
As this optimization technique is based on heuristics, optimization might not always lead to the optima. This is indeed a disadvantage of this optimization technique. While there exist methods that allow approximating the gradient of the cost function, however, the computation involved increases the run time of the algorithm. In the section below, we show that even after using a heuristic optimization, the results are close to the ground truth.
The complete algorithm for obtaining the rigid body transformation is given in (Algorithm \[alg:alignment\]).
{width="\linewidth"}
{width="\linewidth"}
---------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ -------
Method 1 2 3 4 5 6 7 8 9 10
\[0.2ex\] GICP 0.03 0.07 0.11 0.34 1.06 2.49 2.60 3.99 5.42 6.73
3D-NDT 0.29 1.69 3.08 4.12 5.22 6.17 7.17 8.22 9.17 10.29
MI-N 0.05 0.10 0.18 0.43 0.68 1.22 2.09 2.68 2.80 3.23
MI-VARZ 0.06 0.10 0.28 0.30 0.70 0.95 1.37 2.01 2.39 2.70
---------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ -------
---------------- ------ ------ ------ ------ ------ ------- ------- ------- ------- -------
Method 2 4 6 8 10 12 14 16 18 20
\[0.2ex\] GICP 0.41 0.72 1.00 1.14 0.88 0.81 1.23 1.78 3.26 5.05
3D-NDT 0.52 1.82 4.48 5.98 8.96 11.03 13.17 15.24 17.43 19.44
MI-N 1.18 1.90 2.39 4.79 4.19 5.66 2.60 5.61 4.91 1.49
MI-VARZ 0.43 0.70 0.75 0.72 0.92 2.56 0.89 4.11 1.98 1.48
---------------- ------ ------ ------ ------ ------ ------- ------- ------- ------- -------
{width="\linewidth"}
{width="\linewidth"}
[|c|c|c|c|c|c|c|c|c|c|c|c|]{} Method&1&2&3&4&5&6&7&8&9&10\
\[0.2ex\]
GICP&0.03&0.13&0.21&0.60&0.74&1.41&2.53&2.91&4.71&5.42\
3D-NDT&0.24&1.62&2.55&3.92&5.01&5.86&6.91&7.88&8.73&10.01\
MI-N&0.06&0.10&0.22&0.28&0.25&0.47&0.51&0.54&1.90&3.38\
MI-VARZ&0.06&0.06&0.12&0.09&0.21&0.41&0.42&0.36&0.68&1.23\
[|c|c|c|c|c|c|c|]{} Method&2&4&6&8&10&12\
\[0.2ex\]
GICP&0.13&0.34&0.82&2.08&2.70&2.90\
3D-NDT&1.10&3.21&5.43&7.82&9.49&11.47\
MI-N&0.30&0.27&0.51&1.20&1.85&0.31\
MI-VARZ&0.17&0.20&0.31&0.67&0.94&0.36\
[.35]{} {width="\linewidth" height="\linewidth"}
[.35]{} {width="\linewidth" height="\linewidth"}
[.35]{} {width="\linewidth" height="\linewidth"}
Experiments and Results {#evaluation}
=======================
We show results for two types of feature used in our MI framework: (1) Variance of z-height in voxel (MI-VARZ), and (2) Number of points in voxel (MI-N). We have used two point cloud datasets to assess the performance of the proposed approach. Scans from both these datasets are registered using GICP, 3D-NDT, MI-N, and MI-VARZ. The comparison factors were accuracy and runtime for different initial estimates. Datasets used in our experiments were Ford Campus Vision and Lidar dataset [@Pandey:2011:FCV:2049736.2049742] and odometry dataset from the Kitti Vision Benchmark Suite [@6248074]. Both the datasets consist of 3D scans collected from a test vehicle with a Lidar mounted on it and have ground truth pose information available from a highly accurate inertial navigation system (INS).
Implementation of GICP is taken from the open source version available at [@gicp-link] and 3D-NDT is taken from Point Cloud Library (PCL) [@5980567]. Grid resolution ($R_v$) for 3D-NDT, MI-VARZ and MI-N is set as 1m. In our implementation we chose the initial simplex for optimization as \[$s_x=8$, $s_y=8$, $s_z=1$, $s_{roll}=0.1$, $s_{pitch}=0.1$, $s_{yaw}=0.8$\].
Ford Campus Vision and Lidar Dataset
------------------------------------
In this experiment we compare the translation and rotation accuracy of the proposed method with GICP and 3D-NDT on Ford Dataset. To estimate the translation accuracy, we consider equi-spaced reference scan (A) in the downtown test run and sample 10 query scans (B) which are 1-10m from A. Similarly for rotation, all B query scans that have relative rotation upto $20^\circ$ (with translation error less than 5m) from scan A are chosen. In total 3538 scan pairs from Ford dataset were considered to plot error results shown in Figure \[ford\_results\].
Kitti Vision Benchmark Suite
----------------------------
In this experiment, we use the scans from Kitti dataset for testing both translation and rotation accuracy. Here also, we sample scans that have translation 1-10m and rotation upto $12^\circ$ (plot range is reduced here as the number of scan pairs with translation error less than 5m and rotation greater than $12^\circ$ are not enough) resulting in a total of 4919 scan pairs from a single session run. Results are shown in Figure \[kitti\_results\].
Discussion of results
---------------------
It can be seen, that when the initial estimate of transformation parameters are good, all 4 methods perform reasonably good for both the datasets. However, 3D-NDT fails to converge when the initial error is increased: this can be explained by the discontinuity in score function due to the jump caused when the query scan $B$ passes one of the cell boundaries. In [@5152538] and [@4058864], convergence of 3D-NDT and it’s dependence on voxels resolution is explained in detail.
It should be noted that GICP works better when we have a good initial guess for translation parameters. However, it fails to converge to the correct optima when the initial guess is poor. This can be explained from cost surface of GICP algorithm. As seen in Figure \[fig:gicp\_3D\], GICP has several local optima causing the gradient based optimization methods to fail. Figure \[fig:heading\_cost\] also depicts this problem, when the initial heading error is large GICP can converge to incorrect transformation. In contrast, in Figure \[fig:mi\_3D\] it can be seen that our MI based method has a single optima corresponding to the correct rigid-body-transformation for the same range of initial error. It is important to note that this might not always be the case. In scan pairs with multiple dynamic objects and large initial error, the heuristic based optimization in MI-VARZ and MI-N can converge at an optima far from the correct solution. Despite that, we observe that our MI based method performs better than GICP in most cases.
Both MI-VARZ and MI-N have better results for translation as we increase the prior translation error. For the Kitti Dataset, it can be seen that MI-VARZ has average translation error less than 0.5m for the initial translation error up till 8m and MI-N till 6m, which is in contrast to results from GICP where we see an average error of 0.6m at 4m of initial translation error. In the Ford dataset, MI-VARZ and MI-N have error less than 0.5m up till 4m and GICP till 3m.
As for the rotation error, in Kitti Dataset we see a similar trend where MI-VARZ performs better than GICP as we increase the initial error. In the Ford dataset however, both GICP and MI-VARZ produce mixed results. This is primarily due to unstructured areas and multiple dynamic objects in the urban scene. In such cases, the heuristic based search in MI-VARZ leads to outliers (See Figure \[fig:ford\_rotation\_error\_plot\] and \[table:ford\_mean\_rotation\] for initial error $16^{\circ}$) which are relatively farther from the correct rotation parameters than GICP outliers. Yet in most cases (Figure \[fig:ford\_rotation\_error\_plot\]) MI-VARZ has performance better than GICP. We also observe that MI-N doesn’t perform as good as MI-VARZ and GICP. In general too, MI-N has less success rate as compared to MI-VARZ, this is because the number of points in a voxel do not convey any information about the local-structure unlike the case in variance of z.
Runtime analysis
----------------
In this section we compare the runtime of GICP with MI-N and MI-VARZ. We do not consider the 3D-NDT in this analysis as it fails to converge when the initial error is high. The average runtime of GICP, MI-N and MI-VARZ is shown in Table \[table:runtime\]. All the implementations were executed on a system powered by Intel Core i7-7700HQ CPU@ 2.80GHz $\times$ 8 and GeForce GTX 1050Ti. We used [@muja_flann_2009] to optimize the GICP by implementing the point-point search queries in GPU memory. The runtime of GICP after this optimization is also showed in Table \[table:runtime\].
--------------- ------- ---------- ------ ---------
Initial Error GICP GICP-GPU MI-N MI-VARZ
\[0.5ex\] (m) (s) (s) (s) (s)
\[0.5ex\] 1 2.67 1.64 1.26 0.49
3 4.00 2.25 1.27 0.50
5 6.92 4.42 1.26 0.50
7 10.72 7.25 1.23 0.49
9 14.26 10.13 1.22 0.51
--------------- ------- ---------- ------ ---------
: Runtime analysis of GICP (Original), GICP (GPU Search), MI-N and MI-VARZ. MI-N and MI-VARZ have negligible variability in runtime with respect to quality of the initial guess. However, the GICP runtime increases as the initial error is increased.[]{data-label="table:runtime"}
Another important challenge is the variability of runtime with change in the initial error. The runtime of GICP varies when we change the error in translation, whereas MI-N and MI-VARZ have runtime independent of the initial error.
In Table \[table:gpu\_operations\], we show that all major steps in our proposed framework can be easily offloaded to GPU with minimal overhead. This is in contrast to other point-to-point alignment methods like GICP, which require building nearest neighbor structures (like K-D Tree) in GPU. The preprocessing step in GICP for nearest neighbor search takes most of the time in alignment process. Whereas in our MI based framework, the only bottleneck is GPU memory allocation (along with gpu context initialization).
------------------------------- -------- ---------------- --
MI Steps Kernel Total Time(ms)
\[0.2ex\] GPU Allocate memory - 189.00
Data CPU to GPU - 0.5
Voxel Mapping Map 5.87
Find overlap Reduce 33.06
Feature histograms Map 100.10
Entropy Calculation Reduce 8.45
------------------------------- -------- ---------------- --
: GPU profiling for a scan pair while executing MI-VARZ. Second column depicts the type of gpu kernel for the operation and third column is the total time (ms) spent in kernel execution for all iterations in the alignment process.[]{data-label="table:gpu_operations"}
CONCLUSION
==========
In this paper, we report a MI based scan alignment algorithm that maximizes the mutual information between the voxelized features of two partially overlapping scans by calculating a single-dimensional feature in a voxel. Our approach allows us to consider the no-feature voxels in our cost function with the intuition being: that both scans, when aligned, should have same number of unoccupied voxels in the overlapping region. The proposed method is tested and compared with a point-to-point and point-to-distribution method on two real-world datasets covering wide range of dynamic scenes: urban and rural. We see that our method performs relatively better even for large initial errors in transformation. Although, we implemented our method with data from a single sensor (lidar), it can easily be extended to fused-sensor data (lidar-camera). We show that our method has a fast GPU implementation which allows computation to be offloaded to GPU with minimal overhead.
[^1]: \*This work was supported by Ford Motor Company
[^2]: $^{1}$N. Mehta is with the Computer Engineering Department of IIT Kanpur, India. [nikhil@cse.iitk.ac.in]{}
[^3]: $^{2}$G. Pandey and J. R. Mcbride are with Ford Motor Company, Dearborn, USA. [gpandey2@ford.com]{}, [jmcbride@ford.com.]{}
|
---
abstract: 'We analyze the equilibrium transport properties of underscreened Kondo effect in the case of a two-level quantum dot coupled to ferromagnetic leads. Using the numerical renormalization group (NRG) method, we have determined the gate voltage dependence of the dot’s spin and level-resolved spectral functions. We have shown that the polarization of the dot is very susceptible to spin imbalance in the leads and changes sign in the middle of the $S=1$ Coulomb valley. Furthermore, we have also found that by fine-tuning an external magnetic field one can compensate for the presence of ferromagnetic leads and restore the Kondo effect in the case of $S=\frac{1}{2}$ Coulomb valley. However, the underscreened Kondo effect cannot be fully recovered due to its extreme sensitivity with respect to the magnetic field.'
author:
- Ireneusz Weymann
- László Borda
title: Underscreened Kondo effect in quantum dots coupled to ferromagnetic leads
---
Introduction
============
Ever since the observation of the resistivity anomaly in normal metals at low temperatures, [@HaasPhysica36] the Kondo problem [@kondo64] has been investigated perpetually both experimentally and theoretically. [@hewson_book; @goldhaber-gordon_98; @cronenwett_98] As a result of that intensive research, in many respects, the Kondo effect is well understood now. Very recently the interplay between the Kondo effect and other many-body phenomena, such as superconductivity or ferromagnetism, has attracted a lot of interest. That attention was mainly motivated by the recent advances in nanofabrication which have opened the possibility to attach superconducting [@SchoenenbergerPRL02; @TaruchaPRL07] or ferromagnetic [@pasupathy_04; @heerschePRL06; @hamayaAPL07; @hamayaPRB08; @hauptmannNatPhys08; @parkinNL08] leads to molecules or semiconducting quantum dots. In the following we will focus our attention on the case of ferromagnetic leads coupled to a quantum dot exhibiting the Kondo effect.
From the theoretical side, a consensus was found that single level quantum dots (i.e. relatively small dots with level spacing $\delta$ larger than the hybridization $\Gamma$) attached to ferromagnetic leads exhibit finite spin asymmetry as a result of the spin imbalance in the leads, which gives rise to a splitting and suppression of the Kondo resonance. [@LopezPRL03; @martinekPRL03_2; @martinekPRL03; @ChoiPRL04] This spin asymmetry can, however, be compensated by means of an external magnetic field or by tuning the gate voltage properly. [@martinek_PRB05; @sindelPRB07] In that way the strong coupling Kondo fixed point can be reached, with a fully developed Kondo resonance and a somewhat reduced Kondo temperature. This expectation has recently been confirmed experimentally. [@pasupathy_04; @heerschePRL06; @hamayaAPL07; @hamayaPRB08; @hauptmannNatPhys08; @parkinNL08]
While in many cases the quantum dots can successfully be modelled by only a single local level, there are few exceptions when the finite level spacing affects the low energy physics of such a device considerably. An isolated quantum dot with even number of electrons can have spin triplet $S=1$ ground state, if the level spacing between the two single particle levels closest to the Fermi energy is anomalously small, i.e. smaller than the exchange coupling between the two electrons situated on those levels. If such a device is attached to only one lead then the system will exhibit the so-called [*underscreened*]{} Kondo effect. [@NozieresJP80; @LeHurPRB97; @ColemanPRB03; @PosazhennikovaPRL05; @MehtaPRB05] A single lead can only screen half of the local spin below the Kondo temperature $T_K$ while the residual spin-a-half object is left unscreened with a ferromagnetic Kondo coupling to the rest of the conduction electrons. Such a ferromagnetic Kondo coupling is known to be irrelevant in renormalization group sense, as it scales to zero when the energy scale is lowered, resulting in a Fermi liquid plus a decoupled $S=\frac{1}{2}$ object at zero temperature. This behavior is reached in a non-analytic fashion resulting in singular terms in e.g. the single particle scattering rate. Such kind of behavior was recently termed as [*singular Fermi liquid*]{}. [@ColemanPRB03; @MehtaPRB05]
In a real quantum dot subject to a transport experiment the situation is slightly more complex as the dot has to be coupled to two leads to drive current through it. Even if the second lead is just a weakly coupled probe, its presence introduces a second energy scale, $T_K'\ll T_K$, at which the residual spin-a-half is screened by the other lead mode. The screening takes place in two stages and the separation between the energy scales $T_K$ and $T_K'$ can be tuned by the asymmetry between the coupling to the leads. Since the Kondo temperature is exponentially sensitive to the coupling, a realistic value of asymmetry is enough to separate the two stages completely. In such a case, if the other experimentally relevant energy scales, such as temperature $T$ or magnetic field $B$, lie in between the two Kondo temperatures $T_K'\ll B,T < T_K$, then the system is well described by the underscreened Kondo model. In fact, very recently, underscreened Kondo effect was observed experimentally in molecular quantum dots coupled to nonmagnetic leads. [@Roch09]
In the present paper we focus our interest on the interplay of underscreened Kondo effect and itinerant electron ferromagnetism in the leads. To that end we consider a two level quantum dot coupled to a single reservoir of conduction electrons exhibiting spin imbalance. We assume that the second Kondo temperature is always much smaller than the experimental temperature, therefore we consider the second lead as a weakly coupled probe only.
Theoretical description
=======================
The considered system consists of a two-level quantum dot coupled to a ferromagnetic reservoir, see Fig. \[Fig:1\], and its Hamiltonian is given by $$\label{Eq:H}
H = H_{\rm FM} + H_{\rm QD} + H_{\rm tun},$$ where the first term describes the noninteracting itinerant electrons in ferromagnetic lead, $H_{\rm FM} = \sum_{k{\sigma}} {\varepsilon}_{k{\sigma}}
c^\dag_{k{\sigma}} c_{k{\sigma}}$, where $c^\dag_{k{\sigma}}$ is the electron creation operator with wave number $k$ and spin ${\sigma}$, while ${\varepsilon}_{k{\sigma}}$ is the corresponding energy. The second part of the Hamiltonian describes a two-level quantum dot and is given by $$\begin{aligned}
\label{Eq:HQD}
H_{\rm QD} &=& \sum_{j{\sigma}} {\varepsilon}_{j} d^\dag_{j{\sigma}} d_{j{\sigma}} + U
\sum_j n_{j\uparrow} n_{j\downarrow}\nonumber\\
&& + U'\sum_{{\sigma}{\sigma}'} n_{1{\sigma}}n_{2{\sigma}'} + JS^2 + B S^z,\end{aligned}$$ where $n_{j{\sigma}} = d^\dag_{j{\sigma}} d_{j{\sigma}}$ and $d^\dag_{j{\sigma}}$ creates a spin-${\sigma}$ electron in the $j$th level ($j=1,2$), ${\varepsilon}_{j}$ denotes the corresponding energy of an electron in the dot. Here, ${\varepsilon}_1 = {\varepsilon}- \delta/2$ and ${\varepsilon}_2 = {\varepsilon}+ \delta/2$, with $\delta={\varepsilon}_2-{\varepsilon}_1$ being the level spacing. The on-level (inter-level) Coulomb correlations are denoted by $U$ ($U'$), respectively. $J$ is the ferromagnetic exchange coupling ($J<0$) with $\vec{S}=\vec{S}_1+\vec{S}_2$, where $\vec{S}_j =
\frac{1}{2}\sum_{{\sigma}{\sigma}'}d^\dag_{j{\sigma}}\vec{{\sigma}}_{{\sigma}{\sigma}'}d_{j{\sigma}'}$ is the spin operator for electrons in the dot level $j$ and $\vec{{\sigma}}$ denotes the vector of Pauli spin matrices. The last term of $H_{\rm QD}$ describes the Zeeman splitting with $B$ ($g\mu_B\equiv 1$) being the external magnetic field applied along the $z$th direction.
Finally, the tunnel Hamiltonian is given by $$H_{\rm tun} = \sum_{kj{\sigma}} t_{j{\sigma}} \left( d_{j{\sigma}}^\dag c_{k{\sigma}} +
c_{k{\sigma}}^\dag d_{j{\sigma}} \right),$$ where $t_{j{\sigma}}$ describes the spin-dependent hopping matrix elements between the $j$th dot level and ferromagnetic lead. The strength of the coupling between the dot and lead can be expressed as $\Gamma_{j{\sigma}} = \pi \rho |t_{j{\sigma}}|^2$, where $\rho$ is the density of states (DOS) in the lead. In the following, we assume a flat band of width $2D$ and use $D\equiv 1$ as energy unit, as not stated otherwise. Note that by assuming constant DOS the whole spin-dependence has been shifted into the coupling constants. While this assumption may simplify the calculations, it does not affect the low-energy physics we are interested in. [@martinekPRL03; @ChoiPRL04] To parameterize the couplings, it is convenient to introduce the spin polarization of ferromagnetic lead, $p$, defined as $p = (\Gamma_{\uparrow} -
\Gamma_{\downarrow}) / (\Gamma_{\uparrow} + \Gamma_{\downarrow})$, where we have assumed that each dot level is coupled with the same strength to the lead, $\Gamma_{j{\sigma}} = \Gamma_{\sigma}$. Then, the coupling for the spin-${\uparrow}$ (spin-${\downarrow}$) electrons can be written as $\Gamma_{{\uparrow}({\downarrow})} = (1\pm p)\Gamma$, where $\Gamma =
(\Gamma_{\uparrow}+ \Gamma_{\downarrow})/2$. For $p=0$, the couplings do not depend on spin and the system behaves as if coupled to nonmagnetic lead. However, for finite spin polarization, $p\neq 0$, the couplings are spin-dependent, giving rise to an effective exchange field which may spin-split the levels in the dot, suppressing the Kondo resonance. Such behavior has been extensively studied theoretically in the case of single-level quantum dots, [@LopezPRL03; @martinekPRL03_2; @martinekPRL03; @ChoiPRL04; @martinek_PRB05; @sindelPRB07; @UtsumiPRB05; @SwirkowiczPRB06; @MatsubayashiPRB07; @SimonPRB07] where the usual $S=\frac{1}{2}$ Kondo effect develops, while the effect of ferromagnetism on the other types of Kondo effect remains to a large extent unexplored.
![\[Fig:1\] (color online) Schematic of a two-level quantum dot coupled to a ferromagnetic reservoir with spin polarization $p$. The dot level energies are denoted by ${\varepsilon}_1$ and ${\varepsilon}_2$, $\delta$ is the level spacing, while $U'$ and $J$ describe the inter-level Coulomb correlations and spin exchange interaction. The $j$th dot level is coupled to ferromagnetic electrode with strength $\Gamma_{j{\sigma}}$.](Fig1.eps){width="0.5\columnwidth"}
In this paper we thus consider the zero-temperature equilibrium transport properties of two-level quantum dots coupled to ferromagnetic lead in the case of underscreened Kondo effect. In order to perform this analysis in most accurate and exact way, we employ the numerical renormalization group (NRG) method. [@WilsonRMP75; @BullaRMP08] The NRG consists in a logarithmic discretization of the conduction band and mapping of the system onto a semi-infinite chain with the quantum dot sitting at the end of the chain. By diagonalizing the Hamiltonian at consecutive sites of the chain and storing the eigenvalues and eigenvectors of the system, one can accurately calculate the static and dynamic quantities of the system. In particular, numerical results presented here were obtained using the flexible density-matrix numerical renormalization group (DM-NRG) code, which allows to use arbitrary number of Abelian and non-Abelian symmetries. [@Toth_PRB08; @FlexibleDMNRG; @BudapestNRG] In fact, exploiting symmetries as much as possible is crucial in obtaining highly accurate data. Because in the case of ferromagnetic leads the full spin rotational invariance is generally broken, in the case of finite spin polarization we have used the Abelian symmetries for the total charge and spin, while in the case of $p=0$ we have exploited the full spin $SU(2)$ symmetry.
Using the NRG, we have calculated the expectation value of the dot’s spin as well as the spectral function of the dot, $A_{jj'{\sigma}}(\omega) = -\frac{1}{2\pi}{\rm
Im}[G^R_{jj'{\sigma}}(\omega)+G^R_{j'j{\sigma}}(\omega)]$, where $G_{jj'{\sigma}}^R(\omega)$ denotes the Fourier transform of the dot retarded Green’s function, $G_{jj'{\sigma}}^R(t)= -i\Theta(t)
{\langle \{d_{j{\sigma}}(t), d_{j'{\sigma}}^\dag(0)\} \rangle}$. The diagonal elements of the spectral function $A_{j{\sigma}}(\omega) \equiv A_{jj{\sigma}}(\omega)$ are related to the spin-resolved density of states, whereas the off-diagonal elements $A_{jj'{\sigma}}(\omega)$ may be associated with processes of injecting and removing an electron at different sites. The symmetrized spin-resolved spectral function of the dot can be found from $$A_{\sigma}(\omega) = \sum_{jj'}A_{jj'{\sigma}}(\omega).$$ On the other hand, the normalized full spectral function is given by $\pi\sum_{\sigma}\Gamma_{\sigma}A_{\sigma}(\omega)$ and is directly related to the conductivity of the dot.
Numerical results
=================
$n$ ${|Q,S^z\rangle}$ $E_{Q,S^z}$
----- ------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------
1 ${|-2,0\rangle}={|00\rangle}$ $0$
2 ${|-1,-\frac{1}{2}\rangle}={|{\downarrow}0\rangle}$ ${\varepsilon}-\frac{\delta}{2}+\frac{3J}{4}-\frac{B}{2}$
3 ${|-1,-\frac{1}{2}\rangle}={|0{\downarrow}\rangle}$ ${\varepsilon}+\frac{\delta}{2}+\frac{3J}{4}-\frac{B}{2}$
4 ${|-1,\frac{1}{2}\rangle}={|{\uparrow}0\rangle}$ ${\varepsilon}-\frac{\delta}{2}+\frac{3J}{4}+\frac{B}{2}$
5 ${|-1,\frac{1}{2}\rangle}={|0{\uparrow}\rangle}$ ${\varepsilon}+\frac{\delta}{2}+\frac{3J}{4}+\frac{B}{2}$
6 ${|0,0\rangle}={|{\rm d} 0\rangle}$ $2{\varepsilon}-\delta+U$
7 ${|0,0\rangle}={|0{\rm d}\rangle}$ $2{\varepsilon}+\delta+U$
8 ${|0,0\rangle}=\frac{1}{\sqrt{2}}({|{\uparrow}{\downarrow}\rangle}-{|{\downarrow}{\uparrow}\rangle})$ $2{\varepsilon}+U'$
9 ${|0,-1\rangle}={|{\downarrow}{\downarrow}\rangle}$ $2{\varepsilon}+U'+2J-B$
10 ${|0,0\rangle}=\frac{1}{\sqrt{2}}({|{\uparrow}{\downarrow}\rangle}+{|{\downarrow}{\uparrow}\rangle})$ $2{\varepsilon}+U'+2J$
11 ${|0,1\rangle}={|{\uparrow}{\uparrow}\rangle}$ $2{\varepsilon}+U'+2J+B$
12 ${|1,-\frac{1}{2}\rangle}={|{\rm d} {\downarrow}\rangle}$ $3{\varepsilon}-\frac{\delta}{2}+\frac{3J}{4}+U+2U'-\frac{B}{2}$
13 ${|1,-\frac{1}{2}\rangle}={|{\downarrow}{\rm d}\rangle}$ $3{\varepsilon}+\frac{\delta}{2}+\frac{3J}{4}+U+2U'-\frac{B}{2}$
14 ${|1,\frac{1}{2}\rangle}={|{\rm d} {\uparrow}\rangle}$ $3{\varepsilon}-\frac{\delta}{2}+\frac{3J}{4}+U+2U'+\frac{B}{2}$
15 ${|1,\frac{1}{2}\rangle}={|{\uparrow}{\rm d}\rangle}$ $3{\varepsilon}+\frac{\delta}{2}+\frac{3J}{4}+U+2U'+\frac{B}{2}$
16 ${|2,0\rangle}={|{\rm dd}\rangle}$ $4{\varepsilon}+2U+4U'$
: \[tab:energies\] The respective eigenstates ${|Q,S^z\rangle}$ and eigenvalues $E_{Q,S^z}$ of the decoupled quantum dot Hamiltonian. Here, $Q= (\sum_{j{\sigma}} n_{j{\sigma}} - 2)$ and $S^z$ are the charge and $z$th component of the spin in the dot, while ${|\chi_1\chi_2\rangle}$ denotes the local states with $\chi_j=0,{\uparrow},{\downarrow},{\rm d}$ for zero, spin-up, spin-down and two electrons in the level $j$.
Before presenting numerical results, it is instructive to analyze the eigen-spectrum of the decoupled quantum dot Hamiltonian, Eq. (\[Eq:HQD\]). The respective eigenstates ${|Q,S^z\rangle}$ and eigenvalues $E_{Q,S^z}$ of $H_{\rm QD}$ are listed in Table I, where $Q = (\sum_{j{\sigma}} n_{j{\sigma}} - 2)$ denotes the charge in the dot and $S^z$ is the $z$th component of the dot spin. By tuning ${\varepsilon}$ the ground state of the dot changes. In particular, in the absence of magnetic field and for ferromagnetic exchange coupling ($J<0$), when ${\varepsilon}> {\varepsilon}_{-2,-1} = \frac{\delta}{2} - \frac{3J}{4}$, the dot is empty, for ${\varepsilon}={\varepsilon}_{-1,0} = -\frac{\delta}{2} - U' -
\frac{5J}{4}$, the doublet ${|Q=-1,S=\frac{1}{2}\rangle}$ and triplet ${|Q=0,S=1\rangle}$ states become degenerate, where now $S$ is the full dot’s spin, whereas for ${\varepsilon}={\varepsilon}_{0,1} = \frac{\delta}{2} - U'
-U + \frac{5J}{4}$, the states ${|Q=0,S=1\rangle}$ and ${|Q=1,S=\frac{1}{2}\rangle}$ are degenerate. Consequently, for ${\varepsilon}_{-2,-1} > {\varepsilon}> {\varepsilon}_{-1,0}$, the ground state of the decoupled dot is the doublet and the system will exhibit usual $S=\frac{1}{2}$ Kondo effect, whereas for ${\varepsilon}_{-1,0}
> {\varepsilon}> {\varepsilon}_{0,1}$ the dot is in the triplet state and the system will show an underscreened Kondo effect. In calculations we have taken the following parameters: $\delta = 0.1$, $J=-0.2$, $U=U' =
0.4$, and $\Gamma = 0.04$ (in units of $D=1$). One then finds, ${\varepsilon}_{-2,-1} = 0.2$, ${\varepsilon}_{-1,0} = -0.2$ and ${\varepsilon}_{0,1}=1$.
![\[Fig:2\] (color online) The expectation value of the dot spin operator $S^2$ in presence of nonmagnetic (a) and the expectation value of dot spin operator $S^z$ for ferromagnetic (b) lead as a function of the average level position ${\varepsilon}= ({\varepsilon}_1+{\varepsilon}_2)/2$. The solid line corresponds to $\Gamma = 0.04$, while the dashed line corresponds to $\Gamma = 0.015$ (in units of $D=1$). The expectation values separately for the two dot levels are also shown. The parameters are $\delta = 0.1$, $J=-0.2$, $U=U' = 0.4$, $B=0$, and in the ferromagnetic case $p=0.4$. The shadowed regions indicate Coulomb valleys with different electron number in the dot, where $N = \sum_{j{\sigma}} n_{j{\sigma}}$.](Fig2.eps){width="0.8\columnwidth"}
We have calculated the expectation value of the dot’s spin operators $S^2$ and $S^z$ in case of nonmagnetic and ferromagnetic leads, respectively. The results are summarized in Fig. \[Fig:2\]. As the level position is lowered by sweeping the gate voltage, the dot is tuned through the Coulomb blockade valleys with electron number $N=1,2,3$ and spin states $S=1/2,1,1/2$, respectively. When the dot is attached to a ferromagnetic lead, the spin asymmetry in the hybridization results in the spin splitting of the dot level, thus leading to a spin polarized ground state. Precisely in the middle of the $N=2$ Coulomb blockade valley, i.e. for ${\varepsilon}= ({\varepsilon}_{-1,0}+{\varepsilon}_{0,1})/2 =
-U'-\frac{U}{2} = -0.6$, the spin polarization changes sign, as to the right (left) hand side from that point the electron (hole) like virtual processes dominate in the renormalization of the level position.
![\[Fig:3\] (color online) The density plots of the normalized level-resolved spectral function $\pi\Gamma A_{jj'}^{p=0} = \pi\sum_{\sigma}\Gamma_{\sigma}A_{jj'{\sigma}}^{p=0}$ (a)-(c) and the full spectral function $\pi\Gamma A^{p=0} = \sum_{jj'}\pi\Gamma
A_{jj'}^{p=0}$ (d) in the case when the dot is attached to a nonmagnetic electrode ($p=0$). The parameters are the same as in Fig. \[Fig:2\] with $\Gamma=0.04$. Note the logarithmic scale on the frequency axis.](Fig3.eps){width="0.7\columnwidth"}
Note that the abrupt jump of the dot polarization is due to the fact that the underscreened Kondo model is extremely susceptible to even a tiny magnetic field. The reason is that the ground state of the system consists of a Fermi liquid and a decoupled residual spin $S=\frac{1}{2}$ and that residual spin at zero temperature can be polarized by any infinitesimal magnetic field. In the case of a weakly coupled second lead there is a second stage of the Kondo screening at $T_K'$ when the remaining spin degree of freedom is quenched by the weakly coupled lead’s electrons. [@PosazhennikovaPRB07] In that case the sudden step in the polarization turns into a very steep crossover with a width of $\sim \max(T_K',T)$, where $T$ is the experimental temperature.
![\[Fig:4\] (color online) The density plots of the spin-dependent level-resolved spectral function $\pi\Gamma_{\sigma}A_{jj'{\sigma}}^{p=0.4}$ (a)-(f), the spin-dependent full spectral function $\pi\Gamma_{\sigma}A_{{\sigma}}^{p=0.4} = \sum_{jj'}\pi\Gamma A_{jj'\sigma}^{p=0.4}$ (g) and (h), and the full spectral function $\pi\sum_{{\sigma}}\Gamma_{\sigma}A_{{\sigma}}^{p=0.4}$ (i) in the case of ferromagnetic lead with $p=0.4$. The parameters are the same as in Fig. \[Fig:2\] with $\Gamma=0.04$. Note the logarithmic scale on the frequency axis.](Fig4a.eps "fig:"){width="1\columnwidth"} ![\[Fig:4\] (color online) The density plots of the spin-dependent level-resolved spectral function $\pi\Gamma_{\sigma}A_{jj'{\sigma}}^{p=0.4}$ (a)-(f), the spin-dependent full spectral function $\pi\Gamma_{\sigma}A_{{\sigma}}^{p=0.4} = \sum_{jj'}\pi\Gamma A_{jj'\sigma}^{p=0.4}$ (g) and (h), and the full spectral function $\pi\sum_{{\sigma}}\Gamma_{\sigma}A_{{\sigma}}^{p=0.4}$ (i) in the case of ferromagnetic lead with $p=0.4$. The parameters are the same as in Fig. \[Fig:2\] with $\Gamma=0.04$. Note the logarithmic scale on the frequency axis.](Fig4b.eps "fig:"){width="0.65\columnwidth"}
Since the dot spin polarization is rather difficult to detect experimentally, we have computed the level-resolved single particle spectral density $A_{jj'{\sigma}}(\omega) = -\frac{1}{2\pi}{\rm
Im}\left[G^R_{jj'{\sigma}}(\omega)+G^R_{j'j{\sigma}}(\omega)\right]$, where $G_{jj'{\sigma}}^R(\omega)$ denotes the corresponding retarded Green’s function. This quantity contains the information about the transport properties of the dot. Given that the dot is strongly coupled to one of the leads, the conductivity through the setup at voltage bias $eV$ is given by $dI/dV \sim \frac{e^2}{h} \pi
\sum_{jj'{\sigma}} \Gamma_{\sigma}A_{jj'{\sigma}} (\omega=eV)$.
The results for the level-resolved normalized spectral functions $\pi\Gamma A_{jj'}^{p=0} = \pi \sum_{\sigma}\Gamma_{\sigma}A_{jj'{\sigma}}^{p=0}$ in the case of nonmagnetic lead ($p=0$) are shown in Fig. \[Fig:3\]. The spectral functions are plotted as a function of energy $\omega$ and average level position ${\varepsilon}$. To resolved the low-energy behavior of spectral functions logarithmic scale for $\omega$ is used. Furthermore, only the Coulomb blockade valleys with $N=1$ and $N=2$ electrons in the dot are shown. The behavior of spectral functions for blockade valley with $N=3$ electrons is similar to that with a single electron due to the particle-hole symmetry.
First of all, we note that the two Coulomb blockade regimes with $N=1$ and $N=2$ are characterized by fundamentally different fixed points. In the $N=1$ regime ($-0.2<\varepsilon<0.2$) the effective low-energy model is a Fermi liquid with fully screened spin $S=\frac{1}{2}$ ($S=\frac{1}{2}$ Kondo model). On the other hand, in the two-electron regime ($-1<\varepsilon<-0.2$) the effective model is a singular Fermi liquid, [@MehtaPRB05] i.e. a normal Fermi liquid plus a free spin-$\frac{1}{2}$ with weak residual ferromagnetic coupling to the conduction channel. These two distinct ground states of the system are separated by a quantum phase transition, [@PustilnikPRB06; @Logan_arXiv09] which occurs when sweeping the gate voltage through the boundary of $N=1$ ($N=3$) and $N=2$ Coulomb valleys. The quantum phase transition is of Kosterlitz-Thouless type, [@Logan_arXiv09] with exponentially decreasing Kondo temperature when approaching the transition from the normal Fermi liquid side. The different behavior associated with two distinct fixed points can be observed in the dependence of the level-resolved spectral functions. In the $N=1$ Coulomb valley the diagonal elements of the spectral function exceed the unitary value of $\pi\Gamma$, while the off-diagonal elements become negative and their absolute value is also larger than $\pi\Gamma$. On the other hand, in the $N=2$ Coulomb blockade valley, the diagonal and off-diagonal elements of $A^{p=0}$ are rather positive and always smaller than $\pi\Gamma$. In consequence, the full spectral function is properly normalized $A^{p=0} = A^{p=0}_{11} + A^{p=0}_{22} + 2 A^{p=0}_{12} = 2 \pi
\Gamma$ for $\omega = 0$, see Fig. \[Fig:3\](d). Furthermore, the full spectral function clearly exhibits the resonance corresponding to the underscreened Kondo effect in the Coulomb blockade valley $N=2$ ($\varepsilon<-0.2$), while for $N=1$ (${\varepsilon}>-0.2$) the $S=\frac{1}{2}$ Kondo effect develops.
The spin and level-resolved normalized spectral functions $\pi\Gamma A_{jj'{\sigma}}^{p=0.4}$ in the case of ferromagnetic lead with $p=0.4$ are displayed in Fig. \[Fig:4\]. Now the spectral function is different for each spin component due to the spin-dependence of the coupling parameters. Furthermore, when the lead is ferromagnetic, both Kondo resonances become completely suppressed. This is directly associated with spin-dependent level renormalization, which lifts the spin degeneracy in the dot, suppressing the spin-flip cotunneling processes driving the Kondo effect. In other words, the ferromagnetic lead exerts an effective exchange field on the dot, which destroys the Kondo effect. There is only a very narrow resonance at $\varepsilon=-0.6$ when the spin splitting of the dot levels generated by the spin imbalance of the conduction electrons vanishes, which happens exactly in the middle of the triplet valley, see also Fig. \[Fig:2\](b).
![\[Fig:5\] (color online) The spin-up (a), spin-down (b) and the full (c) spectral function in the $S=\frac{1}{2}$ Kondo regime for $\varepsilon=0$ in the presence of external magnetic field $B$, as indicated in the figure. The parameters are the same as in Fig. \[Fig:2\] with $\Gamma=0.04$ and $p=0.4$. The inset in (c) shows the zoom-out full spectral function. The Kondo effect is restored when $B=0.009215$.](Fig5.eps){height="13cm"}
![\[Fig:6\] (color online) The spin-up (a), spin-down (b) and the full (c) spectral function in the underscreened Kondo regime for $\varepsilon=-0.4$ in the presence of external magnetic field $B$, as indicated in the figure. The parameters are the same as in Fig. \[Fig:2\] with $\Gamma=0.04$ and $p=0.4$. The inset in (c) displays the dependence of the spectral function associated with the Kondo peak.](Fig6.eps){height="13cm"}
As mentioned above, the suppression of the Kondo peaks is associated with the exchange field which develops in the presence of spin-dependent couplings. However, as shown in the case of single level quantum dots, [@martinek_PRB05; @sindelPRB07] one may try to compensate for the presence of the exchange field by applying properly tuned external magnetic field. In Fig. \[Fig:5\] we show the spin-resolved spectral functions calculated in the regime of $S=\frac{1}{2}$ Kondo effect for $\varepsilon=0$ for different values of magnetic field $B$. It can be clearly seen that by tuning the magnetic field it is possible to fully restore the Kondo resonance at $\omega=0$. This happens when $B=B_{\rm c}=0.009215$, see Fig. \[Fig:5\], with $B_{\rm
c}$ being the compensating field, i.e. a field at which the previously-split dot level becomes degenerate again.
The situation, however, becomes more complicated for $\varepsilon=-0.4$, which corresponds to the Coulomb valley with $N=2$ and $S=1$, when the dot is described by the underscreened Kondo model. The respective spin-resolved spectral functions are shown in Fig. \[Fig:6\] using the logarithmic scale to emphasize the distinct dependence on magnetic field. Because the dot is coupled only to one conduction channel, only a half of the dot’s spin can be screened by conduction electrons and the ground state consists of a Fermi liquid and a decoupled spin $S=\frac{1}{2}$. Consequently, at $T=0$ an infinitesimally small magnetic field is enough to polarize the residual spin-a-half in the dot. This leads to an extremely high sensitivity of transport properties with respect to magnetic field, as can be seen in Fig. \[Fig:5\]. In consequence, in order to compensate for the presence of exchange field in the case of underscreened Kondo model, one needs to perform a fine-tuning of magnetic field. Furthermore, it turns out that although it is possible to restore the resonance peak at $\omega=0$, the full restoration of the underscreened Kondo effect is not possible, as the height of the peak is below the unitary limit.
Conclusions
===========
In this paper we have analyzed the equilibrium transport properties of a two-level quantum dot asymmetrically coupled to ferromagnetic leads. We have shown that by tuning the position of the dot levels, the ground state of the system changes from a Fermi liquid in the $S=\frac{1}{2}$ Coulomb valley into a Fermi liquid plus residual spin-a-half in the $S=1$ Coulomb valley, the latter being the example of underscreened Kondo effect. The boundary between these two regime is a quantum phase transition. For finite spin polarization of the leads, $p\neq 0$, the Kondo phenomenon becomes suppressed due to an effective exchange field originating from the presence of ferromagnetic leads, and so is the critical behavior.
In the transport regime where the system exhibits underscreened Kondo effect and for $p\neq 0$, we have found that the polarization of the dot abruptly changes sign in the middle of the $S=1$ Coulomb blockade region. This is associated with virtual tunneling processes (either electron-like or hole-like) that give the dominant contribution to the renormalization of the dot levels.
Furthermore, we have also analyzed the effect of an external magnetic field $B$ applied to the dot. It has been shown that an appropriately tuned $B$ can restore the Kondo effect in the case of the $S=\frac{1}{2}$ Coulomb valley. The underscreened Kondo effect, on the other hand, exhibits an extremely high sensitivity towards even a tiny change in magnetic field due to its particular ground state. Consequently, an infinitesimally small magnetic field is enough to polarize the residual spin-a-half in the dot, which definitely hinders a full restoration of the underscreened Kondo effect by tuning the magnetic field.
We acknowledge fruitful discussions with J. von Delft and R. Žitko. The authors acknowledge support from the Alexander von Humboldt Foundation. I.W. acknowledges support from the Foundation for Polish Science and funds of the Polish Ministry of Science and Higher Education as research projects for years 2006-2009 and 2008-2010. Financial support by the Excellence Cluster “Nanosystems Initiative Munich (NIM)” is gratefully acknowledged. L.B. acknowledges the support from Hungarian Grants OTKA through projects K73361 and NNF78842.
[99]{}
W. J. de Haas and G. J. van den Berg, Physica [**3**]{}, 440 (1936).
J. Kondo, Prog. Theor. Phys. 32, 37 (1964).
A. C. Hewson, [*The Kondo Problem to Heavy Fermions*]{} (Cambridge University Press, Cambridge, 1993).
D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. A. Kastner, Nature (London) 391, 156 (1998).
S. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science 281, 182 (1998).
M. R. Buitelaar, T. Nussbaumer, and C. Schönenberger, Phys. Rev. Lett. [**89**]{}, 256801 (2002).
C. Buizert, A. Oiwa, K. Shibatam K. Hirikawa, and S. Tarucha, Phys. Rev. Lett. 99, 136806 (2007).
A. N. Pasupathy, R. C. Bialczak, J. Martinek, J. E. Grose, L. A. K. Donev, P. L. McEuen, and D. C. Ralph, Science 306, 86 (2004).
H. B. Heersche, Z. de Groot, J. A. Folk, L. P. Kouwenhoven, H. S. van der Zant, A. A. Houck, J. Labaziewicz, and I. L. Chuang, Phys. Rev. Lett. 96, 017205 (2006).
K. Hamaya, M. Kitabatake, K. Shibata, M. Jung, M. Kawamura, K. Hirakawa, T. Machida, T. Taniyama, S. Ishida and Y. Arakawa, Appl. Phys. Lett. [**91**]{}, 232105 (2007).
K. Hamaya, M. Kitabatake, K. Shibata, M. Jung, M. Kawamura, S. Ishida, T. Taniyama, K. Hirakawa, Y. Arakawa, and T. Machida, Phys. Rev. B 77, 081302(R) (2008).
J. Hauptmann, J. Paaske, P. Lindelof, Nature Phys. 4, 373 (2008).
H. Yang, S.-H. Yang, S. S. P. Parkin, Nano Lett. [**8**]{}, 340 (2008).
Rosa Lopez and David Sanchez, Phys. Rev. Lett. [**90**]{}, 116602 (2003).
J. Martinek, Y. Utsumi, H. Imamura, J. Barnaś, S. Maekawa, J. König, and G. Schön, Phys. Rev. Lett. [**91**]{}, 127203 (2003).
J. Martinek, M. Sindel, L. Borda, J. Barnaś, J. König, G. Schön, and J. von Delft, Phys. Rev. Lett. [**91**]{}, 247202 (2003).
Mahn-Soo Choi, David Sanchez, and Rosa Lopez, Phys. Rev. Lett. [**92**]{}, 056601 (2004).
J. Martinek, M. Sindel, L. Borda, J. Barnaś, R. Bulla, J. König, G. Schön, S. Maekawa, J. von Delft, Phys. Rev. B [**72**]{}, 121302(R) (2005).
M. Sindel, L. Borda, J. Martinek, R. Bulla, J. König, G. Schön, S. Maekawa, and J. von Delft, Phys. Rev. B [**76**]{}, 045321 (2007).
P. Nozières and A. Blandin, J. Phys. [**41**]{}, 193 (1980).
Karyn Le Hur and B. Coqblin, Phys. Rev. B [**56**]{}, 668 (1997).
P. Coleman and C. Pepin, Phys. Rev. B [**68**]{}, 220405(R) (2003).
A. Posazhennikova and P. Coleman, Phys. Rev. Lett. [**94**]{}, 036802 (2005).
P. Mehta, N. Andrei, P. Coleman, L. Borda, G. Zarand, Phys. Rev. B [**72**]{}, 014430 (2005).
Nicolas Roch, Serge Florens, Theo A. Costi, Wolfgang Wernsdorfer, and Franck Balestro, Phys. Rev. Lett. [**103**]{}, 197202 (2009).
Y. Utsumi, J. Martinek, G. Schön, H. Imamura, and S. Maekawa, Phys. Rev. B [**71**]{}, 245116 (2005).
R. Świrkowicz, M. Wilczyński, M. Wawrzyniak, and J. Barnaś, Phys. Rev. B [**73**]{}, 193312 (2006).
Daisuke Matsubayashi and Mikio Eto, Phys. Rev. B [**75**]{}, 165319 (2007).
P. Simon, P. S. Cornaglia, D. Feinberg, and C. A. Balseiro, Phys. Rev. B [**75**]{}, 045310 (2007).
K. G. Wilson, Rev. Mod. Phys. [**47**]{}, 773 (1975).
R. Bulla, T. A. Costi, and T. Pruschke Rev. Mod. Phys. [**80**]{}, 395 (2008).
A. I. Tóth, C. P. Moca, O. Legeza, and G. Zaránd, Phys. Rev. B [**78**]{}, 245109 (2008).
O. Legeza, C. P. Moca, A. I. Tóth, I. Weymann, G. Zaránd, arXiv:0809.3143 (2008) (unpublished).
We used the open access Budapest NRG code, [http://www.phy.bme.hu/$\sim$dmnrg/]{}.
A. Posazhennikova, B. Bayani, and P. Coleman, Phys. Rev. B [**75**]{}, 245329 (2007).
M. Pustilnik and L. Borda, Phys. Rev. B [**73**]{}, 201301(R) (2006).
D. E. Logan, Ch. J. Wright, and M. R. Galpin, arXiv:0906.3169 (unpblished).
|
---
abstract: 'We show that the Brauer algebra ${\rm Br}_d(\delta)$ over the complex numbers for an integral parameter $\delta$ can be equipped with a grading, in the case of $\delta \neq 0$ turning it into a graded quasi-hereditary algebra. In which case it is Morita equivalent to a Koszul algebra. This is done by realizing the Brauer algebra as an idempotent truncation of a certain level two VW-algebra for some large positive integral parameter $N$. The parameter $\delta$ appears then in the choice of a cyclotomic quotient. This cyclotomic VW-algebra arises naturally as an endomorphism algebra of a certain projective module in parabolic category $\mathcal{O}$ of type $\rm D$. In particular, the graded decomposition numbers are given by the associated parabolic Kazhdan-Lusztig polynomials.'
address:
- 'Department of Mathematics, University of Bonn, 53115 Bonn, Germany'
- 'Department of Mathematics, University of Bonn, 53115 Bonn, Germany'
author:
- Michael Ehrig
- Catharina Stroppel
bibliography:
- 'references.bib'
title: Koszul gradings on Brauer algebras
---
[^1]
Introduction
============
We fix as ground ring the complex numbers $\mathbb{C}$. Given an integer $d\geq 1$ and $\delta\in{\mathbb{C}}$, the associated [*Brauer algebra*]{} ${\rm Br}_d(\delta)$ is a diagrammatically defined algebra with basis all Brauer diagrams on $2d$ points, that is all possible matchings of $2d$ points with $d$ strands, such that each point is the endpoint of exactly one strand. In other words, the basis elements correspond precisely to subdivisions of the set of $2d$ points into subsets of order $2$. Here is an example of such a Brauer diagram (with $d=11$): $$\label{diagram}
\begin{tikzpicture}[thick,>=angle 90]
\begin{scope}[xshift=8cm]
\draw (0,0) -- +(0,1);
\draw (.6,0) -- +(.6,1);
\draw (1.2,0) -- +(-.6,1);
\draw (1.8,0) to [out=90,in=-180] +(.9,.5) to [out=0,in=90] +(.9,-.5);
\draw (1.8,1) to [out=-90,in=-180] +(.6,-.4) to [out=0,in=-90] +(.6,.4);
\draw (2.4,0) -- +(0,1);
\draw (3,0) -- +(.6,1);
\draw (4.2,0) -- +(0,1);
\draw (4.8,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (4.8,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (6,0) -- +(0,1);
\end{scope}
\end{tikzpicture}$$ The multiplication is given on these basis vectors by a simple diagrammatical rule: we fix the positions of the $2d$ points as in the diagram with $d$ points at the bottom and $d$ points at the top. Then the product $bb'$ is equal to $\delta^k (b'\circ b)$, where $b'\circ b$ is the Brauer diagram obtained from the two basis elements $b$ and $b'$ by stacking $b'$ on top of $b$ (identifying the bottom points of $b'$ with the top points of $b$ in the obvious way) and then turning the result into a Brauer diagram by removing all internal circles, and $k$ is the number of such circles removed. For instance: $$\label{multiplication}
\begin{tikzpicture}[thick,>=angle 90]
\begin{scope}
\draw (0,0) -- +(0,1);
\draw (.6,0) -- +(.6,1);
\draw (1.2,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (.6,1) to [out=-90,in=-180] +(.6,-.4) to [out=0,in=-90] +(.6,.4);
\node at (2.4,.5) {$\bullet$};
\end{scope}
\begin{scope}[xshift=3cm]
\draw (0,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (1.2,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (0,0) to [out=90,in=-180] +(.6,.4) to [out=0,in=90] +(.6,-.4);
\draw (.6,0) to [out=90,in=-180] +(.6,.4) to [out=0,in=90] +(.6,-.4);
\node at (2.4,.5) {$=$};
\end{scope}
\begin{scope}[xshift=6.5cm]
\node at (-.5,.5) {$\delta$};
\draw (0,1) to [out=-90,in=-180] +(.6,-.4) to [out=0,in=-90] +(.6,.4);
\draw (.6,1) to [out=-90,in=-180] +(.6,-.4) to [out=0,in=-90] +(.6,.4);
\draw (0,0) to [out=90,in=-180] +(.6,.4) to [out=0,in=90] +(.6,-.4);
\draw (.6,0) to [out=90,in=-180] +(.6,.4) to [out=0,in=90] +(.6,-.4);
\end{scope}
\end{tikzpicture}$$
Brauer algebras form important examples of cellular diagram algebras in the sense of [@GrahamLehrer]. In particular we have [*cell modules*]{} (or Specht modules) $\Delta({\lambda})$ indexed by ${\lambda}\in{\Lambda}_d$. Here $${\Lambda}_d = \bigcup_{m \in \mathbb{Z}_{\geq 0} \cap (d-2\mathbb{Z}_{\geq 0})} {\rm Par}(m),$$ with ${\rm Par}(m)$ denoting the partitions of the integer $m$. We have simple modules $L(\lambda)$ for $\lambda \in {\Lambda}_d^\delta$, where ${\Lambda}_d^\delta = {\Lambda}_d$ in case of $\delta \neq 0$ and ${\Lambda}_d^\delta = {\Lambda}_d \setminus {\rm Par}(0)$ in case of $\delta=0$, see [@CDM].
Although the Brauer algebra can be defined for arbitrary $\delta \in \mathbb{C}$ it turns out that it is always semi-simple for $\delta \notin \mathbb{Z}$, see [@Rui]. For our purposes these cases are trivial, hence we will always assume $\delta \in \mathbb{Z}$.
Brauer algebras were originally introduced by Brauer [@Brauer] in the context of classical invariant theory as centralizer algebras of tensor products of the natural representation of orthogonal and symplectic Lie algebras. More precisely, assuming $d<n$ there is a canonical isomorphism of algebras $$\begin{aligned}
\label{centralizer}
\operatorname{End}_{\mathfrak{g}}(V^{\otimes d})&\cong& {\rm Br}_d(N)\end{aligned}$$ where ${\mathfrak{g}}$ is an orthogonal or symplectic Lie algebra of rank $n$ with vector representation $V$ of dimension $N$ in the orthogonal case and dimension $-N$ in the symplectic case, see e.g. [@CP] or [@GW] for details.
As an algebra, the Brauer algebra is generated by the following elements $t_{i}$, $g_i$ $$\label{diag}
\begin{tikzpicture}[scale=0.7,thick,>=angle 90]
\node at (0,.5) {$t_i$};
\draw (.6,0) -- +(0,1);
\draw [dotted] (1,.5) -- +(1,0);
\draw (2.4,0) -- +(0,1);
\draw (3,0) -- +(.6,1) node[above] {\tiny i+1};
\draw (3.6,0) -- +(-.6,1) node[above] {\tiny i};
\draw (4.2,0) -- +(0,1);
\draw [dotted] (4.6,.5) -- +(1,0);
\draw (6.2,0) -- +(0,1);
\begin{scope}[xshift=8cm]
\node at (0,.5) {$g_i$};
\draw (.6,0) -- +(0,1);
\draw [dotted] (1,.5) -- +(1,0);
\draw (2.4,0) -- +(0,1);
\draw (3,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (3,1) node[above] {\tiny i} to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3) node[above] {\tiny i+1};
\draw (4.2,0) -- +(0,1);
\draw [dotted] (4.6,.5) -- +(1,0);
\draw (6.2,0) -- +(0,1);
\end{scope}
\end{tikzpicture}$$ for $1\leq i < d$, and $t_i$ acts on $V^{\otimes d}$ in by permuting the $i$th and $(i+1)$st tensor factor, and the element $g_{i}$ acts by applying to the $i$th and $(i+1)$st factor the evaluation morphism $V\otimes V=V^*\otimes V\rightarrow \mathbb{C}$ followed by its adjoint.
The realization as centralizers includes the cases ${\rm Br}_d(\delta)$ for $\delta\in\mathbb{Z}$ integral and $\delta$ large enough in comparison to $d$. Hence the Brauer algebra is semisimple in these cases. In fact it was shown by Rui, see [@Rui], that ${\rm Br}_d(\delta)$ is semisimple except for $\delta$ integral of small absolute value, see also [@ES3] for a precise statement and references therein for proofs. For arbitrary $\delta\in\mathbb{Z}$ and $d\geq 1$ the Brauer algebras still appear as centralizers of the form if we replace $\mathfrak{g}$ by an orthosymplectic Lie superalgebra such that its vector representation has super dimension $k|2n$ with $\delta=k-2n$, see [@ESSchurWeyl].\
Whereas the semisimple cases were studied in detail in many papers, including for instance the semiorthogonal form in [@Nazarov], the non-semisimple cases are still not well understood. In the pioneering work of Cox, De Visscher and Martin, [@CDM], it was observed that the multiplicity $[\Delta({\lambda})\,:\,L(\mu)]$ how often a simple module $L(\mu)$ indexed by $\mu$ (that is the simple quotient of $\Delta(\mu)$) appears in a Jordan-Hölder series of the cell module $\Delta({\lambda})$ is given by certain parabolic Kazhdan-Lusztig polynomial $n_{{\lambda},\mu}$ of type ${\rm D}$ with parabolic of type ${\rm A}$, [@Boe], [@LS], i.e. $$\begin{aligned}
\label{KL}
[\Delta({\lambda})\, :\, L(\mu)]&=&n_{{\lambda},\mu}(1).\end{aligned}$$ This result connects the combinatorics of Brauer algebras with Kazhdan-Lusztig combinatorics of type ${\rm D}$ Lie algebras, i.e. multiplicities of simple (possibly infinite) highest weight modules appearing in a parabolic Verma module. Here two obvious questions arise: Is there an interpretation of the variable $q$ in the Kazhdan-Lusztig polynomial $n_{{\lambda},\mu}(q)\in{\mathbb{Z}}[q]$? Is there an equivalence of categories between modules over the Brauer algebra ${\rm Br}_d(\delta)$ for integral $\delta$ and some subcategory of the Bernstein-Gelfand-Gelfand (parabolic) category ${{\mathcal O}}$ for type ${\rm D}$ explaining the mysterious match in the combinatorics? In this paper we will give an answer to both questions.
Let us explain the results in more detail. Given a finite dimensional algebra $A$ we denote by $A-\operatorname{mod}$ its category of finite dimensional modules. If the algebra $A$ is ${\mathbb{Z}}$-graded we denote by $A-\operatorname{gmod}$ its category of finite dimensional graded modules with degree preserving morphisms and by $F:A-\operatorname{gmod}\rightarrow A-\operatorname{mod}$ the grading forgetting functor. For $i\in\mathbb{Z}$ let $\langle i\rangle: M\mapsto M\langle i\rangle$ denote the autoequivalence of $A-\operatorname{gmod}$ which shifts the grading by $i$, i.e. $F(M\langle i\rangle )=FM$ and $(M\langle i\rangle )_j=M_{j-i}$ for any $M\in A-\operatorname{gmod}$. As an application of our main theorem below we obtain the following refinement of , summing up the results obtained in Section \[sec:consequences\]:
\[first\] Let $\delta\in\mathbb{Z}$. The Brauer algebra ${\rm Br}_d(\delta)$ can be equipped with a $\mathbb{Z}$-grading turning it into a $\mathbb{Z}$-graded algebra ${\rm Br}^{\rm gr}_d(\delta)$. Moreover, it satisfies the following:
1. ${\rm Br}^{\rm gr}_d(\delta)$ is Morita equivalent to a Koszul algebra if and only if $\delta\not=0$ or $\delta=0$ and $d$.
2. ${\rm Br}^{\rm gr}_d(\delta)$ is graded cellular.
3. ${\rm Br}^{\rm gr}_d(\delta)$ is graded quasi-hereditary if and only if $\delta\not=0$ or $\delta=0$ and $d$ odd. Moreover, in the quasi-hereditary case
1. We have distinguished graded lifts of standard modules and simple modules in the following sense: For the cell module $\Delta(\lambda)$ of ${\rm Br}_d(\delta)$, $\lambda \in {\Lambda}_d$, there exists a module $\widehat{\Delta}(\lambda)$ for ${\rm Br}^{\rm gr}_d(\delta)$ such that $F\widehat{\Delta}(\lambda) = \Delta(\lambda)$. For a simple module $L(\mu)$ of ${\rm Br}_d(\delta)$, for $\mu \in {\Lambda}_d^\delta$, there exists a module $\widehat{L}(\mu)$ for ${\rm Br}^{\rm gr}_d(\delta)$ such that $F\widehat{L}(\mu) = L(\mu)$. Furthermore $\widehat{L}(\mu)$ is the simple quotient of $\widehat{\Delta}(\mu)$ concentrated in degree $0$, making the choice of these lifts unique.
2. The $\widehat{\Delta}({\lambda})$ form the graded standard modules.
4. The modules $\widehat{\Delta}({\lambda})$ have a Jordan-Hölder series in ${\rm Br}^{\rm gr}_d(\delta)-\operatorname{gmod}$ with multiplicities given by $$\begin{aligned}
\left[\widehat{\Delta}({\lambda})\, :\, \widehat{L}(\mu)<i>\right]&=&n_{{\lambda},\mu,i},\end{aligned}$$ where $n_{{\lambda},\mu}(q)=\sum_{i\geq 0}n_{{\lambda},\mu,i} q^i$.
For instance, ${\rm Br}^{\rm gr}_2(\delta)$ is isomorphic to the algebra $\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}$ in case $\delta\not=0$ whereas it is isomorphic to $\mathbb{C}\oplus \mathbb{C}[x]/(x^2)$ with $x$ in degree $2$, see Section \[sec:example\].\
The above result is based on our main theorem which realizes ${\rm Br}_d(\delta)$ for integral $\delta$ as an [*idempotent truncation*]{} of a level $2$ cyclotomic quotient of a VW-algebra ${\bigdoublevee}_{d}(\Xi)$, see Definition \[defalphabeta\] for the exact parameter set $\Xi$. Here, the [*VW-algebra*]{} ${\bigdoublevee}_{d}(\Xi)$ is as a vector space isomorphic to ${\rm Br}_d(N)\otimes {\mathbb{C}}[y_1,\ldots y_d]$ with both factors being in fact subalgebras. The defining relations imply that there is a unique surjective homomomorphism of algebras $$\begin{aligned}
{{\bigdoublevee}_d}(\Xi)&\longrightarrow&{\rm Br}_d(N),\end{aligned}$$ which extends the identity on ${\rm Br}_d(N)$ and sends $y_1$ to $0$. The polynomial generators $y_k$ are then sent to the famous [*Jucys-Murpy elements*]{} $\xi_k$ in the Brauer algebra, see Proposition \[jucysmurphy\] for a definition. These elements form a commutative subalgebra which plays an important role in the theory of semiorthogonal forms for the Brauer algebras. In this way, the Brauer algebra ${\rm Br}_d(N)$ can be realized naturally as a level 1 cyclotomic quotient of ${\bigdoublevee}_{d}(\Xi)$.
The connection to Lie theory however is based on a more subtle realization of the Brauer algebra as follows: Let $n\in\mathbb{Z}$ be large (say $N=2n\geq 2d$) and consider the the type ${\rm D}_n$ Lie algebra ${\mathfrak{so}}(N)$ of rank $n$ with its vector representation $V$. Let $\varpi_0$ be the fundamental weight corresponding to a spin representation (that is to one of the fork ends in the Dynkin diagram) and let ${\mathfrak{p}}\subset {\mathfrak{so}}(N)$ be the (type $A$) maximal parabolic corresponding to the simple roots orthogonal to $\varpi_0$. For any fixed $\delta\in\mathbb{Z}$ let $M^{\mathfrak{p}}(\delta)$ be the associated parabolic Verma module with highest weight $\delta\omega_0$, see [@Humphreys]. Then [@ES3 Theorem 3.1] gives a natural isomorphism of algebras $$\begin{aligned}
{\operatorname{End}}_{{\mathfrak{so}}(N)}(\MdV)^{{\operatorname}{opp}}\cong {{\bigdoublevee}_d^{\rm cycl}}\end{aligned}$$ where ${{\bigdoublevee}_d^{\rm cycl}}= {{\bigdoublevee}_d}(\Xi)/(y_1-\alpha)(y_1-\beta)$ for $\alpha=\frac{1}{2}(1-\delta)$ and $\beta=\frac{1}{2}(\delta+N-1)$. The finite dimensional algebra ${{\bigdoublevee}_d^{\rm cycl}}$ decomposes into simultaneous generalized eigenspaces with respect to $ {\mathbb{C}}[y_1,\ldots y_d]$. Let ${{\mathbf{f}}}$ be the idempotent of ${{\bigdoublevee}_d^{\rm cycl}}$ which projects onto all common generalized eigenspaces with [*small*]{} eigenvalues $c_j$ with respect to $y_j$, i.e. where $c_j$ satisfy $|c_j|<\beta$ for $1\leq j\leq d$.
Our main result, Theorem \[thm:main\], is then the following:
For any fixed $\delta\in\mathbb{Z}$, there is an isomorphism of algebras $$\begin{aligned}
\Phi_\delta:&& {\rm Br}_d(\delta)\longrightarrow {{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}\end{aligned}$$ given on the standard generators of the Brauer algebra by $$\begin{aligned}
\label{difficult1intro}
t_k \quad\longmapsto\quad -Q_k s_k Q_k + \frac{1}{b_k}{{\mathbf{f}}},&\text{and}& g_k \quad\longmapsto\quad Q_k e_k Q_k.\end{aligned}$$
for certain elements $Q_k$ and $b_k$, defined in and , which can be expressed in term of the polynomial generators $y_j$, $1\leq j \leq d$ and $\beta$.
Note that the idempotent truncation is independent of $N$ (as long $N$ is large enough), but the right hand side as well as the map do depend on $N$. In particular, the parameter $N$ on the right hand side changes into the parameter $\delta$ on the left hand side. Under the isomorphism, the Jucys-Murphy elements $\xi_k$ of the Brauer algebra are sent to $-y_k {{\mathbf{f}}}$ inside ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$, see Proposition \[jucysmurphy\]. In particular, generalized eigenspaces for Jucys-Murphy elements coincide with generalized eigenspaces of the polynomial generators $y_k$.
By general theory on category $\mathcal{O}$, [@BGS], [@Backelin], the algebra ${{\bigdoublevee}_d^{\rm cycl}}$ can be equipped with a positive $\mathbb{Z}$-grading, see [@ES3 Theorem 3.1]. Since the idempotent truncation $\mathbf{f}$ corresponds to successive projections onto blocks, see [@ES3 Section 4.1], $B_d(\delta)$ inherits a grading which is then the grading in Theorem \[first\]. In contrast to a general block in category $\mathcal{O}$, the grading can be made totally explicit in our case using the theory of generalized Khovanov algebras of type $D$, [@ES3]. Note that the theorem implies that all of the combinatorics developed in [@ES3] for ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$ are now applicable to the Brauer algebra.
As an application of our result note that understanding the degree of non-semisimplicity for Brauer algebras and decomposition numbers in the non-semisimple cases gives some first insight into the structure of the tensor product of the natural module for the orthosymplectic Lie superalgebra via the result from [@ESSchurWeyl], or [@LZ].\
The idea and difficulty behind the formulas stems from the fact that we connect directly the semiorthogonal form for ${{\bigdoublevee}_d^{\rm cycl}}$ from [@AMR] and for ${\rm Br}_d(\delta)$ from [@Nazarov] by realizing the latter as obtained from the first via a naive idempotent truncation to small eigenvalues corrected with some extra terms encoded in the rather complicated elements $Q_k$ and $b_k$. The correction terms seriously depend on the generalized weight spaces. In the semisimple case, i.e if all generalized eigenvectors are actually proper eigenvectors, the formulas simplify drastically, since the operators act on an eigenspace just by a certain number which can be expressed combinatorially in terms of contents of tableaux. The point here is that our formulas also work in the non-semisimple cases. Then the definitions of the correction terms involves (inverses) of square roots. It is a nontrivial result, that the operators are well-defined and that the images of the generators of the Brauer algebra satisfy the Brauer algebra relations.
We should mention that a similar result for walled Brauer algebras was obtained in [@SS2]. The two results are independent and, as far as we can see, neither of the two implies the other. In fact, the result [@SS2 Lemma 8.1] seriously simplifies the setup treated there, but doesn’t hold in the Brauer algebra setting. As a result, the Brauer algebra requires a very different treatment.
Another approach defining gradings on Brauer algebras arising from semiorthogonal forms was taken independently in [@Li] and resulted in a KLR-type presentation of Brauer algebras. One can show that the two algebras are actually isomorphic as graded algebras, see [@LiS]. In particular, parabolic category $\mathcal{O}$ of type $D$ should provide a general framework for higher level cyclotomic quotients of VW-algebras provide some KLR-type presentations similar to the beautiful results in type A, see e.g. [@BK], [@BS3], [@HM].
[**Acknowledgment**]{} We like to thank Jonathan Brundan and Antonio Sartori for many helpful discussions.
Brauer algebra and VW algebras
==============================
We start with the definition of the Brauer algebra in terms of generators and relations. Then we recall the definition of its degenerate affine analogue, the so-called VW-algebra with its cyclotomic quotients. By an algebra we always mean an associative unitary algebra with unit $1$.
Let $d \in {\mathbb{N}}$ and $\delta \in \mathbb{C}$. The [*Brauer algebra*]{} ${\rm Br}_d(\delta)$ is the associative $\mathbb{C}$-algebra generated by elements $t_i$, $g_i$, $1\leq i\leq d-1$ subject to the relations $$\begin{array}{lll}
t_i^2=1, & t_i t_j= t_j t_i \text{ for } |i-j|>1, & t_i t_{i+1} t_i = t_{i+1} t_i t_{i+1}, \\
g_i^2=\delta g_i, & g_i g_j= g_j g_i \text{ for } |i-j|>1, & g_i g_{i+1} g_i = g_i, \\
t_i g_i= g_i = g_i t_i, & g_i t_j= t_j g_i \text{ for } |i-j|>1, &g_{i+1} g_{i} g_{i+1}= g_{i+1}, \\
t_i g_{i+1} g_i = t_{i+1} g_i, & t_{i+1} g_{i} g_{i+1} = t_{i} g_{i+1}.&
\end{array}$$ whenever the terms in the expressions are defined.
\[rem:different\_rel\] Since $g_i g_{j} g_i = g_i t_i g_{j} g_i = g_i t_j g_i$ one can replace the relation $g_i t_{j} g_i = g_i$ for $|i-j| = 1$ by the relation $g_{i} g_{j} g_{i}= g_{i}$.
\[def:VW\] Let $d \in {\mathbb{N}}$ and $\Xi = (\omega_i)_{i \in \mathbb{N}}$ with $\omega_i \in
\mathbb{C}$ for all $i$. Then the associated *VW-algebra* ${{\bigdoublevee}_d}(\Xi)$ is the algebra generated by $$\begin{aligned}
s_i, e_i, y_j&&1 \leq i \leq d-1, 1 \leq i \leq d, k \in {\mathbb{N}},\end{aligned}$$ subject to the following relations (for $1
\leq a,b \leq d-1$, $1 \leq c < d-1$, and $1 \leq i,j \leq d$):
1. $s_a^2 = 1$ \[1\]
2. 1. $s_as_b = s_bs_a$ for $\mid a-b \mid > 1$ \[2a\]
2. $s_c s_{c+1} s_c = s_{c+1} s_c s_{c+1}$ \[2b\]
3. $s_ay_i = y_is_a$ for $i \not\in \{a,a+1\}$ \[2c\]
3. $e_a^2 = \omega_0 e_a$ \[3\]
4. $e_1y_1^ke_1 = \omega_k e_1$ for $k \in {\mathbb{N}}$ \[4\]
5. 1. $s_ae_b = e_bs_a$ and $e_ae_b = e_be_a$ for $\mid a-b \mid > 1$ \[5a\]
2. $e_ay_i = y_ie_a$ for $i \not\in \{a,a+1\}$ \[5b\]
3. $y_iy_j = y_jy_i$ \[5c\]
6. 1. $e_as_a = e_a = s_ae_a$ \[6a\]
2. $s_ce_{c+1}e_c = s_{c+1}e_c$ and $e_ce_{c+1}s_c = e_cs_{c+1}$ \[6b\]
3. $e_{c+1}e_cs_{c+1} = e_{c+1}s_c$ and $s_{c+1}e_ce_{c+1} =
s_ce_{c+1}$ \[6c\]
4. $e_{c+1}e_ce_{c+1} = e_{c+1}$ and $e_ce_{c+1}e_c = e_c$ \[6d\]
7. $s_ay_a - y_{a+1}s_a = e_a - 1$ and $y_as_a - s_ay_{a+1} = e_a - 1$ \[7\]
8. 1. $e_a(y_a+y_{a+1}) = 0$ \[8a\]
2. $(y_a+y_{a+1})e_a = 0$ \[8b\]
The VW-algebra ${{\bigdoublevee}_d}$ is a degeneration of the affine BMW-algebra [^2], [@DRV], hence plays the analogue role for the Brauer algebra as the degenerate affine Hecke algebra for the symmetric group. It was introduced originally by Nazarov in [@Nazarov] under the name generalized Wenzl-algebra [^3].
Finally we introduce, following [@AMR], the cyclotomic quotients of ${{\bigdoublevee}_d}$ of level $\ell$:
Given $\mathbf{u}=(u_1, u_2,\ldots, u_\ell)\in \mathbb{C}^\ell$ we denote by ${{\bigdoublevee}_d}(\Xi;{\bf u})$ the quotient $$\begin{aligned}
{{\bigdoublevee}_d}(\Xi,{\bf u})&=&{{\bigdoublevee}_d}(\Xi)/\prod_{i=1}^\ell(y_1-u_i)\end{aligned}$$ and call it the *cyclotomic VW-algebra of level $\ell$* with parameters $\bf{u}$.
As explained in [@AMR], the tuple $\Xi$ must satisfy some admissibility condition for the algebra ${{\bigdoublevee}_d}(\Xi)$ to have a nice basis. Furthermore, the $\Xi$ must satisfy some $\mathbf{u}$-admissibility condition for this basis to be compatible with the quotient, see [@AMR Theorem A, Prop. 2.15].
Inside the VW-algebra ${{\bigdoublevee}_d}(\Xi)$, the elements $\{y_k | 1 \leq k \leq d \}$ generate a free commutative subalgebra, hence we can consider the simultaneous generalized eigenspace decompositions for these elements. Any finite dimensional ${{\bigdoublevee}_d}(\Xi)$-module $M$ has a decomposition $$\begin{aligned}
\label{eigspaces}
M &=& \bigoplus_{\mathbf{i} \in \mathbb{C}^d} M_{\mathbf{i}},\end{aligned}$$ where $M_{\textbf{i}}$ is the generalized eigenspace with eigenvalue $\mathbf{i}$, i.e., $(y_k - \mathbf{i}_k)^N M_{\mathbf{i}} = 0$ for $N \gg 0$ sufficiently large. We first describe how the generators $e_k$ and $s_k$ interact with this eigenspace decomposition.
\[lem:eigenvalue\_and\_e\] For all $1 \leq k < d$ the following holds $$e_k M_{\mathbf{i}} \subset \left\lbrace
\begin{array}{ll}
\{0\} & \text{if } \mathbf{i}_k + \mathbf{i}_{k+1} \neq 0, \\
\bigoplus_{\mathbf{i}' \in {\rm I}} M_{\mathbf{i}'} & \text{if } \mathbf{i}_k + \mathbf{i}_{k+1} = 0,
\end{array} \right.$$ where ${\rm I} = \{\mathbf{i}' \in \mathbb{C}^d | \mathbf{i}_j' = \mathbf{i}_j \text{ for } j \neq k,k+1 \text{ and } \mathbf{i}_k' + \mathbf{i}_{k+1}' = 0\}$.
Assume first that $a := \mathbf{i}_k+\mathbf{i}_{k+1} \neq 0$. Then the endomorphism induced by $(y_k+y_{k+1} - a)$ is nilpotent on $M_\mathbf{i}$ and hence $y_k+y_{k+1}$ induces an automorphism of $M_\mathbf{i}$. Hence $e_k M_\mathbf{i} = e_k (y_k + y_{k+1})M_\mathbf{i} = \{0\}$, where for the last equality Relation (VW.\[8a\]) was used.
Now assume $\mathbf{i}_k+\mathbf{i}_{k+1} = 0$. Since $(y_k + y_{k+1})e_k = 0$ by Relation (VW.\[8b\]) we know that on the image of $e_k$ the endomorphism induced by $y_k+y_{k+1}$ has eigenvalue $0$, hence $y_k$ and $y_{k+1}$ have eigenvalues that add up to $0$.
The situation for $s_k$ is more complicated and we need the following preparation:
\[lem:eigenvalue\_and\_s\] Let $\psi_k = s_k(y_k - y_{k+1}) +1$, then for all $1 \leq k \leq d-1$ it holds $$\psi_k M_{\mathbf{i}} \subset \left\lbrace
\begin{array}{ll}
M_{s_k \mathbf{i}} & \text{if } \mathbf{i}_k + \mathbf{i}_{k+1} \neq 0, \\
\bigoplus_{\mathbf{i}' \in {\rm I}} M_{\mathbf{i}'} & \text{if } \mathbf{i}_k + \mathbf{i}_{k+1} = 0,
\end{array} \right.$$ In case $\mathbf{i}_k+\mathbf{i}_{k+1}\neq 0$ and additionally $|\mathbf{i}_k-\mathbf{i}_{k+1}| \neq 1$, the map $\psi_k$ defines an isomorphism of vector spaces $M_\mathbf{i} \cong M_{s_k(\mathbf{i})}$.
From Relation (VW.\[7\]) we obtain $y_k \psi_k = \psi_k y_{k+1} + e_k (y_k-y_{k+1})$ and by Relation (VW.\[2c\]) then $y_j \psi_k = \psi_k y_j$ for $|j-k|>1$.
Assume $\mathbf{i}_k + \mathbf{i}_{k+1} \neq 0$ and let $m \in M_{\mathbf{i}}$. Then by Lemma \[lem:eigenvalue\_and\_e\] we have $y_k \psi_k m = \psi_k y_{k+1} m$ and $y_{k+1} \psi_k = \psi_k y_{k} m$, hence $(y_i - \mathbf{i}_{s_k(i)})^N \psi_k m = \psi_k (y_{s_k(i)} - \mathbf{i}_{s_k(i)})^N m$ for all $N$ and all $i$. Thus $\psi_k M_\mathbf{i} \subset M_{s_k(\mathbf{i})}$.
Relation (VW.\[7\]) implies $(y_k+y_{k+1})\psi_k = \psi_k(y_k+y_{k+1})$. Assuming $i_k + i_{k+1} = 0$ we have $(y_k+y_{k+1})^N M_\mathbf{i} = \{0\}$ and hence $ (y_k+y_{k+1})^N \psi_k M_\mathbf{i} = \psi_k (y_k+y_{k+1})^N M_\mathbf{i} = \{0\}$. Thus, on the image of $\psi_k$, the endomorphism induced by $y_k+y_{k+1}$ has eigenvalue $0$ and therefore $y_k$ and $y_{k+1}$ have eigenvalues adding up to $0$.
Assuming now $\mathbf{i}_k + \mathbf{i}_{k+1} \neq 0$ and furthermore $|\mathbf{i}_k-\mathbf{i}_{k+1}| \neq 1$, then thanks to Relation (VW.\[7\]) and Lemma \[lem:eigenvalue\_and\_e\] we have $\psi_k^2=-(y_k-y_{k+1})^2+1$ as an endomorphisms of $M_\mathbf{i}$. Setting $c=\mathbf{i}_k - \mathbf{i}_{k+1}$ it follows $\left( (y_k-y_{k+1}) - c \right)^N M_\mathbf{i} = \{ 0 \}$ for $N \geq 0$ by definition. In particular, as endomorphisms of $M_\mathbf{i}$, this implies $$\begin{aligned}
\psi_k^2 &=& 1-\left((y_k-y_{k+1})-c+c\right)^2 \\
&=&1- c^2 - \left((y_k-y_{k+1})-c \right)^2 - c (y_k-y_{k+1}-c) = 1-c^2-z\end{aligned}$$ for some nilpotent endomorphism $z$. Since $c^2 \neq 1$ by assumption, $\psi_k^2$ is invertible and therefore also $\psi_k$. Note that the concrete form of the inverse depends on $\mathbf{i}$.
\[cor:permute\_eigenvalues\] Assume $M_\mathbf{i} \neq \{0\}$ for some $\mathbf{i} = (\mathbf{i}_1,\dots, \mathbf{i}_{d-1}, \mathbf{i}_d)$ such that $\mathbf{i}_k + \mathbf{i}_d \neq 0$ and $|\mathbf{i}_k - \mathbf{i}_d| \neq 1$ for all $k < d$. Let $\mathbf{i}'=(\mathbf{i}_d, \mathbf{i}_1,\dots, \mathbf{i}_{d-1})$, then it holds $$(y_d - \mathbf{i}_d)^N M_\mathbf{i}=\{0\} \Longleftrightarrow (y_1-\mathbf{i}_d)^N M_{\mathbf{i}'}=\{0\}$$ for all positive integers $N$.
By Lemma \[lem:eigenvalue\_and\_s\] the element $\psi:=\psi_1 \cdots \psi_{d-1}$ as an isomorphism between $M_{\mathbf{i}}$ and $M_{\mathbf{i}'}$ intertwining the actions of $y_1$ and $y_d$. i.e. $\psi(y_1m)=y_d\psi(m)$ for any $m\in M_{\mathbf{i}}$.
\[cor:eigenvalue\_and\_s\] Assume that $\mathbf{i}_k \neq \mathbf{i}_{k+1}$, then $$s_k M_{\mathbf{i}} \subset \left\lbrace
\begin{array}{ll}
M_{s_k \mathbf{i}} \oplus M_{\mathbf{i}} & \text{if } \mathbf{i}_k + \mathbf{i}_{k+1} \neq 0, \\
\bigoplus_{\mathbf{i}' \in {\rm I}} M_{\mathbf{i}'} & \text{if } \mathbf{i}_k + \mathbf{i}_{k+1} = 0.
\end{array} \right.$$
Under the assumption that $\mathbf{i}_k \neq \mathbf{i}_{k+1}$ we have that $y_k-y_{k+1}$ is an automorphism of $M_\mathbf{i}$ and the statement follows then directly from Lemma \[lem:eigenvalue\_and\_s\].
Cyclotomic quotients and category $\mathcal{O}$ {#sec:cycl_and_cat_O}
===============================================
Fix $\delta \in {\mathbb{Z}}$. Let $\mathfrak{g}=\mathfrak{so}(2n)$ be the complex special orthogonal Lie algebra corresponding to the Dynkin diagram $\Gamma$ of type ${\rm D}_n$ and fix a triangular decomposition $\mathfrak{g}= \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Fix $\mathfrak{l}$, a Levi subalgebra obtained from an embedding of the type ${\rm A}_{n-1}$ Dynkin diagram into $\Gamma$ and denote by $\mathfrak{p} = \mathfrak{l} \oplus \mathfrak{n}^+$ the corresponding parabolic subalgebra. Denote by $\varepsilon_1,\ldots,\varepsilon_n$ the standard basis of $\mathfrak{h}^*$.
By $\mathcal{O}^\mathfrak{p}(n) = \mathcal{O}^\mathfrak{p}_{\rm int}(\mathfrak{so}(2n))$ we denote the integral parabolic BGG category $\mathcal{O}$, i.e., the full subcategory of $\mathcal{U}(\mathfrak{g})$-modules consisting of finitely generated $\mathcal{U}(\mathfrak{g})$-modules, semisimple over $\mathfrak{h}$ with integral weights, and locally finite for $\mathfrak{p}$, see [@Humphreys Chapter 9]. Let $$X_n^\mathfrak{p} = \{{\lambda}\in{\mathfrak{h}}^*\text{ integral}\mid{\lambda}+\rho=\sum_{i=1}^n{\lambda}_i\epsilon_i \text{ where } {\lambda}_1< {\lambda}_2<\cdots< \lambda_n\},
\label{eqn:DefLa}$$
where $\rho$ denotes the half-sum of the positive roots, $\rho=\sum_{i=1}^n(i-1)\epsilon_i$. Then $X_n^\mathfrak{p}$ is precisely the set of highest weights of simple objects in $\mathcal{O}^\mathfrak{p}(n)$, see [@Humphreys]. For an element $\lambda \in X_n^\mathfrak{p}$ we denote by $+M^{\mathfrak{p}}({\lambda})$ the parabolic Verma module with highest weight $\lambda$. Note that a weight ${\lambda}=({\lambda}_1,{\lambda}_2,\cdots,{\lambda}_n)$ written in the $\epsilon$-basis is integral, if either ${\lambda}_i\in\mathbb{Z}$, i.e. $2{\lambda}_i$ is even for all $i$, or ${\lambda}_i\in\frac{1}{2}+\mathbb{{\mathbb{Z}}}$, i.e. $2{\lambda}_i$ is odd for all $i$. As in [@ES3], a crucial player in the following will be the parabolic Verma module $M^\mathfrak{p}(\underline{\delta})$ of highest weight $$\begin{aligned}
\label{eqn:delta}
\underline{\delta}& =&\frac{\delta}{2}(\varepsilon_1 + \ldots + \varepsilon_n),\end{aligned}$$ i.e., a multiple of the fundamental weight $\frac{1}{2}(\varepsilon_1 + \ldots + \varepsilon_n)$. With an appropriate choice of parameters $\Xi_\delta$, there is , see [@ES3], a natural (right) action of ${{\bigdoublevee}_d}(\Xi_\delta)$ on $\Mde \otimes V^{\otimes d}$ by $\mathfrak{g}$-endomorphisms. Hence we have an algebra homomorphism ${{\bigdoublevee}_d}(\Xi_\delta)\rightarrow {\operatorname{End}}_{\mathfrak{g}}(\MdV)^{{\operatorname}{opp}}$. The parameter set $\Xi_\delta = (\omega_a)_{a \geq 0}$ appear as part of the following definition’:
\[defalphabeta\] For $N = 2n$, we define the *cyclotomic parameters* as follows $$\begin{aligned}
&\omega_0=N, \quad \omega_1=N\frac{N-1}{2}, \quad \omega_a=(\alpha+\beta)\omega_{a-1}-\alpha\beta \omega_{a-2} \quad \text{for } a \geq 2,&\label{NN}\\
&\text{where we set}&\nonumber\\
&\alpha=\frac{1}{2}(1-\delta),\quad \quad\quad \beta=\frac{1}{2}(\delta+N-1).\label{ab}&\end{aligned}$$
Then the following important result holds:
\[iso\] If $n \geq 2d$, then the ${{\bigdoublevee}_d}(\Xi_\delta)$-action from above induces an isomorphism of algebras $$\begin{aligned}
\label{Psi}
\Psi({{\underline\delta}}):\quad{{\bigdoublevee}_d}(\Xi_\delta;\alpha,\beta)&\longrightarrow&{\operatorname{End}}_{\mathfrak{g}}(\MdV)^{{\operatorname}{opp}}.\end{aligned}$$
In the following we abbreviate ${{\bigdoublevee}_d^{\rm cycl}}={{\bigdoublevee}_d}(\Xi_\delta;\alpha,\beta)$.\
Note that via , the space $\MdV$ becomes a module for ${{\bigdoublevee}_d}(\Xi_\delta;\alpha,\beta)$ with the action preserving the finite dimensional ${\mathfrak{g}}$-weight spaces, hence we have a simultaneous generalized eigenspace decomposition . We describe now this decomposition Lie theoretically and then combinatorially using the notion of a up-down bitableaux and bipartitions. We start with the following well-known fact:
\[flags\] Let $\mu \in {\Lambda}$. Then $M^\mathfrak{p}(\mu)\otimes V$ has a filtration (called Verma flag) with sections isomorphic to precisely the $M^\mathfrak{p}(\mu\pm\epsilon_j)$ for all $j=1,\dots n$ such that $\mu\pm\epsilon_j \in {\Lambda}$. The sections are pairwise not isomorphic.
This is a standard consequence of the tensor identity; see e.g. [@Humphreys Theorem 3.6], noting that $V$ has precisely the weights $\pm\epsilon_j$ for $1\leq j\leq n$.
In particular, $M^\mathfrak{p}(\underline{\delta})\otimes V^{\otimes{d}}$ has a Verma flag with sections isomorphic to precisely the $M^\mathfrak{p}({\lambda})$, where ${\lambda}-{{\underline\delta}}=\sum_{j=i}^d a_i$, where $a_i\in\{\pm\epsilon_j\mid1\leq j\leq n\}$. Given such a weight ${\lambda}$ we can write ${\lambda}-{{\underline\delta}}=\sum_{i=1}^n m_i\epsilon_i$ with $m_i\leq 0$ for $1\leq i\leq s$ and $m_i>0$ for $i>s$ for some (uniquely defined) $s$. Then we assign to ${\lambda}$ the bipartition $\varphi({\lambda})=({\lambda}^{(1)},{\lambda}^{(2)})$ with ${\lambda}^{(1)}_i=|m_i|$ and ${\lambda}^{(2)}_i=m_{s+i}$. This will be seen as a pair of [*Young diagrams*]{}, which are both consisting of arrangements of boxes with left-justified rows and number of boxes per row weakly decreasing from top to bottom. Each box $b$ in such a pair of Young diagrams has a content $c(b)$, let $b$ be in the $r$-th row of its diagram and in the $c$-th column (counting from top to bottom and from left to right, starting with $1$ in both cases), then $c(b) = r-c$ if $b$ is in ${\lambda}^{(1)}$ and $c(b) = c-r$ if $b$ is in ${\lambda}^{(2)}$. Below, the pair of Young diagrams attached to the bipartition $(3,2,1,1), (2,2,1)$ is shown with the contents of each box written in the box
$$\begin{aligned}
\label{bipartition}
\begin{minipage}{6cm}
$\left({\lambda}^{(1)}={\young(0{$-1$}{$-2$},10,2,3)}, {\lambda}^{(2)}=\young(01,{$-1$}0,{$-2$})\right)$
\end{minipage}\end{aligned}$$
It will be convenient to draw such a pair of Young diagrams $\varphi({\lambda})$ as a [*double Young diagram*]{} with a total of $d$ boxes in the following way.
Consider an infinite strip with $n$ columns and a horizontal line $o$. This horizontal line will split the strip into two regions, an upper and a lower part. A double Young diagram consists of two Young diagrams, one placed in the upper half with center of gravity on the lower right point of that region and a second one places in the lower part with center of gravity on the upper left such that no column contains boxes above and below the line $o$.
The double Young diagram attached to the weight ${\lambda}$ is constructed as follows: the Young diagram at the bottom is just the the Young diagram for ${\lambda}^{(1)}$ transposed; the Young diagram at the top is obtained from the Young diagram for ${\lambda}^{(2)}$ by transposing the diagram and then rotating it by 180 degrees. Denote the result $(({\lambda}^{(1)})^t,{}^t({\lambda}^{(2)}))$. The contents for the boxes are transposed respectively rotated accordingly. Below, the double Young diagram attached to the bipartition $(3,2,1,1), (2,2,1)$ is displayed:
$$\begin{aligned}
\label{displaybipartition}
&
\begin{minipage}{6cm}
$\left({\lambda}^{(1)}={\young(0{$-1$}{$-2$},10,2,3)}, {\lambda}^{(2)}=\young(01,{$-1$}0,{$-2$})\right)$
\end{minipage}
\begin{minipage}{6cm}
\begin{tikzpicture}[scale=1, thick]
\draw[thin,red] (-.25,0) -- +(5.5,0);
\node at (.25,1.5) {\tiny $1$};
\node at (.75,1.5) {\tiny $2$};
\node at (1.25,1.5) {\tiny $3$};
\node at (1.75,1.5) {\tiny $4$};
\node at (2.75,1.5) {\tiny $\cdots$};
\node at (3.75,1.5) {\tiny n-2};
\node at (4.25,1.5) {\tiny n-1};
\node at (4.75,1.5) {\tiny n};
\draw[dotted] (0,1.75) -- +(0,-4);
\draw[dotted] (0.5,1.75) -- +(0,-4);
\draw[dotted] (1,1.75) -- +(0,-4);
\draw[dotted] (1.5,1.75) -- +(0,-4);
\draw[dotted] (2,1.75) -- +(0,-4);
\draw[dotted] (3.5,1.75) -- +(0,-4);
\draw[dotted] (4,1.75) -- +(0,-4);
\draw[dotted] (4.5,1.75) -- +(0,-4);
\draw[dotted] (5,1.75) -- +(0,-4);
\draw rectangle (.5,-.5);
\draw (.5,0) rectangle +(.5,-.5);
\draw (0,-.5) rectangle +(.5,-.5);
\fill[fill=lightgray, draw=black] (0,-1) rectangle +(.5,-.5);
\fill[fill=lightgray, draw=black] (.5,-.5) rectangle +(.5,-.5);
\fill[fill=lightgray, draw=black] (1,0) rectangle +(.5,-.5);
\fill[fill=lightgray, draw=black] (1.5,0) rectangle +(.5,-.5);
\node at (.25,-.25) {$0$};
\node at (.75,-.25) {$1$};
\node at (1.25,-.25) {$2$};
\node at (1.75,-.25) {$3$};
\node at (.25,-.75) {$-1$};
\node at (.75,-.75) {$0$};
\node at (.25,-1.25) {$-2$};
\draw (2,-.025) rectangle +(.5,.05);
\draw (2.5,-.025) rectangle +(.5,.05);
\draw (3,-.025) rectangle +(.5,.05);
\fill[fill=lightgray, draw=black] (3.5,0) rectangle +(.5,.5);
\draw (4,0) rectangle +(.5,.5);
\fill[fill=lightgray, draw=black] (4,.5) rectangle +(.5,.5);
\draw (4.5,0) rectangle +(.5,.5);
\fill[fill=lightgray, draw=black] (4.5,.5) rectangle +(.5,.5);
\node at (3.75,.25) {$-2$};
\node at (4.25,.25) {$-1$};
\node at (4.75,.25) {$0$};
\node at (4.25,.75) {$0$};
\node at (4.75,.75) {$1$};
\\
\end{tikzpicture}
\end{minipage}
&\nonumber\\\end{aligned}$$
In addition we will modify the [**content**]{} of a box in such a double Young diagram as follows. If a box $b$ is inside the lower part of the diagram, then $c_\delta(b)=c(b) + \alpha$, while the content of a box $b$ in the upper part of the diagram is defined to be $c_\delta(b)=c(b) + \beta$.
To read off the weight ${\lambda}$ from the corresponding double Young diagram we consider the boundary boxes responsible for the shape (i.e. the boxes shaded in the example ). Let $b_i$ denote the number of boxes in column $i$, multiplied with $-1$ if the boxes are in the lower half. Then $$\begin{aligned}
\lambda &=& \underline{\delta} + \sum_{i=1}^n b_i \varepsilon_i.\end{aligned}$$ More generally, the [*weight*]{}, ${\rm wt}(Y)$, of a double diagram $Y$ is defined as $$\begin{aligned}
{\rm wt}(Y) &=& \sum_{i=1}^n b_i \varepsilon_i.\end{aligned}$$ An [*up-down bitableaux*]{} of length $d$ is a sequence $\mathbf{Y}=(\mathbf{Y}_1, \mathbf{Y}_2,\ldots, \mathbf{Y}_d)$ of double Young diagrams such that $\mathbf{Y}_1$ corresponds to the trivial bipartition $(\emptyset,\emptyset)$ and two consecutive double diagrams differ just by one box (added or removed). Let ${\rm wt}(\mathbf{Y})=({\rm wt}(\mathbf{Y}))_{1\leq i\leq d}$ be the weight sequence attached to $\mathbf{Y}$. The set of all such up-down bitableaux of length $d$ is denoted by $\mathcal{T}_d$.\
The module $M:=\Mde \otimes V^{\otimes d}$ decomposes into a direct sum of submodules $$\begin{aligned}
\label{stupid}
M&=&\bigoplus_{\mathbf{a} \in \mathbb{C}^d/S_d} M_{S_d\mathbf{a}},\end{aligned}$$ where $a$ runs through a fixed set of representatives for the $S_d$-orbits on $\mathbb{C}^d$ and $M_{S_d\mathbf{a}}=\oplus_{w\in S_d} M_{w(\mathbf{a})}$, with the summands defined as in . These are then the subspaces of $M$ where the multiset of occurring generalized eigenvalues of all the individual $y_i$’s is fixed.
\[surjectivity\] Assume $n \geq 2d$. There is a canonical bijection between $\mathcal{T}_d$ and the parabolic Verma modules appearing as sections in a Verma filtration of $\Mde \otimes V^{\otimes d}$ counted with multiplicities such that the following holds
1. The bijection is given by assigning to a up-down bitableau $\mathbf{Y}=(\mathbf{Y}_1, \mathbf{Y}_2,\ldots, \mathbf{Y}_d)$ the parabolic Verma module $M(\underline{\delta} + {\rm wt}(\mathbf{Y}_d))$.
2. For $1 \leq j \leq d$, let $\eta_j=1$ if $\mathbf{Y}_j$ was obtained from $\mathbf{Y}_{j-1}$ by adding a box $B_j$ and $\eta_j=-1$ if $\mathbf{Y}_j$ was obtained from $\mathbf{Y}_{j-1}$ by removing a box $B_j$. Then the parabolic Verma module $M^\mathfrak{p}(\underline{\delta} + {\rm wt}(\mathbf{Y}_d))$ associated to $\mathbf{Y}$ appears as a subquotient of the summand in containing the generalized eigenspace of the operator $y_k$ for eigenvalue $\eta_k c_\delta(B_k)$.
In case $\delta\geq 0$ this follows directly from [@ES3] and the bijection between up-down bitableaux and Verma path describing the Verma modules appearing in a Verma filtration and their eigenvalues. For $\delta<0$ the arguments are totally analogous.
We call an eigenvalue of $y_k$ *small* if it corresponds to the content of a box in the lower half and we call it *large* if it corresponds to the content of a box in the upper half.
The idempotent we define now projects onto the generalized common eigenspaces of the elements $y_k$ that are small.
For $1 \leq k \leq d$ we denote by $\eta_k \in {{\bigdoublevee}_d^{\rm cycl}}$ the idempotent projecting onto the generalized eigenspace of $y_k$ with eigenvalue different from $\beta$. Furthermore let $\mathbf{f}_k = \eta_1 \cdots \eta_k$ and $\mathbf{f} = \mathbf{f}_d$.
It is clear from the definition resp. the calculus of up-down sequences that if $y_k$ has a large eigenvalue on some composition factor, then there exists a $j \leq k$ such that $y_j$ has eigenvalue $\beta$ on that composition factor. Hence the element $\mathbf{f}$ projects onto the common generalized eigenspaces of small eigenvalues for the commutative subalgebra generated by the $y_j$’s.\
The idempotent ${{\mathbf{f}}}$ not central, but we have at least the following formulas:
\[prop:kill\_f\] In ${{\bigdoublevee}_d^{\rm cycl}}$ the following equalities hold:
1. For $1 \leq k \leq d-1$ $$\begin{array}{rccrc}
i) & c_{k} {{\mathbf{f}}}s_k {{\mathbf{f}}}= c_{k} s_k {{\mathbf{f}}}, & \qquad & iii) & c_k {{\mathbf{f}}}e_k {{\mathbf{f}}}= c_k e_k {{\mathbf{f}}},\\
ii) &b_{k+1} {{\mathbf{f}}}s_k {{\mathbf{f}}}= b_{k+1} s_k {{\mathbf{f}}}, & \qquad & iv) &b_{k+1} {{\mathbf{f}}}e_k {{\mathbf{f}}}= b_{k+1} e_k {{\mathbf{f}}}.
\end{array}$$
2. For $1 \leq k < d-1$ $$\begin{array}{rccrc}
i) & e_k {{\mathbf{f}}}s_{k+1} {{\mathbf{f}}}= e_k s_{k+1} {{\mathbf{f}}}, & \qquad & iv) & {{\mathbf{f}}}s_k {{\mathbf{f}}}s_{k+1} {{\mathbf{f}}}= {{\mathbf{f}}}s_k s_{k+1} {{\mathbf{f}}},\\
ii) & e_k {{\mathbf{f}}}e_{k+1} {{\mathbf{f}}}= e_k e_{k+1} {{\mathbf{f}}}, & \qquad & v) & {{\mathbf{f}}}s_k {{\mathbf{f}}}e_{k+1} {{\mathbf{f}}}= {{\mathbf{f}}}s_k e_{k+1} {{\mathbf{f}}},\\
iii) & {{\mathbf{f}}}e_k {{\mathbf{f}}}s_{k+1} {{\mathbf{f}}}e_{k} {{\mathbf{f}}}= {{\mathbf{f}}}e_k s_{k+1} e_k {{\mathbf{f}}}.
\end{array}$$
as well as all of these equalities with $k$ and $k+1$ swapped.
We start with part *1.)*. For case *i)* we claim that $$c_{k} {{\mathbf{f}}}s_k {{\mathbf{f}}}= c_{k} s_k {{\mathbf{f}}}- c_{k} (1 - {{\mathbf{f}}}) s_k {{\mathbf{f}}}= c_{k} s_k {{\mathbf{f}}}.$$ We only have to justify the last equality. But this holds due to Lemma \[lem:eigenvalue\_and\_s\]; the image of $(1-{{\mathbf{f}}})s_k {{\mathbf{f}}}$ is either 0 or consists of generalized eigenvectors for $y_{k}$ with eigenvalue $\beta$. By Corollary \[cor:eigenvalue\_and\_s\] and the fact that due to the diagram calculus and the assumptions on $n$, $y_1$ always acts by a scalar these are honest eigenvectors and thus $c_k$ acts by zero. For case *ii)* we calculate $$\begin{aligned}
b_{k+1}{{\mathbf{f}}}s_k {{\mathbf{f}}}&=& b_{k+1}{{\mathbf{f}}}s_k \frac{c_{k+1}}{c_{k+1}}{{\mathbf{f}}}\\
&\overset{(a)}{=}& b_{k+1}{{\mathbf{f}}}(c_k s_k + e_k - 1)\frac{1}{c_{k+1}}{{\mathbf{f}}}\\
&\overset{(b)}{=}& b_{k+1} c_k s_k \frac{1}{c_{k+1}}{{\mathbf{f}}}+ b_{k+1} e_k \frac{1}{c_{k+1}}{{\mathbf{f}}}- \frac{b_{k+1}}{c_{k+1}}{{\mathbf{f}}})\\
&\overset{(c)}{=}& b_{k+1} (s_k c_{k+1} - e_k + 1) \frac{1}{c_{k+1}}{{\mathbf{f}}}+ b_{k+1} e_k \frac{1}{c_{k+1}}{{\mathbf{f}}}- \frac{b_{k+1}}{c_{k+1}}{{\mathbf{f}}}= b_{k+1} s_k {{\mathbf{f}}}\end{aligned}$$ Here equalities $(a)$ and $(c)$ hold by Lemma \[lem:commute\_b\], while equality $(b)$ is due to the other cases of this lemma. For case *iii)* we have $$c_k e_k {{\mathbf{f}}}= c_k {{\mathbf{f}}}_{k-1} e_k {{\mathbf{f}}}= c_k {{\mathbf{f}}}_k {{\mathbf{f}}}_{k-1} e_k {{\mathbf{f}}}+ c_k (1-{{\mathbf{f}}}_k){{\mathbf{f}}}_{k-1} e_k {{\mathbf{f}}}= c_k {{\mathbf{f}}}_k e_k {{\mathbf{f}}}.$$ The final equality holds because the image of $(1-{{\mathbf{f}}}_k){{\mathbf{f}}}_{k-1}$ consists of eigenvectors for $y_k$ with eigenvalue $\beta$, hence are annihilated by $c_k$. That the image consists of eigenvectors follows as in case *i)*. Furthermore, due to Lemma \[lem:eigenvalue\_and\_e\], ${{\mathbf{f}}}_k e_k = {{\mathbf{f}}}_{k+1} e_k$ and the statement follows. Case *iv)* is the same since $b_{k+1}e_k = c_k e_k$ by Relation (VW.\[8b\]).\
For part *2.)*, we note that all of these are more or less proven in the same way using Lemma \[lem:eigenvalue\_and\_e\] and Corollary \[cor:eigenvalue\_and\_s\]. We will argue for $e_k {{\mathbf{f}}}s_{k+1} {{\mathbf{f}}}= e_k s_{k+1} {{\mathbf{f}}}$ and leave the others to the reader. It holds $$e_k s_{k+1} {{\mathbf{f}}}= e_k {{\mathbf{f}}}s_{k+1} {{\mathbf{f}}}+ e_k(1-{{\mathbf{f}}})s_{k+1}{{\mathbf{f}}}.$$ If we now look at a generalized eigenspace $M_\textbf{i}$ in the image of $(1-{{\mathbf{f}}})s_{k+1}{{\mathbf{f}}}$, we see that, due to the diagram combinatorics, this can only be non-zero if $\textbf{i}_{k+1} = \beta$ and $\textbf{i}_{k+2} = -\beta$ while all other eigenvalues are small. Applying $e_k$ to this eigenspace is zero, due to Lemma \[lem:eigenvalue\_and\_e\] since $\textbf{i}_{k}$ is small and thus cannot be $-\beta$. Hence only the first summand survives which proofs the claim. Similar arguments have to be applied to the other cases.
Relation (VW.\[7\]) from Definition \[def:VW\] on the nose or multiplied with the idempotent ${{\mathbf{f}}}$ from both sides yields the following equalities:
\[lem:commute\_b\] In ${{\bigdoublevee}_d^{\rm cycl}}$ the following equalities hold for $1\leq k\leq d-1$:
1. We have $b_{k+1}s_k = s_kb_k - e_k + 1,$ and ${{\mathbf{f}}}s_k \frac{1}{b_k} {{\mathbf{f}}}= \frac{1}{b_{k+1}} {{\mathbf{f}}}s_k {{\mathbf{f}}}- \frac{1}{b_{k+1}} {{\mathbf{f}}}e_k \frac{1}{b_{k}}{{\mathbf{f}}}+ \frac{1}{b_k b_{k+1}} {{\mathbf{f}}},$
2. We have $s_k b_{k+1}= b_ks_k - e_k + 1,$ and $\frac{1}{b_k} {{\mathbf{f}}}s_k {{\mathbf{f}}}= {{\mathbf{f}}}s_k \frac{1}{b_{k+1}}{{\mathbf{f}}}- \frac{1}{b_{k}} {{\mathbf{f}}}e_k \frac{1}{b_{k+1}}{{\mathbf{f}}}+ \frac{1}{b_k b_{k+1}} {{\mathbf{f}}}$.
3. We have $c_{k+1}s_k = s_kc_k + e_k - 1,$ and ${{\mathbf{f}}}s_k \frac{1}{c_k} {{\mathbf{f}}}= \frac{1}{c_{k+1}} {{\mathbf{f}}}s_k {{\mathbf{f}}}+ \frac{1}{c_{k+1}} {{\mathbf{f}}}e_k \frac{1}{c_{k}}{{\mathbf{f}}}- \frac{1}{c_k c_{k+1}} {{\mathbf{f}}},$
4. We have $s_k c_{k+1}= c_ks_k + e_k - 1,$ and $\frac{1}{c_k} {{\mathbf{f}}}s_k {{\mathbf{f}}}= {{\mathbf{f}}}s_k \frac{1}{c_{k+1}}{{\mathbf{f}}}+ \frac{1}{c_{k}} {{\mathbf{f}}}e_k \frac{1}{c_{k+1}}{{\mathbf{f}}}- \frac{1}{c_k c_{k+1}} {{\mathbf{f}}}$.
Note moreover that by Lemma \[lem:eigenvalue\_and\_e\], Corollaries \[cor:permute\_eigenvalues\] and \[cor:eigenvalue\_and\_s\], the definition of ${{\mathbf{f}}}$, and our diagrammatic calculus we have that the expressions $$\begin{aligned}
\frac{1}{b_k} s_k {{\mathbf{f}}}\text{ and } \frac{1}{b_k} e_k {{\mathbf{f}}}.\end{aligned}$$ are well-defined. This will be used in many of the proofs to come.
The isomorphism theorem {#sec:invers-square-roots}
=======================
In the isomorphism of the main theorem will appear some square roots. We start by recalling their definition and the required setup from [@SS2]. The crucial result is the following fact from [@SS2 Section 4]:
\[prop:2\] Let $f(x) \in \mathbb{C}[x]$. Assume $B$ is a finite-dimensional algebra, and let $x_0 \in B$. Suppose that $(x-a) \nmid f(x)$ for all $a \in \mathbb{C}$ which are generalized eigenvalues for the action of $x_0$ on the regular representation. Then $f(x_0)\in B$ has a (unique) inverse and a (non-unique) square root.
Let as above $x_0 \in B$ be an element of a finite-dimensional algebra, and let $a \in \mathbb{C}$. If $a$ is not a generalized eigenvalue of $x_0$ then by Proposition \[prop:2\] we can write expressions like $$\frac{1}{x_0-a}, \qquad \sqrt{x_0-a}, \qquad \sqrt\frac{1}{x_0 -a}.\label{eq:52}$$ The square root is not unique, but we make one choice once and for all, so that for example $ \sqrt{x_0-a} \sqrt\frac{1}{x_0 -a} = 1 $. For more details we refer to [@SS2].
Let $1\leq k\leq d$. We fix the following elements of ${{\bigdoublevee}_d^{\rm cycl}}$ $$\label{19}
b_k = \beta + y_k \qquad \text{and}\qquad c_k = \beta - y_k$$ and set (for a choice of square root) $$\label{20}
Q_k = \sqrt{\frac{b_{k+1}}{b_k}} \mathbf{f}.$$ Note that ${\frac{b_{k+1}}{b_k}} \mathbf{f}$ is well-defined (in contrast to $\frac{b_{k+1}}{b_k}$) by definition of ${b_k}$ and $ \mathbf{f}$. By the above arguments, also the square root makes sense.
For $1\leq k \leq d-1$ define $$\begin{aligned}
\tilde{s}_k = - Q_k s_k Q_k + \frac{1}{b_k} \mathbf{f}&\text{and}&
\tilde{e}_k = Q_k e_k Q_k.\end{aligned}$$
We finally can state our main result:
\[thm:main\] The map $$\Phi_\delta :\quad {\rm Br}_d(\delta) \longrightarrow \mathbf{f} {{\bigdoublevee}_d^{\rm cycl}}\mathbf{f}.$$ given on the standard generators by $$\begin{aligned}
\label{difficult1}
t_k \quad\longmapsto\quad -Q_k s_k Q_k + \frac{1}{b_k}{{\mathbf{f}}},&\text{and}& g_k \quad\longmapsto\quad Q_k e_k Q_k.\end{aligned}$$ for $1 \leq k \leq d-1$ defines an isomorphism of algebras.
That the map is well-defined follows from a series of statements in Section \[sec:proof\_thm:main\]. Namely altogether the Lemmas \[lem:s2=f\] and \[lem:sisj=sjsi\], Proposition \[lem:si\_braid\].
Lemmas \[lem:e2=deltae\], \[lem:eiej=ejei\], \[lem:eee=e\], \[lem:es=e\], \[lem:se=es\], and \[lem:see=se\] prove that the elements $\tilde{s}_k$ and $\tilde{e}_k$ for $1 \leq k < d$ satisfy all the defining relations of the Brauer algebra ${\rm Br}_d(\delta)$. It suffices to prove surjectivity of $\Phi_\delta$, since the algebras have the same dimension, namely $(2d-1)!!$, by [@ES3 Proposition 4.4].
To prove surjectivity we use the description of a basis of ${{\bigdoublevee}_d^{\rm cycl}}$ from [@AMR Theorem 5.5], see [@ES3 Corollary 2.25] for our special case which says in particular that any element in ${{\bigdoublevee}_d^{\rm cycl}}$ is a linear combination of elements of the form $p_1wp_2$, where $p_1,p_2\in{\mathbb{C}}[y_1,\ldots,y_d]$ with degree $\leq 1$ in each variable and $w=x_1 \cdots x_r$ where $x_j\in\{s_i,e_i\mid 1\leq i\leq d-1\}$ for $1\leq j\leq r$. We will call such a presentation $x_1 \cdots x_r$ for $w$ a [*reduced word*]{} if $r$ is chosen minimally to present $w$ in such a form.
Since by Lemma \[lem:jucysmurphy\] all the elements $y_k {{\mathbf{f}}}$ are in the image $I$ of $\Phi_\delta$ it suffices to show that ${{\mathbf{f}}}x_1 \cdots x_r {{\mathbf{f}}}\in I$ for any reduced word $x_1 \cdots x_r$.
We show this by two inductions on the sum of the length of the word and the number of $s_i$’s occurring in the expression. For $r=0,1$ the claim is clear, since $y_k {{\mathbf{f}}}\in I$ for all $k$ and so are all polynomial expressions in the $y$’s, e.g. $Q_k^{-1} {{\mathbf{f}}}$ and $\frac{1}{b_k}{{\mathbf{f}}}$, and thus also the elements of the form ${{\mathbf{f}}}s_k {{\mathbf{f}}}$ and ${{\mathbf{f}}}e_k {{\mathbf{f}}}$ for all $k$. Hence the claim is true for $r\leq 1$.
For $r > 1$, we first assume that the expression ${{\mathbf{f}}}x_1 \cdots x_r {{\mathbf{f}}}$ contains no $s_i$’s. In this case assume that $x_r = e_l$ for some $l$, then by induction we know that $${{\mathbf{f}}}x_1 \cdots x_{r-1} {{\mathbf{f}}}c_{l} {{\mathbf{f}}}x_r {{\mathbf{f}}}$$ is in the image of $\Phi_\delta$, since all three factors are in the image by induction. By Proposition \[prop:kill\_f\] we have $$\begin{aligned}
{{\mathbf{f}}}x_1 \cdots x_{r-1} {{\mathbf{f}}}c_{l} {{\mathbf{f}}}x_r {{\mathbf{f}}}& =& {{\mathbf{f}}}x_1 \cdots x_{r-1} c_{l} x_r {{\mathbf{f}}}\nonumber\\
&=& \beta {{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}+ {{\mathbf{f}}}x_1 \cdots x_{r-1} y_{l+1} x_r {{\mathbf{f}}}\nonumber\\
&=& \beta {{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}\pm \left\lbrace
\begin{array}{ll}
y_j {{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}& \text{for some } j \text{ or } \\
{{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}y_j & \text{for some } j.
\end{array}\right.\end{aligned}$$ The last equality is possible because the word was assumed to be reduced, thus the element $y_{l+1}$ can be moved to the outside by using repeatedly Relation (VW.\[8a\]) and (VW.\[8b\]) - only creating a possible sign change. Since $\frac{1}{\beta \pm y_j} {{\mathbf{f}}}\in I$, it follows that ${{\mathbf{f}}}x_1 \cdots x_r {{\mathbf{f}}}\in I$. Assume now that the reduced word ${{\mathbf{f}}}x_1 \cdots x_r {{\mathbf{f}}}$ for some $w$ contains a positive number, say $m$, of $s$’s. Let $l$ be such that $x_r \in \{e_l,s_l\}$. Then again by induction we know that $${{\mathbf{f}}}x_1 \cdots x_{r-1} {{\mathbf{f}}}c_{l} {{\mathbf{f}}}x_r {{\mathbf{f}}}\in I.$$ Using again Proposition \[prop:kill\_f\] we obtain a similar expression as before, namely $$\begin{aligned}
{{\mathbf{f}}}x_1 \cdots x_{r-1} {{\mathbf{f}}}c_{l} {{\mathbf{f}}}x_r {{\mathbf{f}}}& =& {{\mathbf{f}}}x_1 \cdots x_{r-1} c_{l} x_r {{\mathbf{f}}}\\
&=& \beta {{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}+ {{\mathbf{f}}}x_1 \cdots x_{r-1} y_{l+1} x_r {{\mathbf{f}}}\\
&=& \beta {{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}\pm \left\lbrace
\begin{array}{ll}
y_j {{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}& \text{for some } j \text{ or } \\
{{\mathbf{f}}}x_1 \cdots x_{r-1} x_r {{\mathbf{f}}}y_j & \text{for some } j
\end{array}\right.\\
&&\hspace{3cm}+ \text{ smaller summands.}\end{aligned}$$ The smaller summands do not contain any $y_j$’s in this case, see Relation (VW.\[7\]), and are moreover either of length smaller than $r$, or of length $r$, but then with strictly less than $m$ letters $s$’s. Since by induction all these smaller terms are contained in $I$, the claim follows also in this case. Thus $\Phi_\delta$ is surjective and the theorem follows.
\[theremark\] The main feature of our isomorphism is the change of the parameter $N$ for ${{\bigdoublevee}_d^{\rm cycl}}$ to the corresponding parameter $\delta$ of ${\rm Br}_d(\delta)$; the most important relation we have to prove is $$\begin{aligned}
(Q_k e_k Q_k)^2= \delta \, Q_k e_k Q_k.
\end{aligned}$$ Essentially, this amounts to check that $$\begin{aligned}
\label{eq:30}
Q_k e_k \frac{b_{k+1}}{b_k} {{\mathbf{f}}}e_k Q_k
&=& (2\beta + 1) Q_k e_k Q_k - N Q_k e_k Q_k = \delta \, Q_k e_k Q_k.\end{aligned}$$ see Lemma \[lem:e2=deltae\]. By [@Nazarov] we can take a formal variable $u$ and write $$\begin{aligned}
\label{eq:55}
e_k \frac{1}{u-y_k} e_k {{\mathbf{f}}}&=& \frac{W_k(u)}{u} e_k {{\mathbf{f}}},
\end{aligned}$$ where $W_k(u)$ is a formal power series in $u^{-1}$ as in [@Nazarov]. We may now be tempted to replace $u =
-\beta$ and be able to compute $W_k(-\beta)=\beta_1$, and hence obtain from . Now, while this can be formalized in the semisimple case (by using the eigenvalues of $y_k$, as done several times in [@Nazarov]), it gets much more tricky in the non-semisimple case. Hence we need to take another way using the formalism from Section \[sec:invers-square-roots\]. \[rem:5\]
The algebra $\mathbf{f} {{\bigdoublevee}_d^{\rm cycl}}\mathbf{f}$ is generated by the elements ${{\mathbf{f}}}s_i {{\mathbf{f}}}$, ${{\mathbf{f}}}e_i {{\mathbf{f}}}$, and $ y_k {{\mathbf{f}}}$ for $1 \leq i < d$ and $1 \leq k \leq d$.
This follows immediately from the proof of Theorem \[thm:main\].
Before the proof the theorem we describe the preimages of the polynomial generators $y_k$.
The map $\Phi_\delta$ and Jucys-Murphy elements {#JM}
-----------------------------------------------
The [*Jucys-Murphy elements*]{} $\xi_k$, for $1\leq k\leq d$ in the Brauer algebra ${\rm Br}_d(\delta)$ are defined as follows: $$\begin{aligned}
\label{jucysmurphy}
\xi_1 = 0 &\text{ and } &\xi_{k+1} = t_k \xi_{k} t_k + t_k - g_k \text{ for all } 1 < k < d. \end{aligned}$$
\[lem:jucysmurphy\] The map $\Phi_\delta$ from Theorem \[thm:main\] maps $\xi_k - \alpha$ to $-y_k {{\mathbf{f}}}$.
We prove this by induction on $k$, for $k=1$ we have $$\Phi_\delta(\xi_1 - \alpha) = \Phi_\delta(-\alpha) = -\alpha {{\mathbf{f}}}= -y_1 {{\mathbf{f}}}.$$ For $k+1 > 1$ we calculate $$\begin{aligned}
\Phi_\delta(\xi_{k+1} - \alpha) &=& \Phi_\delta(t_k \xi_k t_k + t_k - g_k - \alpha)\quad =\quad \Phi_\delta(t_k (\xi_k - \alpha) t_k + t_k - g_k)\\
&=& - \tilde{s}_k y_k \tilde{s}_k + \tilde{s}_k - \tilde{e}_k\quad =\quad Q_k s_k Q_k y_k \tilde{s}_k + \tilde{s}_k - \tilde{e}_k - \frac{1}{b_k} {{\mathbf{f}}}y_k \tilde{s}_k\\
&=& Q_k (y_{k+1} s_k + e_k -1 )Q_k \tilde{s}_k + \tilde{s}_k - \tilde{e}_k - \frac{1}{b_k} {{\mathbf{f}}}y_k \tilde{s}_k\\
&=& y_{k+1} Q_k s_kQ_k\tilde{s}_k + \tilde{e}_k \tilde{s}_k - \frac{b_{k+1}}{b_k}{{\mathbf{f}}}\tilde{s}_k + \tilde{s}_k - \tilde{e}_k - \frac{1}{b_k} {{\mathbf{f}}}y_k \tilde{s}_k\\
&=& -y_{k+1} \tilde{s}_k \tilde{s}_k + \frac{y_{k+1}}{b_k}{{\mathbf{f}}}\tilde{s}_k- \frac{b_{k+1}}{b_k}{{\mathbf{f}}}\tilde{s}_k + \tilde{s}_k - \frac{1}{b_k} {{\mathbf{f}}}y_k \tilde{s}_k\\
&=& -y_{k+1} {{\mathbf{f}}}\end{aligned}$$ The proposition is proved.
Consequences: Koszulity and graded decomposition numbers {#sec:consequences}
========================================================
We deduce now some non-trival consequences of the main theorem. For the whole section $d$ is a positive integer and $\delta \in \mathbb{Z}$. First, one of the main results of [@ES3] allows us to equip the Brauer algebra with a grading. To state the result we need some additional notation for graded modules. Let $A$ be a $\mathbb{Z}$-graded algebra. For $M \in A-\operatorname{gmod}$ we denote its *graded endomorphism ring* by $$\operatorname{end}_A(M) = \bigoplus_{r \in \mathbb{Z}} \operatorname{Hom}_{A-\operatorname{gmod}}(M,M \left\langle r \right\rangle),$$ which becomes a graded ring by putting $\operatorname{end}_A(M)_r = \operatorname{Hom}_{A-\operatorname{gmod}}(M,M \left\langle r \right\rangle)$. The composition of $f \in \operatorname{Hom}_{A-\operatorname{gmod}}(M,M \left\langle r \right\rangle)$ and $g \in \operatorname{Hom}_{A-\operatorname{gmod}}(M,M \left\langle s \right\rangle)$ is given by $g \left\langle r \right\rangle \circ f$ in the category of graded modules. Note that for a graded lift $\widehat{M} \in A-\operatorname{gmod}$ of $M \in A-\operatorname{mod}$ it holds $${\rm End}_{A-\operatorname{mod}}(M) \cong \operatorname{end}_A(\widehat{M})$$ as (ungraded) algebras.
\[prop:brauergrading\] The Brauer algebra ${\rm Br}_d(\delta)$ can be equipped with a $\mathbb{Z}$-grading turning it into a $\mathbb{Z}$-graded algebra ${\rm Br}^{\rm gr}_d(\delta)$.
Recall from Section \[sec:cycl\_and\_cat\_O\] the parabolic category $\mathcal{O}^\mathfrak{p}(n)$. Consider the endofunctor $\mathcal{F} = ? \otimes V$ of $\mathcal{O}^\mathfrak{p}(n)$. Following [@ES3 Sections 4] we have the summand $\mathcal{G}$ of this functor that corresponds to projecting onto blocks with small weights (in [@ES3] this functor was denoted by $\widetilde{\mathcal{F}}$) such that $${\rm Br}_d(\delta) \, \cong \, {{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}\, \cong \, {{\mathbf{f}}}{\rm End}_{\mathfrak{g}}( \mathcal{F}^d M^\mathfrak{p}(\underline{\delta})) {{\mathbf{f}}}\, \cong \, {\rm End}_{\mathfrak{g}}( \mathcal{G}^d M^\mathfrak{p}(\underline{\delta})),$$ as algebras. By choosing a minimal projective generator $P$ of $\mathcal{O}^\mathfrak{p}(n)$ we have an equivalence of categories $$\mathcal{O}^\mathfrak{p}(n) \cong {\rm mod}-{\rm End}_\mathfrak{g}(P).$$ Following [@ES2] we equip $A:={\rm End}_\mathfrak{g}(P)$ with a Koszul grading and denote by $\widehat{\mathcal{O}}^\mathfrak{p}(n):=A-\operatorname{gmod}$ its associated category of graded modules. In [@ES3 Section 5] a graded lift $\widehat{\mathcal{F}}$ of $\mathcal{F}$ is constructed by choosing graded lifts for each summand obtained by projecting onto blocks, see [@ES3 Lemma 5.3]. Thus it also yields a graded lift $\widehat{\mathcal{G}}$ of $\mathcal{G}$ and gives $${\rm Br}_d^{\rm gr}(\delta) \, := \, {\rm end}_A( \widehat{\mathcal{G}}^d \widehat{M^\mathfrak{p}(\underline{\delta})}),$$ where $\widehat{M^\mathfrak{p}(\underline{\delta})}$ is the standard graded lift of the parabolic Verma module.
With this grading one can establish Koszulity.
\[thm:koszul\] The Brauer algebra ${\rm Br}^{\rm gr}_d(\delta)$ is Morita equivalent to a Koszul algebra if and only if $\delta \neq 0$ or $\delta=0$ and $d$ odd.
This follows directly from our main Theorem \[thm:main\] together with [@ES3 Theorem 5.1].
\[thm:cellularity\] The Brauer algebra ${\rm Br}^{\rm gr}_d(\delta)$ is graded cellular.
It follows from [@Koenig-Xi] that the idempotent truncation of the quasi-hereditary algebra ${{\bigdoublevee}_d^{\rm cycl}}$ is always cellular.
\[thm:quasihereditary\] The Brauer algebra ${\rm Br}^{\rm gr}_d(\delta)$ is graded quasi-hereditary if and only if $\delta\not=0$ or $\delta=0$ and $d$ odd.
By [@ES3 Theorem 4.13 and Remark 4.14] the highest weight structure of ${{\bigdoublevee}_d^{\rm cycl}}$ induces a highest weight structure on ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$ if and only if $\delta\not=0$ or $\delta=0$ and $d$ odd. Since by [@ES3 Definition 4.11] the labelling posets of standard modules for ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$ agrees with the one for the Brauer algebra from [@CDM] the isomorphism from Theorem \[thm:main\] is an isomorphism of quasi-hereditary algebras.
The result then follows from the general theory of graded category $\mathcal{O}$ (see [@Stroppel]) using [@ES3 Theorem 4.9].
Denote by $\Delta(\lambda)$ for $\lambda \in \Lambda_d$ the standard module for ${\rm Br}_d(\delta)$ and by $L(\lambda)$ for $\lambda \in \Lambda^\delta_d$ the corresponding simple quotient, see [@CDM]. As in the introduction denote by $F$ the grading forgetting functor from ${\rm Br}^{\rm gr}_d(\delta)-\operatorname{gmod}$ to ${\rm Br}_d(\delta)-\operatorname{mod}$.
\[thm:liftmodules\] Assume that $\delta \neq 0$ or $\delta = 0$ and $d$ is odd, i.e. the case where ${\rm Br}^{\rm gr}_d(\delta)$ is graded quasi-hereditary. For any ${\lambda}\in{\Lambda}_d$ there exists a unique modules $\widehat{\Delta}({\lambda}) \in {\rm Br}^{\rm gr}_d(\delta)-\operatorname{gmod}$ such that $F\widehat{\Delta}({\lambda})\cong {\Delta}({\lambda})$ and for $\lambda \in \Lambda_d^\delta$ there exists a unique $\widehat{L}({\lambda}) \in {\rm Br}^{\rm gr}_d(\delta)-\operatorname{gmod}$ such that $F\widehat{L}({\lambda})\cong L({\lambda})$, and both modules are concentrated in non-negative degrees with non-vanishing degree zero.
From [@Stroppel Lemma 1.5] it follows that graded lifts, if they exist, are unique up to isomorphism and grading shifts. With the assumption on the degree of the modules the graded lifts will be unique up to isomorphism in our case.
For ${{\bigdoublevee}_d^{\rm cycl}}$ the existence of graded lifts follows from [@ES3 Theorem 4.9] and general theory of category $\mathcal{O}$, see [@Stroppel].
The existence of the graded lifts for ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$ then follows from [@ES3 Theorem 4.13] via a quotient functor construction.
We now want to match the multiplicities of simple modules occurring in a standard module with the coefficients of certain Kazhdan-Lusztig polynomials. Denote by $W$ the Weyl group of $\mathfrak{g}$ and by $W_\mathfrak{p}$ the parabolic subgroup generated by all simple roots except $\alpha_0$, i.e., the one corresponding to the parabolic $\mathfrak{p}$ from the introduction and Section \[sec:cycl\_and\_cat\_O\]. By $W^\mathfrak{p}$ we denote the shortest coset representatives in $W_\mathfrak{p} \backslash W$. For $x,y \in W^\mathfrak{p}$ let $n_{x,y}(q) \in \mathbb{Z}[q]$ be the parabolic Kazhdan-Lusztig polynomial of type $({\rm D}_n,{\rm A}_{n-1})$, see [@Boe] and for this special case [@LS].
Given now $\nu \in X_n^\mathfrak{p}$ there is a unique $\nu_{\rm dom} \in X_n^\mathfrak{p}$ and $x_\nu \in W^\mathfrak{p}$ such that $\nu_{\rm dom} + \rho$ is dominant and $$x_\nu(\nu_{\rm dom} + \rho) = \nu + \rho.$$
We now give a dictionary how to translate between the labelling set of standard modules and Weyl group elements. To a partition $\lambda$ we associate a double Young diagram $Y(\lambda)$ via the bipartition $(\lambda,\emptyset)$ and a weight ${\rm wt}(\lambda)=\underline{\delta} + {\rm wt}(Y(\lambda))$. For $\lambda,\mu \in {\Lambda}_d$ we put $$n_{\lambda,\mu}(q) = \left\lbrace \begin{array}{ll}
n_{x_{{\rm wt}(\lambda)}, x_{{\rm wt}(\mu)}}(q) & \text{if } {\rm wt}(\lambda) + \rho \in W \cdot ({\rm wt}(\mu)+\rho), \\
0 & \text{otherwise.}
\end{array}\right.$$
\[thm:multiplicities\] For $\lambda \in {\Lambda}_d$, the module $\widehat{\Delta}({\lambda})$ has a Jordan-Hölder series in ${\rm Br}^{\rm gr}_d(\delta)-\operatorname{gmod}$ with multiplicities given by $$\begin{aligned}
\left[\widehat{\Delta}({\lambda})\, :\, \widehat{L}(\mu)<i>\right]&=&n_{\lambda,\mu,i},\end{aligned}$$ where $n_{\lambda,\mu}(q)=\sum_{i\geq 0}n_{\lambda,\mu,i} q^i$ and $\mu \in {\Lambda}_d^\delta$.
Denote by $\widetilde{\Delta}({\lambda})$ the standard module in ${{\bigdoublevee}_d^{\rm cycl}}-\operatorname{gmod}$ that is sent to $\widehat{\Delta}({\lambda})$ via the quotient functor $\mathcal{Q}$ to ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}-\operatorname{gmod}$ and analogously $\widetilde{L}(\mu)$ the simple module in ${{\bigdoublevee}_d^{\rm cycl}}-\operatorname{gmod}$. Then the following equality holds $$n_{\lambda,\mu,i} \overset{(a)}{=} \left[\widetilde{\Delta}({\lambda})\, :\, \widetilde{L}(\mu)<i>\right] \overset{(b)}{=} \left[\mathcal{Q}\widetilde{\Delta}({\lambda})\, :\, \mathcal{Q}\widetilde{L}(\mu)<i>\right] \overset{(c)}{=} \left[\widehat{\Delta}({\lambda})\, :\, \widehat{L}(\mu)<i>\right],$$ where $(a)$ is due to [@ES3 Theorem 4.9], $(b)$ is due to [@Donkin p.136 (4)(vii)], and $(c)$ is due to [@Donkin p.136 (4)(iv)] for the simple module and [@Koenig-Xi Prop. 4.3] for the standard module.
Well-definedness of $\Phi_\delta$ {#sec:proof_thm:main}
=================================
In this section we establish the remaining part of Theorem \[thm:main\], namely the map $\Phi_\delta$ being well-defined. In other words, we have to verify that the $\tilde{s}_i$ and $\tilde{e}_i$ satisfy the Brauer relations. We will commence with a few lemmas that will help in the calculations, allowing us to simplify various expressions.
It holds $e_k \eta_{k+1} = e_k \eta_k$ for all $k$, hence $e_k \mathbf{f}_{k+1} = e_k \mathbf{f}_k$.
Let ${{\bigdoublevee}_d^{\rm cycl}}$ act on itself by the regular representation. By Lemma \[lem:eigenvalue\_and\_e\] it follows that if either side of the equation acts non-trivially then the eigenvalues of $y_k$ and $y_{k+1}$ are opposite. Hence if one is small so is the other.
We often need to simplify expressions involving fractions, such as in the definitions of $\tilde{s}_k$ and $\tilde{e}_k$. The following proposition collects a few useful formulas for this.
\[prop:reduce\_fracs\] In ${{\bigdoublevee}_d^{\rm cycl}}$ the following equalities hold for $1 \leq k \leq d-1$: $$\begin{array}{rccrc}
i) & e_k \frac{1}{b_k} s_k {{\mathbf{f}}}= \frac{1}{2\beta}e_k \frac{1}{b_k}{{\mathbf{f}}}, & \qquad & iii) & e_k \frac{1}{b_k} e_k {{\mathbf{f}}}= (1 + \frac{1}{2\beta})e_k {{\mathbf{f}}},\\
ii) & {{\mathbf{f}}}s_k \frac{1}{b_k} e_k {{\mathbf{f}}}= \frac{1}{2\beta} \frac{1}{b_k} {{\mathbf{f}}}e_k {{\mathbf{f}}}.
\end{array}$$
Note first that $\frac{1}{b_k}s_k{{\mathbf{f}}}$ and $\frac{1}{b_k}e_k{{\mathbf{f}}}$ are well-defined. To see this we look at the action of these elements on the module $M=\MdV$. For both, $s_k{{\mathbf{f}}}$ and $e_k{{\mathbf{f}}}$ it holds that their image is contained in those subspaces $M_{\textbf{i}}$ such that $i_k \neq - \beta$, due to Lemma \[lem:eigenvalue\_and\_e\] and Lemma \[lem:eigenvalue\_and\_s\] in conjunction with the diagram calculus, hence $\frac{1}{b_k}$ is defined as an endomorphism of these images. Let us first assume $iii)$ is proven already. To verify $i)$ we calculate $$\begin{aligned}
e_k \frac{1}{b_k}s_k {{\mathbf{f}}}&\overset{(a)}{=}& e_k s_k \frac{1}{b_{k+1}}{{\mathbf{f}}}- e_k \frac{1}{b_k}e_k \frac{1}{b_{k+1}}{{\mathbf{f}}}+ e_k \frac{1}{b_k b_{k+1}}{{\mathbf{f}}}\\
&\overset{(b)}{=}& e_k s_k \frac{1}{b_{k+1}}{{\mathbf{f}}}- (1 + \frac{1}{2\beta}) e_k \frac{1}{b_{k+1}}{{\mathbf{f}}}+ e_k \frac{1}{b_k b_{k+1}}{{\mathbf{f}}}\\
&=& e_k \left( \frac{2\beta - b_k}{2\beta b_k b_{k+2}} \right) = e_k \left( \frac{c_k}{2\beta b_k b_{k+1}} \right)
= e_k \left( \frac{b_{k+1}}{2\beta b_k b_{k+1}} \right) \\
&=& \frac{1}{2\beta} e_k \frac{1}{b_k}{{\mathbf{f}}}\end{aligned}$$ where equality $(a)$ holds by Lemma \[lem:commute\_b\] and equality $(b)$ is valid thanks to part $iii)$ of this proposition.
Formula $ii)$ is shown analogously, but note that since $\frac{1}{b_{k+1}}e_k$ is in general not defined we have to multiply the whole equation by ${{\mathbf{f}}}$ from the left to make it well-defined.
Finally let us consider the formula $iii)$ which we will prove by induction on $k$. If $k=1$ then note that by Definition \[defalphabeta\] the element $y_1$ has exactly two eigenvalues, namely $\alpha$ and $\beta$ as in , with the projections $\frac{y_1-\beta}{\alpha-\beta}$ respectively $\frac{y_1-\alpha}{\beta-\alpha}$ onto the eigenspaces. Then we obtain $$\begin{aligned}
&&e_1 \frac{1}{b_1} e_1{{\mathbf{f}}}\\
& =& e_1\left (\frac{1}{\alpha+\beta}\frac{y_1-\beta}{\alpha-\beta}+ \frac{1}{2\beta}\frac{y_1-\alpha}{\beta-\alpha}\right)e_1{{\mathbf{f}}}\quad=\quad \frac{1}{\alpha-\beta}e_1\left(\frac{y_1-\beta}{\alpha+\beta}+\frac{\alpha-y_1}{2\beta}\right)e_1 {{\mathbf{f}}}\\
&=&
e_1\frac{1}{(\alpha+\beta)2\beta}\left(-y_1+\alpha+2\beta\right)e_1 {{\mathbf{f}}}\quad\stackrel{\eqref{ab}}{=}
\quad\frac{1}{N\beta}\left(-e_1y_1e_1+(\alpha+2\beta) e_1^2\right){{\mathbf{f}}}\\
&=&\frac{1}{2N\beta}\left(N(1-N)+N^2+2\beta\right){{\mathbf{f}}}=\left(1 + \frac{1}{2\beta}\right)e_1 {{\mathbf{f}}}. \end{aligned}$$ Now assume the formula holds for $k$ and, by applying Lemma \[lem:commute\_b\] repeatedly, we obtain $$\begin{aligned}
\label{puh}
&& e_{k+1} \frac{1}{b_{k+1}} e_{k+1}{{\mathbf{f}}}\quad=\quad e_{k+1}s_{k}(s_{k} \frac{1}{b_{k+1}}) e_{k+1}{{\mathbf{f}}}\nonumber\\
&{=}&
e_{k+1}s_{k}\left(\frac{1}{b_k}\right)s_{k}e_{k+1}{{\mathbf{f}}}\
+ e_{k+1}s_k\left(\frac{1}{b_k}e_k\frac{1}{b_{k+1}}\right)e_{k+1}{{\mathbf{f}}}-
e_{k+1}s_k\left(\frac{1}{b_k}\frac{1}{b_{k+1}}\right)e_{k+1}{{\mathbf{f}}}\end{aligned}$$ by Lemma \[lem:commute\_b\]. Now the first summand in equals, by (VW.\[6c\]), $$\begin{aligned}
e_{k+1}e_{k}s_{k+1}\left(\frac{1}{b_k}\right)s_{k+1}e_{k}e_{k+1}{{\mathbf{f}}}&=&e_{k+1}e_{k}\frac{1}{b_k}e_{k}e_{k+1}{{\mathbf{f}}}=\left(1+\frac{1}{2\beta}\right)e_{k+1}e_ke_{k+1}{{\mathbf{f}}}\\
&=&\left(1+\frac{1}{2\beta}\right)e_{k+1}{{\mathbf{f}}}\end{aligned}$$ by induction, whereas the second summand equals $$\begin{aligned}
&&e_{k+1}s_k\frac{1}{b_k}e_k\frac{1}{b_{k+1}}e_{k+1}{{\mathbf{f}}}\\
&=&e_{k+1}\left(s_k\frac{1}{b_k}\right)e_ke_{k+1}\frac{1}{c_k}{{\mathbf{f}}}\\
&=&e_{k+1}\left(\frac{1}{b_{k+1}}s_k\right)e_ke_{k+1}\frac{1}{c_k}{{\mathbf{f}}}-e_{k+1}\left(\frac{1}{b_{k+1}}e_k\frac{1}{b_k}\right) e_ke_{k+1}\frac{1}{c_k}{{\mathbf{f}}}+e_{k+1}\left(\frac{1}{b_{k+1}}\frac{1}{b_k}\right) e_ke_{k+1}\frac{1}{c_k}{{\mathbf{f}}}\\
&=&
\frac{1}{c_k}e_{k+1}e_ke_{k+1}\frac{1}{c_{k}}{{\mathbf{f}}}-\left(1+\frac{1}{2\beta}\right)\frac{1}{c_{k}} e_{k+1}e_ke_{k+1}\frac{1}{c_{k}}{{\mathbf{f}}}+\frac{1}{b_{k}c_k} e_{k+1}e_ke_{k+1}\frac{1}{c_k}{{\mathbf{f}}}\\
&=&-\frac{1}{2\beta} \frac{1}{c_k^2}e_{k+1}{{\mathbf{f}}}+ \frac{1}{b_kc_k^2}e_{k+1}{{\mathbf{f}}}\\\end{aligned}$$ by (VW.\[8a\]) and (VW.\[5a\]), Lemma \[lem:commute\_b\] and induction hypothesis. Finally the third summand in equals $$\begin{aligned}
&&
-e_{k+1}\left(s_k\frac{1}{b_{k+1}}\right)e_{k+1}\frac{1}{b_k}{{\mathbf{f}}}\\
&=&
-e_{k+1}\left(\frac{1}{b_{k}s_k}\right)e_{k+1}\frac{1}{b_k}{{\mathbf{f}}}-e_{k+1}\left(\frac{1}{b_{k}}e_k\frac{1}{b_{k+1}}\right)e_{k+1}\frac{1}{b_k}{{\mathbf{f}}}+e_{k+1}\left(\frac{1}{b_kb_{k+1}}\right)e_{k+1}\frac{1}{b_{k}}{{\mathbf{f}}}\\
&=&-\frac{1}{b_{k}^2}e_{k+1}{{\mathbf{f}}}-\frac{1}{b_{k}^2c_k} e_{k+1}{{\mathbf{f}}}+
\frac{1}{b_{k}^2} e_{k+1} \frac{1}{b_{k+1}} e_{k+1}{{\mathbf{f}}}.\end{aligned}$$ Hence altogether we obtain $$\begin{aligned}
&&\left(1-\frac{1}{b_{k}^2}\right) e_{k+1} \frac{1}{b_{k+1}} e_{k+1} {{\mathbf{f}}}=\left(1+\frac{1}{2\beta}\right)e_{k+1}{{\mathbf{f}}}+ \left( -\frac{1}{2\beta} \frac{1}{c_k^2} + \frac{1}{b_kc_k^2}-\frac{1}{b_{k}^2}-\frac{1}{b_{k}^2c_k}\right) e_{k+1}{{\mathbf{f}}}\\
&=&\left(1-\frac{1}{b_{k}^2}\right)\left(1+\frac{1}{2\beta}\right)e_{k+1}{{\mathbf{f}}}+ \left(\frac{1}{2\beta}\frac{1}{b_{k}^2} -\frac{1}{2\beta} \frac{1}{c_k^2} + \frac{1}{b_kc_k^2}-\frac{1}{b_{k}^2c_k} \right) e_{k+1}{{\mathbf{f}}}.\end{aligned}$$ Now one easily checks that the last coefficient in front of the final $e_{k+1}{{\mathbf{f}}}$ is zero and since $\left(1-\frac{1}{b_{k}^2}\right)$ is invertible on the image of $e_{k+1}{{\mathbf{f}}}$ we obtain $$\begin{aligned}
e_{k+1} \frac{1}{b_{k+1}} e_{k+1} {{\mathbf{f}}}&=&\left(1+\frac{1}{2\beta}\right)e_{k+1} {{\mathbf{f}}}\end{aligned}$$ which finishes the proof.
The key relation ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$
-----------------------------------------------------------------------------
The following is the most crucial point of the proof (see also Remark \[theremark\]):
\[lem:e2=deltae\] We have $\tilde{e}_k^2 = \delta \tilde{e}_{k}$ for $1 \leq k \leq d-1$.
We compute $$\label{eqn:e2=e}
\begin{aligned}
\tilde{e}_k^2 &= Q_k e_k \frac{b_{k+1}}{b_k} {{\mathbf{f}}}e_k Q_k \overset{(a)}{=} Q_k e_k \frac{c_k}{b_k} {{\mathbf{f}}}e_k Q_k \overset{(b)}{=} Q_k e_k \frac{c_k}{b_k} e_k Q_k \\
&\overset{(c)}{=} 2 \beta Q_k e_k \frac{1}{b_k}e_k Q_k - Q_k e_k^2 Q_k \overset{(d)}{=} (2\beta + 1) Q_k e_k Q_k - N Q_k e_k Q_k = \delta \tilde{e_k}.
\end{aligned}$$ Where equality $(a)$ follows from Relation (VW.\[8b\]), equality $(b)$ holds by Proposition \[prop:kill\_f\], equality $(c)$ just expands $c_k$ as $2\beta - b_k$ and equality $(d)$ is valid thanks to Proposition \[prop:reduce\_fracs\].
Symmetric group relations in ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$
-----------------------------------------------------------------------------------------
In this part we show that the $\tilde{s}_j$’s satisfy the defining relation of the symmetric group.
\[lem:s2=f\] We have $\tilde{s}_k^2 = \mathbf{f}$ for $1 \leq k \leq d-1$.
We compute $$\begin{aligned}
\tilde{s}_k^2 =\,& Q_k \left( s_k \frac{b_{k+1}}{b_k}{{\mathbf{f}}}s_k - s_k \frac{1}{b_k} - \frac{1}{b_k} {{\mathbf{f}}}s_k \right) Q_k + \frac{1}{b_k^2}{{\mathbf{f}}}\\
\overset{(a)}{=}&Q_k \left( s_k \frac{b_{k+1}}{b_k} s_k - s_k \frac{1}{b_k} - \frac{1}{b_k} s_k \right) Q_k + \frac{1}{b_k^2}{{\mathbf{f}}}\\
\overset{(b)}{=}& Q_k \left( s_k b_{k+1} s_k \frac{1}{b_{k+1}} - s_k \frac{b_{k+1}}{b_k} e_k \frac{1}{b_{k+1}} - \frac{1}{b_k} s_k \right) Q_k + \frac{1}{b_k^2}{{\mathbf{f}}}\\
\overset{(c)}{=}& Q_k \left( \frac{b_k}{b_{k+1}} + (s_k - e_k) \frac{1}{b_{k+1}} - s_k \frac{b_{k+1}}{b_k} e_k \frac{1}{b_{k+1}} - \frac{1}{b_k} s_k \right) Q_k + \frac{1}{b_k^2}{{\mathbf{f}}}\\
\overset{(d)}{=}& Q_k \left( \frac{b_k}{b_{k+1}} + (s_k - e_k) \frac{1}{b_{k+1}} - b_k s_k \frac{1}{b_k} e_k \frac{1}{b_{k+1}} + e_k \frac{1}{b_k} e_k \frac{1}{b_{k+1}} - \frac{1}{b_k} e_k \frac{1}{b_{k+1}} - \frac{1}{b_k} s_k \right) Q_k\\
& + \frac{1}{b_k^2}{{\mathbf{f}}}\\
\overset{(e)}{=}& Q_k \left( \frac{b_k}{b_{k+1}} + (s_k - e_k) \frac{1}{b_{k+1}} - \frac{1}{2\beta} e_k \frac{1}{b_{k+1}} + (1+\frac{1}{2\beta}) e_k \frac{1}{b_{k+1}} - \frac{1}{b_k} e_k \frac{1}{b_{k+1}} - \frac{1}{b_k} s_k \right) Q_k\\
& + \frac{1}{b_k^2}{{\mathbf{f}}}\\
\overset{(f)}{=}& Q_k \left( \frac{b_k}{b_{k+1}} - \frac{1}{b_kb_{k+1}} \right) {{\mathbf{f}}}Q_k + \frac{1}{b_k^2}{{\mathbf{f}}}= {{\mathbf{f}}}\\\end{aligned}$$ where $(a)$ follows from Proposition \[prop:kill\_f\] and the fact that ${{\mathbf{f}}}$ commutes with $\frac{1}{b_k}$ on the image of $s_k$, $(b)$ is due to Lemma \[lem:commute\_b\], $(c)$ and $(d)$ are applications of Relation (VW.\[7\]), $(e)$ uses Proposition \[prop:reduce\_fracs\], and finally $(f)$ uses again Lemma \[lem:commute\_b\].
Since ${{\mathbf{f}}}s_i {{\mathbf{f}}}$ commutes with $Q_j$ and $\frac{1}{b_{j}} {{\mathbf{f}}}$ when $| i - j | > 1$ the following lemma holds.
\[lem:sisj=sjsi\] We have $\tilde{s}_i \tilde{s}_j= \tilde{s}_j \tilde{s}_i$ for $1 \leq i,j \leq d-1$ with $|i-j| > 1$.
We verify now the braid relations, which is a surprisingly non-trivial task.
\[lem:si\_braid\] The braid relation $\tilde{s}_i \tilde{s}_{i+1} \tilde{s}_i= \tilde{s}_{i+1} \tilde{s}_i \tilde{s}_{i+1}$ holds for $1 \leq i < d-1$.
We expand both sides of the equality below and compute\
$$\begin{aligned}
& \tilde{s}_k \tilde{s}_{k+1} \tilde{s}_k = \nonumber \\
& -Q_k s_k Q_k Q_{k+1} s_{k+1} Q_{k+1} Q_k s_k Q_k \label{eqn:lhs1} \\
& + \frac{1}{b_k} {{\mathbf{f}}}Q_{k+1} s_{k+1} Q_{k+1} Q_k s_k Q_k \label{eqn:lhs2}\\
& + Q_k s_k Q_k \frac{1}{b_{k+1}} {{\mathbf{f}}}Q_k s_k Q_k \label{eqn:lhs3}\\
& + Q_k s_k Q_k Q_{k+1} s_{k+1} Q_{k+1} \frac{1}{b_k} {{\mathbf{f}}}\label{eqn:lhs5}\end{aligned}$$
$$\begin{aligned}
& \notag \\
& - \frac{1}{b_k b_{k+1}} {{\mathbf{f}}}Q_k s_k Q_k \label{eqn:lhs4} \\
& - \frac{1}{b_k} {{\mathbf{f}}}Q_{k+1} s_{k+1} Q_{k+1} \frac{1}{b_k} {{\mathbf{f}}}\label{eqn:lhs6}\\
& - Q_k s_k Q_k \frac{1}{b_k b_{k+1}} {{\mathbf{f}}}\label{eqn:lhs7}\\
& + \frac{1}{b_k^2 b_{k+1}} {{\mathbf{f}}}\label{eqn:lhs8}.\end{aligned}$$
\
and\
$$\begin{aligned}
&\tilde{s}_{k+1} \tilde{s}_{k} \tilde{s}_{k+1} = \nonumber\\
& -Q_{k+1} s_{k+1} Q_{k+1} Q_{k} s_{k} Q_{k} Q_{k+1} s_{k+1} Q_{k+1} \label{eqn:rhs1} \\
& + \frac{1}{b_{k+1}} {{\mathbf{f}}}Q_{k} s_{k} Q_{k} Q_{k+1} s_{k+1} Q_{k+1} \label{eqn:rhs2}\\
& + Q_{k+1} s_{k+1} Q_{k+1} \frac{1}{b_{k}} {{\mathbf{f}}}Q_{k+1} s_{k+1} Q_{k+1} \label{eqn:rhs3}\\
& + Q_{k+1} s_{k+1} Q_{k+1} Q_{k} s_{k} Q_{k} \frac{1}{b_{k+1}} {{\mathbf{f}}}\label{eqn:rhs5}\end{aligned}$$
$$\begin{aligned}
& \nonumber \\
& - \frac{1}{b_k b_{k+1}} {{\mathbf{f}}}Q_{k+1} s_{k+1} Q_{k+1} \label{eqn:rhs4}\\
& - \frac{1}{b_{k+1}} {{\mathbf{f}}}Q_{k} s_{k} Q_{k} \frac{1}{b_{k+1}} {{\mathbf{f}}}\label{eqn:rhs6}\\
& - Q_{k+1} s_{k+1} Q_{k+1} \frac{1}{b_{k} b_{k+1}} {{\mathbf{f}}}\label{eqn:rhs7}\\
& + \frac{1}{b_k b_{k+1}^2} {{\mathbf{f}}}\label{eqn:rhs8}.\end{aligned}$$
\
For the following calculations we abbreviate $A:=Q_{k+1} \frac{1}{\sqrt{b_k}} {{\mathbf{f}}}$. To improve readability we highlight those terms that are modified in each step, using either Lemma \[lem:commute\_b\] to move terms past $s_j$’s, Proposition \[prop:kill\_f\] to eliminate ${{\mathbf{f}}}$’s or Proposition \[prop:reduce\_fracs\] to modify terms involving fractions.
We first simplify some parts of the right hand side of the braid. $$\begin{aligned}
\label{eqn:14g+14h}
& \text{(\ref{eqn:rhs7})}+\text{(\ref{eqn:rhs8})} = - A {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} \frac{1}{b_{k+1}}$};\phantom{s_{k+1} \frac{1}{b_{k+1}}}} {{\mathbf{f}}}A + \frac{1}{b_kb_{k+1}^2} {{\mathbf{f}}}\notag \\
=& - A \left( \frac{1}{b_{k+2}}{{\mathbf{f}}}s_{k+1} - \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} \frac{1}{b_{k+1}}+ \frac{1}{b_{k+1}b_{k+2}}\right) A + \frac{1}{b_k b_{k+1}^2} {{\mathbf{f}}}\\
=& A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} \frac{1}{b_{k+1}}A - A \frac{1}{b_{k+2}}{{\mathbf{f}}}s_{k+1} A\notag \end{aligned}$$
$$\begin{aligned}
\label{eqn:14c+14d}
& \text{(\ref{eqn:rhs3})}+\text{(\ref{eqn:rhs4})} = A \left( s_{k+1} \frac{b_{k+2}}{b_{k+1}} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} - \frac{1}{b_{k+1}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} \right) A \notag \\
=& A \left( {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1}b_{k+2}$};\phantom{s_{k+1}b_{k+2}}} \frac{1}{b_{k+1}} s_{k+1} - \frac{1}{b_{k+1}}s_{k+1} \right) A \notag \\
=& A \left( \left(b_{k+1}s_{k+1} - e_{k+1} + 1 \right) \frac{1}{b_{k+1}} s_{k+1} - \frac{1}{b_{k+1}}s_{k+1} \right) A \notag \\
=& A \left( b_{k+1}s_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle \frac{1}{b_{k+1}} s_{k+1}$};\phantom{\frac{1}{b_{k+1}} s_{k+1}}} - {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle e_{k+1}\frac{1}{b_{k+1}} s_{k+1}$};\phantom{e_{k+1}\frac{1}{b_{k+1}} s_{k+1}}} \right) A\notag \\
=& A \left( b_{k+1}s_{k+1} \left( s_{k+1} \frac{1}{b_{k+2}} - \frac{1}{b_{k+1}} e_{k+1} \frac{1}{b_{k+2}} + \frac{1}{b_{k+1}b_{k+2}} \right)- \frac{1}{2\beta} e_{k+1}\frac{1}{b_{k+1}} \right) A\notag \\
=& A \left( \frac{b_{k+1}}{b_{k+2}} - b_{k+1}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1}\frac{1}{b_{k+1}} e_{k+1}$};\phantom{s_{k+1}\frac{1}{b_{k+1}} e_{k+1}}} \frac{1}{b_{k+2}} + b_{k+1}s_{k+1}\frac{1}{b_{k+1}b_{k+2}} - \frac{1}{2\beta} e_{k+1}\frac{1}{b_{k+1}} \right) A\\
=& A \left( \frac{b_{k+1}}{b_{k+2}} - \frac{1}{2\beta} e_{k+1} \frac{1}{b_{k+2}} + b_{k+1}s_{k+1}\frac{1}{b_{k+1}b_{k+2}} - \frac{1}{2\beta} e_{k+1}\frac{1}{b_{k+1}} \right) A\notag \\
=& A \left( \frac{b_{k+1}}{b_{k+2}} - e_{k+1} \frac{1}{b_{k+1}b_{k+2}} + {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+1}s_{k+1}$};\phantom{b_{k+1}s_{k+1}}}\frac{1}{b_{k+1}b_{k+2}} \right) A\notag \\
=& A \left( \frac{b_{k+1}}{b_{k+2}} - e_{k+1} \frac{1}{b_{k+1}b_{k+2}} + \left(s_{k+1} b_{k+2} + e_{k+1} - 1 \right) \frac{1}{b_{k+1}b_{k+2}} \right) A\notag \\
=& A \left( \frac{b_{k+1}}{b_{k+2}} + s_{k+1} \frac{1}{b_{k+1}} - \frac{1}{b_{k+1}b_{k+2}} \right) A = \frac{1}{b_{k}} {{\mathbf{f}}}+ A {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} \frac{1}{b_{k+1}}$};\phantom{s_{k+1} \frac{1}{b_{k+1}}}} A - \frac{1}{b_{k}b_{k+1}^2} {{\mathbf{f}}}\notag \\
=& \frac{1}{b_{k}} {{\mathbf{f}}}+ A \left(\frac{1}{b_{k+2}} {{\mathbf{f}}}s_{k+1} \right)A - A \left(\frac{1}{b_{k+2}}{{\mathbf{f}}}e_{k+1} \frac{1}{b_{k+1}} \right)A\notag \end{aligned}$$
Adding and we obtain: $$\label{eqn:14c+14d+14g+14h}
\text{(\ref{eqn:rhs3})}+\text{(\ref{eqn:rhs4})} + \text{(\ref{eqn:rhs7})}+\text{(\ref{eqn:rhs8})}= \frac{1}{b_{k}} {{\mathbf{f}}}$$
On the left hand side we simplify the following. $$\begin{aligned}
\label{eqn:-13c-13d}
& -\text{(\ref{eqn:lhs3})}-\text{(\ref{eqn:lhs4})} = A \left( -\frac{b_{k+1}}{b_{k+2}} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_k \frac{1}{b_k} {{\mathbf{f}}}s_k b_{k+1} + \frac{1}{b_kb_{k+2}} {{\mathbf{f}}}s_k b_{k+1} \right) A \notag \\
=& A \left( - {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+1} s_k$};\phantom{b_{k+1} s_k}} \frac{1}{b_k} {{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} + \frac{1}{b_k} {{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} \right) A \notag \\
=& A \left( - \left( s_k b_k - e_k + 1 \right) \frac{1}{b_k} {{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} + \frac{1}{b_k} {{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} \right) A \\
=& A \left( - s_k {{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} + e_k \frac{1}{b_k}{{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} \right) A \notag \\
=& A \left( - \frac{b_{k+1}}{b_{k+2}} + s_k (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}} + e_k \frac{1}{b_k} s_k \frac{b_{k+1}}{b_{k+2}} - e_k \frac{1}{b_k} (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}}\right) A \notag \\
=& - \frac{1}{b_{k}} + A s_k (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}}A + A \frac{1}{2\beta} e_k \frac{b_{k+1}}{b_k b_{k+2}} A - A e_k \frac{1}{b_k} (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}}A \notag\end{aligned}$$
Furthermore, we simplify the following.
$$\label{eqn:14f-13g-13h}
\begin{aligned}
& \text{(\ref{eqn:rhs6})}-\text{(\ref{eqn:lhs7})}-\text{(\ref{eqn:lhs8})} = - \frac{1}{b_{k+1}} {{\mathbf{f}}}Q_{k} s_{k} \frac{Q_{k}}{b_{k+1}}{{\mathbf{f}}}+ Q_k {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_k \frac{1}{b_k}$};\phantom{s_k \frac{1}{b_k}}} \frac{Q_{k}}{b_{k+1}} {{\mathbf{f}}}- \frac{1}{b_k^2b_{k+1}}{{\mathbf{f}}}\\
=& - \frac{1}{b_{k+1}} {{\mathbf{f}}}Q_{k} s_{k} \frac{Q_{k}}{b_{k+1}}{{\mathbf{f}}}+ Q_k \left( \frac{1}{b_{k+1}}{{\mathbf{f}}}s_k - \frac{1}{b_{k+1}}{{\mathbf{f}}}e_{k} \frac{1}{b_k} + \frac{1}{b_k b_{k+1}} \right)\frac{Q_k}{b_{k+1}} {{\mathbf{f}}}- \frac{1}{b_k^2b_{k+1}}{{\mathbf{f}}}\\
=& Q_k \left( - \frac{1}{b_{k+1}}{{\mathbf{f}}}e_{k} \frac{1}{b_k} + \frac{1}{b_k b_{k+1}} \right)\frac{Q_k}{b_{k+1}} {{\mathbf{f}}}- \frac{1}{b_k^2b_{k+1}}{{\mathbf{f}}}= - Q_k \frac{1}{b_{k+1}}{{\mathbf{f}}}e_{k} \frac{Q_k}{b_k b_{k+1}} {{\mathbf{f}}}\\
=& - A e_{k} \frac{1}{b_k b_{k+2}} A
\end{aligned}$$
To summarize, we have: $$\label{eqn:rest1}
\begin{aligned}
& \text{(\ref{eqn:rhs3})}+\text{(\ref{eqn:rhs4})} + \text{(\ref{eqn:rhs7})}+\text{(\ref{eqn:rhs8})} -\text{(\ref{eqn:lhs3})}-\text{(\ref{eqn:lhs4})} + \text{(\ref{eqn:rhs6})}-\text{(\ref{eqn:lhs7})}-\text{(\ref{eqn:lhs8})}\\
=& A s_k (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}}A + A \frac{1}{2\beta} e_k \frac{b_{k+1}}{b_k b_{k+2}} A - A e_k \frac{1}{b_k} (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}}A \\
& - A e_{k} \frac{1}{b_k b_{k+2}} A
\end{aligned}$$
We consider the following expressions $$\label{eqn:14b-13e-13f}
\begin{aligned}
\text{(\ref{eqn:rhs2})}-\text{(\ref{eqn:lhs5})}-\text{(\ref{eqn:lhs6})} =& A \left( s_k {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} - b_{k+1} s_k {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} \frac{1}{b_k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} + \frac{1}{b_k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} \right) A \\
=& A \left( s_k s_{k+1} - {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+1} s_k$};\phantom{b_{k+1} s_k}} s_{k+1} \frac{1}{b_k} + \frac{1}{b_k} s_{k+1} \right) A \\
=& A \left( s_k s_{k+1} - (s_k b_k - e_k + 1) s_{k+1} \frac{1}{b_k} + \frac{1}{b_k} s_{k+1} \right) A \\
=& A e_k s_{k+1} \frac{1}{b_k} A
\end{aligned}$$ and $$\label{eqn:14e-13b}
\begin{aligned}
& \text{(\ref{eqn:rhs5})}-\text{(\ref{eqn:lhs2})} = Q_{k+1} s_{k+1} Q_{k+1} Q_k s_k Q_k \frac{1}{b_{k+1}} {{\mathbf{f}}}- \frac{1}{b_k}{{\mathbf{f}}}Q_{k+1} s_{k+1} Q_{k+1}Q_k s_k Q_k \\
=& A \left( s_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_k - s_{k+1} \frac{1}{b_k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_k b_{k+1}\right) A = A \left( s_{k+1} s_k - s_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle \frac{1}{b_k} s_k$};\phantom{\frac{1}{b_k} s_k}} b_{k+1}\right) A \\
=& A \left( s_{k+1} s_k - s_{k+1} \left( s_k \frac{1}{b_{k+1}} - \frac{1}{b_k}{{\mathbf{f}}}e_k \frac{1}{b_{k+1}} + \frac{1}{b_kb_{k+1}} \right) b_{k+1}\right) A \\
=& A \left( s_{k+1} \frac{1}{b_k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} e_k - s_{k+1} \frac{1}{b_k} \right) A = A s_{k+1} \frac{1}{b_k} e_k A - A s_{k+1} \frac{1}{b_k} A.
\end{aligned}$$
The only terms left are (\[eqn:lhs1\]) and (\[eqn:rhs1\]), which we need to expand into a large number of terms. We will do so separately, starting with the one from the left hand side.
$$\begin{aligned}
& \text{(\ref{eqn:rhs1})} = -A \left( s_{k+1} {{\mathbf{f}}}s_{k} b_{k+2} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} \right) A \nonumber = -A \left( \frac{1}{b_{k+2}} {{\mathbf{f}}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+2} s_{k+1}$};\phantom{b_{k+2} s_{k+1}}} {{\mathbf{f}}}s_{k} b_{k+2} s_{k+1} \right) A \nonumber\\
=& -A \frac{1}{b_{k+2}} {{\mathbf{f}}}s_{k+1} b_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k} b_{k+2} s_{k+1} A + A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k} b_{k+2} s_{k+1} A - A s_{k} s_{k+1} A\nonumber\\
=& -A \frac{1}{b_{k+2}} {{\mathbf{f}}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} b_{k+1}$};\phantom{s_{k+1} b_{k+1}}} s_{k} b_{k+2} s_{k+1} A + A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+2} s_{k+1}$};\phantom{b_{k+2} s_{k+1}}} A - A s_{k} s_{k+1} A\nonumber\\
=& -A s_{k+1} s_{k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+2} s_{k+1}$};\phantom{b_{k+2} s_{k+1}}} A -A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+2} s_{k+1}$};\phantom{b_{k+2} s_{k+1}}} A +A s_{k} s_{k+1} A\nonumber\\
&+A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} s_{k+1} b_{k+1} A -A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} e_{k+1} A +A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} A\nonumber\\
&- A s_{k} s_{k+1} A\nonumber\\
=& -A s_{k+1} s_{k} s_{k+1} b_{k+1} A +A s_{k+1} s_{k} e_{k+1} A -A s_{k+1} s_{k} A \nonumber\\
&-A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} s_{k+1} b_{k+1} A +A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} e_{k+1} A -A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} A\nonumber\\
& +A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} s_{k+1} b_{k+1} A -A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} e_{k+1} A +A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_{k+1} s_{k} A \nonumber\\
=& -A s_{k+1} s_{k} s_{k+1} b_{k+1} A\label{eqn:14a1}\\
&+A s_{k+1} s_{k} e_{k+1} A\label{eqn:14a2}\\
&-A s_{k+1} s_{k} A\label{eqn:14a3}\end{aligned}$$
For the other term we obain
$$\begin{aligned}
&-\text{(\ref{eqn:lhs1})} = A \left( b_{k+1} s_k {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_{k+1} \frac{1}{b_k}{{\mathbf{f}}}s_k b_{k+1} \right) A \nonumber \\
=& A \left( \frac{b_{k+1}}{b_{k+2}} {{\mathbf{f}}}s_k {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+2} s_{k+1}$};\phantom{b_{k+2} s_{k+1}}} \frac{1}{b_k}{{\mathbf{f}}}s_k b_{k+1} \right) A \nonumber\\
=& A \frac{b_{k+1}}{b_{k+2}}{{\mathbf{f}}}\left( s_k \frac{1}{b_k} s_{k+1} b_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_k b_{k+1} - s_k \frac{1}{b_k} e_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} s_k b_{k+1} + s_k \frac{1}{b_k} {{\mathbf{f}}}s_k b_{k+1}\right) A \nonumber\\
=& A \frac{b_{k+1}}{b_{k+2}}{{\mathbf{f}}}s_k \frac{1}{b_k} s_{k+1} b_{k+1} s_k b_{k+1} A \label{eqn:13a1}\\
& - A \frac{b_{k+1}}{b_{k+2}}{{\mathbf{f}}}s_k \frac{1}{b_k} e_{k+1} s_k b_{k+1}A\label{eqn:13a2}\\
& + A \frac{b_{k+1}}{b_{k+2}}{{\mathbf{f}}}s_k \frac{1}{b_k} {{\mathbf{f}}}s_k b_{k+1} A \label{eqn:13a3}.\end{aligned}$$
We examine the terms (\[eqn:13a1\]), (\[eqn:13a2\]), (\[eqn:13a3\]) separately. $$\begin{aligned}
& \text{(\ref{eqn:13a1})} = A \frac{b_{k+1}}{b_{k+2}}{{\mathbf{f}}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_k \frac{1}{b_k}$};\phantom{s_k \frac{1}{b_k}}} s_{k+1} b_{k+1} s_k b_{k+1} A \nonumber\\
=& A \left( \frac{1}{b_{k+2}}{{\mathbf{f}}}s_k {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} b_{k+1}$};\phantom{s_{k+1} b_{k+1}}} s_k - \frac{1}{b_{k+2}}{{\mathbf{f}}}e_k \frac{1}{b_k} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} b_{k+1}$};\phantom{s_{k+1} b_{k+1}}} s_k + \frac{1}{b_{k+2}b_k}{{\mathbf{f}}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} b_{k+1}$};\phantom{s_{k+1} b_{k+1}}} s_k \right) b_{k+1} A \nonumber\\
=& A s_k s_{k+1} s_k b_{k+1} A \label{eqn:13a1-1}\\
&+ A \frac{1}{b_{k}}{{\mathbf{f}}}s_k e_{k+1} s_k b_{k+1} A \label{eqn:13a1-2}\\
&- A \frac{b_{k+1}}{b_{k+2}} A \label{eqn:13a1-3}\\
& -A e_k \frac{1}{b_k} s_{k+1} s_k b_{k+1} A\label{eqn:13a1-4}\\
& -A \frac{1}{b_{k+2}} e_k \frac{1}{b_k} e_{k+1} s_k b_{k+1} A\label{eqn:13a1-5}\\
& +A \frac{1}{2\beta}\frac{1}{b_{k+2}} e_k \frac{b_{k+1}}{b_k} A\label{eqn:13a1-6}\\
& +A \frac{1}{b_k}{{\mathbf{f}}}s_{k+1} s_k b_{k+1} A\label{eqn:13a1-7}\\
& +A \frac{1}{b_{k+2}b_k}{{\mathbf{f}}}e_{k+1} s_k b_{k+1} A\label{eqn:13a1-8}\\
& -A \frac{1}{b_{k+2}b_k}{{\mathbf{f}}}s_k b_{k+1} A\label{eqn:13a1-9}\end{aligned}$$
$$\begin{aligned}
& \text{(\ref{eqn:13a2})} = - A \left( \frac{b_{k+1}}{b_{k+2}} {{\mathbf{f}}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_k \frac{1}{b_k}$};\phantom{s_k \frac{1}{b_k}}} e_{k+1} s_k b_{k+1} \right) A \nonumber\\
=& - A \frac{1}{b_{k+2}} {{\mathbf{f}}}s_k e_{k+1} s_k b_{k+1} A \label{eqn:13a2-1}\\
& + A \frac{1}{b_{k+2}} {{\mathbf{f}}}e_k \frac{1}{b_k} e_{k+1} s_k b_{k+1} A \label{eqn:13a2-2}\\
& - A \frac{1}{b_k b_{k+2}} {{\mathbf{f}}}e_{k+1} s_k b_{k+1} A \label{eqn:13a2-3}\end{aligned}$$
The sum of (\[eqn:13a3\]) and (\[eqn:rest1\]) simplifies to $$\begin{aligned}
& \text{(\ref{eqn:13a3})} + \text{(\ref{eqn:rest1})} \nonumber \\
=& A \left( \frac{1}{b_{k+2}}{{\mathbf{f}}}{\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle b_{k+1}s_k$};\phantom{b_{k+1}s_k}} \frac{1}{b_k} {{\mathbf{f}}}s_k b_{k+1}\right. +s_k (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}}A + A \frac{1}{2\beta} e_k \frac{b_{k+1}}{b_k b_{k+2}} \nonumber\\
&- \left. e_k \frac{1}{b_k} (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}} A - A e_{k} \frac{1}{b_k b_{k+2}} \right) A\nonumber\\
=& A \left( \frac{1}{b_{k+2}}{{\mathbf{f}}}s_k {{\mathbf{f}}}s_k b_{k+1} - \frac{1}{b_{k+2}}{{\mathbf{f}}}e_k \frac{1}{b_k}{{\mathbf{f}}}s_k b_{k+1} + \frac{1}{b_{k+2}b_k}{{\mathbf{f}}}s_k b_{k+1}\right. \nonumber\\
&+s_k (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}} + \frac{1}{2\beta} e_k \frac{b_{k+1}}{b_k b_{k+2}} - \left. e_k \frac{1}{b_k} (1-{{\mathbf{f}}}) s_k \frac{b_{k+1}}{b_{k+2}} - e_{k} \frac{1}{b_k b_{k+2}} \right) A\nonumber\\
=&A \left( \frac{b_{k+1}}{b_{k+2}} - {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle e_k \frac{1}{b_k} s_k$};\phantom{e_k \frac{1}{b_k} s_k}} \frac{b_{k+1}}{b_{k+2}} + \frac{1}{b_{k+2}b_k}{{\mathbf{f}}}s_k b_{k+1} + \frac{1}{2\beta} e_k \frac{b_{k+1}}{b_k b_{k+2}} - e_{k} \frac{1}{b_k b_{k+2}}\right) A\nonumber\\
=&\frac{1}{b_{k}} \label{eqn:rest2-1}\\
& + A\frac{1}{b_k}{{\mathbf{f}}}s_k \frac{b_{k+1}}{b_{k+2}} A \label{eqn:rest2-2}\\
& - A e_{k} \frac{1}{b_k b_{k+2}} A\label{eqn:rest2-3}.\end{aligned}$$
We now compare all the results and see that the following summands cancel each other: (\[eqn:13a2-1\]) and (\[eqn:13a1-2\]), (\[eqn:13a2-2\]) and (\[eqn:13a1-5\]), (\[eqn:13a2-3\]) and (\[eqn:13a1-8\]), (\[eqn:rest2-1\]) and (\[eqn:13a1-3\]), (\[eqn:rest2-2\]) and (\[eqn:13a1-9\]), (\[eqn:14a1\]) and (\[eqn:13a1-1\]).
From the remaining summands we add the following: $$\label{eqn:finalrest}
\begin{aligned}
&\text{(\ref{eqn:13a1-4})} + \text{(\ref{eqn:14a2})} + \text{(\ref{eqn:14b-13e-13f})}\\
=&A \left(- e_k s_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle \frac{1}{b_k} s_k$};\phantom{\frac{1}{b_k} s_k}} b_{k+1} + {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle s_{k+1} s_{k} e_{k+1}$};\phantom{s_{k+1} s_{k} e_{k+1}}} + e_k s_{k+1} \frac{1}{b_k} \right) A\\
=&A \left( - {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle e_k s_{k+1} s_k$};\phantom{e_k s_{k+1} s_k}} + e_k s_{k+1} \frac{1}{b_k} e_k - e_k s_{k+1} \frac{1}{b_k}+ e_{k} e_{k+1} + e_k s_{k+1} \frac{1}{b_k} \right) A\\
=&A \left(- e_k e_{k+1} + e_k s_{k+1} \frac{1}{b_k} e_k + e_{k} e_{k+1}\right) A = A e_k s_{k+1} \frac{1}{b_k} e_k A\\
\end{aligned}$$ A number of terms cancel each other: $$\begin{aligned}
&\text{\eqref{eqn:13a1-7}} + \text{\eqref{eqn:14a3}} + \text{\eqref{eqn:14e-13b}}\\
=& A \left( s_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle \frac{1}{b_k} s_k$};\phantom{\frac{1}{b_k} s_k}} b_{k+1} - s_{k+1} s_{k} + s_{k+1} \frac{1}{b_k} e_k - s_{k+1} \frac{1}{b_k} \right) A\\
=& A \left( s_{k+1} s_k - s_{k+1} \frac{1}{b_k}e_k + s_{k+1} \frac{1}{b_k} - s_{k+1} s_{k} + s_{k+1} \frac{1}{b_k} e_k - s_{k+1} \frac{1}{b_k} \right) A =0
\end{aligned}$$ Collecting all the remaining summands we get the following expression: $$\begin{aligned}
&\text{\eqref{eqn:13a1-6}} + \text{\eqref{eqn:rest2-3}} + \text{\eqref{eqn:finalrest}}\\
=& A \left(\frac{1}{2\beta}\frac{1}{b_{k+2}} e_k \frac{b_{k+1}}{b_k}
- e_{k} \frac{1}{b_k b_{k+2}}
+ {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle e_k s_{k+1}$};\phantom{e_k s_{k+1}}} \frac{1}{b_k} e_k \right) A\\
=& A \left( \frac{1}{2\beta}\frac{1}{b_{k+2}} e_k \frac{b_{k+1}}{b_k} - e_{k} \frac{1}{b_k b_{k+2}} + e_k e_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}s_{k} \frac{1}{b_k} e_k$};\phantom{{{\mathbf{f}}}s_{k} \frac{1}{b_k} e_k}} \right) A \\
=&A \left( \frac{1}{2\beta}\frac{1}{b_{k+2}} e_k \frac{b_{k+1}}{b_k} - e_{k} \frac{1}{b_k b_{k+2}} + \frac{1}{2\beta}\frac{1}{b_{k+2}} e_k e_{k+1} {\tikz[overlay]\node[fill=blue!20,inner sep=1pt, anchor=text, rectangle, rounded corners=1mm,]{$\displaystyle {{\mathbf{f}}}$};\phantom{{{\mathbf{f}}}}} e_k \right) A \\
=& A \left(e_k \frac{b_{k+1} - 2 \beta + b_k}{2\beta b_{k+2}b_k} \right)A = A \left(e_k \frac{y_{k+1} + y_k}{2\beta b_{k+2}b_k} \right)A = 0.
\end{aligned}$$ This proves the claim of the proposition.
Relations involving only $\tilde{e}_k$’s in ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$
--------------------------------------------------------------------------------------------------------
We continue with the defining relations that will only involve the $\tilde{e}_k$’s. The key relation that $\tilde{e}_k$ squares to $\delta \tilde{e}_k$ was already proven in Lemma \[lem:e2=deltae\]. We continue with the remaining relations:
\[lem:eiej=ejei\] We have $\tilde{e}_i \tilde{e}_j= \tilde{e}_j \tilde{e}_i$ for $1 \leq i,j < d$ with $|i-j| > 1$.
Since ${{\mathbf{f}}}e_i {{\mathbf{f}}}$ commutes with $Q_j$ when $| i - j | > 1$ the statement follows.
\[lem:eee=e\] We have $\tilde{e}_k \tilde{e}_{k+1} \tilde{e}_k= \tilde{e}_{k}$ and $\tilde{e}_{k+1} \tilde{e}_{k} \tilde{e}_{k+1}= \tilde{e}_{k+1}$ for $1 \leq k < d-1$.
We only prove the first equality, the second is done in an analogous way. We compute $$\begin{aligned}
\tilde{e}_k \tilde{e}_{k+1} \tilde{e}_k &=& Q_k e_k Q_k Q_{k+1}e_{k+1} Q_{k+1} Q_k e_k Q_k\\
&=& Q_k \sqrt{b_{k+2}}{{\mathbf{f}}}e_k {{\mathbf{f}}}e_{k+1} \frac{1}{b_k} {{\mathbf{f}}}e_k \sqrt{b_{k+2}} {{\mathbf{f}}}Q_k \\
&\overset{(a)}{=}& Q_k \sqrt{b_{k+2}}{{\mathbf{f}}}e_k {{\mathbf{f}}}e_{k+1} \frac{1}{c_{k+1}} {{\mathbf{f}}}e_k \sqrt{b_{k+2}} {{\mathbf{f}}}Q_k\\
&\overset{(b)}{=}& Q_k \sqrt{b_{k+2}}{{\mathbf{f}}}e_k {{\mathbf{f}}}e_{k+1} \frac{1}{b_{k+2}} {{\mathbf{f}}}e_k \sqrt{b_{k+2}} {{\mathbf{f}}}Q_k\\
&=& Q_k \sqrt{b_{k+2}}{{\mathbf{f}}}e_k {{\mathbf{f}}}e_{k+1} {{\mathbf{f}}}e_k \frac{\sqrt{b_{k+2}}}{b_{k+2}} {{\mathbf{f}}}Q_k \overset{(c)}{=} \tilde{e}_k\end{aligned}$$ Where $(a)$ is due to Relation (VW.\[8b\]), since $\frac{1}{b_k} {{\mathbf{f}}}e_k = \frac{1}{c_{k+1}} {{\mathbf{f}}}e_k$, and $(b)$ follows from $e_{k+1} \frac{1}{c_{k+1}} {{\mathbf{f}}}= e_{k+1}\frac{1}{b_{k+2}} {{\mathbf{f}}}$ due to Relation (VW.\[8a\]). Finally $(c)$ is again a consequence of Lemma \[lem:commute\_b\] and Relation (VW.\[6d\]).
Mixed relations in ${{\mathbf{f}}}{{\bigdoublevee}_d^{\rm cycl}}{{\mathbf{f}}}$
-------------------------------------------------------------------------------
We are now left with proving the relations involving both $\tilde{s}_j$’s and $\tilde{e}_j$’s.
\[lem:es=e\] We have $\tilde{e}_{k} \tilde{s}_{k} = \tilde{e}_{k}= \tilde{s}_{k} \tilde{e}_{k}$ for $1 \leq k < d$.
We compute $$\begin{aligned}
\tilde{e}_k \tilde{s}_k &=& -Q_k e_k \frac{b_{k+1}}{b_k} {{\mathbf{f}}}s_k Q_k + Q_k e_k Q_k \frac{1}{b_k}{{\mathbf{f}}}\\
&\overset{(a)}{=}& -Q_k e_k \frac{b_{k+1}}{b_k} s_k Q_k + Q_k e_k Q_k \frac{1}{b_k}{{\mathbf{f}}}\\
&\overset{(b)}{=}& -Q_k e_k \frac{1}{b_k} s_k b_{k}Q_k + Q_k e_k \frac{1}{b_k} e_k Q_k\\
&\overset{(c)}{=}& - \frac{1}{2\beta} Q_k e_k Q_k + (1+\frac{1}{2\beta}) Q_k e_k Q_k = \tilde{e}_k\end{aligned}$$ where equality $(a)$ is due to Proposition \[prop:kill\_f\], $(b)$ is due to Relation (VW.\[7\]), and finally $(c)$ is due to Proposition \[prop:reduce\_fracs\]. The second equality in the claim follows analogously.
\[lem:se=es\] We have $\tilde{s}_i \tilde{e}_j= \tilde{e}_j \tilde{s}_i$ for $1 \leq i,j < d$ with $|i-j| > 1$.
This follows by the same arguments as for Lemmas \[lem:sisj=sjsi\] and \[lem:eiej=ejei\].
\[lem:see=se\] We have $\tilde{s}_k \tilde{e}_{k+1} \tilde{e}_k= \tilde{s}_{k+1}\tilde{e}_{k}$ and $\tilde{s}_{k+1} \tilde{e}_{k} \tilde{e}_{k+1}= \tilde{s}_{k}\tilde{e}_{k+1}$ for $1 \leq k < d-1$.
We only prove the first equality, the second is done analogously, $$\begin{aligned}
\tilde{s}_k \tilde{e}_{k+1} \tilde{e}_k &= -Q_k s_k Q_k Q_{k+1}e_{k+1} Q_{k+1} Q_k e_k Q_k + \frac{1}{b_{k}}{{\mathbf{f}}}Q_{k+1} e_{k+1} Q_{k+1} Q_k e_k Q_k \\
&\overset{(a)}{=} - Q_k \sqrt{b_{k+2}}{{\mathbf{f}}}s_k \frac{1}{b_k} e_{k+1} e_k {{\mathbf{f}}}\sqrt{b_{k+2}} {{\mathbf{f}}}Q_k + \frac{\sqrt{b_{k+2}}}{b_k\sqrt{b_{k}}\sqrt{b_{k+1}}}{{\mathbf{f}}}e_{k+1} e_k {{\mathbf{f}}}\sqrt{b_{k+2}} {{\mathbf{f}}}Q_k\\
&\overset{(b)}{=} - Q_{k+1} \frac{1}{\sqrt{b_{k}}}{{\mathbf{f}}}s_k e_{k+1} e_k {{\mathbf{f}}}\sqrt{b_{k+2}} {{\mathbf{f}}}Q_k + Q_{k+1} \frac{1}{\sqrt{b_{k}}}{{\mathbf{f}}}e_{k} \frac{1}{b_k} e_{k+1} e_k {{\mathbf{f}}}\sqrt{b_{k+2}} {{\mathbf{f}}}Q_k\\
&\overset{(c)}{=} - Q_{k+1} \frac{1}{\sqrt{b_{k}}}{{\mathbf{f}}}s_{k+1} e_k {{\mathbf{f}}}\sqrt{b_{k+2}} {{\mathbf{f}}}Q_k + \frac{1}{b_{k+2}} Q_{k+1} \frac{1}{\sqrt{b_{k}}}{{\mathbf{f}}}e_{k} {{\mathbf{f}}}\sqrt{b_{k+2}} {{\mathbf{f}}}Q_k\\
&\overset{(d)}{=} - Q_{k+1} s_{k+1} Q_{k+1} Q_k e_k Q_k + \frac{1}{b_{k+1}} Q_{k} e_{k} Q_k = \tilde{s}_{k+1} \tilde{e}_k.
\end{aligned}$$ Where equality $(a)$ follows from Proposition \[prop:kill\_f\], while the second equality is a consequence of Lemma \[lem:commute\_b\] and again Proposition \[prop:kill\_f\]. Equality $(c)$ follows by using relations (\[8a\]) and (\[8b\]) to rewrite the second summand and then applying relations (\[6b\]) and (\[6d\]). Finally equality $(d)$ is using Proposition \[prop:kill\_f\] and reordering the factors afterwards.
Example: the graded Brauer algebras ${\rm Br}^{\rm gr}_2(\delta)$ {#sec:example}
=================================================================
In this section we will illustrate explicitly the construction of the isomorphism for the Brauer algebras ${\rm Br}_2(\delta)$ and describe their graded version ${\rm Br}^{\rm gr}_2(\delta)$.
Case $\delta\neq 0$
-------------------
We first consider the case ${\rm Br}_2(\delta)$ for $\delta \neq 0$. By [@Rui] this Brauer algebra is semisimple with basis $1$, $t=t_1$, and $g=g_1$. The set of orthogonal idempotents is $$\begin{aligned}
\left\{\frac{1+t}{2}-\frac{1}{\delta}g,\, \frac{1-t}{2}+\frac{1}{\delta}g,\, \frac{1}{\delta}g \right\}\end{aligned}$$ which gives rise to an isomorphism $${\rm Br}_2(\delta) \cong \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C}.$$ Note that the grading on ${\rm Br}_2(\delta)$ needs to be trivial since all idempotents have to have degree $0$. We now want to illustrate the idempotent truncation of the level $2$ cyclotomic quotient ${\bigdoublevee}_2(\Xi)$ with parameters from Definition\[defalphabeta\].
We describe ${\bigdoublevee}_2^{\rm cycl}$ in terms of the seminormal representation of ${\bigdoublevee}_2^{\rm cycl}$ from [@AMR Theorem 4.13] by an action of ${\bigdoublevee}_2^{\rm cycl}$ on the vector space with basis given by all up-down bitableaux of length $2$. Explicitly, this basis consists of $$\begin{xy}
\xymatrix@R=4pt@C=2pt{
v_1 \ar@{=}[d] & v_2 \ar@{=}[d] & v_3 \ar@{=}[d] & v_4 \ar@{=}[d] & v_5 \ar@{=}[d] & v_6 \ar@{=}[d] & v_7 \ar@{=}[d] & v_8 \ar@{=}[d] \\
\left( \emptyset , \emptyset \right) \ar@{-}[d] & \left( \emptyset , \emptyset \right) \ar@{-}[d] &\left( \emptyset , \emptyset \right) \ar@{-}[d] &\left( \emptyset , \emptyset \right) \ar@{-}[d] &\left( \emptyset , \emptyset \right) \ar@{-}[d] &\left( \emptyset , \emptyset \right) \ar@{-}[d] &\left( \emptyset , \emptyset \right) \ar@{-}[d] &\left( \emptyset , \emptyset \right) \ar@{-}[d]\\
\left( \yng(1), \emptyset \right) \ar@{-}[d] & \left( \yng(1), \emptyset \right) \ar@{-}[d] & \left( \yng(1), \emptyset \right) \ar@{-}[d] & \left( \emptyset, \yng(1) \right) \ar@{-}[d] & \left( \yng(1), \emptyset \right) \ar@{-}[d] & \left( \emptyset, \yng(1) \right) \ar@{-}[d] & \left( \emptyset, \yng(1) \right) \ar@{-}[d] & \left( \emptyset, \yng(1) \right) \ar@{-}[d]\\
\left( \yng(2) , \emptyset \right) & \left( \yng(1,1) , \emptyset \right) & \left( \emptyset , \emptyset \right) & \left( \emptyset , \emptyset \right) & \left( \yng(1) , \yng(1) \right) & \left( \yng(1) , \yng(1) \right) &\left( \emptyset , \yng(1,1) \right) & \left( \emptyset , \yng(2) \right)
}
\end{xy}$$ all of which are common eigenvectors for $y_1$ and $y_2$. The corresponding pairs of eigenvalues are the following: $$\begin{array}{cccccccc}
v_1 & v_2 & v_3 & v_4 & v_5 & v_6 & v_7 & v_8\\
(\alpha,\alpha-1) & (\alpha, \alpha+1) & (\alpha,-\alpha) & (\beta,-\beta) & (\alpha,\beta) & (\beta, \alpha) & (\beta,\beta-1) & (\beta,\beta+1)
\end{array}$$ where the first entry denotes the eigenvalue for $y_1$ and the second for $y_2$. Using these eigenvalues one can calculate via [@AMR Theorem 4.13] the matrix of $s_1$ $$s_1=\left(\begin{array}{cccccccc}
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & \frac{\alpha+\beta-1}{\alpha-\beta} & \frac{\sqrt{-(2\beta-1)(2\alpha-1)}}{\alpha-\beta} & 0 & 0 & 0 & 0\\
0 & 0 & \frac{\sqrt{-(2\beta-1)(2\alpha-1)}}{\alpha-\beta} & -\frac{\alpha+\beta-1}{\alpha-\beta} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & \star & \star & 0 & 0\\
0 & 0 & 0 & 0 & \star & \star & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \star & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \star
\end{array} \right).$$ The $\star$’s indicate some further non-zero entries which are irrelevant for the construction. Similarly one obtains the matrix for $e_1$ as $$e_1=\left(\begin{array}{cccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & (2\alpha-1)\frac{\alpha+\beta}{\alpha-\beta} & \sqrt{-(2\beta-1)(2\alpha-1)}\frac{\alpha+\beta}{\alpha-\beta} & 0 & 0 & 0 & 0\\
0 & 0 & \sqrt{-(2\beta-1)(2\alpha-1)}\frac{\alpha+\beta}{\alpha-\beta} & -(2\beta-1)\frac{\alpha+\beta}{\alpha-\beta} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array} \right).$$ Although tedious, it is of course straightforward to check that these matrices together with the action of $y_1,y_2$ satisfy all the defining relations of ${\bigdoublevee}_2^{\rm cycl}$.
For the isomorphism in Theorem \[thm:main\] we first need to apply the idempotent ${{\mathbf{f}}}$ from both sides. In the chosen basis this amounts to truncation with respect to the basis vectors where the second partition is always empty (i.e. $v_1$, $v_2$ and $v_3$) that means we look at the submatrices consisting of the first three columns in the first three rows. Clearly, the submatrices ${{\mathbf{f}}}s_1 {{\mathbf{f}}}$ and ${{\mathbf{f}}}e_1 {{\mathbf{f}}}$ do not even satisfy the most basic Brauer algebra relations, i.e. the relation for squares. This deficiency is overcome by the correction term $Q=Q_1=\sqrt{\frac{b_2}{b_1}}{{\mathbf{f}}}$ and $\frac{1}{b_1}{{\mathbf{f}}}$ from the definition of the isomorphism in Theorem \[thm:main\] as we show now explicitly. We have $$Q=\left(\begin{array}{ccc}
\sqrt{\frac{\alpha+\beta-1}{\alpha+\beta}}& 0 & 0\\
0 & \sqrt{\frac{\alpha+\beta+1}{\alpha+\beta}} & 0\\
0 & 0 & \sqrt{\frac{\beta-\alpha}{\alpha+\beta}}
\end{array}\right) \text{ and }
\frac{1}{b_1}{{\mathbf{f}}}=\left(\begin{array}{ccc}
\frac{1}{\alpha+\beta}& 0 & 0\\
0 & \frac{1}{\alpha+\beta} & 0\\
0 & 0 & \frac{1}{\alpha+\beta}
\end{array}\right).$$ By multiplying ${{\mathbf{f}}}e_1 {{\mathbf{f}}}$ from both sides with $Q$ we obtain $$Q {{\mathbf{f}}}e_1 {{\mathbf{f}}}Q =\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & \delta
\end{array}\right),$$ which is obviously a matrix that squares to $\delta$ times itself. The analogous construction for ${{\mathbf{f}}}s_1 {{\mathbf{f}}}$ needs an extra correction term (as given in Theorem \[thm:main\]): $$Q {{\mathbf{f}}}s_1 {{\mathbf{f}}}Q =\left(\begin{array}{ccc}
-\frac{\alpha+\beta-1}{\alpha+\beta} & 0 & 0\\
0 & \frac{\alpha+\beta+1}{\alpha+\beta} & 0\\
0 & 0 & -\frac{\alpha+\beta-1}{\alpha+\beta}
\end{array}\right) \text{ and } -Q {{\mathbf{f}}}s_1 {{\mathbf{f}}}Q + \frac{1}{b_1}{{\mathbf{f}}}= \left(\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 1
\end{array}\right).$$
Case $\delta\neq 0$
-------------------
If we try to do the same for the Brauer algebra ${\rm Br}_2(0)$ we immediately encounter a problem, since the algebra is not semisimple and so we cannot apply the formulas and constructions from [@AMR]. The whole difficulty of our proof is to show that the formulas in Theorem \[thm:main\] still make sense and give the required correction terms even in the non-semisimple case.
One can show that via the orthogonal idempotents $\{\frac{1+t}{2},\frac{1-t}{2}\}$ one obtains $$\begin{aligned}
\label{Schluss}
{\rm Br}_2(0) &\cong&\mathbb{C} \oplus \mathbb{C}[x]/(x^2), \end{aligned}$$ with the element $x$ corresponding to the element $g \in {\rm Br}_2(0)$ It becomes graded in the obvious way by putting the idempotents in degree $0$ and $x$ in degree $2$.
There is in fact a generalization of the up-down tableaux basis in the graded setting via the diagram calculus developed in [@ES3]. The explicit isomorphism between this description and the Brauer algebra itself is being worked out in [@LiS].
To make the connection from to [@ES3] we note that ${\rm Br}^{\rm gr}_2(0)$ can be realized as the subalgebra of the generalized type ${\rm D}$ Khovanov algebra from [@ES2] (using the notation from there) with the following diagrams as basis $$\begin{tikzpicture}[thick,scale=0.5]
\begin{scope}[xshift=0cm]
\draw[thin,gray,dashed] (0,0) -- +(3,0);
\node at (0,-.04) {$\mathbf{\circ}$};
\node at (1,.1) {$\mathbf{\down}$};
\draw (1,1) -- +(0,-1);
\node at (2,0) {$\mathbf{\times}$};
\node at (3,.1) {$\mathbf{\down}$};
\draw (3,1) -- +(0,-1);
\draw (1,0.1) -- +(0,-1.1);
\draw (3,0.1) -- +(0,-1.1);
\end{scope}
\begin{scope}[xshift=5cm]
\draw[thin,gray,dashed] (0,0) -- +(3,0);
\node at (0,.1) {$\mathbf{\down}$};
\node at (1,-.1) {$\mathbf{\up}$};
\draw (0,0) .. controls +(0,.75) and +(0,.75) .. +(1,0);
\draw (0,0) .. controls +(0,-.75) and +(0,-.75) .. +(1,0);
\draw (2,1) -- +(0,-1);
\node at (2,.1) {$\mathbf{\down}$};
\draw (3,1) -- +(0,-1);
\node at (3,.1) {$\mathbf{\down}$};
\draw (2,0.1) -- +(0,-1.1);
\draw (3,0.1) -- +(0,-1.1);
\end{scope}
\begin{scope}[xshift=10cm]
\draw[thin,gray,dashed] (0,0) -- +(3,0);
\node at (0,-.1) {$\mathbf{\up}$};
\node at (1,.1) {$\mathbf{\down}$};
\draw (0,0) .. controls +(0,.75) and +(0,.75) .. +(1,0);
\draw (0,0) .. controls +(0,-.75) and +(0,-.75) .. +(1,0);
\draw (2,1) -- +(0,-1);
\node at (2,.1) {$\mathbf{\down}$};
\draw (3,1) -- +(0,-1);
\node at (3,.1) {$\mathbf{\down}$};
\draw (2,0.1) -- +(0,-1.1);
\draw (3,0.1) -- +(0,-1.1);
\end{scope}
\end{tikzpicture}$$ Here the first diagram spans the copy of $\mathbb{C}$ iunder the identification with , while the other two span a copy of $\mathbb{C}[x]/(x^2)$, with the second diagram being the unit and the third one being the element of degree $2$, i.e. it corresponds to $x$.
[^1]: M.E. was financed by the DFG Priority program 1388, C.S. thanks the MPI in Bonn for excellent working conditions and financial support.
[^2]: as every car lover can probably imagine easily ...
[^3]: which translated to German is [*Verallgemeinerte Wenzl Algebra*]{}, abbreviated as $\bigdoublevee$. It is also sometimes called [*Nazarov-Wenzl algebra*]{} in the literature. Hence $\bigdoublevee$ can be viewed as composed of the letters $N$, $W$ and $V$ as well.
|
---
abstract: |
We study the clustering properties of groups and of galaxies in groups in the DEEP2 Galaxy Redshift Survey dataset at $z\sim1$ in three separate fields covering a total of 2 degrees$^2$. Four measures of two-point clustering in the DEEP2 data are presented: 1) the group correlation function for 460 groups with estimated velocity dispersions of $\sigma\ge200$ , 2) the galaxy correlation for the full DEEP2 galaxy sample, using a flux-limited sample of 9800 objects between $0.7\leq z\leq1.0$, 3) the galaxy correlation for galaxies in groups or in the field, and 4) the group-galaxy cross-correlation function. Our results are compared with mock group and galaxy catalogs produced from simulations. Using the observed number density and clustering amplitude of the DEEP2 groups, the estimated minimum group dark matter halo mass is $M_{min}\sim 6 \times 10^{12} h^{-1} M_{\Sun}$ for a flat cosmology with $\sigma_8=0.9$. Groups are more clustered than galaxies in the DEEP2 data, with a relative bias of $b=1.17 \pm0.04$ on scales $r_p=0.5-15$ . Galaxies in groups are also more clustered than the full galaxy sample, with a scale-dependent relative bias which falls from $b\sim2.5 \pm0.3$ at $r_p=0.1$ to $b\sim1
\pm0.5$ at $r_p=10$ . The correlation function of galaxies in groups has a steeper slope ($\gamma\sim2.12 \pm0.06$) than for the full galaxy sample ($\gamma\sim1.74 \pm0.03$), and both samples can be fit by a power-law on scales $r_p=0.05-20$ . We empirically measure the contribution to the projected correlation function, , for galaxies in groups from a ‘one-halo’ term and a ‘two-halo’ term by counting pairs of galaxies in the same or in different groups. The projected cross-correlation between group centers and the full galaxy sample, which is sensitive to the radial distribution of galaxies in and around groups, shows that red galaxies are more centrally concentrated in groups than blue galaxies at $z\sim1$. DEEP2 galaxies in groups appear to have a shallower radial distribution than that of mock galaxy catalogs made from N-body simulations, which assume a central galaxy surrounded by satellite galaxies with an NFW profile. Using simulations with different halo model parameters, we show that the clustering of galaxies in groups can be used to place tighter constraints on the halo model than can be gained from using just the usual galaxy correlation function alone.
author:
- 'Alison L. Coil, Brian F. Gerke, Jeffrey A. Newman, Chung-Pei Ma, Renbin Yan, Michael C. Cooper, Marc Davis, S. M. Faber, Puragra Guhathakurta, David C. Koo,'
title: 'The DEEP2 Galaxy Redshift Survey: Clustering of Groups and Group Galaxies at $z\sim1$'
---
Introduction
============
Groups of galaxies populate an intermediate range in density-contrast between galaxies and clusters and occupy a regime that is critical to understanding hierarchical galaxy formation in models. Merger events between galaxies likely occur within groups rather than clusters due to their lower velocity dispersions [e.g., @Ostriker80; @Barnes85]. Galaxy groups should also be more easily related to dark matter halos than galaxies themselves, which can have a complicated halo-occupation function that depends significantly on galaxy properties. In order to understand and test galaxy formation and evolution models it is useful to relate galaxies to observable groups as a proxy for their parent dark matter halos. The current halo model paradigm [e.g., @Seljak00; @Ma00; @Peacock00; @Cooray02; @Kravtsov04] provides a statistical analytic measure for relating galaxies to their dark matter halos. A key statistic in the halo model is the halo occupation distribution (HOD) which measures the probability of a halo of a given mass hosting $N$ galaxies. The halo model also naturally explains the small deviations seen in the clustering of galaxies at $z\sim0$ from a power-law model, where there is a transition from galaxies within a single halo and between different halos. The clustering of groups and group galaxies depends not only on halo model parameters but also on the nature of bias and the details of hierarchical structure formation, as well as cosmological parameters such as $\Omega_{m}$ and $\sigma_8$.
Observationally, clusters of galaxies have been shown to be very strongly clustered [@Bahcall88], with the clustering strength depending on the richness of the cluster. [@Kaiser84] show that the large clustering scale-length of massive and rare Abell clusters can be explained by a simple model in which these clusters formed in regions where the primordial density enhancement was unusually high. Objects forming in the densest peaks would naturally be biased tracers of the underlying dark matter field such that massive clusters would have a higher correlation amplitude than that of galaxies. Since their masses are intermediate, galaxy groups are therefore expected to have clustering properties between those of galaxies and clusters.
The first papers analyzing the clustering of groups in local redshift surveys at $z\sim0$ present conflicting results and were hampered by small samples and cosmic variance [e.g., @Jing88; @Maia90; @Ramella90]. [@Trasarti97] investigated the effect of changing the linking-length parameters in the Friends-of-Friends (FoF) algorithm and its effect on the clustering signal and found that these early papers used too large a linking length, which led to a diminished clustering strength due to the presence of interlopers. [@Trasarti97] found in their data from the Perseus-Pisces redshift survey, using two fields and $\sim50$ and 200 groups in each field, that groups are approximately twice as clustered as galaxies, but with significant error bars. All of these papers showed that determining the clustering properties of groups is a tricky endeavor, which can depend quite sensitively on the volume and magnitude depth of the survey, the handling of the selection function and varying completeness, and the method used to identify groups. In addition, these analyses all suffered from significant uncertainties, from both Poisson statistics, due to the small number of groups in the surveys, and cosmic variance, due to the small volumes surveyed, which was not quantified in any of these papers.
Significant advances have recently been made with much larger datasets. [@Girardi00], with the combined CfA2 and SSRS2 surveys, have a sample of 885 groups in a volume of $\sim3 \times 10^5 h^{-3}
{\rm Mpc}^3$, much larger than earlier surveys. With this large sample size, they are able to construct volume-limited subsamples and investigate the dependence of clustering on group properties, finding that groups with more members and/or larger internal velocity dispersion are more strongly clustered. [@Merchan00] use a sample of 517 groups from the Updated Zwicky Catalog and 104 groups from the SSRS2 to show that groups are at least twice as clustered as galaxies, and that more massive groups have larger clustering strength.
Most recently, the 2dF Galaxy Redshift Survey has provided a vast dataset with which to study large-scale structure at $z\lesssim0.2$. Using data from the 100k release, [@Zandivarez03] measure the clustering of groups as a function of virial mass and find that more massive groups are more clustered and that their measurements match the clustering of dark matter halos in a N-body simulation. Using the completed 2dF survey, [@Padilla04] analyze group clustering as a function of luminosity and show that while the least luminous groups actually cluster less than the galaxies in the survey, there is a strong relation between group luminosity and correlation length, and the most luminous groups (with $L\sim4\times10^{11} h^{-2}
L_\sun$) are $\sim10$ times more clustered as the least luminous groups (with $L\sim2\times10^{10} h^{-2} L_\sun$). The relation between clustering scale length and mean group separation that they find continues the trend seen on larger scales for clusters. They find very good agreement between their data and mock catalogs constructed from simulations and semi-analytic galaxy evolution recipes. These same conclusions are reached by [@Yang05a], who also use the 2dF data to measure the clustering of groups as a function of luminosity. It appears that at $z\sim0$ there is now convergence among group clustering analyses.
The extensive 2dF group catalogs have now also allowed studies of groups beyond simple measures of the correlation function of groups. Several authors measure the radial profile of galaxies in groups using 2dF data [@Collister05; @Diaz05; @Yang05b], and analyze the cross-correlation between group centers and galaxies [@Yang05b]. These papers find that galaxies in groups are less concentrated than dark matter particles in simulations, and also find that, locally, the centers of groups are preferentially populated by red galaxies compared to blue galaxies. The HOD has now also been measured directly at $z\sim0$ using counts of galaxies in groups of different masses [@Collister05; @Yang05HOD], constraining halo model parameters locally. Clustering measures of the correlation function of all galaxies (not just those in groups) at intermediate redshifts indicate that the HOD does not change significantly between $z\sim1$ and $z\sim0$ [@Yan03b; @Phleps05].
These measurements have only been performed with local samples; group catalogs have not been available at intermediate- or high-redshift. In this paper we present the first analysis of group clustering at $z\sim1$, using group catalogs from the DEEP2 Galaxy Redshift Survey [@Gerke05]. The high resolution of the DEEP2 data allows us to identify groups in three dimensions regardless of their galaxy properties, using only their overdensity in space. We focus here on the clustering of groups and galaxies within groups at $z\sim1$, as these measures, when combined with similar measures at $z\sim0$, will provide constraints on galaxy evolution and structure formation models. We also show how these measures can constrain the HOD at $z\sim1$.
Four measures of two-point clustering in the DEEP2 dataset between $0.7\leq z\leq1.0$ are analyzed: 1) the group correlation function, 2) the galaxy correlation function for DEEP2 galaxies, 3) the galaxy correlation function for galaxies in groups, and 4) the group-galaxy cross-correlation function. These clustering measures can be used to constrain the halo model parameters by comparing the data to mock catalogs with different HODs. The first clustering measure, when combined with the observed number density of groups in our sample, is used to estimate the typical dark matter masses of the halos the groups studied reside in. The second measure provides constraints on the HOD, though not in an entirely unique way; further constraints on the HOD are provided by the last two measures. For the clustering of galaxies in groups, we empirically distinguish between ‘one-halo’ and ‘two-halo’ terms using pairs of galaxies within the same group and pairs in different groups, respectively. The clustering of galaxies in groups and the group-galaxy cross-correlation function can also constrain the radial distribution of galaxies within groups when compared with mock catalogs.
Lastly, groups may be used to constrain cosmological parameters like the dark energy equation of state, $w$, if their bivariate distribution in redshift $z$ and velocity dispersion $\sigma$ can be accurately measured [@Newman02]. This test, however, requires that the relation between group velocity dispersion and dark matter halo mass be known and accurately calibrated. It has recently been suggested [@Majumdar04; @Lima04] that the clustering properties of galaxy clusters may be used for “self-calibration”, since the clustering properties of halos can be predicted as a function of mass and compared to the measured clustering. The group correlation function results presented here should be useful as such a self-calibration procedure for future DEEP2 studies.
An outline of the paper is as follows: §2 briefly describes the DEEP2 Galaxy Redshift Survey and the sample of galaxy groups used here, as well as the mock galaxy catalogs constructed for the survey. §3 discusses the methods used to calculate the two-point correlation functions. We present clustering results for groups in §4, where we compare with simulations and estimate the minimum dark matter halo mass for our groups. In §5 we analyze the clustering of the full galaxy sample and galaxies in groups in the DEEP2 data and in mock catalogs and show the contribution to the correlation function for galaxies in groups from the ‘one-halo’ and ‘two-halo’ terms. §6 presents the cross-correlation between the full galaxy sample and group centers, which depends upon the radial profile of galaxies in groups. The relative biases between galaxies in groups and all galaxies and between groups and galaxies are presented in §7. Mock catalogs with different halo models are used to illustrate how these various clustering measures can be used to constrain parameters of the halo model in §8. We conclude in §9.
Data Sample and Mock Catalogs
=============================
In this paper we use data from the DEEP2 Galaxy Redshift Survey, which is an ongoing project using the DEIMOS spectrograph [@Faber02] on the 10-m Keck II telescope to survey optical galaxies at $z\simeq1$ in a comoving volume of approximately 5$\times$10$^6$ $h^{-3}$ Mpc$^3$. Using $\sim1$ hr exposure times, the survey will measure redshifts for $\sim40,000$ galaxies in the redshift range $0.7\sim z\sim1.5$ to a limiting magnitude of $R_{\rm AB}=24.1$ [@Coil03xisp; @Faber05]. Spectroscopic targets are pre-selected using a color cut in $B-R$ - $R-I$ space to ensure that most galaxies lie beyond $z\sim0.75$. This color-cut results in a sample with $\sim$90% of the targeted objects at $z>0.75$, missing only $\sim$3% of the $z>0.75$ galaxies which meet our magnitude limit [@Davis02]. Due to the high dispersion ($R\sim5,000$) of our spectra, our redshift errors, determined from repeated observations, are $\sim30$km s$^{-1}$. Restframe $(U-B)_0$ colors have been derived as described in [@Willmer05]. Details of the observations, catalog construction and data reduction can be found in @Davis02 [@Coil04; @Davis04; @Faber05].
The completed survey will cover 3 square degrees of the sky over four widely separated fields to limit the impact of cosmic variance. Each field is comprised of two to four contiguous photometric ’pointings’ of size $0.5$ by $0.67$ degrees. Here we use data from six of our most complete pointings to date, in three separate fields. We use data from pointings 1 and 2 in the DEEP2 fields 2, 3 and 4. Each DEEP2 pointing corresponds to a volume of comoving dimensions $\sim20 \times 27 \times 550$ in a model for $0.7\leq z \leq1.0$. The total volume of the sample used in this paper is thus $\sim1.8 \times 10^6 \ h^{-3} {\rm Mpc}^3$. To convert measured redshifts to comoving distances along the line of sight we assume a flat cosmology with $\Omega_{\rm m}=0.3$ and $\Omega_{\Lambda}=0.7$. We define $h \equiv
{\rm {\it H}_0/(100 \ km \ s^{-1} \ Mpc^{-1}})$ and quote correlation lengths, , in comoving . Throughout the paper, we quote empirical errors calculated from the variance across the six pointings. In so doing we have treated the six DEEP2 pointings as being entirely independent, and they are not: there are two adjacent pointings in each of three independent fields. We estimate from Monte Carlo simulations using the methods outlined in [@Newman02] that, for the distribution of pointings used here, the measured standard error should be increased by $17 \pm5$%, for errors which are dominated by cosmic variance; this is a conservative assumption, as (especially for the group-group correlations) other contributions such as shot noise are significant.
A description of the methods employed to detect groups is presented in [@Gerke05], along with details of the group catalog. A Voronoi-Delaunay Method (VDM) group-finder [@Marinoni02] is used to identify galaxy groups. This method searches for galaxy overdensities in redshift space, using an asymmetric search window to account for redshift-space distortions. The advantage of the method over traditional FoF group-finding methods lies in the fact that it has no fixed length scale, but instead uses an adaptive search radius based on estimated group richness. The VDM group-finder thus avoids a common problem of FoF methods, in which clustered groups are merged together along filaments.
We use three galaxy samples in this paper; all are drawn from the same volume as the group sample. The full galaxy sample includes a total of 9787 galaxies in the redshift range $0.7\leq z \leq1.0$ in the same six pointings used for the group analysis. We also split the full galaxy sample into subsamples of field and group galaxies, with 5947 and 3840 objects in each. Throughout this paper we use the terms “all galaxies” and “the full galaxy sample” interchangeably; they include both the field and group galaxy samples.
Our main group sample consists of all DEEP2 groups with an estimated velocity dispersion $\sigma\ge200$ , with a total of 460 groups and $\sim50-100$ groups per pointing. We do not make a distinction between groups and clusters - clusters are simply the larger groups; they are included in all analyses, but have minimal impact on this work due to their rarity. The $\sigma\ge200$ group sample has similar completeness ($74
\pm5$%) and purity ($57 \pm3$%) as samples with higher $\sigma$ cutoffs, as estimated using mock catalogs described below. Here completeness is defined as the fraction of real groups (defined as a group of galaxies belonging to the same parent halo, where halos are identified using FoF in real space on the dark matter particles in the simulations) that are successfully identified in the recovered group catalog created by the group-finder, and purity is the fraction of recovered groups that correspond to real groups (see [@Gerke05] for more details). As shown in Fig. 8 of [@Gerke05], these statistics are nearly independent of group velocity dispersion. There are more recovered groups than real groups in these mock catalogs by a factor of 1.4. The $\sigma\ge200$ group sample has a relatively high “galaxy success rate” of $70 \pm1$%, defined as the fraction of galaxies in real groups that are identified as group galaxies in our recovered group catalog, and an interloper fraction of $43 \pm1$%. We do not use the $\sigma\ge350$ cut that was used in [@Gerke05], as that cut was used to define a sample that recovered different physical properties than the ones relevant here; instead of the distribution of groups in redshift and velocity dispersion, the relevant parameters here are the positions of groups and the interloper fraction (the fraction of identified group galaxies that are not actually in groups).
The observed richness distribution of our groups in the DEEP2 data is roughly a power-law, with most groups having two observed galaxies; the largest groups have $\sim10$ galaxies, though there are only a few groups this large. Fig. \[mockrich\] shows the observed richness distribution for groups in the DEEP2 data as well as for real and recovered groups in our mock catalogs, for groups with $0.7\leq z \leq1.0$ and $\sigma\ge200$ .
A histogram of the redshift distribution of our group sample is shown in Fig. \[sf\]. We restrict the analyses here to the redshift range $0.7\leq z\leq1.0$ to minimize systematic effects. While groups are found in the survey to higher redshifts, those groups are likely to be more massive and hence not as representative of groups in our sample as a whole. Additionally, the $R$-band target selection of the survey corresponds to a bluer restframe color-selection at higher redshift; this results in fewer red galaxies being targeted at higher redshift compared to blue galaxies [@Willmer05], which could systematically affect the group richness and velocity dispersion estimates at $z\gtrsim1.0$. The spatial distribution of DEEP2 groups in three of the six pointings used is shown in Fig. \[conegroup\]. Shown are pointings 2 in the DEEP2 fields 2, 3 and 4.
The mock galaxy catalogs used throughout the paper are described in [@Yan03]; relevant details are repeated here. The mock catalogs are constructed from N-body simulations of $512^3$ dark matter particles with a particle mass of $m=1 \times 10^{10} h^{-1}$ $M_{\Sun}$ in a box with dimensions 256 on a side, for a cosmology with $\sigma_8=0.9$. Dark matter halos were identified using FoF in real space and galaxies were placed within halos using a halo model prescription. To populate dark matter halos with galaxies, two functions need to be specified. The first is the halo occupation distribution function (HOD), which is the probability that a halo of mass M hosts N galaxies, $P(N|M)$. The second is the spatial distribution of galaxies within halos. The first moment of the HOD function, the average number of galaxies as a function of halo mass M, is shown in Fig. 1 in [@Yan03] and in Fig. 12 of this paper (labeled as “B256”) for the HOD used in the mock galaxy catalogs here. This HOD is constrained at $z\sim 0$ with the 2dF luminosity function and luminosity dependent two-point correlation function of galaxies and at $z\sim 1$ with the DEEP2 $\xi(r)$ and the COMBO-17 luminosity functions. In §8 of this paper we compare clustering results for mock catalogs with two different halo models; for the bulk of the paper, however, we use lightcones with the HOD as published in [@Yan03]. Once the number of galaxies in each halo and their corresponding luminosities are known, the most luminous galaxy is assigned to the center of mass of the halo, and positions and velocities for the other galaxies are drawn randomly from those of the dark matter particles. No other radial or velocity bias is included in assigning galaxies to particles, so that galaxies trace the mass and velocity distributions of the dark matter particles in the halo. The spatial distribution of galaxies follow the dark matter density profile, which on average is an NFW profile. Although no velocity bias is included, the velocity dispersions of the galaxies are systematically smaller than those of the mass (see Fig.8 in Yan, White & Coil 2004), due to the fact that the most luminous galaxy is always assigned to the particle at the center of mass. Here we use a set of twelve independent catalogs which each have the same spatial extent as a single DEEP2 pointing ($\sim0.5$ by $0.67$ degrees).
The center of each DEEP2 group (needed for the clustering measures performed here) is measured as the median of the positions (in comoving x, y, and z) of the galaxies identified in that group. Errors on the positions of the recovered groups are estimated using the differences in the mock catalogs between the centers of real groups and recovered groups. As stated above, there is a factor of 1.4 more recovered groups in the mock catalogs than real groups for $\sigma\ge200$ . In the mock catalogs 74% of recovered groups have a real group within $r_p=1$ , while 48% have a real group within $r_p=0.2$ and 41% have a real group within $r_p=0.1$ . Here $r_p$ is the projected distance on the plane of the sky, which is the relevant distance for the projected clustering used in §5 and later in the paper; in §4 we use the redshift-space correlation function, but only measure it on scales $s>2$ . We therefore believe that on scales larger than 1 our results are robust to errors in the group positions, while on smaller scales, where we will measure the cross-correlation between group centers and galaxies, there is likely to be some degradation in the signal due to position errors; this is discussed more in §9. However, the mock catalogs have been treated in an identical manner as the data so that comparisons between the data and mock catalogs are unaffected.
Methods
=======
The two-point correlation function is defined as a measure of the excess probability above Poisson of finding an object in a volume element $dV$ at a separation $r$ from another randomly chosen object, $$dP = n [1+\xi(r)] dV,$$ where $n$ is the mean number density of the object in question [@Peebles80].
Measuring requires constructing a random catalog with the same selection criteria and observational effects as the data, to serve as an unclustered distribution with which to compare. For each data sample we create a random catalog with the same overall sky coverage and redshift distribution as the data. This is achieved by first applying the two-dimensional window function of our data in the plane of the sky to the random catalog. Our overall redshift success rate is $gtrsim$70% and is not entirely uniform across the survey; some slitmasks are observed under better conditions than others and therefore yield a slightly higher completeness. This spatially-varying redshift success rate is taken into account in the spatial window function which is applied to both the random catalog and the mock catalogs, such that regions of the sky with a higher completeness have a correspondingly higher number of random points or more objects in the mock galaxy catalogs. This ensures that there is no bias introduced when computing correlation statistics or comparing the data to the mock catalogs. We also mask the regions of the random and mock catalogs where the photometric data have saturated stars and CCD defects.
We then apply a selection function, $\phi(z)$, defined as the probability of observing a group as a function of redshift, so that the random catalog has the same overall redshift distribution as the data. The selection function for groups in the DEEP2 survey is determined by smoothing the observed redshift distribution of groups in the catalog used here, as shown in Fig. \[sf\]. The apparent overdensity at $z\sim0.85$ seen in this figure is due to structures in several pointings and does not significantly affect our results, which are averaged over all pointings and a wide redshift range. As the group catalog includes three separate fields, the data are combined when calculating $\phi(z)$, which reduces the effects of cosmic variance. Using a smoothed redshift distribution to estimate the selection function can cause a systematic bias, but we estimate that this bias is less than the errors on . In what follows, for both the data and the mock catalogs the redshift range over which we compute is limited to $0.7\leq z \leq1.0$, the redshift range over which the selection function varies by less than a factor of two. Each of the six pointings has between $\sim50-100$ groups, and each of the twelve independent mock catalogs contains $\sim100$ groups. We use $\sim3,000$ random points in each pointing to calculate the correlation function for groups and $\sim20,000$ random points to calculate the correlation function for galaxies.
The two-point correlation function is measured for both groups and galaxies using the @Landy93 estimator, $$\xi=\frac{1}{RR}\left[DD \left(\frac{n_R}{n_D}
\right)^2-2DR\left(\frac{n_R}{n_D} \right)+RR\right],$$ where $DD, DR$, and $RR$ are pair counts in a given separation range in the data-data, data-random, and random-random catalogs, and $n_D$ and $n_R$ are the mean number densities in the data and random catalogs. This estimator has been shown to perform as well as the Hamilton estimator [@Hamilton93] but is preferred as it is relatively insensitive to the size of the random catalog and handles edge corrections well [@Kerscher00]. As the magnitude limit of our survey results in a non-uniform selection function, a standard $J_3$-weighting scheme is applied [@Davis82den], which attempts to weight each volume element equally while minimizing the variance on large scales. As the redshift range is limited to $0.7\leq z \leq1.0$ for all analyses in this paper, this effect is not large ($\lesssim20$%).
Measurements of the cross-correlation between two samples (presented in §6) use a symmetrized version of Eqn. 2. Each data sample, with pair counts labeled $D_1$ and $D_2$, has an associated random catalog, with pair counts $R_1$ and $R_2$, with the same selection function as the data. After normalizing each data and random catalog by its number density, the cross-correlation is estimated using $$\xi=\frac{1}{R_1R_2}\left[D_1D_2-D_1R_2-D_2R_1+R_1R_2\right].$$
Redshift-space distortions due to peculiar velocities along the line-of-sight will introduce systematic effects to the estimate of . At small separations, random motions within a virialized overdensity cause an elongation along the line-of-sight (“fingers of God”), while on large scales, coherent infall of galaxies into forming structures causes an apparent contraction of structure along the line-of-sight (the “Kaiser effect”). What is actually measured then is , where $s$ is the redshift-space separation between a pair of galaxies. As we will show in the next section using mock group catalogs, for galaxy groups over the scales relevant here, the systematic effects of redshift-space distortions are of order 20%. Mock catalogs are used to correct for redshift-space distortions and infer from by multiplying the observed for DEEP2 groups by the ratio of to for groups in the mock catalogs.
However, redshift-space distortions are more significant when measuring $\xi$ for galaxies, as smaller scales are probed. We are able to measure $\xi$ on small scales for the galaxy sample both because of the larger sample size, which allows us to stably measure the clustering on small scales, and also because there is no exclusion radius, unlike with groups, which typically have a radius of $R\sim1$ . To uncover the real-space clustering properties of galaxies we measure $\xi$ in two dimensions, both perpendicular to and along the line of sight. Following [@Fisher94], ${\bf v_1}$ and ${\bf
v_2}$ are the redshift positions of a pair of galaxies, ${\bf s}$ is the redshift-space separation (${\bf v_1}-{\bf v_2}$), and ${\bf l}
=\frac{1}{2}$(${\bf v_1}+{\bf v_2})$ is the mean distance to the pair. The separation between the two galaxies across ($r_p$) and along ($\pi$) the line of sight are defined as $$\pi=\frac{{\bf s} \cdot {\bf l}}{{\bf |l|}},$$ $$r_p=\sqrt{{\bf s} \cdot {\bf s} - \pi^2}.$$ In applying the @Landy93 estimator to galaxies, pair counts are computed over a two-dimensional grid of separations to estimate .
To recover is projected along the $r_p$ axis. As redshift-space distortions affect only the line-of-sight component of , integrating over the $\pi$ direction leads to a statistic , which is independent of redshift-space distortions. Following [@Davis83], $$w_p(r_p)=2 \int_{0}^{\infty} d\pi \ \xi(r_p,\pi)=2 \int_{0}^{\infty}
dy \ \xi(r_p^2+y^2)^{1/2},
\label{eqn}$$ where $y$ is the real-space separation along the line of sight. If is modeled as a power-law, $\xi(r)=(r/r_0)^{-\gamma}$, then and $\gamma$ can be readily extracted from the projected correlation function, , using an analytic solution to Equation \[eqn\]: $$w_p(r_p)=r_p \left(\frac{r_0}{r_p}\right)^\gamma
\frac{\Gamma(\frac{1}{2})\Gamma(\frac{\gamma-1}{2})}{\Gamma(\frac{\gamma}{2})},
\label{powerlawwprp}$$ where $\Gamma$ is the gamma function. A power-law fit to will then recover and $\gamma$ for the real-space correlation function, .
Group Clustering Results
========================
This section presents results on the clustering of groups at $z\sim1$. The two-point correlation function for groups in the DEEP2 data is measured. The clustering properties of groups in our mock catalogs are analyzed, which allows us to quantify our systematic errors and to correct the observed clustering in the DEEP2 data for redshift-space distortions and the effects of the group-finder. The number density of groups in the DEEP2 data is then used to infer the minimum dark matter halo mass which hosts our groups; we then compare the clustering of halos with this mass to the clustering of our group sample.
Clustering of Groups in DEEP2 Data and Mock Catalogs
----------------------------------------------------
The left panel of Fig. \[mocks\] shows the observed two-point correlation function in redshift space, , for our $\sigma\ge200$ group sample, for both the full sample (solid line) with two or more galaxies in each group ($N\ge2$) and for a subsample with four or more galaxies in each group ($N\ge4$). Also shown is for the recovered group sample in the mock catalogs with $\sigma\ge200$ and $N\ge2$ (dotted line). We show the standard error across the six pointings; this empirical error therefore includes both cosmic variance and Poisson error. The $N\ge4$ data sample shows a stronger clustering amplitude than the $N\ge2$ data sample. This is likely due to the $N\ge4$ sample containing more massive groups, on average, than the $N\ge2$ sample. The $N\ge4$ sample is also less likely to have interlopers, which may also increase the observed clustering.
Power-law fits to over the range $s=3-20$ are given in Table 1. Below $s\sim3$ does not continue to rise as a single power-law in our data; this lack of pairs on scales $s<3$ is likely due to the finite physical extent of groups. To take into account the covariance between $s_0$ and $\gamma$, we perform a $\chi^2$ minimization and marginalize over each parameter separately. This procedure leads to errors which are roughly a factor of two larger than if we neglected this covariance.
The mock galaxy catalogs described in §2 are used to quantify systematic errors in our group clustering analysis. Effects that may be introduced by the group-finder are tested by comparing in redshift space for both real and recovered groups. The results are shown in Table 1. The recovered groups have a slightly higher clustering amplitude (5%) than the real groups. Redshift-space distortions are quantified by measuring in redshift space and in real space for real groups. Redshift-space distortions appear to enhance the clustering properties of groups by $\sim20$% on scales $s\sim2-15$ and decrease the clustering amplitude on smaller scales. The clustering scale length increases by $\sim20$% when measured in redshift space, while there is no change to the slope over the scales used here. The effects of our slitmask target selection algorithm on the clustering of groups are also tested using the mock catalogs. Our target selection code determines which galaxies would be observed on slitmasks; because spectra from neighboring galaxies can not overlap on the CCD, not all galaxies can be selected to be observed. In particular, in overdense regions on the plane of the sky the number of galaxies which can be observed decreases in a non-trivial way. By running our slitmask target selection code on the mock catalogs we can quantify the effect on the measured .
Under the assumption that corrections for the above effects should be proportional to the clustering strength, these results from the mock catalogs can now be used to correct the observed for groups in the DEEP2 data for target selection effects, redshift space distortions, and the group-finder, in order to estimate in real space for real groups. We can either apply corrections to the power-law fits themselves or to the data points as a function of scale. If we apply the corrections to $s_0$ and $\gamma$ as measured for the DEEP2 groups, the corrections would infer that $r_0=6.8\pm0.6$ and $\gamma=1.4 \pm0.2$ for the real-space correlation function of real groups in the DEEP2 data with $N\ge2$. If we explicitly correct the observed for the DEEP2 groups as a function of scale, using the ratio of / in the mock catalogs, the resulting has a power-law fit of $r_0=6.2 \pm0.4$ and $\gamma=1.5 \pm0.2$, within the 1$\sigma$ error of the inferred values. The corrected is shown as the solid line in the right panel of Fig. \[mocks\], for both the $N\ge2$ and $N\ge4$ samples. The clustering of groups in the DEEP2 data is therefore 1-3$\sigma$ lower than the clustering of real groups before target selection in the mock catalogs, where $r_0=7.4 \pm0.4$ and $\gamma=1.6
\pm0.2$, for $N\ge2$.
Minimum Group Mass Derived from Clustering Results
--------------------------------------------------
The minimum dark matter halo mass that our groups reside in can be inferred from the observed number density and clustering strength of our groups using either analytic formulations of the dark matter halo mass function or by comparing directly to simulations. Here we estimate a minimum dark matter halo mass from the observed number density using analytic theory and then compare the corresponding predicted clustering strength of those halos in both theory and simulations with the observed clustering strength of our groups.
The $\sigma\ge200$ group sample has an observed density of $n=4.5 \times 10^{-4} \ h^3 \ {\rm Mpc}^{-3}$, calculated from the observed number of groups between $z=0.75-0.85$, where the group selection function is the highest, divided by the comoving volume occupied by the groups. The mock catalogs are used to estimate the effects on the observed number density due to our slitmask target selection (which decreases the number of observed groups, as most groups have only two or three observed galaxies) and the false interloper rate due to the group-finder. Correcting for these effects, the actual comoving number density is estimated to be $n=6 \times 10^{-4} \ h^3 \
{\rm Mpc}^{-3}$. This corresponds to a mean inter-group spacing of $d=11.8$ . For comparison to the sample used by [@Gerke05], the number density of groups with $\sigma\ge350$ is $n=2.4 \times 10^{-4} \ h^3 \ {\rm Mpc}^{-3}$, after applying the above corrections, which corresponds to an intergroup spacing of $d=16$ . For a cosmology with $\Omega_b=0.05$, $\Lambda=0.7$, $\Omega_m=0.3$, $h=0.7$, $\sigma_8=0.9$, a comoving number density of $n=6
\times 10^{-4} h^3 {\rm Mpc}^{-3}$ results in $M_{min}=5.9 \times 10^{12} h^{-1} M_{\Sun}$ at $z=0.8$ and $M_{min}=5.5 \times 10^{12} h^{-1} M_{\Sun}$ at $z=1$ for a [@Sheth99] mass function. A comoving density of $n=2.4 \times 10^{-4} \ h^3 \ {\rm Mpc}^{-3}$, estimated for the $\sigma\ge350$ sample, corresponds to $M_{min}=1.2 \times 10^{13} h^{-1} M_{\Sun}$ at $z=0.8$ and $M_{min}=1.1 \times 10^{13} h^{-1} M_{\Sun}$ at $z=1$. These masses are only approximate as the group number and volume are both just estimates. The errors on the minimum dark matter halo masses inferred from the observed number densities are likely $\sim50$%, given cosmic variance errors and the uncertainties in the corrections made to the number densities from the mock catalogs.
We check for consistency between the observed and predicted clustering of these halos. [@Mo02] use the [@Sheth99] model to predict the evolution in the clustering of dark matter halos and find that halos of mass $M_{min}=5.5 \times 10^{12} h^{-1} M_{\Sun}$ will have a clustering amplitude of $\sigma_8=1.0$ at $z=1$. Here $\sigma_8$ is defined as the standard deviation of halo count fluctuations in a sphere of radius 8 ; it can be preferable to quoting a scale-length, $r_0$, as it removes the significant covariance with $\gamma$. $\sigma_8$ can be calculated from a power-law fit to using the formula, $$(\sigma_8)^2 \equiv J_2(\gamma) \left(\frac{r_0}{8 \ h^{-1} {\rm Mpc}}\right)^\gamma,
\label{sig8eqn}$$ where $$J_2(\gamma) = \frac{72}{(3-\gamma)(4-\gamma)(6-\gamma) 2^\gamma}$$ [@Peebles80]. Using the power-law fits to for groups in the DEEP2 data, we find $\sigma_8=1.0$, in agreement with the predicted value of [@Mo02].
[@Kravtsov04] find using dark matter simulations with the same concordance cosmology that a number density of $n=6 \times 10^{-4} \ h^3 \ {\rm Mpc}^{-3}$ at $z=1$ corresponds to a minimum mass of $M_{min}=5.9 \times 10^{12}
h^{-1} M_{\Sun}$. This value is comparable to the [@Sheth99] value quoted above. They predict a clustering amplitude of $r_0=5.2$ and $\gamma=2.16$ for these halos, which corresponds to $\sigma_8=1.05$, in good agreement with our observed value of $\sigma_8$ for the DEEP2 groups.
We verify from the mock catalogs that the actual minimum dark matter halo masses for these group samples are similar to those estimated above. For the group sample defined as having an estimated $\sigma\ge200$ , the mass distribution has a rough minimum dark matter halo mass of $M_{min}=2-3 \times 10^{12} h^{-1} M_{\Sun}$ and the $\sigma\ge350$ sample has a rough minimum dark matter halo mass of $M_{min}=4-5 \times 10^{12} h^{-1} M_{\Sun}$. The mass distributions for the group samples in the mock catalogs do not have a very clearly defined lower-mass cutoff, however, and these values are roughly half of the values quoted above. Even though the galaxies in the mock catalogs are randomly drawn from the velocity distribution of individual dark matter particles, the fact that only a small number of galaxies are observed in a single group leads to significant scatter between the estimated $\sigma$ from the observed galaxies and the actual $\sigma$ and therefore the mass of the underlying dark matter particles; this scatter likely accounts for the factor of two discrepancy between the halos in the mock catalogs and the predictions of [@Sheth99].
Clustering of Galaxies in Different Environments
================================================
In this section the clustering properties for galaxies in groups relative to the full galaxy population and to field galaxies are investigated. Unlike for the group sample, which has larger Poisson errors and can only be measured on scales $r\ge3$ , where redshift-space distortions are small, here we are interested in measuring clustering properties of galaxies on small scales where redshift-space distortions are much more significant. Instead of inferring in real space from measurements of in redshift space, we measure the projected two-point correlation function, , from which can be more directly inferred. Color information additionally allows us to divide the sample of group galaxies into red and blue populations and measure for each. We test the effects of our slitmask target selection algorithm and group-finder on these results using mock catalogs. Finally, we are able to empirically separate the observed for galaxies in groups into a ‘one-halo’ and a ‘two-halo’ term, by keeping pair counts where both galaxies are in the same or in different groups.
Clustering of Full Galaxy Sample and Galaxies in Groups in DEEP2 data
---------------------------------------------------------------------
The full galaxy sample here refers to all DEEP2 galaxies between $0.7\leq z
\leq1.0$ in the same six pointings for which we have group catalogs. The group and field galaxy samples are identified using the $\sigma\ge200$ group catalog, and these samples combined make up the full galaxy sample. Fig. \[coneplot\] shows the spatial distribution of galaxies in three DEEP2 pointings, with different symbols for group (open triangles) and field (crosses) galaxies. In the DEEP2 data 39 +/-4% of all galaxies are identified as belonging in recovered groups which have $\sigma\ge 200$ in the redshift range $0.7\leq z \leq1.0$, where the error quoted is the standard error across the six pointings. This rate is artificially increased by false interlopers, but it is also decreased by our slitmask target selection by roughly the same amount, as estimated using mock galaxy catalogs. In the mock catalogs, 27% of all galaxies are identified as belonging in recovered groups after target selection (24% are in real groups before target selection), significantly less than in the DEEP2 data. This difference is most likely due to the free parameters in the group-finding algorithm having been tuned to reproduce the observed $n(\sigma,z)$ for $\sigma\ge350$ , not the $\sigma\ge200$ cutoff in the present sample (see Fig. 6 in [@Gerke05] for details). The clustering results shown here do not depend on the absolute number of galaxies in each of the samples (all, group, and field galaxies).
Fig. \[mockfield\] presents for the full galaxy sample (solid lines), galaxies in groups (dashed lines) and for field galaxies (dotted lines), in both the DEEP2 data (top) and in the mock catalogs (bottom). The top left panel compares as measured in the observed DEEP2 data (thin lines with no error bars) with after correcting for effects due to target selection and the group-finder (thick lines with error bars). To make this correction, we have used the ratio of as a function of scale in the mock catalogs for group galaxies, field galaxies, and all galaxies separately, identified using real groups before target selection (thick lines with error bars in the bottom right panel), to for galaxies identified using recovered groups after target selection (thick lines with error bars in the bottom left panel). Power-law fits to the corrected points are shown in the top right panel of Fig. \[mockfield\] and are listed in table \[r0table\], along with fits to the observed points. Field galaxies are well fit by a power-law on scales $r_p=1-20$ . On smaller scales the correlation function is negative, as field galaxies are not found within $\sim0.6$ of each other. Pairs of galaxies within that distance are likely to be part of a group.
We will present updated results for for galaxies in the DEEP2 dataset as a function of galaxy color, luminosity, redshift, etc. in a future paper (Coil et al. 2005, in preparation). Here we focus on the difference between the clustering of galaxies in groups relative to the full galaxy sample (throughout this paper we use the terms “all galaxies” and “the full galaxy sample” interchangeably; they include both field and group galaxies). The full galaxy sample used here includes a total of 9787 galaxies in the redshift range $0.7\leq z \leq1.0$ in the same six pointings used for the group analysis. This represents a great advance over the sample used in [@Coil03xisp], which contained 2219 galaxies in one pointing over the redshift range $0.7\leq z\leq1.35$. The fits for $r_0$ and $\gamma$ here agree with our earlier results, but the errors, both Poisson and cosmic variance, are much smaller in the current sample. For example, we can now address whether the significant rise in slope of on small scales predicted by @Kravtsov04 for galaxies at $z=1$ based upon their simulations is present in the DEEP2 data. They find that for a galaxy sample with a comoving number density of $n=1.5 \times 10^{-2} \ h^3 \ {\rm Mpc}^{-3}$ (similar to the DEEP2 sample at $z\sim0.8$; see [@LLin04]) that the slope of changes from $\gamma\sim1.65$ when measured on scales of $r=0.3-10$ , where $r_0$ is measured to be $\sim4$ , to $\gamma\sim1.9$ over scales $r\sim0.1-10$ , where $r_0\sim3.5$ . Fitting our corrected for the full galaxy sample over the same range in $r_p$ results in $r_p=0.3-10$ , $r_0=3.64
\pm0.07$ and $\gamma=1.73 \pm0.04$, while for scales $r_p=0.01-10$ , $r_0=3.64 \pm0.07$ and $\gamma=1.73 \pm0.03$, with no change at all in either amplitude or slope. Therefore no evidence is found for a rise in the slope on small scales as predicted by [@Kravtsov04]. We also do not find a significant difference in the slope on scales $r_p<1$ compared to scales $r_p>1$ , as is seen for the galaxies in groups. Fitting for a power-law on scales $r_p=0.05-1$ , $r_0=3.52 \pm0.16$ and $\gamma=1.78 \pm0.06$, while on scales $r_p=1-20$ , $r_0=3.70 \pm0.10$ and $\gamma=1.77 \pm0.06$. The full galaxy sample therefore appears to be consistent with a single power-law slope over the range $r_p=0.1-20$ . We note that in the mock catalogs, the slope measured for is consistent with the data, though the amplitude of $r_0$ is $\sim7$% higher. As stated before, the mock catalogs were designed to match the previously published measurements of for the full DEEP2 galaxy sample.
The correlation function for galaxies in groups is relatively well-fit by a power-law over all scales; a broken power-law fit with a break at $r_p=1$ results in a steeper slope on small scales, with low significance. The slope for a single power-law is $\gamma=2.12 \pm0.06$ for scales $r_p=0.05-20$ , while it increases to $\gamma=2.16 \pm0.11$ for scales $r_p=0.05-1$ and decreases to $\gamma=2.02 \pm0.15$ for scales $r_p=1-20$ . Note the significantly different shape of for group galaxies in the mock catalogs, which exhibit a strong break at $r_p\sim1$ . This sharp rise on small scales is not seen in the DEEP2 data. As we will show in the next subsection, this difference between the mock catalogs and the data is not due to any systematic effect from our slitmask algorithm or in our group identification, as the general shape of the correlation function in the mock catalogs is not changed by these. This difference in the clustering of galaxies in groups between the mock catalogs and the data is the first indication that the mock catalogs, which were constructed to match the $z\sim1$ luminosity function and clustering of all galaxies in the DEEP2 data, do not reproduce additional properties of the data. It appears that the spatial distribution of galaxies in groups is less concentrated in the data than in the mock catalogs. We discuss the implications of this in §9.
We also investigate for group galaxies as a function of color. Fig. \[grpgalcolor\] shows for galaxies in the DEEP2 data which are identified as belonging to groups and have red or blue colors, defined by the minimum in the color bi-modality in restframe $(U-B)_0$, at $(U-B)_0=1.05$. In the group galaxy sample, 20% of group galaxies are red, while for the full galaxy sample 15% of galaxies are red. Corrections for target selection and our group-finder are made using the mock catalogs as above. Red galaxies in groups have a steeper slope in and a higher correlation length than blue galaxies; power-law fits result in $r_0=4.77 \pm0.20$ and $\gamma=2.15 \pm0.05$ for blue group galaxies and $r_0=5.81
\pm0.45$ and $\gamma=2.27 \pm0.11$ for red group galaxies. The steeper slope for the red galaxy sample implies that red galaxies are more centrally concentrated in their parent dark matter halos; we investigate this more directly in §6. Colors are not currently included in the mock catalogs and so we can not present this measurement for the mock catalogs.
Effects of Slitmask Target Selection and Group-finder
-----------------------------------------------------
As for the clustering of groups in the DEEP2 data, mock catalogs are used to quantify systematic errors in our measurements of for each galaxy sample, where real groups are used to define galaxies in groups or in the field. The bottom left panel of Fig. \[mockfield\] shows measured for all galaxies, group galaxies, and field galaxies in the mock catalogs, before (thin lines without error bars) and after (thick lines with error bars) target selection. Separate power-law least-squares fits to in the mock catalogs are performed for the full galaxy sample and for galaxies in groups and in the field; the results before and after target selection are shown in Table \[r0table\]. Field galaxies are only affected by the target selection on the very smallest scales, $r_p\lesssim 0.2$ , and there is no significant change to the correlation function of field galaxies on the scales over which we measure a power-law, $r_p=1-20$ . For the full galaxy sample, our target selection algorithm causes to be slightly underestimated on small scales ($r_p\lesssim 2$ ) due to our inability to target all close neighbors. Before target selection is applied, a power-law fits well for the full galaxy sample on scales $r_p=1-20$ , with a steeper slope on small scales, $r_p=0.1-1$ . This change in slope on small scales in the mock catalogs is more significant (2.5$\sigma$ vs 1.3$\sigma$) for the sample before target selection than after, due to the smaller error bars for the larger sample. We note that the scale at which the slope changes is larger than predicted by [@Kravtsov04] in their simulations, and that this difference in slope is not seen in the DEEP2 data, as discussed above.
For galaxies in groups, however, the effects of target selection are more complicated; it [*increases*]{} their observed clustering on all scales. In further tests, we find that target selection does not affect the measured clustering of galaxies in mock catalogs known to belong in real groups, so long as the group membership is identified before target selection. However, galaxies can be identified as belonging to a group after target selection only if two or more observed galaxies are in that group. We can identify only half of the groups after target selection that were detectable before target selection, and the groups which are preferentially lost are those with a low richness. This causes for galaxies in groups to be greater when groups are identified after target selection, as only galaxies in the richest, and presumably highest mass and most clustered, groups will be included in the observed sample.
The mock catalogs are also used to investigate the effect of our group-finder on the measured clustering of galaxies in groups, as we want to be sure that our group-finding algorithm is not imposing (even indirectly) a preferred scale for groups, which could potentially artificially cause a change in slope at small scales in for group galaxies. This is tested by comparing the clustering of galaxies in real and recovered groups in the mock catalogs, where both samples have had the DEEP2 target selection algorithm applied. The results are shown in the bottom right panel of Fig. \[mockfield\], where both real and recovered groups are seen to have an inflection in such that the slope rises on small scales ($r_p<1$ ). the group-finder, then, is not imposing this scale on the clustering results. We also find that the group-finder effectively cancels much of the effect of our slitmask target selection, in that for galaxies in recovered groups after target selection is similar to for galaxies in real groups before target selection. The overall correction applied to the observed for group galaxies in the DEEP2 data is therefore small.
Throughout the paper we compare for the DEEP2 data with mock catalogs by applying corrections to the observed correlation function to account for effects of our slitmask target selection and group-finder and compare to results in the mock catalogs before target selection for real groups. None of our results change if instead we compare results for the observed in the data to results in the mock catalogs before target selection for real groups.
One- and Two-Halo Terms of the Group Galaxy Correlation Function
----------------------------------------------------------------
As it is known which group each of the group galaxies belongs to, we can empirically measure the contribution to from pairs of galaxies in the same or in different groups. This is akin to determining the ‘one-halo’ and ‘two-halo’ terms of in the halo model language, where each group is identified with a single dark matter halo. The total correlation function is then the sum of these two terms: $$\xi(r)=[1+\xi_{1h}(r)]+\xi_{2h}(r).$$ We calculate the ‘one-halo’ and ‘two-halo’ correlation functions using the following estimators: $$\xi_1=\frac{1}{RR}\left[DD_1 \left(\frac{n_R}{n_D}
\right)^2-2DR\left(\frac{n_R}{n_D} \right)+RR\right]$$ $$\xi_2=\frac{1}{RR}\left[DD_2 \left(\frac{n_R}{n_D}
\right)^2-2DR\left(\frac{n_R}{n_D} \right)+RR\right],$$ where $DD_1$ and $DD_2$ are pair counts of galaxies within the same group and in different groups, respectively. We then sum these along the line of sight (in the $\pi$ direction, to $\pi_{max}$) to obtain $w_{p,1h}$ and $w_{p,2h}$. The projected correlation functions sum such that $$w_p(r_p)=[\pi_{max}+w_{p,1h}(r_p)]+w_{p,2h}(r_p).$$ Fig. \[mockseparate\] shows $w_{p,1h}(r_p)$ and $w_{p,2h}(r_p)$ for galaxies in groups in both the DEEP2 data (left) and for real groups in the mock catalogs (right). The data have been corrected for effects due to our target slitmask and group-finder algorithms.
In the DEEP2 data the scale at which the ‘one-halo’ and ‘two-halo’ terms intersect is $r_p=1.0$ ; the scale in the mock catalogs is $r_p=0.5$ . Exactly where the break occurs between the ‘one-halo’ and ‘two-halo’ terms will presumably depend on the type of groups we are probing; larger groups may have this break at a larger radius. The change in slope seen in for group galaxies in the mock catalogs is easily understood as the scale at which the ‘one-halo’ and ‘two-halo’ terms equally contribute to . However, the ‘one-halo’ term has a very different shape in the DEEP2 data than in the mock catalog, such that galaxies in the data which belong to the same group are not as clustered on small scales as in the mock catalogs. This is likely due the mock catalogs having an incorrect spatial distribution of galaxies within dark matter halos on small scales; we discuss this further in §9.
[@Yang05a] measure for group galaxies in 2dF Galaxy Redshift Survey data at $z\sim0$ and find that it is not well fit by a single power-law; the ‘one-halo’ term is enhanced relative to the ‘two-halo’ term and there is a rise in on scales $r_p\sim1-2$ . The strength of the rise depends on the abundance and luminosities of the groups; galaxies in more luminous (and presumably more massive) groups have a larger ‘one-halo’ term and a stronger rise in the slope of on small scales. It is only for the full galaxy population (including field galaxies) that they find a single power-law fit to . For our group sample at $z\sim1$, the shape for for galaxies in groups is similar to what is seen by [@Yang05a] for groups with a comparable number density. [@Yang05a] find a small rise in the slope of on scales below $r_p=1$ but do not quantify this. Our results at $z\sim1$ appear to be similar to their findings at $z\sim0$, though with larger errors due to our smaller sample size.
Group-Galaxy Cross-Correlation Function
=======================================
In this section we present the cross-correlation function between group centers and the full galaxy sample, which is sensitive to the radial profile of galaxies in and around groups. As with the group and galaxy correlation functions, to avoid redshift-space distortions we measure the projected cross-correlation, . As discussed in §2, errors in the positions of group centers will have some effect on scales $r_p<1$ ; for this reason we do not plot results for $r_p<0.3$ . However, comparisons between data and mock catalogs are unaffected, as the mock catalogs have been treated in an identical manner as the data.
Fig. \[crossdata\] shows the projected cross-correlation between group centers and the full galaxy sample in the DEEP2 data (left) and the mock catalogs (right). The left panel shows the observed as a dashed line and the corrected as a solid line, where corrections for target selection and the group-finding algorithm as a function of scale are made using the ratio of in the mock catalogs between real group centers and all galaxies before (solid line, right panel) target selection and between recovered group centers and all galaxies after (dashed line, right panel) target selection. The dotted line in the right panel shows the cross-correlation between real groups and all galaxies after target selection. The target selection algorithm has the effect of increasing the cross-correlation on scales $r_p>0.4$ , while decreasing the amplitude on smaller scales. The small-scale decrease is due to our inability to target galaxies which are in close projection on the plane of the sky; this causes us to undersample close neighbors. On large scales, the effect is due to the slitmask target algorithm affecting [*which*]{} groups we identify; after target selection we lose many of the pairs of galaxies which were identified as groups before such that we preferentially identify the groups with more observed members, which are presumably more massive and therefore more clustered. The effect of the group-finding algorithm is to increase the cross-correlation on scales $r_p>0.4$ , where the group-finder has by definition targetted overdensities in the galaxy distribution.
Comparing the solid lines in the two panels of Fig. \[crossdata\], which shows for real groups before target selection to the corrected data, the overall shape of the cross-correlation agrees reasonably well, though the amplitude is somewhat higher in the mock catalogs on both small and large scales. This is consistent with what is seen for the correlation function of galaxies in groups shown in Fig. \[mockfield\], which are more strongly clustered in the mock catalog than in the DEEP2 data.
We also investigate the dependence of the radial distribution of galaxies in groups on galaxy color. Fig. \[redcross\] shows the projected cross-correlation function between either red or blue galaxies and group centers, where again the galaxy sample has been split at the bi-modality in the restframe $(U-B)_0$ color distribution at $(U-B)_0=1.05$. Within groups, on small scales, $r_p\lesssim0.5$ , red galaxies are much more strongly clustered than blue galaxies, i.e., red galaxies are preferentially found near the centers of groups. In the DEEP2 data 20% of group galaxies in our sample are red, while 13% of field galaxies and 15% of the full galaxy sample (used in this cross-correlation) are red.
Similar trends are seen at $z\sim0$ by [@Yang05b], who measure the cross-correlation between group centers and all galaxies in 2dF and SDSS data. They also find a difference between the radial distribution of red and blue galaxies, though it is only apparent for groups with masses $M\gtrsim 10^{13} h^{-1} M_\sun$, and the differences are smaller than those found here at $z\sim1$.
Relative Bias Between Groups and Galaxies
=========================================
Measuring the clustering properties of groups, all galaxies, and galaxies in groups in the DEEP2 data allows us to measure the relative bias between galaxies in groups and all galaxies and between groups and galaxies. Fig. \[grpgalbias\] plots the relative bias of group galaxies to the full galaxy sample, which we define as the square root of for group galaxies (dashed lines in Fig. 6) divided by for the full galaxy sample (solid lines in Fig. 6), as a function of scale for $r_p=0.1-20$ in both the DEEP2 data (top left panel) and the the mock catalogs (bottom left panel), after correcting for target selection and the group-finder. The bias seen between group galaxies and all galaxies is not surprising, as group galaxies reside in more massive dark matter halos. There is a clear scale-dependence to the relative bias between group galaxies and all galaxies in the DEEP2 data, which falls from $b_{rel}\sim2.5 \pm0.3$ at $r_p=0.1$ to $b_{rel}\sim1 \pm0.5$ at $r_p=10$ . The mock catalogs have a much higher relative bias on small scales ($r_p\lesssim1$ ) which does not match the bias seen in the data. This reflects the strong rise in slope of the correlation function of galaxies in groups seen on small scales in the mock catalogs.
The ratio of the group center-full galaxy sample cross-correlation function (Fig. 9) to the galaxy correlation function (solid lines in Fig. 6) provides a measure of the relative bias of groups to galaxies, which is shown on the right side of Fig. \[grpgalbias\]. We have corrected for slitmask target selection and the group-finder. There is some scale-dependence to the relative bias between groups and galaxies in the DEEP2 data (top right panel) and the weighted mean relative bias is $b_{rel}=1.17
\pm0.04$ over scales $r_p=0.5-15$ . The mock catalogs have a mean value of $b_{rel}=1.23 \pm0.02$ for $r_p=0.5-15$ , in reasonable agreement with the data, though the mock catalogs again show a higher bias on small scales, below $r_p\sim1$ , and the agreement with the data is better on scales $r_p>1$ . These measures of the relative biases of groups to galaxies and galaxies in groups to all galaxies at $z\sim1$ are further constraints which simulations and galaxy evolution models must match, in addition to measures of for all galaxies and for groups. We discuss the implications of these differences between the data and mock catalogs in the next two sections.
Effect of Varying the Halo Model Parameters
===========================================
The differences seen between the clustering of group galaxies on small scales in the DEEP2 data and the mock catalogs could be due to the mock catalogs having the wrong spatial distribution for galaxies within their parent dark matter halos and/or the wrong HOD, which specifies the probability that a dark matter halo of mass M hosts N galaxies, $P(N|M)$. To illustrate how much these differences may be due to the HOD used to create the mock catalogs, we investigate the clustering of group galaxies in two mock catalogs with similar number densities and different HODs. In addition to the mock catalog used throughout this paper (labeled as “B256”), we also analyze a mock catalog in which a different HOD was applied to the same dark matter simulation; this model is labeled as “C256” and was chosen as one of the most discrepant HOD models that has an observed for the full galaxy sample that, by design, matches the results for the DEEP2 data published in [@Coil03xisp]. The HODs for galaxies with $L>L*$ for these two models are shown in the upper left of Fig. \[grpgalhod\], where model B256 is seen to have a lower minimum halo mass hosting a single galaxy, and a steeper slope on larger mass scales ($\sim0.5$ compared to $\sim0.26$ for the C256 model), which results in having a greater fraction of galaxies residing in massive halos. The curves for galaxies with lower luminosity thresholds have a similar shape and higher amplitude than what is shown here (see Fig. 1 in [@Yan03]). Mock catalogs made with the C256 model have 35% of galaxies in recovered groups after target selection, similar to what is found for the DEEP2 data (39%), and higher than the value found in the B256 model (27%), even though the C256 model has relatively fewer galaxies in more massive halos. This is due to the parameters of the group-finding algorithm having been tuned to match the observed $n(\sigma,z)$ of the DEEP2 data for $\sigma\ge350$ ; we have not re-tuned the group-finder to the C256 mock catalogs or our $\sigma\ge200$ cutoff. Both the B256 and C256 mock catalogs have the same number density for the full galaxy sample. The clustering measures shown here have all been corrected for slitmask target effects and the group-finder. Fig. \[grpgalhod\] shows the correlation function for all galaxies (top right panel) and for group galaxies (bottom left panel) in each of the two halo model mock catalogs. The for the full galaxy sample is very similar in the two catalogs; the only differences are on scales less than $r_p\sim0.5$ , where the C256 model has a slightly higher clustering amplitude. For for group galaxies, the overall shape is similar for the two models but the amplitude in model B256 is higher at all scales, as this model has more galaxies in massive halos, such that the group galaxies will be more clustered. These figures show that the amplitude of the group galaxy correlation function adds an additional constraint on the HOD, which is not gained from the correlation function of all galaxies alone. It also shows that the general shape of the group galaxy correlation function, and in particular, the rise in slope on small scales, is [*not*]{} sensitive to the parameters of the HOD used.
The bottom right panel of Fig. \[grpgalhod\] presents the cross-correlation function of group centers and all galaxies in both mock catalogs. Here there is a difference in the shape of the cross-correlation on small scales for the different HODs. The C256 model has a lower amplitude than the B256 model over almost all scales but shows a distinct rise on the smallest scales, $r_p\lesssim0.5$ , which is not seen in the B256 model. Indeed, both the correlation function for all galaxies and the group-galaxy cross-correlation function in the C256 model show a rise on small scales that is not seen in the B256 model; this results from the C256 model having preferentially more galaxies in smaller mass halos which dominate the pair counts at small separations.
Results from the DEEP2 data are also compared to the different halo model mock catalogs in Fig. \[grpgalhod\]. By design, for all galaxies matches both mock catalogs well, though the data do not show the rise on the smallest scales that is seen for the C256 model. The shape of the correlation function for group galaxies in the DEEP2 data does not match either of the halo model mock catalogs; the data show a significantly shallower slope on small scales. The amplitude of the cross-correlation function agrees better with the B256 model than the C256 model, and the shape of the cross-correlation disagrees with the C256 model on small scales. The significant difference in for group galaxies on small scales is presumably due to a difference in the spatial distribution of galaxies in groups in the data and the mock catalogs, as it does not appear to be reconcilable by altering the HOD. This implies that the spatial distribution of galaxies in groups in the mock catalogs is incorrect. This will be discussed further in the next section.
We note that if the C256 mock catalogs had been used to correct the observed for the full DEEP2 galaxy sample and galaxies in groups as presented in Fig. \[mockfield\] for slitmask target effects, none of our conclusions in the paper would change, as the relative differences before and after target selection are similar in the two mock catalogs. The differences for all DEEP2 galaxies if using the C256 mock catalogs to correct for target selection effects are negligable, well within the $1 \sigma$ errors quoted in Table \[r0table\]. The differences for group galaxies are within the $2 \sigma$ errors, with both the corrected $r_0$ and $\gamma$ being lower ($r_0=4.72 \pm0.23$ and $\gamma=2.05 \pm0.06$), and there is still no significant difference in the slope of on small and large scales in the DEEP2 data.
Discussion and Conclusions
==========================
Groups bridge the gap between galaxies and clusters in both mass and scale, and are also the likely locations of galaxy mergers. Measurements of the clustering of groups can constrain cosmological parameters, and measurements of the clustering of galaxies in groups can constrain both halo model parameters and the spatial profile of galaxies in their parent dark matter halos. Here we present the first results on the clustering of groups and galaxies in groups at $z\sim1$. We measure four types of correlation function statistics in the DEEP2 dataset: 1) the group correlation function, 2) the galaxy correlation for the full galaxy sample, 3) the galaxy correlation function for galaxies in groups, and 4) the group-galaxy cross-correlation function. The first clustering measure probes the dark matter halo-halo correlation function on mass scales of galaxy groups, which is well-understood from dark matter simulations alone. The clustering of groups in the DEEP2 data at $z\sim1$ matches predictions for a cosmology and is used to estimate the typical dark matter masses of the halos the groups studied reside in.
The second clustering measure, the galaxy correlation function for the full DEEP2 galaxy sample, is an update on results using early DEEP2 data presented in [@Coil03xisp]. Here we present this measurement for the full galaxy sample using a much larger dataset, with over four times as many galaxies covering three fields in the sky; the statistical error on $r_0$ is now 2%. We show that for the full galaxy sample provides constraints on the halo occupation distribution (HOD), the number of galaxies that reside in a dark matter halo of a given mass), though not in an unique way. There is some leeway in how halos can be populated which results in a correlation function that matches our measurements, as shown in the upper right panel of Fig. 12.
The third clustering measure, the correlation function of galaxies in groups, is similar to the second but is restricted to galaxies in more massive halos, as measuring the clustering of galaxies in groups is sensitive to a higher halo mass range than for the full galaxy sample. The contribution from the ‘one-halo’ term is necessarily higher for galaxies in groups as these galaxies are identified as belonging in halos with several other galaxies. This provides further constraints on the halo model parameters than those from the measurement of for all galaxies alone. We also find that red galaxies in groups have a steeper slope and higher clustering amplitude than blue galaxies in groups.
The fourth clustering measure, group-galaxy cross-correlation function, reflects the spatial distribution of galaxies within dark matter halos above a given mass, and depends as well upon the parameters of the halo model. We find using the group center-galaxy cross-correlation function that red galaxies are found preferentially in the centers of groups compared to blue galaxies, which has also been seen locally [@Collister05; @Yang05b]. We find that this trend is in place at $z\sim1$.
All four of these measurements are compared to mock catalogs constructed from N-body simulations and depend differently on, and can therefore simultaneously constrain, both parameters of the halo model and the spatial profile of galaxies within halos. Comparing these clustering measurements in the DEEP2 data with the mock catalogs of [@Yan03], three of the four measures agree fairly well with the simulations, with the exception of the correlation function for galaxies in groups. The clustering amplitude for the full galaxy sample roughly matches the mock catalogs; this is by design: the catalogs were constructed with an HOD that is consistent with earlier DEEP2 clustering results for all galaxies. The HOD used is not uniquely determined however; the observed for the full DEEP2 galaxy sample can be matched with substantially different HODs (two samples are shown in the upper left panel of Fig. 12). We leave an improved HOD reconstruction from for the full galaxy sample for a future paper, where we will study the clustering as a function of galaxy properties such as luminosity, color, redshift, etc., in volume-limited samples; here we focus on comparing the clustering of galaxies in groups to all galaxies and the different constraints they provide on the HOD. We do note that for all galaxies is fit by a single power-law on scales $r_p=0.05-20$ , with $r_0=3.63 \pm0.07$ and $\gamma=1.74 \pm0.03$.
While the mock catalogs have similar projected clustering for the full galaxy population to the DEEP2 data by design, there is a strong discrepancy in the clustering of galaxies in groups. The DEEP2 data do not show a significant rise in the slope of on small scales, for either group galaxies or the full galaxy sample (upper panels of Fig. 6). In contrast, our mock catalogs have a very strong rise on small scales for for group galaxies, though not for the full galaxy sample (bottom panels of Fig. 6). To test whether this discrepancy can be accounted for by the halo model parameters used, we analyze mock catalogs constructed with a different HOD but similar clustering for the full galaxy sample (Fig. 12). We find that there is still a rise in slope for the clustering of galaxies in groups in the second mock catalogs (model C256) which is not seen in the data. This result is unaffected by our definition of the group center.
The slope of for group galaxies on small scales should depend quite sensitively on the spatial distribution of galaxies within dark matter halos. We therefore conclude that there is a difference in the spatial distribution of galaxies within their parent dark matter halos in the DEEP2 data and our mock catalogs. The mock catalogs assume no spatial bias, except for the assumption of a central galaxy; the most luminous galaxy in a halo is placed at the center of the halo, while all subsequent galaxies are assigned to random dark matter particles, following an NFW profile. Assuming that the brightest galaxy occupies the very center of the halo is likely not to be correct, as groups at $z=1$ are not expected to have a large, dominant, bright galaxy in their centers. This assumption will result in a higher correlation function on small scales.
In our mock catalogs the satellite galaxies are assumed to follow the same NFW profile as the dark matter particles; this appears to not be the correct spatial profile for the galaxy population at $z=1$. There is evidence in both simulations and data at $z\sim0$ that satellite galaxies do not follow the same spatial profile as the dark matter particles. Simulations have found that subhalos have a shallower spatial profile than the dark matter particles at $z\sim0$ [e.g., @Gao04a; @Diemand04; @Nagai05], though exactly how galaxies are related to subhalos is still not entirely known. Observationally, several authors have measured the spatial profiles of galaxies in groups in data at $z\sim0$. Using the Two Micron All Sky Survey, [@Lin04] stack groups and clusters to measure the radial mass-to-light profile and find that galaxies are less concentrated in the centers of groups and clusters than the dark matter. [@Collister05] use the 2dF 2PIGG group catalog to directly measure the radial profile of galaxies within groups and find that galaxies are less centrally concentrated than what is seen for dark matter particles in simulations. Similar results are found by [@Hansen05] for clusters in SDSS, and by [@Diaz05] and [@Yang05b] for groups in SDSS and 2dF.
The cross-clustering between groups and galaxies matches the mock catalogs well on scales $r_p>0.5$ ; on smaller scales the mock catalogs have a slightly steeper slope. The cross-correlation between groups and galaxies is linearly proportional to the radial distribution of galaxies in groups, while the correlation of group galaxies is proportional to the second power of the radial distribution; this may be why the shape agreement between the data and the mock catalogs is better for the cross-correlation than for the correlation of group galaxies. The group-galaxy cross-correlation function can also be affected by uncertainties in the location of the center of each group which can dilute the signal on small scales, unlike for the correlation function of galaxies in groups. [@Yang05b] find that the cross-correlation between group centers and galaxies at $z=0.1$ in 2dF and SDSS data is lower on scales $r_p<0.1$ than in their mock catalogs, which do not have a spatial bias with respect to the dark matter distribution. They create a series of mock catalogs with NFW profiles for the galaxies with lower concentration parameters, $c$, than in the dark matter and find that catalogs with concentration values of about one-third the value for the dark matter halos match their data well. These mock catalogs show the same trend which is required here, namely a lower cross-correlation on small scales of $r_p<0.1$ .
We show here that the clustering properties of galaxies in groups can be used to break degeneracies among different HODs that can not be distinguished by the clustering of all galaxies alone. We find that galaxies in the DEEP2 data do not have the same spatial profile as in our mock catalogs, which assumes a central galaxy surrounded by satellite galaxies following an NFW profile. Using the clustering statistics presented in this paper will allow us to now construct more realistic mock catalogs for the DEEP2 survey that have a better constrained HOD and radial profile for galaxies within dark matter halos.
We would like to thank the anonymous referee for helpful comments and Zheng Zheng for useful discussions. This project was supported by the NSF grant AST-0071048. J.A.N. acknowledges support by NASA through Hubble Fellowship grant HST-HF-01165.01-A awarded by the Space Telescope Science Institute, which is operated by AURA Inc. under NASA contract NAS 5-26555. C.-P. Ma is supported in part by NASA grant NAG5-12173 and NSF grant AST-0407351. S.M.F. would like to acknowledge the support of a Visiting Miller Professorship at UC Berkeley. The DEIMOS spectrograph was funded by a grant from CARA (Keck Observatory), an NSF Facilities and Infrastructure grant (AST92-2540), the Center for Particle Astrophysics and by gifts from Sun Microsystems and the Quantum Corporation. The DEEP2 Redshift Survey has been made possible through the dedicated efforts of the DEIMOS staff at UC Santa Cruz who built the instrument and the Keck Observatory staff who have supported it on the telescope. The data presented herein were obtained at the W.M. Keck Observatory, which 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 DEEP2 team and Keck Observatory acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community and appreciate the opportunity to conduct observations from this mountain.
[lrcccc]{} & & & & &\
$N\ge2$ & 460 &$7.3 \pm0.6$ & $1.4 \pm0.2$ &$6.2 \pm0.4$ & $1.5 \pm0.2$\
$N\ge4$ & 204 &$9.5 \pm0.9$ & $1.6 \pm0.3$ &$8.3 \pm0.8$ & $1.6 \pm0.3$\
[**Mock Catalog**]{}& & & & &\
Recovered & 395 &$7.9 \pm0.5$ & $1.6 \pm0.2$ & &\
Real & 281 &$7.5 \pm0.8$ & $1.6 \pm0.3$ &$6.0 \pm0.7$ & $1.8 \pm0.3$\
Before Target Selection & 527 & & &$7.4 \pm0.4$ & $1.6 \pm0.2$\
\[grptable\]
[lrccccc]{} & & & & & &\
Group galaxies & 3840 & $0.05-20$ & $5.68 \pm0.26$ & $2.12 \pm0.05$ & $5.26 \pm0.27$ & $2.12 \pm0.06$\
& & $0.05-1$ & $7.21 \pm0.73$ & $1.96 \pm0.06$ & $5.08 \pm0.62$ & $2.16 \pm0.11$\
& & $1-20$ & $5.50 \pm0.33$ & $1.95 \pm0.19$ & $5.34 \pm0.30$ & $2.02 \pm0.15$\
Red group galaxies & 794 & $0.05-20$ & $6.31 \pm0.44$ & $2.29 \pm0.09$ & $5.81 \pm0.45$ & $2.27 \pm0.11$\
Blue group galaxies & 3046 & $0.05-20$ & $5.10 \pm0.20$ & $2.12 \pm0.04$ & $4.77 \pm0.20$ & $2.15 \pm0.05$\
All galaxies & 9787 & $0.05-20$ & $3.46 \pm0.09$ & $1.68 \pm0.04$ & $3.63 \pm0.07$ & $1.74 \pm0.03$\
Field galaxies & 5947 & $1-20$ & $2.55 \pm0.26$ & $1.68 \pm0.12$ & $2.76 \pm0.20$ & $1.72 \pm0.12$\
&&&&&&\
[**Mock Catalogs**]{} (B256) & & & & & &\
Group galaxies & 3020 & $0.05-1$ & $3.63 \pm0.35$ & $2.84 \pm0.12$ & $3.02 \pm0.25$ & $2.99 \pm0.13$\
& & $1-20$ & $5.57 \pm0.37$ & $1.75 \pm0.16$ & $5.43 \pm0.29$ & $1.84 \pm0.11$\
All galaxies & 11262 & $0.05-20$ & $3.75 \pm0.09$ & $1.69 \pm0.03$ & $3.75 \pm0.09$ & $1.69 \pm0.03$\
& & $0.05-1$ & $3.55 \pm0.18$ & $1.76 \pm0.07$ & $3.52 \pm0.14$ & $1.85 \pm0.05$\
& & $1-20$ & $3.71 \pm0.15$ & $1.65 \pm0.07$ & $3.85 \pm0.11$ & $1.68 \pm0.05$\
Field galaxies & 8242 & $1-20$ & $3.01 \pm0.14$ & $1.57 \pm0.06$ & $3.35 \pm0.12$ & $1.63 \pm0.05$\
\[r0table\]
|
---
abstract: 'We present exact results for the optical response in the one-dimensional Holstein model. In particular, by means of a refined kernel polynomial method, we calculate the ac and dc electrical conductivities at finite temperatures for a wide parameter range of electron phonon interaction. We analyze the deviations from the results of standard small polaron theory in the intermediate coupling regime and discuss non-adiabaticity effects in detail.'
author:
- 'G. Schubert'
- 'G. Wellein'
- 'A. Wei[ß]{}e'
- 'A. Alvermann'
- 'H. Fehske'
title: Optical absorption and activated transport in polaronic systems
---
Introduction
============
The investigation of transport properties has been playing a central role in condensed matter physics for a long time. In recent years optical spectroscopy, for example, has contributed a lot to unravel the complex physics of highly correlated many-body systems, such as the one-dimensional (1d) MX chains,[@BS98] the quasi-2d high-temperature superconducting cuprates,[@Da94] or the 3d colossal magneto-resistive manganites.[@JTMFRC94] That way, optical measurements proved the importance of electron-phonon (EP) interactions in all these materials and, in particular, corroborated polaronic scenarios for modeling their electronic transport properties at least at high temperatures.[@Emi93; @AM95; @WMG98]
Polarons are quasi-particles composed of an electron and the surrounding ions which in a polar solid, provided the electron lattice interaction is sufficiently strong, are displaced from their equilibrium positions due to the presence of the electron. This bootstrap relation between electron and lattice displacement makes the particle heavy, because it has to drag with it the potential well of the phonons. Polaron motion is largely understood and has been worked out theoretically in two important limits: In the first case the electronic bandwidth is large and there is a only a slight change in the particle’s effective mass due to the EP coupling. These quasi-particles are called large polarons or Fröhlich polarons. In the second case it is assumed that the bandwidth is small, whereas the EP interaction is strong and short-ranged. Now polaronic effects trap the electron at a certain lattice site and the size of the quasi-particle becomes comparable to the inter-atomic lattice spacing. Thermally activated hopping will necessarily be the dominant transport process of such small or Holstein-type polarons. Although there are experimental systems with clear large and small polaron characteristics, most of the above-mentioned novel materials belong to the transition region between these two limiting cases. Here the relevant energy scales are not well-separated and perturbative approaches cannot describe the complicated transport mechanisms adequately.
A recent non-perturbative dynamical mean-field study of the Holstein model in infinite dimensions[@FC03] reports quantitative discrepancies of the temperature dependence of the resistivity from standard polaron formulas, but the dynamical mean-field approach inherently does not account for vertex corrections to the conductivity and for longer ranged hopping processes induced by the EP interaction. On the other hand, exact numerical investigations of the Holstein model provided quite a number of reliable results for the zero-temperature optical conductivity in the 1d and 2d (extended) Holstein models.[@FLW97; @ZJW99]
Motivated by this situation, in the present work we use a recently developed extension of the Kernel Polynomial Method (KPM),[@SRVK96; @WWAF05] a refined numerical Chebyshev expansion technique, to compute both the dc and ac hopping conductivities at finite temperatures without any serious approximation.
Model and Method
================
Our starting point is the 1d tight-binding Holstein Hamiltonian,[@Ho59a] $$H = -t \sum_{{\langle}i,j{\rangle}} c^\dag_i c^{{\phantom{\dag}}}_j
-\sqrt{{\varepsilon_p}{\omega}_0} \sum_i (b^\dag_i + b^{{\phantom{\dag}}}_i) n_i^{{\phantom{\dag}}}
+\hbar{\omega}_0\sum_i b^\dag_i b^{{\phantom{\dag}}}_i\,,
\label{homo}$$ describing a single electron coupled locally to a dispersion-less optical phonon mode, where $c^\dag_i$ ($b^\dag_i$) denotes the corresponding fermionic (bosonic) creation operator, and $n_i=c^\dag_i
c^{{\phantom{\dag}}}_i$. Setting $\hbar=1$ and measuring all energies in units of the nearest-neighbor hopping integral $t$, the physics of the model is determined by the dimensionless EP coupling constants $$\lambda={\varepsilon_p}/2t\;\;\; \mbox{and}\;\;\;g^2={\varepsilon_p}/{\omega}_0
\label{coco}$$ in the adiabatic (${\omega}_0/t \ll 1$) and anti-adiabatic (${\omega}_0/t \gg
1$) regimes, respectively. In 1d the crossover from large to small polaron behavior takes place at $\lambda \simeq 1$ ($g^2 \simeq 1$) in the former (latter) case.[@WF97; @CSG97] Whether large polarons form in the Holstein model for $\text{d}>1$ is still under debate.[@Emi86; @FRWM95; @CFI99]
Addressing the linear response of our system to an external (longitudinal) electric field we consider the Kubo formula for the electrical conductivity at finite temperatures, which is[@Mah00] $$\operatorname{Re}\sigma(\omega) = \pi
\sum_{m,n}^{\infty} \frac{\operatorname{e}^{-\beta E_n} - \operatorname{e}^{-\beta E_m}}{Z L \omega}
\,|\langle n|\hat{\jmath}|m\rangle|^2 \,
\delta(\omega - \omega_{mn})\,.
\label{si_1}$$ Here $Z = \sum_{n}^{\infty} \operatorname{e}^{-\beta E_n}$ is the partition function and $\beta=T^{-1}$ denotes the inverse temperature. Since the Holstein Hamiltonian (\[homo\]) involves bosonic degrees of freedom, the Hilbert space even of a finite $L$-site system has infinite dimension. In practice, however, the contribution of highly excited phonon states is negligible at the relevant temperatures, and the system is well approximated by a truncated phonon space with at most $M(\lambda,g,\omega_0;T)$ phonons.[@BWF98] Then $|n\rangle$ and $|m\rangle$ are the eigenstates of $H$ within our truncated $D$-dimensional Hilbert space, $E_n$ and $E_m$ are the corresponding eigenvalues, and $\omega_{mn} = E_m - E_n$. In Eq. (\[si\_1\]) the current operator has the standard hopping form, $\hat{\jmath}=\operatorname{i}\, e t \sum_i (c^\dag_i c^{{\phantom{\dag}}}_{i+1} - c^{\dag}_{i+1} c^{{\phantom{\dag}}}_i)\,,$ and connects states with different parity.[@Ba02] Thus, assuming that the ground state is non-degenerate, the expectation value $\langle
0|\hat{\jmath}|0\rangle$ vanishes in the absence of an external electric field. The limit ${\omega}\to 0$ of (\[si\_1\]) yields the dc conductivity, whereas the optical absorption is given by the finite frequency data. For small polarons both results are interesting and have been evaluated analytically at an early stage.[@LF62] At $T=0$, the regular part of the optical conductivity, $\sigma^{\text{reg}}(\omega) = \frac{\pi}{L} \sum_{n>0}
\omega_{n0}^{-1} |\langle n|\hat{\jmath}|0\rangle|^2 \delta(\omega -
\omega_{n0})$, was calculated for finite 1d and 2d lattices with periodic boundary conditions (PBC) in a wide parameter range of the Holstein model, using a combination of the Lanczos algorithm and the KPM.[@FLW97]
At finite temperatures, a similar straight-forward expansion of the conductivity is spoiled by the presence of the Boltzmann factors and the contribution of all matrix elements between eigenstates of the system. Instead, it turns out that a new generalised KPM scheme [@WWAF05; @We04] can be based upon a current operator density $$j(x,y) = \sum_{m,n} |\langle n|\hat{\jmath}|m\rangle|^2
\ \delta(x-E_n)\ \delta(y-E_m)\,.
\label{j_xy}$$ Being a function of two variables, $j(x,y)$ can be expanded by a two-dimensional KPM, $$\tilde{\jmath}(x,y) = \sum\limits_{k,l=0}^{N-1}\frac{
\mu_{kl} w_{kl} g_k^J g_l^J T_k(x) T_l(y)
}{\pi^2 \sqrt{(1-x^2)(1-y^2)}}\,,
\label{jt_xy}$$ where the tilde refers to a rescaling of energy ($H\to\tilde{H}$) that maps the spectrum of $H$ into the domain $[-1,1]$ of the Chebyshev polynomials of first kind $T_k(x)$. The finite order $N$ of the expansion leads to Gibbs oscillations which can be damped by introducing appropriate damping factors. Here we use $g_n^J$ derived from the Jackson kernel[@Ja12; @SRVK96], and $N=512$ throughout the paper. The core of the numerical work is the iterative calculation of the moments $\mu_{kl} = \operatorname{Tr}(T_k(\tilde H) \hat{\jmath} T_l(\tilde H) \hat{\jmath})$, where the trace can be replaced by an average over a relatively small number of random vectors $|r\rangle$.[@DS93] Finally, the factors $1/w_{kl} = (2-\delta_{k0})(2-\delta_{l0})$ account for the correct normalisation. Given the operator density $j(x,y)$ we find the optical conductivity by integration $$\operatorname{Re}\sigma(\omega) \!= \!
\frac{\pi}{ZL\omega} \int\limits_{-\infty}^{\infty}
j(y+\omega,y)
\big[\operatorname{e}^{-\beta y} - \operatorname{e}^{-\beta (y+\omega)}\big]\, dy\,.
\label{si_2}$$ The partition function $Z=\int_{-\infty}^{\infty} \rho(E)\exp (-\beta
E)$ is easily obtained by integrating over the density of states $\rho(E) = \sum_{n=0}^{D-1} \delta (E-E_n)$, which can be expanded in parallel to $\tilde{\jmath}(x,y)$. Note the main advantage of this approach: The current operator density that enters the conductivity is the same for all temperatures, i.e., it needs to be expanded only once. Figure \[f\_od\] exemplifies the different form of $\tilde{\jmath}(x,y)$ in the weak and strong EP coupling regimes.
![(Color online) Renormalized current operator density $\tilde{\jmath}(x,y)$ used in the 2d KPM. Data obtained for the 1d Holstein model with $\lambda=0.2$ (left panel) and $\lambda=2.0$ (right panel) at ${\omega}_0/t=0.4$ ($L=6$, $M=50$; PBC).[]{data-label="f_od"}](fc1.eps){width="\linewidth"}
![(Color online) Schematic setup for the calculation of the finite-temperature optical conductivity (left panel). Lowest eigenvalues of the Holstein Hamiltonian for $L=6$, $M=25$, and PBC (right panel). The shaded area marks the six lowest eigenvalues to be separated from the rest of the spectrum.[]{data-label="f_setup"}](fc2.eps){width="0.98\linewidth"}
At very low temperatures, the numerical evaluation of expression (\[si\_2\]) requires some caution, since the Boltzmann factors heavily amplify small numerical errors in $j(y+{\omega},y)$. We can avoid these problems, occurring mainly at the lower bound of the spectrum, by treating the contributions of the ground state and some of the lowest excitations separately. This is illustrated in Fig. \[f\_setup\]. We split the optical conductivity into three parts, $$\operatorname{Re}\sigma(\omega) =
\operatorname{Re}\sigma^{\text{ED}}(\omega) +
\operatorname{Re}\sigma^{\text{1d}}_{\text{KPM}}(\omega) +
\operatorname{Re}\sigma^{\text{2d}}_{\text{KPM}}(\omega)\,,
\label{si_3}$$ where the first contribution describes the transitions (matrix elements) between the $S$ separated eigenstates, the second part those between the separated states and the rest of the spectrum, which can be expressed as standard 1d KPM expansions, and finally the transitions within the remaining $D-S$ states of the spectrum, handled by a 2d expansion. Using the projection operator $P = 1 - \sum_{s=0}^{S-1}
|s\rangle\langle s|$, the moments for the contributions $\operatorname{Re}\sigma_{\text{1d}}(\omega)$ read $\mu_k^n = \langle
n|\hat{\jmath}PT_k(\tilde H)P\hat{\jmath}|n\rangle$. Of course, the number of states one has to separate depends on the physical situation. The right panel of Fig. \[f\_setup\] gives the lowest eigenvalues of the Holstein model at various coupling strengths. In the strong-coupling regime ($\lambda \gg 1$) states belonging to the lowest small polaron band have almost the same energy as the ground state and therefore should be treated separately (cf. the curves for $\lambda=1.5$ and 2). Obviously the situation is far less dramatic at weak EP couplings.
![(Color online) Optical conductivity in the 1d Holstein model at $T=0$ (in units of $\pi e^2 t^2$) compared to the analytical small polaron result Eq. (\[aspt\]) \[dashed blue lines\]. Exact diagonalization data (ED) are obtained for a system with $L=6$ and $M=45$; $\sigma_0$ is determined to give the same integrated spectral weight of the $\omega>0$ (regular part) of $\operatorname{Re}\sigma$.[]{data-label="acsigma_kpm_reik"}](fc3.eps){width="0.9\linewidth"}
Numerical results and discussion
================================
ac concductivity
----------------
We now apply our numerical scheme to the calculation of the optical absorption in the 1d Holstein model. The results for $\operatorname{Re}\sigma({\omega})$ and possible deviations from established polaron theory are important for relating theory and experiment. The standard description of small polaron transport[@RH67; @Emi93] provides the ac conductivity at $T=0$ as $$\operatorname{Re}\sigma (\omega) =
\frac{\sigma_0}{\sqrt{{\varepsilon_p}\omega_0}} \frac{1}{\omega}
\exp \left[- \frac{(\omega-2 {\varepsilon_p})^2}{4 {\varepsilon_p}\omega_0}\right]
\; .
\label{aspt}$$ For sufficiently strong coupling this formula predicts a weakly asymmetric Gaussian absorption peak centered at $\omega = 2 {\varepsilon_p}$. A similiar analytical formula can be derived for finite temperatures.[@BB85; @Mah00]
Starting at [*zero temperature*]{}, Fig. \[acsigma\_kpm\_reik\] shows $\operatorname{Re}\sigma(\omega)$ for various EP-parameters. For $\lambda=2$ and ${\omega}_0/t=0.4$, i.e., at rather large EP coupling, but not in the extreme small polaron limit, we find also a pronounced maximum in the low-temperature optical response, which, however, is located, somewhat below $2{\varepsilon_p}=2g^2{\omega}_0$, being the value for small polarons at $T=0$. At the same time, the line-shape is more asymmetric than in standard polaron theory, with a weaker decay at the high-energy side, which fits even better the experimental behavior observed in polaronic materials such as TiO$_2$.[@KMF69] Varying the parameters significant discrepancies to a Gaussian-like absorption are found. Then the polaron motion is not adequately described as hopping of a self-trapped carrier almost localized on a single lattice site.
![(Color online) Optical absorption by Holstein polarons at finite temperatures in the adiabatic regime ($L=6$, $M=45$). Dashed curves give the analytical result for finite-temperature small polaron transport.[@BB85; @Mah00] The deviations observed for high excitation energies at very large temperatures are caused by the necessary truncation of the phonon Hilbert space in ED.[]{data-label="acsigma_sc_lf"}](fc4.eps){width="0.9\linewidth"}
At [*finite temperature*]{} two different transport mechanism can be distinguished. Clearly, coherent transport, which for large EP couplings is related to diagonal (zero-phonon) transitions within the lowest extremely narrow polaron band, will be negligible at high temperatures. For instance, the amplitude of the current matrix elements between the degenerate states with momentum $K=\pm\pi/3$ ($K=0,\, \pm\pi/3,\,
\pm2\pi/3$, and $\pi$ are the allowed wave numbers of a 6-site system with PBC) is of the order of $10^{-7}$ only. Whereas phase coherence is maintained during a diagonal transition, the particle loses its phase coherence if its motion is triggered by (multi-) phonon absorption and emission processes. These so-called non-diagonal transitions which, of course, can take place also with zero energy transfer, become more and more important as the temperature increases. Accordingly the main transport mechanism is thermally activated hopping, where each hop becomes a statistically independent event. In the small polaron limit, where the polaronic sub-bands are roughly separated by the bare phonon frequency (cf. Fig. \[f\_setup\], right panel), this happens for $T\gtrsim {\omega}_0$. Let us consider the activated regime in more detail (cf. Fig. \[acsigma\_sc\_lf\]: With increasing temperatures we observe a substantial spectral weight transfer to lower frequencies, and an increase of the zero-energy transition probability in accordance with previous results.[@Na63]
In addition, we find a strong resonance in the absorption spectra at about ${\omega}\sim 2t$, which can be easily understood using a configurational coordinate picture. Placing a homogeneous lattice distortion $u$ at $L_u$ consecutive sites by applying the unitary transformation $S^\dagger(u)=\prod_i^{L_u}S^\dagger_i(u)=\prod_i^{L_u}
\exp[u(b_i^\dagger -b_i^{{\phantom{\dag}}})]$, the transformed Holstein Hamiltonian takes the form $\bar{H}= \langle
0|S^\dagger(u)HS(u)|0\rangle_{\rm ph}= -t \sum_{{\langle}i,j{\rangle}} c^\dag_i
c^{{\phantom{\dag}}}_j -2\sqrt{{\varepsilon_p}{\omega}_0} u \sum_i^{L_u} n_i + {\omega}_0u^2L_u$. In the adiabatic strong-EP-coupling regime, the ground state can be approximated as an electron localized at a certain single site with the lattice being in a shifted oscillator state ($L_u=1$). That is, $|\Psi_0\rangle= |1\rangle_{el}\otimes S^{\dag}_1(g) |0\rangle_{ph}$ and $E_0=-{\varepsilon_p}$, in accordance with the Lang-Firsov approximation. Now let us consider excitations from this ground-state, where the lattice distortion spreads over two neighboring sites ($L_u=2$) and the electron is in a symmetric or antisymmetric linear combination of $|1\rangle_{el}$ and $|2\rangle_{el}$, i.e., the particle is mainly located at sites 1 and 2 but delocalized between these sites. We then find $|\Psi_{1,\pm} \rangle= (|1\rangle_{el}\pm |2\rangle_{el})\otimes
S^{\dag}_2(g/2) S^{\dag}_1(g/2) |0\rangle_{ph}$ and $E_{1,\pm} =\mp
t-{\varepsilon_p}/2$. Whereas the (potential) energy related to the displacement field is reduced to $-{\varepsilon_p}/L_u$, the kinetic energy comes into play since hopping processes between 1 and 2 are allowed. The current operator $\hat{\jmath}$ connects these different-parity states with perfect overlap $|\langle \Psi_{1,+}|\hat{\jmath} | \Psi_{1,-}\rangle|=(et)^2$, giving rise to a strong signal in the optical absorption. Note that the excitation energy ${\omega}_{1-,1+}=2t$ is independent of ${\varepsilon_p}$. In order to activate these transitions thermally, the electron has to overcome the “adiabatic” barrier $\Delta=E_{1+}-E_0={\varepsilon_p}/2-t$. A finite phonon frequency will relax this condition. From Fig. \[acsigma\_sc\_lf\], we find the signature to occur above $T\gtrsim 0.5t$. Obviously this feature is absent in the standard small-polaron transport description which essentially treats the polaron as a quasiparticle without resolving its internal structure. Owing to the infinite number of neighboring sites it is also absent in the DMFT calculation. Of course, one could also extend this scenario to excitations where the electron is delocalized over more than two distorted lattice sites, but for the present parameters the signature of these weakly bound states would be rather small.
![(Color online) Optical absorption by Holstein polarons at finite temperatures in the non-adiabatic regime ($L=6$, $M=30$). Note that now the abscissa is scaled with respect to the phonon frequency.[]{data-label="acsigma_sc_hf"}](fc5.eps){width="0.9\linewidth"}
![(Color online) Optical absorption in the adiabatic intermediate EP-coupling regime (the notation is the same as in Fig. \[acsigma\_sc\_lf\]; again we use $L=6$, $M=40$). The inset illustrates the finite-size dependence of $\operatorname{Re}\sigma(\omega)$ for $T\sim 0$ (the vertical dotted line gives the phonon absorption threshold). It demonstrates that the gap observed at low frequencies and temperatures is clearly a finite-size effect, i.e., at weak-to-intermediate couplings the discrete electronic levels of our finite system show up in the conductivity spectra. These effects, of course, are of minor importance at larger EP couplings, where the polaronic bandwidth is strongly reduced, as well as for high temperatures.[]{data-label="acsigma_ic_lf"}](fc6.eps){width="0.9\linewidth"}
Entering the non-adiabatic regime of large phonon frequencies at fixed $\lambda=2$, the pattern of sub-bands separated roughly by ${\omega}_0$ becomes more pronounced, but is, of course, washed out at higher temperatures (see Fig. \[acsigma\_sc\_hf\] for ${\omega}_0/t=0.8$). In Fig. \[acsigma\_sc\_hf\] the average number of phonons contained in the ground state ($\propto g^2$) is smaller ($g^2=5$) than in the previous case where $g^2=10$ (${\omega}_0/t=0.4$). This also concerns the activated region ${\omega}_0/t\lesssim
T\lesssim\Delta$ but for these parameters the simple adiabatic picture anticipated above breaks down anyway.
Now let us decrease the EP coupling strength at small phonon frequencies ${\omega}_0/t=0.2$ keeping $g^2=10$ fixed. Results for the optical response in the vicinity of the large to small polaron crossover ($\lambda=1$) are depicted in Fig. \[acsigma\_ic\_lf\]. Here the small polaron maximum has almost disappeared and the $2t$-absorption feature can be activated at very low temperatures ($\Delta \to 0$ for the two-site model with $\lambda=1$). The overall behavior of $\operatorname{Re}\sigma(\omega)$ resembles that of polarons of intermediate size. At high temperatures these polarons will dissociate readily and the transport properties are equivalent to those of electrons scattered by thermal phonons. Let us emphasize that many-polaron effects become increasingly important in the large-to-small polaron transition region.[@Hoea04p] As a result, polaron transport might be changed entirely compared to the one-particle picture discussed so far.
Sum rules
---------
Before we consider the temperature dependence of the dc conductivity, it is useful to test the sum rules for the real part of the optical response. First we have the so-called $f$-sum rule, $$S_{\rm tot}:=\int_{-\infty}^\infty \operatorname{Re}\sigma(\omega) d{\omega}=
- \frac{\pi e^2}{L} E_{\rm kin}\,,
\label{fsum1}$$ which relates the $\omega$-integrated $\operatorname{Re}\sigma(\omega)$ to the kinetic energy $E_{\rm kin}=-t\sum_{\langle i,j \rangle} \langle
c_i^{\dag}c_j^{{\phantom{\dag}}}\rangle_T$. Note that the $\omega=0$ (Drude) contribution is included in (\[fsum1\]). The second sum-rule for $\operatorname{Re}\sigma(\omega)$ is $$\int_{0}^\infty \omega \operatorname{Re}\sigma(\omega) d{\omega}=
\frac{\pi}{L} \langle \hat{\jmath}^{2}\rangle_T\,.
\label{fsum2}$$ Throughout our calculations Eq. (\[fsum2\]) was fulfilled within numerical accuracy, where the thermal average $\langle
\hat{\jmath}^{2}\rangle_T$ was determined again using a 1d KPM. Testing sum rule (\[fsum1\]) we gain important information about finite-size effects.
Figure \[fsum1rule\] compares $E_{\rm kin}$ and $S_{\rm tot} L/ \pi e^2$ obtained from our finite-cluster calculation. At weak EP couplings the kinetic energy is a strictly monotonic increasing function of temperature and becomes strongly suppressed at high temperatures due to scattering of the electron by thermal phonons. Whereas the $f$-sum rule is almost perfectly fulfilled for smaller values of $\beta$, we found pronounced deviations at low temperatures which, without doubt, can be assigned to the finite size of our Holstein ring (cf. the $L$ dependence of $S_{\rm tot}$ shown in the upper panel of Fig. \[fsum1rule\], and also the discussion of the dc conductivity below). Clearly finite-size effects become important when the temperature is comparable to the energy gaps in the spectrum of $H$ (see Fig. \[f\_setup\]). At strong EP couplings the transport is hopping-dominated and the kinetic energy exhibits a maximum at a finite temperature that can be related to the thermal activation energy of polarons. Now, in view of the narrow small polaron band (cf. Fig. \[f\_setup\]), finite size gaps are small and the $f$-sum rule is fulfilled down to very low temperatures. Small deviations appear to vanish rapidly with increasing system size (see inset, lower panel of Fig. \[fsum1rule\]). In order to analyze the contributions of different transport processes to $E_{\rm kin}$ in some more detail, we have decomposed $S_{\rm tot}=S_{\rm tot}^{\text{ED}}+\sum_{s=0}^{S-1}
S_{\rm tot}^{\text{1d} (s)} +S_{\rm tot}^{\text{2d}}$ in analogy to Eq. (\[si\_3\]). Figure \[fsum1rule\] shows that the coherent (intra-band) contribution ($\propto S_{\rm tot}^{\text{ED}}$) is almost negligible at the temperatures considered. Inter-band transitions connecting eigenstates of the lowest polaron band to higher excited states (1d KPM) are the determining factor at low $T$. If the renormalised polaron bandwidth is small enough, all states in the band are equally populated, leading to pretty much the same values of $S_{\rm tot}^{\text{1d} (s)}$. At high temperatures, of course, the transitions covered by the 2d KPM are predominant.
![(Color online) Conformance of the $f$-sum rule (\[fsum1\]) in the weak (upper panel) and strong (lower panel) EP coupling regimes of the 1d Holstein model. Results for $S_{\rm tot}$, partial $S_{\rm tot}$’s and $E_{\rm kin}$ are given by solid, dot-dashed and dashed lines, respectively. For further explanation see text.[]{data-label="fsum1rule"}](fc7a.eps "fig:"){width="0.9\linewidth"}\
![(Color online) Conformance of the $f$-sum rule (\[fsum1\]) in the weak (upper panel) and strong (lower panel) EP coupling regimes of the 1d Holstein model. Results for $S_{\rm tot}$, partial $S_{\rm tot}$’s and $E_{\rm kin}$ are given by solid, dot-dashed and dashed lines, respectively. For further explanation see text.[]{data-label="fsum1rule"}](fc7b.eps "fig:"){width="0.9\linewidth"}
![(Color online) Top row: DC conductivity as a function of temperature at weak (left panel) and strong (right panel) EP coupling. The bottom panel shows, for strong coupling, a comparison with recent DMFT results[@FC03] and standard polaron theory[@Mah00]. []{data-label="sigma_dc"}](fc8.eps "fig:"){width="0.93\linewidth"}\
dc conductivity
---------------
The dc conductivity is obtained by taking the limit ${\omega}\to 0$ of (\[si\_1\]), which with $\lim_{{\omega}\to 0} (1-\operatorname{e}^{-\beta{\omega}})/{\omega}=\beta$ yields $$\operatorname{Re}\sigma_{\rm dc} =
\frac{\pi\beta}{ZL}
\sum_{n,m}^{D-1}\operatorname{e}^{-\beta E_n}|\langle n|\hat{\jmath}|m\rangle|^2
\, \delta(E_m-E_n)\,.
\label{si_dc}$$ $\operatorname{Re}\sigma_{\rm dc}$ essentially counts the number of thermally accessible current carrying (degenerate) states. Since we have $\langle 0|\hat{\jmath}|0\rangle=0$ the conductivity almost vanishes for small $T$ below the finite size gap between the ground state and the first excited state. In the thermodynamic limit, $L\to\infty$, $\operatorname{Re}\sigma_{\rm dc}$ is related to the charge stiffness $D_c$, or the so-called Drude weight (at $T=0$).[@SS90]
The temperature dependence of the dc conductivity is illustrated in Fig. \[sigma\_dc\]. Again the weak coupling results appear to have a rather strong finite-size dependence (note that $\operatorname{Re}\sigma_{\rm dc }$ depicted in Fig. \[sigma\_dc\] is an intensive quantity). When $L$ increases a pronounced peak develops at low temperatures. This peak can be attributed to a normal “metallic-like” behavior. Independent of the coupling strength ${\varepsilon_p}$ the polaron is an itinerant quasiparticle thus, for $T\to0$, always leading to band conduction. When ${\varepsilon_p}>0$, we expect that $\operatorname{Re}\sigma({\omega}\to 0)$ is finite for $T > 0$ in contrast to an ideal conductor.
The shoulder observed for the 10 and 12 site systems at $T/t\gtrsim 1$, again is an artifact of our phonon truncation procedure as can be seen by comparing the data obtained for $L=6$ and different numbers of the phonon cut-off $M$. In the strong-EP-coupling polaronic regime, band-like transport becomes extensively suppressed (the Drude weight is exponentially small). Nevertheless quantum zero-point phonon fluctuations cause polaron delocalization at $T=0$. At higher temperatures incoherent polaron hopping transport manifests in the temperature dependence, leading to the well-known absorption maximum in $\operatorname{Re}\sigma_{\rm dc}(T)$ (cf. Fig. \[sigma\_dc\], lower panel). Since this signature is related to (rather local) polaron excitation processes the position of the maximum is almost independent of the system size. In comparison to the $d=\infty$ (DMFT) results [@FC03] we find the same qualitative behavior in the relevant temperature regime $\omega_0 \lesssim T \sim 2 {\varepsilon_p}$ but a different location of the conductivity maximum. Generally in DMFT the activation energy for polaron hopping turns out to be lower than expected from commonly accepted arguments for finite-d systems. Increasing the lattice size, our 1d Holstein data indicates that this discrepancy does not necessarily imply the failure of standard theory of hopping conduction [@BB85] but may partly arise from dimensionality effects on polaron transport in infinite dimensions. Conversely, and in light of the deviations found for the ac conductivity (cf. Fig. \[acsigma\_sc\_lf\]), the standard (anti-adiabatic) strong-coupling description can only be supposed to provide estimates on relevant energy scales in the intermediate adiabatic EP-coupling regime.
Summary
=======
In this work, we have investigated the motion of a charge carrier in response to ac and dc external fields for strongly correlated 1d electron-phonon systems. The combination of Lanczos diagonalization and the Kernel Polynomial Method has enabled us to calculate for the first time quasi-exactly the temperature dependence of the optical absorption spectra and the dc conductivity in the framework of the one-dimensional Holstein model. Besides the well-known polaron maximum a pronounced absorption feature at about $2t$ is found in the optical conductivity. Finite-size effects were identified and assessed, e.g., on the basis of the $f$-sum rule. In the physically most interesting range of intermediate coupling strengths and phonon frequencies, we find that the conductivity deviates from the standard small polaron results.
We are grateful to M. Hohenadler and J. Loos for helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft through SPP1073, by KONWIHR and by the Australian Research Council. H. F. acknowledges the hospitality at the University of New South Wales sponsored by the Gordon Godfrey Bequest. Special thanks go to NIC Jülich and HLRN Berlin for granting access to their supercomputer facilities.
[31]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, (); , , , , ****, ().
, ****, (); , , , , , , , , ****, ().
, , , , , , ****, (); , ****, ().
, ****, ().
, ** (, , ).
, , , ****, ().
, , ****, (); , ****, ().
, , , ****, (); , ****, (); , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , (), <http://arXiv.org/abs/cond-mat/0504627>.
, ****, (); [*ibid*]{} ****, ().
, ****, ().
, , , ****, ().
, ****, (); , ****, ().
, , , , ****, (); , , , ****, ().
, , , ****, (); [*ibid*]{} ****, ().
, ** (, , ).
, , , ****, ().
, ****, (); , , , , ****, ().
, ****, (); , ** (, , ); , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ** (, , ).
, , , ****, ().
, ****, ().
, , , , , , ****, ().
, ****, (); , , , ****, ().
|
---
abstract: 'We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen hamiltonian acting within the lowest Landau level. The resulting “energy projection” is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach[@Girvin84], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.'
author:
- 'M. Fremling$^{1,2}$, J. Fulsebakke$^{2}$, N. Moran$^{2}$, J. K. Slingerland$^{2,3,4}$'
bibliography:
- 'corrsorted.bib'
title: Energy projection and modified Laughlin states
---
INTRODUCTION {#sec:Introduction}
============
Much of our understanding of the fractional quantum Hall effect (FQHE) and other strongly correlated systems comes from trial wave functions which describe the ground state and low lying excitations of the various phases of these systems. A crucial step in constructing most of these trial wave functions is restriction of the Hilbert space of the system so that only single particle states in a low lying band can be occupied. In the case of the FQHE, the single particle states are usually restricted to the lowest Landau level (LLL). Sometimes it is possible to get a simple closed form expression for the trial states which satisfies this requirement, as in the case of Laughlin’s trial wave functions[@Laughlin83], for the Hall conductance plateau at filling $\nu=\frac{1}{3}$. However, more often, trial wave functions are constructed from some physical intuition without taking this restriction into account and then explicitly projected onto the LLL. This is most famously the case for Jain’s composite fermion (CF) wave functions[@Jain89; @Jain_CF], which give a good description of the physics of the Hall effect at fillings throughout the region of the LLL where it is observed. The simplest of these wave functions are of the form $$\label{eq:CF_basic}
P_{LLL}\left(\chi^{(CF)}(z_1,...,z_{N})\prod_{i<j}(z_{i}-z_{j})^{2p}\right).$$ Here, $z_1,...,z_{N}$ are the complex coordinates of the $N$ electrons in the two-dimensional space. The Jastrow factor $\prod_{i<j}(z_{i}-z_{j})^{2p}$ can be thought of as attaching $2p$ magnetic flux quanta to each of the electrons, or more naively, we can think of it as just lowering the correlation energy by keeping the electrons well separated. The factor $\chi^{(CF)}$ is a Slater determinant built from single particle wave functions for a system at effective flux $N_{\Phi}^{(eff)}=N_{\Phi}- 2pN$. This factor usually has nonzero occupation of states in higher Landau levels which means its polynomial part will depend on $\bar{z}_{i}$ as well as $z_{i}$. The explicit orthogonal projection $P_{LLL}$ onto the LLL is then needed to bring the trial state back into the LLL Fock space.
The projection $P_{LLL}$ is hard to implement exactly. The simplest method is to calculate the overlap of the unprojected trial states with the LLL Fock states by Monte Carlo integration, but this is only feasible for small system sizes. Alternatively, one may use algebraic methods based on normal ordering and converting occurrences of $\bar{z}_{i}$ to derivatives $\partial_{z_{i}}$ (see [*e.g.* ]{}[Ref. ]{}), but this too can only be done for small system sizes. In fact, considerable effort has been devoted to the development of approximate projection methods for these wave functions[@Jain97_PRB_55] and for the related CF wave functions with reverse flux attachment[@Moller_05_PRB; @Davenport_2012_PRB]. Only after the introduction of these methods has it been possible to probe even the largest system sizes which can be accessed by numerical diagonalization of the exact hamiltonian and then only for composite fermion type wave functions on a plane or sphere. There is considerable interest now in studying systems on a torus, because this allows for a more direct examination of the topological order, for example through calculation of the Hall viscosity[@Fremling_14_PRB]. Toroidal systems also lend themselves well to numerical study by density matrix renormalization group methods [@Shibata01; @Zaletel13]. On a torus, until recently, there was not even a consensus on the correct form of the CF trial wave functions, due to the fact that taking products of wave functions, as in , does not satisfy the toroidal boundary conditions. A number of recent works, [*e.g.* ]{}[Refs. ]{}, have introduced natural CF trial wave functions on the torus, but so far there is no efficient way of evaluating the required LLL-projection. Looking beyond the CF paradigm, it is easy to write down a great many trial wave functions by employing intuitive reasoning followed by explicit LLL-projection. Many realizations of Haldane Halperin hierarchy wave functions [@Haldane83; @Halperin84] which do not lie in the main CF series fall into this class, but much more is possible. A comprehensive overview of hierarchy constructions can be found in [Ref. ]{}.
The main aims of this paper are first of all, to introduce and test a projection method which will allow any proposed projected wave function to be studied up to the system sizes which can be reached by exact diagonalization of a reasonable local hamiltonian ([*e.g.* ]{}the Coulomb hamiltonian), as long as the real space form of the unprojected wave function can be easily evaluated. We call this method the *Energy Projection (EP)* and it simply consists of replacing the projection $P_{LLL}$ onto the lowest Landau level by projection onto a much lower dimensional space generated by low energy eigenstates of some hamiltonian.
Secondly, as a first application of this method, we study a set of trial wave functions which attempt to improve on Laughlin’s wave function at filling $\nu=\frac{1}{q}$. These “modified Laughlin states” lower the correlation energy by inclusion of a factor which pushes the electrons further apart without changing the flux, or at least this is the naive intuition before projection. Such wave functions were already proposed by Girvin and Jach in 1984[@Girvin84] and on the disk they take the form $$\label{eq:alt-L_basic}
P_{LLL}\left(\prod_{i<j}(z_{i}-z_{j})^{q}\prod_{i<j}|z_{i}-z_{j}|^{2d} e^{-\frac{q+2d}{4q}\sum_k |z_{k}|^2}\right).$$ When $d=1$, this can actually be interpreted as a state of composite fermions at CF filling $\nu_{CF}=1$, with $q+1$ fluxes attached to each CF in the direction opposite to that of the external field. However, for other values of $d$, there is no such interpretation and the projection is not straightforward to perform even on the plane. A torus version of these states can also be constructed and has been examined for up to $N=4$ particles in [Ref. ]{}. Using EP, we are able to study these wave functions in any geometry (we focus on sphere and torus here) and at much larger sizes.
Energy Projection
=================
In our projection scheme, we find a number of low energy eigenstates of some reasonable hamiltonian (most commonly the Coulomb hamiltonian) acting within the LLL. We then project the trial wave function onto this set of low energy states. The idea is that for any reasonable trial wave function, we will find nearly the entire projection onto the LLL using a number of eigenstates of the hamiltonian which is very small compared to the size of the LLL. Before we analyze whether this approach really works, a rough analysis of the computation time involved is useful. Particularly, we should compare the time it takes to calculate the energy projection of a state to the time taken to calculate the exact projection.
We will assume that the real space form of the trial wave function $\psi$ before LLL-projection can be easily calculated. The exact LLL-projection of $\psi$ can then be found by calculating its overlaps ${\left\langle \phi_{\vec{n}} | \psi \right\rangle}$ with the Fock states $\phi_{\vec{n}}$ labeled by the occupation numbers $\vec{n}$ of the LLL-orbitals. Calculation of these overlaps can be done by Monte Carlo integration. The number of orbitals in the LLL equals $N_{\Phi}$ (up to small geometry dependent corrections) and hence for a fermionic system, the number of Fock states spanning the LLL is $\binom{N_{\Phi}}{N}$. The evaluation of a single Fock state (using Gaussian elimination to evaluate the determinant) scales as $N^3$, where $N$ is the number of particles. If we denote by $N_{MC}$ the average number of evaluations per Fock state necessary to find the overlaps with $\psi$ to the desired accuracy, then the total time needed for the LLL-projection of $\psi$ scales as $N^3 N_{MC} \binom{N_{\Phi}}{N}$. The factor $\binom{N_{\Phi}}{N}$ clearly increases very fast with both $N$ and $N_{\phi}$. At fixed filling fraction $\nu$, we have $N_{\phi}\approx\nu^{-1}N$ and, defining $\xi=\nu^{-1}$ we see that $$\binom{N_{\Phi}}{N}\sim \frac{1}{\sqrt{2 \pi N}} \sqrt{\frac{\xi}{\xi-1}}\left(\frac{\xi^{\xi}}{(\xi-1)^{(\xi-1)}}\right)^{N},$$ so the size of the Hilbert space increases exponentially in $N$. Unfortunately, $N_{MC}$ also tends to grow quickly with $N$. Wave functions for strongly interacting systems tend to have nonzero overlaps of comparable size with a significant fraction of the Fock states in the LLL. Hence, if we define $f=|{\left\langle \psi | P_{LLL}\psi \right\rangle}|^2$, then the Monte Carlo integration has to resolve overlaps of typical size $\sqrt{f/\binom{N_{\Phi}}{N}}$. This means that the allowable error on a given overlap should also be very small. In fact, if the typical error on the overlaps is $\epsilon$, then if we are very optimistic and take the errors on the overlaps to be independent of each other, we expect an overall error on the projection of $\psi$ which is of order $\sqrt{\binom{N_{\Phi}}{N}}\epsilon$. To bring this back to something of order $\sqrt{f}$, we require that $\epsilon\sim \sqrt{f/\binom{N_{\Phi}}{N}}$. The statistical error in the Monte Carlo integration will normally be inversely proportional to the square root of the number of independent Monte Carlo samples generated, and we expect that the number of independent samples will be (at most) of order $\frac{N_{MC}}{N}$. Hence for the desired accuracy, we require that $N_{MC}\sim N\binom{N_{\Phi}}{N}/f$. This yields an (optimistic) estimate of the scaling of computation effort for exact LLL-projection as $f^{-1}N^4 \binom{N_{\Phi}}{N}^2$.
Clearly, this is problematic in studying large systems. Nevertheless naively, it would seem that obtaining exact states for comparison to these trial states may be even more onerous, as numerical diagonalization scales naively as the third power the Hilbert space dimension, [*i.e.* ]{}$\binom{N_{\Phi}}{N}^3$. In practice however, the hamiltonians of interest are relatively sparse, usually have a high degree of symmetry and we are normally only interested in a small number of low lying eigenstates. As a result, the LLL-projection of the trial wave functions as described here nearly always becomes impossible at system sizes considerably smaller than those accessible to exact diagonalization methods.
Now consider the computational effort needed to perform energy projection. We will assume first of all that we can get a good approximation of the exact projection using the lowest $M$ eigenstates of the chosen hamiltonian. Here $M$ should be a number that does not grow quickly with $N$, in particular, $M\ll \binom{N_{\Phi}}{N}$. We will ignore the computational effort needed to obtain the $M$ lowest eigenstates of the hamiltonian – we will usually work at system sizes where this is not the bottleneck of the computation. The energy projection is simply the projection onto the $M$-dimensional subspace of the Hilbert space spanned by the $M$ lowest eigenstates. It involves calculating the overlaps ${\left\langle i | \psi \right\rangle}$, where ${\left|i\right\rangle}$, $i\in\{1,...M\}$ labels the lowest $M$ eigenstates of the hamiltonian. Each state ${\left|i\right\rangle}$ involves up to $\binom{N_{\Phi}}{N}$ Fock states. Hence a single evaluation of the state ${\left|i\right\rangle}$ in real space takes effort of order $N^3\binom{N_{\Phi}}{N}$. Once the Fock states have all been evaluated, we can keep their values (subject to memory constraints) and use them to evaluate the other $M-1$ eigenstates of the hamiltonian. Evaluation of the $M$ overlaps needed for the energy projection then takes effort of order $(N^3 +M-1)\binom{N_{\Phi}}{N} N'_{MC}$, where $N'_{MC}$ is the average number of evaluations of the states ${\left|i\right\rangle}$ needed to get good Monte Carlo estimates of the overlaps. The main advantage of energy projection is that the individual overlaps involved should now be much larger than the overlaps with individual Fock states. We now expect the average overlap to be $\sqrt{f/M}$ and to get the error on the projection of $\psi$ to be of order $\sqrt{f}$, we will need $N'_{MC}\sim N M/f$, giving an estimate for the total computational effort involved in energy projection as $f^{-1}M N (N^3 + M-1) \binom{N_{\Phi}}{N}$. This is clearly much better scaling than exact projection, as long as $M\ll \binom{N_{\Phi}}{N}$ and $M<N^3$ which usually the case, or in any event if $M^2\ll \binom{N_{\Phi}}{N}$. This scaling can be further improved if the trial wave function is an eigenstate of some symmetry of the hamiltonian, because in that case, we only need to consider overlaps with eigenstates with the same symmetry, reducing $M$. It turns out that energy projection usually allows us to work with trial wave functions at system sizes close to the largest sizes accessible to exact diagonalization. Of course, it must be remembered that energy projection is an approximation, as we are throwing away components of the trial wave function at higher energies. For reasonable trial wave functions we can hope that these components are small. In the next sections we will investigate in some detail whether this is actually the case.
Testing the projection
======================
-------- -------
Sphere Torus
-------- -------
To establish how well the energy projection method works we perform a number of tests. A first test would be to select some trial wave function, calculate the exact projection using Monte Carlo evaluation of overlaps with the Fock basis and compare it to the energy projection. However, for system sizes which are small enough to allow for accurate calculation of the exact projection in this way, we can also calculate the full spectrum of the Coulomb hamiltonian in the LLL and use energy projection using the full spectrum, which is in effect also exact projection and moreover with a smaller error than the projection using the Fock basis. Therefore we have tested the energy projection first of all on trial wave functions which are fully in the LLL. This means the projection is redundant, but allows us to see how well the energy projection reproduces the full state. Here we present results for the $\nu=1/3$ Laughlin wave-function $\psi_{L}$, on the sphere and torus, energy projected using the Coulomb hamiltonian, see [FIG. \[fig:d\_0\_Ef\]]{}.
The upper panels of [FIG. \[fig:d\_0\_Ef\]]{} show the exact and approximate overlaps of $\psi_L$ with the $n$:th energy eigenstate of the Coulomb hamiltonian. The exact wave function $\psi_{L}$ was computed by diagonalizing the short range hamiltonian based on Haldane’s pseudopotentials[@Haldane83] for which it is the unique ground state. Since this give us $\psi_{L}$ in the same basis as the Coulomb eigenstates, it is easy to obtain numerically exact overlaps also. The approximate overlaps are marked by $d=0$, in reference to the later use of nonzero values of $d$ when we modify the Laughlin wave function. These overlaps were obtained directly by performing Monte Carlo integration in real space. We find that on both sphere and torus all overlaps larger than $10^{-3}$ can be well resolved by the MC estimates. This can obviously be improved by taking more Monte Carlo samples. In the torus plot we see many small overlaps at a level just above $10^{-4}$. Most of these overlaps are actually exactly zero; the system on a square torus has a $C_4$ symmetry and hence the eigenstates of the hamiltonian with symmetry behavior different from $\psi_{L}$ have zero overlaps. The nonzero values observed give us a useful idea of the accuracy of the Monte Carlo overlaps. Note that – as expected – the (nonzero) overlaps are diminishing as a function of $n$, and the declining trend is clearly visible even on the logarithmic scale. This confirms the physical intuition that most of the projected state is captured by the low energy excitations and higher energy excitations become less and less important.
The middle panels show the cumulative square overlap $$f_n=\sum_{j=1}^n |c_j|^2,$$ and the reconstructed energy $$E_n=\frac{1}{f_n}\sum_{j=1}^n \epsilon_j |c_j|^2.$$ Here $c_j={\left\langle \phi_j | \psi \right\rangle}$ is the overlap between energy eigenstate no. $j$ and the unprojected wave function $\psi$, and $\epsilon_j$ is the energy of that $j$:th eigenstate. When $n$ approaches the total number of states in the LLL, $f_{n}$ represents the LLL content of the state (in this case we know this equals $1$) and $E_{n}$ becomes the variational energy of the state. For comparison and guidance, the exact the Coulomb energy of the Laughlin state, $E_L$ and the limit value $f=1$ are included. Since $f_n$ measures the $LLL$ content, it increases monotonically with $n$ and saturates quickly for $n \ll N$ (note that these panels do not use a logarithmic scale). As the energy levels are ordered ( $\epsilon_i>\epsilon_j$ for $i>j$ ) and $|c_j|\geq 0$, $E_n$ is also monotonic, and we note again that $E_n$ converges fast as a function of $n$.
In the lower panels we show the differences $|E_n-E_L|$ and $|f_n-1|$ on a logarithmic scale. We see that 99.9% of $\psi_L$ is already captured by using $n\approx50$ states on the torus and $n\approx15$ states on the sphere. The energy is reproduced to within four decimals by taking $n\approx 20$ states on the torus and $n\approx30$ states on the sphere. The stepwise behavior of these graphs is explained by considering their dependence on $c_{i}$. For example, the difference between two consecutive energy estimates is $E_{n+1}-E_n=\frac{|c_{n+1}|^2}{f_{n+1}}(\epsilon_{n+1}-E_n)\geq0$ and will thus jump when $|c_{n+1}|^2$ is large, which is at the same time that $f$ jumps.
--------------- --------------
Sphere, $d=0$ Torus, $d=0$
$d=1$ $d=1$
$d=7$ $d=6$
--------------- --------------
-------- -------
Sphere Torus
-------- -------
Next we test whether the energy projection is stable against changing the hamiltonian. For the energy projection to be generically useful, its success should not depend crucially on which hamiltonian is used. While the low energy sector of the hamiltonian should capture the state that is being projected, the detailed structure of the low energy states should not be important. We would expect that hamiltonians with completely different ground states should be viable, as long as they incorporate [*e.g.* ]{}repulsive interactions between the particles.
Here, we compare the energy projection using the lowest LL and second LL Coulomb hamiltonians obtained by extracting pseudopotentials a la Haldane[@Haldane83] from the real space Coulomb hamiltonian using the lowest and second LL orbital wave functions. It is well known that these hamiltonians provide a completely different set of ground states at most accessible values of the flux and electron number. In particular, $\psi_{L}$ is an excellent trial wave function for the ground state of the LLL but not for the second LL Coulomb hamiltonian, where it has squared overlap of order at most $0.4$ with the ground state for systems of up to $15$ particles[@Ambrumenil88; @Balram13]. Significant efforts have recently been made to determine whether the ground state of the SLL Coulomb hamiltonian even represents the same topological order as the Laughlin wave function[@Zaletel15; @Johri14; @Balram13] and there is also recent work on alternative wave functions which may improve the overlap[@Jeong16]. In addition to $\psi_{L}$ we show the $d=1$, $d=2$ and $d=6$ (torus) and $d=7$ (sphere) modified Laughlin states introduced in [Eq. ]{}. Explicit expressions for the modified wave functions on the sphere and torus are given in and [Eq. ]{}. Note that these $d\neq0$ wave functions are not entirely contained within the LLL so the energy projection is not redundant for these.
In [FIG. \[fig:LLL\_v\_SLL\_torus\_d\_0126\_f\]]{} we show results for $N_e=6$ particles on the torus (right panels, the small system size is chosen for illustrative purposes) and for $N_e=10$ particles on a sphere (left panels). The upper panels in each subfigure again show the overlap for the $n$:th eigenstates and the lower panels show the cumulative content $f$ for the LLL (red) and SLL (blue) hamiltonians. Looking first at $d=0$ (or $\psi_L$, upper panels) we see that, just as in [FIG. \[fig:d\_0\_Ef\]]{}, the LLL overlap falls off rapidly with increasing $n$ and that $f$ converges to good precision with only a few terms. Comparing this to the SLL hamiltonian (blue), we see that the SLL ground state and $\psi_L$ have a small overlap (zero within error on the torus and squared overlap of less than $0.4$ on the sphere). On the other hand, practically all of $\psi_L$ is still captured by the low energy states. In the toroidal system, after including as little as $n=15$ states (out of a total of $1038$ states with the same total momentum), projection using the LLL and SLL hamiltonians both give $f=1$ to within $10^{-3}$. The spherical system also clearly gives projections from the two hamiltonians which are in close agreement, although at this system size more eigenstates of the SLL hamiltonian are needed.
Turning our attention to $d=1$ (middle panels), we see qualitatively the same behavior. However – as for all $d\neq0$ – the modified Laughlin wave functions are not contained within the LLL anymore, so $f<1$ and we find limiting values of $f\approx0.8$ (torus) and $f\approx0.74$ (sphere). Nevertheless, it is clear from the figures that the energy projection still works, both with the LLL hamiltonian and with the SLL hamiltonian, as the $c_i$ decrease with $n$ and the low lying $c_i$ are still large enough to be accurately determined. If the LLL content of the unprojected wave function is very small, then the accuracy will also be reduced as the smaller $c_j$ incur larger relative errors in the MC-projection. Nevertheless, it is still possible to extract perfectly viable energy projections for considerably higher values of $d$. [*E.g.* ]{}the lower left panel shows the $d=6$ system on the torus, where $f$ is only about $0.12$ and the lower right panel shows the $d=7$ system on the sphere, with $f\approx0.022$. Both of these panels show that the SLL hamiltonian is more competitive with the LLL hamiltonian at larger values of $d$. For the $d=6$ torus state, the SLL hamiltonian actually manages to capture the LLL content faster than the LLL hamiltonian at larger $n$. In fact $f^{(\mathrm{SLL})}_n\gtrsim f^{(\mathrm{LLL})}_n$ for $3<n<10$, which shows that it is not always best to have good overlap with the ground state, since it may sacrifice weight in the other low energy states and lead to a lower value of the total weight $f$. For the $d=7$ system on the sphere, the LLL hamiltonian wins out over the SLL hamiltonian throughout, but we can nevertheless observe that the overlaps are considerably closer than at $d=0$ or $d=1$.
-------- -------
Sphere Torus
-------- -------
All in all [FIG. \[fig:LLL\_v\_SLL\_torus\_d\_0126\_f\]]{} shows that the energy projection method is stable to appreciable changes of the projecting hamiltonian as long as the low energy content is preserved and the state has reasonable weight in the LLL. We can also comment briefly on the suitability of the modified Laughlin wave functions as improved trial wave functions for the $\nu=7/3$ quantum Hall plateau. We find that the overlap with the ground state of the $SLL$ Coulomb hamiltonian on the sphere does improve when $d>0$ (as compared to $d=0$), but the improvement is not spectacular. The highest overlap we obtained was $0.603(2)$ for a system of $N=10$ particles on a sphere at $d=7$ (shown in [FIG. \[fig:LLL\_v\_SLL\_torus\_d\_0126\_f\]]{}). On the torus the overlap is always found to be zero within error, which appears to signal that the ground state of the SLL hamiltonian is in a different $C_4$ symmetry sector from that of the LLL hamiltonian.
Next we examine the speed of convergence for modified Laughlin states, as well as the LLL–content $f$ of these states, using the LLL Coulomb hamiltonian. Results for the limiting value of $f$ as a function of $d$ and $N$ are shown in figure [FIG. \[fig:LLL\_content\_vs\_d\_N\]]{}. As expected, the value of $f$ decreases both with increasing $N$ and with increasing $d$. However, this decrease is perhaps not as fast as might be naively expected, with appreciable LLL–content still remaining even at the largest system sizes probed, especially at low values of $d$. Results on the convergence of $f$ and of the variational energy for $d=1$ and $d=2$ with $N=10$ electrons are shown in [FIG. \[fig:d\_1\_d\_2\_Ef\]]{}. On the sphere (left panels), both $f$ and the energy stabilize very quickly at $f\approx0.74$ for $d=1$ and $f\approx0.425$ for $d=2$. Similarly, on the torus, we find $f\approx0.72$ for $d=1$ and $f\approx0.397$ for $d=2$. Table \[tab:convergence\] shows how fast the energy estimate and cumulative overlap converge for $d=1$ and $d=2$. It is clear from the table that in both cases these values converge rapidly, but that for $d=1$, the convergence is faster.
The greatest impact of the fact that $f<1$ is that the overall scale of the overlaps $c_j$ is lower for these states than for the $d=0$ Laughlin state; overlaps for $d=0$ are included in the figures for comparison. Note that the $c_n$ still fall off rapidly as a function of $n$, so that the bulk of the $d=1$ state is captured using as few as 20 states on the torus and fewer still on the sphere. The energies of the $d=1$ states stabilize in a similar manner to the LLL–content. For $d=2$ (lower panels) more states are needed before $f$ has converged. On the torus, as many as $80\sim100$ states are now needed to reach stable values of $f$ and $E$. Nevertheless the number of states needed to capture the $d=2$ state at high accuracy is clearly much smaller than the full Hilbert space dimension of $10^6$ states. Similar plots for higher $d$ reveal lower limiting values for $f$ (see [FIG. \[fig:LLL\_content\_vs\_d\_N\]]{}), but interestingly, the number of states needed for stability of $E$ and $f$ does not increase much beyond what is shown for $d=2$.
Note in these plots, as is generic for the method, that we are only able to resolve overlaps down to some finite size set by the number of MC samples. This scale is set at overlaps of [*e.g.* ]{}size $10^{-4}$ on the torus for $N=10$ and $2\times 10^7$ MC samples, whereas it is at [*e.g.* ]{}$10^{-5}$ on the sphere for $N=10$ and $1.2\times10^8$ MC samples. On the torus, this can be directly observed from the band of low overlaps in the plots. These represent the zero overlaps of states with $C_4$ symmetry different from the modified Laughlin wave functions. These states could be excluded from the analysis, but only for the square torus. In other geometries these states would all have non-zero overlap and could contain important information on the reconstruction of the state being projected.
Many other tests of the energy projection could be devised. Most obviously one may apply it to other classes of well known wave functions. We have done this for example for a number of composite fermion or hierarchy states and the results are qualitatively similar to those for the (modified) Laughlin wave functions. One may also calculate the overlap of the unprojected trial wave function with high energy eigenstates of the hamiltonian, to make sure no important components of the LLL-projection at high energy are missed. Clearly for large systems this can only be done for a number of eigenstates that is much smaller than Hilbert space dimension, so one would need to have an idea where to look for the potential missing overlap. Generally one would observe the behavior for smaller systems to see if there is such high energy overlap and hope that, if there is none, it does not appear in large systems either.
-------- -------
Sphere Torus
-------- -------
Modified Laughlin states as trial wave functions
================================================
We now turn from testing the energy projection to using it as a tool to analyze the modified Laughlin states as trial wave functions for the LLL Coulomb problem. Since we will be working on the sphere and torus we give explicit expressions for the sphere and torus versions of the states below. We then go on and study the variational energies and overlaps with the exact Coulomb ground state as a function of $d$ and $N$, as well as two-point correlation functions and entanglement spectra. We will find that by letting $1<d<2$, we can significantly improve on the $d=0$ Laughlin wave function.
The explicit form of the modified Laughlin wave functions on the sphere are obtained by directly generalizing the planar wave function from [Ref. ]{} to the spherical geometry introduced in [Ref. ]{}. The wave functions on the sphere are $$\psi^{(q,d)}=\prod_{i<j}(u_iv_j-u_jv_i)^q|u_iv_j-u_jv_i|^{2d}, \label{eq:Mod_Laughlin_Sphere}$$ written in terms of spinor coordinates $u=\cos(\frac\theta2) \exp(i\frac\phi2)$ and $v=\sin(\frac\theta2) \exp(-i\frac\phi2)$. Here the spherical coordinates are (radius, polar, azimuthal) $ = (R,\theta,\phi)$, with $R=\sqrt{N_{\Phi}/2}$.
The explicit form of the modified Laughlin wave functions on the torus was introduced in [Ref. ]{} and is a natural generalization of the toroidal Laughlin wave function constructed by Haldane and Rezayi in [Ref. ]{}. The wave functions on the torus (in Landau gauge) are $$\begin{aligned}
&&\psi_n^{(q,d)}=e^{-\frac{q+2d}{2q}\sum_{i}y_{i}^{2}}\nonumber\\
&&\quad\times\prod_{i<j}
{\vartheta_{1}\!\left(\left.z_{ij}\vphantom{\tau}\right|\tau\right)}^q
|{\vartheta_{1}\!\left(\left.-\bar{z}_{ij}\vphantom{-\bar{\tau}}\right|-\bar{\tau}\right)}|^{2d}\label{eq:Mod_Laughlin_Torus}\\
&&\quad\times{\vartheta\left[\begin{array}{c} \frac nq+\alpha\\ \alpha\end{array}\right]\left(\left.\left(q+d\right)Z-d\bar Z\vphantom{\tau\left(q+d\right)-\bar{\tau}d}\right|\tau\left(q+d\right)-\bar{\tau}d\right)}.\nonumber\end{aligned}$$ Here $n=1,\ldots,q$ enumerates the different momentum sectors, and $\alpha=\frac 12(N_e-1)$ is chosen for periodic boundary conditions. We have defined $z_{ij}=\frac{z_i-z_j}{L_x}$ and $Z=\sum_{j=1}^{N_e}\frac{z_i}{L_x}$ to be the relative and center of mass coordinates respectively. The area of the torus is $L_x L_y=L_x^2\tau_2=2\pi N_\Phi\ell_B$ and the modular parameter $\tau=\tau_1+i\tau_2$ encodes the geometry of the torus. The torus version of the Jastrow factor consists of ${\vartheta_{1}\!\left(\left.z\vphantom{\tau}\right|\tau\right)}={\vartheta\left[\begin{array}{c} \frac12\\ \frac12\end{array}\right]\left(\left.z\vphantom{\tau}\right|\tau\right)}$ where
$${\vartheta\left[\begin{array}{c} a\\ b\end{array}\right]\left(\left.z\vphantom{\tau}\right|\tau\right)} = \sum_{k=-\infty}^\infty e^{i\pi\tau(k+a)^2}e^{i2\pi(k+a)(z+b)},$$ is a generalized Jacobi theta function. Since, at small $|z|$, ${\vartheta_{1}\!\left(\left.z\vphantom{\tau}\right|\tau\right)}\approx z\cdot{\vartheta_{1}^{\prime}\!\left(\left.0\vphantom{\tau}\right|\tau\right)}$, the short distance correlations of are the same as those of the planar version in .
### Coulomb overlap
We start by considering the overlap with the Coulomb ground state as a function of $d$ and $N$. In [FIG. \[fig:ground\_state\_overlap\_scaling\]]{} this is shown for both the sphere (left) and torus (right). The main feature of interest is that the overlap of the standard Laughlin state is systematically improved for all system sizes by tuning $d>0$. Values of $d$ between $d=1$ and $d=2$ give the best overlap with the Coulomb ground state. While the optimal value of $d$ is not completely independent of system size, this dependence is weak (especially on the sphere) and we note that near the optimal value of $d$ the overlap decreases only very slowly with increasing system size. For values of $d$ with lower overlaps (and notably for the standard Laughlin wave function at $d=0$), the overlap also decreases much faster with system size. The optimal squared overlap is above $0.998$ on the torus for all system sizes considered (up to $N=10$). On the sphere the system sizes go even to $N=11$ and we still obtain optimal squared overlap of $0.999$.
Note that these figures show results for many fractional values of $d$. That we are able to project wave functions that have a fractional value of $d$ is a powerful feature of the energy projection method. In many other methods this kind of projection would be difficult as it would be unclear how to handle fractional powers on the Jastrow factors. Here the projection is no more difficult than that of integer $d$, and the only extra effort lies in generating the unprojected wave functions.
### Variational energy
Another measure of the quality of a trial ground state is its variational energy. Results for the variational energy of the modified Laughlin states for various values of $d$ and $N$ are shown in [FIG. \[fig:energy\_scaling\]]{}. The energy per particle is plotted against $1/N$ to detect scaling behavior for $N\rightarrow \infty$. To obtain the correct scaling we perform the usual background subtractions and density corrections on the sphere (see [*e.g.* ]{}[Ref. ]{}, appendix I) and background subtractions on the torus[@Yoshioka83; @Bonsall59].
On the sphere we find energies lower than that of the Laughlin state in the region $0<d<3$ (for all $N$). The minimal energies at these finite system sizes are found around $d=1.3$. The energies for $0<d<3$ appear to be in a scaling region for systems from size $N=7$ upwards, enabling an attempt at computing the thermodynamic ground state energy density. For the modified Laughlin state at $d=1.3$ we thus obtain a variational energy per particle $E_{d=1.3}=-0.410149(6)$. This should be compared to the scaled Coulomb energy at $E_C=-0.410179(3)$ and the scaled Laughlin energy at $E_L=-0.40984(1)$. Limiting values of the variational energy for other values of $d$ are given in the figure. The reported errors give one standard deviation, and only reflect the uncertainty that comes from MC estimation and the linear fit. This leaves out effects from the cutoff in energy eigenstates used in the projection and more importantly any finite size effects which may still occur at larger system sizes. Nevertheless, it is clear that the modified Laughlin state at $d=1.3$ has an excess energy which is an order of magnitude smaller than that of the standard Laughlin state at all finite sizes considered and we expect this will continue to be true at larger sizes.
On the torus, finite size effects are larger and we have not performed fits of the energy for $N\rightarrow \infty$, except for the Coulomb and standard Laughlin states, which we calculated out to larger sizes than the energy projected modified Laughlin states. Because the points jump around more we have only included $d=1$, $d=3$ and $d=5$ results in this plot, to keep it readable. Nevertheless, the general picture is more or less the same as on the sphere, in that modified Laughlin states with $1<d<2$ can very significantly reduce the variational energy from that of the standard Laughlin wave function at all system sizes examined. For example at $N=10$, the energy obtained for $d=1.3$ is $E_{d=1.3}=-0.41061$ as compared to the Coulomb energy at $E_C=-0.41063$ and the Laughlin energy of $E_L=-0.4104$.
-------- -------
Sphere Torus
-------- -------
In [FIG. \[fig:GS\_energy\_scan\_of\_d\]]{}, we show the variational energy on the sphere (left) and torus (right) as a function of $d$, for $N=10$ (upper panels). It is clear that the energy is a smooth function of $d$ which is very well fit by a low order polynomial. We use the value of $d$ where this fit takes its minimum as a good estimate for the optimal $d$ at a given $N$. These optimal values of $d$ were plotted against $1/N$ (lower panels) to get an idea of the best possible value of $d$ in the thermodynamic limit. On the sphere there is again what appears to be excellent scaling behavior from $N=7$ upwards, leading to an estimated limit value $d_{\infty}=1.487\pm 0.009$. On the torus, finite size effects again appear larger, but a linear scaling fit can still be attempted leading in this case to $d_{\infty}=1.655\pm 0.12$. Again, the errors on these numbers represent a single standard deviation and do not take into account finite size effects which may manifest when considering larger sizes. It is encouraging that there appears to be proper scaling behavior of $d$, as this supports the idea that $d$ is a physical parameter of the system in the thermodynamic limit.
-------- -------
Sphere Torus
-------- -------
-------- -------
Sphere Torus
-------- -------
### Two-particle correlation functions
The intuition which led Girvin and Jach[@Girvin84] to introduce the modified Laughlin wave functions was that a nonzero $d$ would “discourage close encounters of the particles". While this seems obvious for the unprojected wave function, it is less obvious after projection. For example we may observe that the planar wave functions (\[eq:alt-L\_basic\]) are all the same for $N=2$ (after projection). Also, our results on the variational energy show that while close encounters may be discouraged for $1<d<3$, this is not so clear for large $d$, where the variational energy increases again. To directly investigate the matter we have calculated the $2$-particle correlation functions of the modified Laughlin states.
Correlation functions are shown in [FIG. \[fig:correlation\]]{} both for the sphere (left panels) and for the torus (right panels). The correlation function on the sphere depends only on the distance between the particles. We can think of it as a density plot for a system with one particle fixed at the north pole. The plot for the (square) torus has the first particle fixed at $z=0$ and is showing a diagonal cut to $z=\frac{1+i}2L$ (the diametrically opposed point of the square), with a density plot of the full 2D correlation function in the inset. In the upper panels of the figures, the correlations for the unprojected wave functions are shown as dashed lines for $d\in\{1,2,3,4\}$. In these plots we see two very clear trends with increasing $d$: the correlation hole around $z=0$ widens, showing directly that close encounters are discouraged before projection, and the oscillations at larger distances increase, showing increasing signs of the local onset of crystallization. We note that the wave length of the oscillations appears to be fairly independent of $d$ and is approximately $1.5\ell_B$. The correlation functions after projection are shown as solid lines in the same figures. We see that almost the entire effect observed before projection is reversed. This is likely due to the fact that the basis functions of the LLL only allow particles to be localized to within about a magnetic length, which limits the sharpness of any peaks in the correlation function. In order to see the remaining modifications clearly, we plot the difference between the correlation functions for the same $d$’s and the standard Laughlin wave function in the lower panels, as well as the difference between the correlation function for the Coulomb ground state and the Laughlin state. We see that the remaining effects still echo the effects observed before projection. As $d$ increases the correlation functions have increasing oscillatory behavior at the same wave length as before projection. The widening of the correlation hole is now seen to be simply part of this oscillatory behavior. On the sphere we note that the strongest effects of the modification are at longer distances with the correlation hole less affected. Note that the use of chord length in [FIG. \[fig:correlation\]]{}, instead of arc length, makes the oscillations at larger $r_{12}$ on the sphere appear to have shorter wavelength than is actually the case. We also see, in good agreement with what we know from the overlap and energy, that the best fit to the Coulomb correlation function lies somewhere in between $d=1$ and $d=2$. The Coulomb ground state clearly has stronger long range oscillations than $\psi_L$, which fits with intuition, since $\psi_{L}$ is the ground state of an ultra short ranged interaction, while the Coulomb interaction is long ranged. We can think of the introduction of a nonzero $d$ as a way to reintroduce these longer range oscillations.
For the sphere we have also added the curve for $g_{d=1.3}$. We see that $g_{d=1.3}\approx g_C$ to very good accuracy, especially at shorter distances, where the difference is imperceptible in the plot. At the longest distances, $g_C$ is closer to $g_{d=2}$. Perhaps this is related to the fact that $d$ drifts towards $1.5$ at large $N$, where longer distances can be probed. Finally, we note that while close encounters may be discouraged for $1<d<3$, For large d, the correlation functions start to show an increased probability to find pairs of particles at a distance of approximately $1$ magnetic length (where each particle would be inside the other’s “correlation hole”).
Entanglement spectra sphere
-- --
-- --
Entanglement spectra torus
------------------------------------------------------------- --------------------------------------------------------- ----------------------------------------------------------
{width="6cm"} {width="6cm"} {width="6cm"}
------------------------------------------------------------- --------------------------------------------------------- ----------------------------------------------------------
### Entanglement spectra
Finally, we consider the entanglement spectrum[@Li08] (ES) of the modified Laughlin states, using the orbital cut introduced in [Ref. ]{}. The entanglement spectrum is a powerful tool for the determination of the topological order of gapped systems. For Hall states, the low lying part of the ES of a system on the sphere often resembles the spectrum of the chiral conformal field theory (CFT) describing the modes propagating along the circular edge of the corresponding state on a disk[@Li08]. For Hall systems on the torus, the ES will resemble the edge spectrum on a cylinder, where the edge consists of two circles governed by counterpropagating versions of the same chiral CFT[@Lauchli10]. The Laughlin state has the special property that *all* states in its orbital ES correspond to states in its edge CFT. The ES of the exact Coulomb ground state on the other hand has a clearly identifiable low lying branch corresponding to the ES of the Laughlin state, but in addition has many other states in higher branches. These states can be attributed to components of the Coulomb ground state which can be thought of as neutral bulk excitations of the Laughlin state[@Sterdyniak11_NewJPhys].
In [FIG. \[fig:ES\_sphere\]]{} we show the orbital ES of a number of systems of $N=10$ particles on a sphere. In all four panels, we show the ES of the Coulomb ground state (blue dashes) with superimposed on it the ES of the energy projected state (red crosses). The top left figure shows the ES of the Laughlin state as determined from its energy projection (we can think of it as the $d=0$ state). We clearly see from the graph that the Laughlin state indeed reproduces the lowest branch of the Coulomb ES but not the higher ones. We also see that at values of the entanglement energy $\xi$ above $20$ there are many spurious states in the $d=0$ ES which would not appear if we had used the exact Laughlin state rather than its energy projection. These states appear purely due to the error of the energy projection. The scale at which they first appear can in principle be shifted upward by taking more MC samples. It is clear that with the amount of MC samples we have taken here, states with $\xi>20$ can be safely discarded as noise and we have therefore cut off the scale at this level in the other panels of [FIG. \[fig:ES\_sphere\]]{}. In the upper right panel, we consider the ES of the $d=0.5$ modified Laughlin state. We see that there is still a good fit to the $d=0$ branch of the ES but additional branches of states are swooping down from above as a result of setting $d>0$. In the bottom left panel we consider $d=1.3$ which gives more or less the optimal fit to the Coulomb energy as well as the highest overlap at this system size. We see that the branches of the ES have now settled very closely to the location where they are in the Coulomb ES. The fit of the low lying ES (in the Laughlin branch) is also noticeably improved for $d=1.3$ for entanglement energies up to $\xi\approx10$. While the detailed positioning of the individual levels within the higher branches does not always match very well, we stress that the structure of these branches, [*i.e.* ]{}the counting of levels at each angular momentum, is identical to that of the Coulomb ES, even though this may not always be obvious from the plot. We also notice that while the overall trend in raising $d$ has been to bring levels down out of the noise, there are some exceptions at low angular momentum, where the entanglement energies of some levels have risen from values below to values above those of the Coulomb ES. These trends continue for higher values of $d$. The ES for $d=5$ is shown in the bottom right panel. Even at $d=5$ the lower part of the Laughlin branch of the ES is still mostly in place, except for some levels at low angular momentum which seem to have migrated up into the noise. On the other hand the higher branches visible at large angular momenta have now all descended well below the corresponding Coulomb branches.
Entanglement spectra for $N=10$ electrons on the torus are shown [FIG. \[fig:ES\_torus\]]{}. In the left panel we again compare the energy projected $d=0$ state to the exact Laughlin state to give an idea of the Monte Carlo noise on the data for the modified Laughlin wave functions. Any disagreement between the $d=0$ and exact Laughlin states’ ES is due to the error in the determination of the $d=0$ state, which would be exactly equal to the Laughlin state if this error was zero. We see that the noise in the ES in this case becomes severe above entanglement energy $\xi\approx 12$. This more severe noise, as compared to the sphere ES, is due to the fact that the data is based on fewer MC samples. In the middle panel, we show the exact Laughlin ES superimposed on the ES for the exact Coulomb ground state. As on the sphere, there is good matching of the low lying levels, but many higher lying levels from $\xi\approx 8$ upwards are completely missing from the Laughlin ES. In the right hand panel, we show the ES for $d=1.5$, which is close to optimal. We find that all levels in the Coulomb ES are now reproduced with excellent matching of the entanglement energies. The entanglement energies obtained for levels that were already present in the Coulomb ES is also visibly improved. Overall, the ES for the torus shows similar feature to those for the sphere. As $d$ is increased, levels which come from the higher branches of the Coulomb ES come down. There is also a tendency for levels that are far from the center of the conformal towers of states to be shifted up, which is analogous to the shifting up of high angular momentum states on the sphere.
Discussion {#sec:discussion}
==========
We have introduced the energy projection (EP) as a method for projecting quantum Hall trial wave functions to the lowest Landau level. In effect we replace the LLL projection by projection onto the low lying spectrum of a suitable hamiltonian acting in the LLL, carefully checking convergence of this projection to what should be the full LLL projection. The method works well for all states we have considered, up to system sizes where a few hundred states are accessible by exact diagonalization. We have shown this here in some detail for the Laughlin state and for the modified Laughlin states proposed by Girvin and Jach in [Ref. ]{}.
We have also applied EP to investigate the modified Laughlin states as trial wave functions for the Coulomb ground state at filling $\nu=\frac{1}{3}$. It turns out that these states allow for significant improvements over the standard Laughlin state. For example the squared overlap with the Coulomb ground state of a system of $N=11$ electrons on the sphere is improved from $\sim0.98$ for the Laughlin state to $\sim0.999$ for the modified state at $d=1.3$. On the torus at $N=10$, there is a similar improvement from $\sim0.97$ for the Laughlin state to values above $0.998$ for the modified states with $1.4<d<1.9$. The variational energy per particle can also be improved from that of the Laughlin state at the finite sizes we considered and likely also in the thermodynamic limit as can be seen from the scaling results in [FIG. \[fig:energy\_scaling\]]{}. We also investigated the two particle correlation functions of the modified states and found that, compared to the standard Laughlin state, the states at $d>0$ exhibit more pronounced medium range oscillations, which allows them to better mimic the Coulomb ground state. While close encounters of the particles are to some extent discouraged at $d>0$ (as expected by Girvin and Jach), the much improved matching with the Coulomb ground state’s longer range oscillations is at least as striking. Turning to the entanglement spectrum, we find that introducing even a small nonzero $d$ brings forward the branches of the Coulomb ES at higher entanglement energies that are completely missing from the Laughlin ES. Using the optimal values of $d$ allows for a very good qualitative fit of the entire Coulomb ES as well as a good quantitative fit at low entanglement energy.
There can be little argument that the modified Laughlin states describe the same universality class as the usual Laughlin state. All observables we have calculated show very smooth behavior as a function of $d$. Of course we are limited to small system sizes, but, especially on the sphere, we appear to nevertheless reach the scaling region at least for the energy and for the optimal value of $d$ (see [FIG. \[fig:GS\_energy\_scan\_of\_d\]]{}). The entanglement spectra also show a stable low lying Laughlin type branch for a broad range of $d$-values.
A natural extension of this work is a study of modified Laughlin states with excitations, such as quasiholes, quasiparticles and excitons. Trial wave functions for these can be constructed by applying modification factors similar to those in to the Laughlin state with excitations. However, this is not the only possible way. One may also introduce additional variational parameters modifying the quasihole profile or construct excitations using a CF construction based on reverse flux attachment ([*e.g.* ]{}at $d=1$). We can also consider different filling fractions, especially $\nu=1/5$. Early indications are that improvements over the Laughlin state similar to those at $\nu=1/3$ can be obtained there but at substantially higher values of $d$. The energy projection can be used to study all these possibilities and we intend to report on a number of them shortly[@FremlingWIP].
The fact that the energy projection is a controlled approximation allows one also to use it to test the Jain-Kamilla (JK) type projections used for numerical work on composite fermion and BS–hierarchy[@Bonderson08] wave functions against exact projection for larger system sizes than were possible up to now. We are in the process of doing this as part of a larger study of reverse flux CF wave functions[@FulsebakkeWIP]. The EP can also be used to evaluate the CF wave functions on the torus. Work on these was recently done by Hermanns[@Hermanns13_PRB] but the wave functions could only be evaluated for a very small number of particles. Using EP the wave functions could be tested at larger system sizes. We have some hope that the EP may also help alleviate computational difficulties other than the LLL-projection, notably explicit symmetrization and antisymmetrization of trial wave functions.
Other ways to improve the Laughlin wave function include the fixed phase quantum Monte Carlo method of [Ref. ]{}, which can find the optimal wave function when the phase of the function is given. It would be interesting to compare the results from this method to the best results obtained using the single parameter family of states considered here, and also potentially to try and further improve the modified Laughlin states using this method. We have checked by direct analytic calculation for small systems that the phase of the modified Laughlin wave functions does depend on $d$ and in particular that it is not the same as the phase of the standard Laughlin wave function. Recently there has been much interest also in modifications of Hall states (including the Laughlin state) by the introduction of geometric anisotropy[@Haldane11; @Qiu12; @Yang12] and it would be interesting to generalize the modified Laughlin states to this context also.
Going beyond the Laughlin states, modifications similar to those in can be made to any planar or spherical trial wave function. This could thus be used to massage the CF wave functions of the Jain series, but can also be applied to more exotic wave functions such as [*e.g.* ]{}the Moore-Read Pfaffian wave function[@Moore91] at $\nu=5/2$ or its generalizations such as the Read-Rezayi[@Read99] or BS–hierarchy[@Bonderson08] wave functions.
The modification made to the Laughlin wave functions can also be easily generalized. In fact, the modified Laughlin states are only the simplest type of modified states in a large class of wave functions which can be described using Wen’s $K$-matrix formalism[@Wen92a]. For any such wave function, one may split the $K$-matrix into a holomorphic and an anti-holomorphic part, writing $K=\kappa-\bar{\kappa}$, where $\kappa$ and $\bar{\kappa}$ are both positive definite[@Suorsa11_PRB; @Suorsa11_NewJPhys; @Hansson16]. For the Laughlin state at filling $\nu=\frac{1}{3}$, we simply have the $1\times 1$ matrix $K=3$, with the modified states obtained using $\kappa=3+d$ and $\bar{\kappa}=d$. For multi-layer states or states based on CF constructions with multiple Landau levels, the $K$-matrix will be higher dimensional and many more modifications become possible. States with counterpropagating edge modes must be realized with nonzero $\kappa$ and $\bar{\kappa}$ and in such cases the EP may be the only way to evaluate them at reasonable system sizes. Such non-chiral states would include for example the $\nu=2/3$ state, especially on the torus where other approximate projection methods are not available.
We would like to stress that the division of the K-matrix into holomorphic and anti-holomorphic parts does not introduce any extra (counter-propagating) edge modes. The chiralities of the edge modes are given by the signs of eigenvalues of the full K-matrix[@Wen92b], and are independent of this decomposition. Thus we expect to get only one edge mode for all the modified Laughlin states, in the same way that we expect one chiral and one anti-chiral edge mode in the case of $\nu=2/3$. This is also supported by the entanglement spectra in [FIG. \[fig:ES\_sphere\]]{} and [FIG. \[fig:ES\_torus\]]{}.
Computations {#sec:computations}
============
This project entailed significant numerical computations using a mix of freely available codes as well as codes developed in-house by the authors.
A set of codes, christened Hammer[^1] were developed by the authors and were used for the majority of the computations. These codes have many notable features. They provide a diagonalisation code with the ability to accurately resolve large numbers of eigenstates for large sparse matrices by employing the Krylov subspace methods provided by SLEPc [@Hernandez:2005:SSF] and taking advantage of large distributed memory machines. This code can exploit many symmetries of the hamiltonian to reduce the computational effort and provide additional quantum numbers. It includes utilities to calculate many other quantities including entanglement spectra, correlation functions and Hall viscosities. There is a significant functionality for performing Monte Carlo (MC) simulations on both the sphere and torus, with utilities that can efficiently evaluate many trial wave-functions.
The DiagHam package [^2] is a freely available set of utilities for performing calculations of FQH systems. This package was used for the following computations on the sphere: initial diagonalization calculations for small systems, real space evaluation of Fock space wave-functions and the calculation of entanglement spectrum and correlation functions.
We thank Hans Hansson for pointing us in the direction of the modified Laughlin wave functions and Steve Simon and Andrei Bernevig for useful discussions. JKS thanks Nordita for its hospitality during the program on Physics of Interfaces and Layered Structures (2015), when part of this work was done. NM thanks the organisers of the TOPO2015 workshop held at the Institut d’Etudes Scientifiques de Cargèse where part of this work and related discussions took place. All authors acknowledge financial support through SFI Principal Investigator Award 12/IA/1697. We also wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support.
[^1]: http://www.thphys.nuim.ie/hammer
[^2]: http://nick-ux.lpa.ens.fr/diagham/wiki
|
---
abstract: |
We consider the totally asymmetric simple exclusion process (TASEP) on a periodic one-dimensional lattice of $L$ sites. Using Bethe ansatz, we derive parametric formulas for the eigenvalues of its generator in the thermodynamic limit. This allows to study the curve delimiting the edge of the spectrum in the complex plane. A functional integration over the eigenstates leads to an expression for the density of eigenvalues in the bulk of the spectrum. The density vanishes with an exponent $2/5$ close to the eigenvalue $0$.\
[**PACS numbers:** 02.30.Ik 02.50.Ga 05.40.-a 05.60.Cd]{}\
[**Keywords:** TASEP, non-Hermitian operator, complex spectrum, Bethe ansatz, functional integration]{}
address: |
Laboratoire de Physique Théorique, IRSAMC, UPS, Université de Toulouse, France\
Laboratoire de Physique Théorique, UMR 5152, Toulouse, CNRS, France
author:
- Sylvain Prolhac
title: 'Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues'
---
[Introduction]{} \[section introduction\] Markov processes [@S70.1] form a class of mathematical models much studied in relation with non-equilibrium statistical physics. Their evolution in time is generated by an operator $M$, the Markov matrix, whose non-diagonal entries represent the rates at which the state of the system changes from a given microstate to another.\
For processes verifying the detailed balance condition, which forbids probability currents between the different microstates at equilibrium, the operator $M$ is real symmetric, up to a similarity transformation. It implies that its eigenvalues are real numbers. An example is the Ising model with *e.g.* Glauber dynamics. Processes that do not satisfy detailed balance, on the other hand, generally have a complex spectrum. This is the case for the asymmetric simple exclusion process (ASEP) [@D98.1; @S01.1; @GM06.1; @D07.1; @S07.1; @FS11.1; @M11.1; @CMZ11.1], which consists of classical hard-core particles hopping between nearest neighbour sites of a lattice with a preferred direction.\
In one dimension, ASEP is known to be exactly solvable by means of Bethe ansatz. This has allowed exact calculations of the gap of the spectrum [@GS92.1; @GS92.2; @K95.1; @GM04.1; @GM05.1] and of the fluctuations of the current, both in the infinite line setting [@J00.1; @PS02.1; @BFPS07.1; @TW09.1; @SS10.1; @ACQ11.1] and on a finite lattice with either periodic [@DL98.1; @DA99.1; @PM08.1; @P08.1; @PM09.1; @P10.1] or open [@LM11.1; @GLMV12.1; @L13.1] boundary conditions. The exponents and scaling functions obtained in these articles are universal: they characterize not only driven diffusive systems [@KLS84.1; @SZ98.1] far from equilibrium, to which ASEP belongs, but also interface growth models [@BS95.1; @M98.1; @HHZ95.1] and directed polymers in a random medium [@HHZ95.1; @BD00.1; @D10.1; @CLDR10.1]. This forms the Kardar-Parisi-Zhang universality class [@KPZ86.1; @KK10.1; @SS10.4].\
We focus in this article on the special case of ASEP with unidirectional hopping of the particles called totally asymmetric simple exclusion process (TASEP). We consider the system with periodic boundary conditions, for which the number of microstates is finite: the spectrum is then a discrete set of points in the complex plane. The aim of the present article is to obtain a large scale description of these points. We obtain explicit expressions for the curve delimiting the edge of the spectrum in the complex plane. This curve is singular near the eigenvalue $0$, with an imaginary part scaling as the real part to the power $5/3$. By a functional integration over the eigenstates, we also derive expressions for the density of eigenvalues. Near the eigenvalue $0$, the density vanishes with an exponent $2/5$.\
The paper is organized as follows: in section \[section TASEP\], we recall a few known things about TASEP and present our main results. In section \[section eigenvalues\], we derive parametric expressions for the eigenvalues of the Markov matrix of TASEP in the thermodynamic limit, and apply this to the curve delimiting the edge of the spectrum in section \[section envelope\]. In section \[section density\], we study the density of eigenvalues. Finally in section \[section trace\] we consider a generating function for the cumulants of the eigenvalues.
[Totally asymmetric exclusion process on a ring]{} \[section TASEP\] We consider in this article the totally asymmetric simple exclusion process (TASEP) with $N$ particles on a periodic one-dimensional lattice of $L$ sites (see fig. \[fig TASEP\]). A site is either empty or occupied by one particle. We call $\Omega$ the set of all microstates (or *configurations*), which has cardinal $|\Omega|={{L \choose N}}$. A particle at site $i$ can hop to site $i+1$ if the latter is empty. The hopping rate for any particle is equal to $1$, *i.e.* each particle allowed to move has a probability $\delta t$ in a small time interval $\delta t$. In the following, we call $\rho=N/L$ the density of particles.
(70,50) (35,25) (35,25) (35,45)[(0,1)[5]{}]{} (42.6,43.48)[(5,12)[1.92]{}]{} (49.2,39.2)[(1,1)[3.54]{}]{} (53.48,32.6)[(12,5)[4.62]{}]{} (55,25)[(1,0)[5]{}]{} (53.48,17.4)[(12,-5)[4.62]{}]{} (49.2,10.8)[(1,-1)[3.54]{}]{} (42.6,6.52)[(5,-12)[1.92]{}]{} (35,5)[(0,-1)[5]{}]{} (27.4,6.52)[(-5,-12)[1.92]{}]{} (20.8,10.8)[(-1,-1)[3.54]{}]{} (16.52,17.4)[(-12,-5)[4.62]{}]{} (15,25)[(-1,0)[5]{}]{} (16.52,32.6)[(-12,5)[4.62]{}]{} (20.8,39.2)[(-1,1)[3.54]{}]{} (27.4,43.48)[(-5,12)[1.92]{}]{} (53.8,37.4) (57.1,29.3) (13,20.7) (62,28)(65,25)(62,22) (62,22)[(-1,-1)[0.5]{}]{} (8,22)(5,25)(8,28) (8,28)[(1,1)[0.5]{}]{} (65,24.5)[$1$]{} (3.5,25)[$1$]{}
[Master equation]{} \[subsection master equation\] We call $P_{t}(\mathcal{C})$ the probability to observe the system in the microstate $\mathcal{C}$ at time $t$. The probabilities evolve in time by the *master equation* $$\label{master eq}
\frac{dP_{t}(\mathcal{C})}{dt}=\sum_{\mathcal{C}'\neq\mathcal{C}}\Big[w(\mathcal{C}\leftarrow\mathcal{C}')P_{t}(\mathcal{C}')-w(\mathcal{C}'\leftarrow\mathcal{C})P_{t}(\mathcal{C})\Big]\;.$$ The rate $w(\mathcal{C}'\leftarrow\mathcal{C})$ is equal to $1$ if it is possible to go from configuration $\mathcal{C}$ to configuration $\mathcal{C}'$ by moving one particle to the next site, and $0$ otherwise.\
The master equation (\[master eq\]) can be conveniently written as a matrix equation by defining the vector $|P_{t}\rangle=\sum_{\mathcal{C}\in\Omega}P_{t}(\mathcal{C})|\mathcal{C}\rangle$ of the configuration space $V$ with dimension $|\Omega|$, where $|\mathcal{C}\rangle$ is the canonical vector of $V$ corresponding to the configuration $\mathcal{C}$. Then, calling $M$ the matrix with non-diagonal entries $\langle\mathcal{C}|M|\mathcal{C}'\rangle=w(\mathcal{C}\leftarrow\mathcal{C}')$ and diagonal $\langle\mathcal{C}|M|\mathcal{C}\rangle=-\sum_{\mathcal{C}'\neq\mathcal{C}}w(\mathcal{C}'\leftarrow\mathcal{C})$, one has $$\label{master eq matrix}
\frac{d}{dt}|P_{t}\rangle=M|P_{t}\rangle\;,$$ which is formally solved in terms of the time evolution operator $\rme^{tM}$ as $|P_{t}\rangle=\rme^{tM}|P_{0}\rangle$.\
The graph of allowed transitions for TASEP presents an interesting cyclic structure with period $L$: let us consider the observable $X$ such that $X(C)$ is the sum of the positions of the particles (we take a fixed arbitrary site to be the origin of positions). We see that each time a particle hops to the next site, $X(C)$ increases of $1$ modulo $L$. It is then possible to split the configurations in $L$ sectors according to the value of $X$ modulo $L$. The only allowed transitions between configurations are then transitions from configurations of a sector $r$ to configurations of sector $r+1$ (modulo $L$), see fig. \[fig graph dynamics TASEP\]. We emphasize that this cyclic structure is not a consequence of the periodic boundary conditions. Indeed, a similar cyclic structure exists for ASEP on an open segment of $L$ sites connected to reservoirs of particles, with periodicity $L+1$ instead of $L$.
(130,115) (67.5,92.5) (72.5,92.5) (62.5,102.5) (77.5,102.5) (52.5,112.5) (57.5,112.5) (50,90)(0,10)[3]{}[ (0,0)(0,5)[2]{}[(1,0)[30]{}]{} (0,0)(5,0)[7]{}[(0,1)[5]{}]{} ]{} (82.5,92.5)[(1,0)[16.3]{}]{} (82.5,101.5)[(5,-2)[18.3]{}]{} (82.5,103.5)[(9,-2)[25.3]{}]{} (82.5,112.5)[(2,-1)[26.3]{}]{} (112.5,77.5) (118.5,69.5) (111.5,95.5) (117.5,87.5) (100,90)(8,6)[2]{}[ (0,0)(4,3)[2]{}[(3,-4)[18]{}]{} (0,0)(3,-4)[7]{}[(4,3)[4]{}]{} ]{} (119.8,65.2)[(-4,-9)[7.2]{}]{} (121.8,66.2)[(-1,-10)[2.3]{}]{} (127,71)[(-2,-9)[6.2]{}]{} (128.8,72.2)[(0,-1)[35.2]{}]{} (120.5,26.5) (123.5,30.5) (109.5,28.5) (118.5,40.5) (95.5,26.5) (98.5,30.5) (108,14)(-8,6)[3]{}[ (0,0)(4,-3)[2]{}[(3,4)[18]{}]{} (0,0)(3,4)[7]{}[(4,-3)[4]{}]{} ]{} (93,23.5)[(-1,-1)[10.3]{}]{} (100,18)[(-3,-1)[17.3]{}]{} (102,17)[(-9,-5)[20.3]{}]{} (108,12)[(-3,-1)[25.3]{}]{} (62.5,2.5) (72.5,2.5) (57.5,12.5) (77.5,12.5) (50,0)(0,10)[2]{}[ (0,0)(0,5)[2]{}[(1,0)[30]{}]{} (0,0)(5,0)[7]{}[(0,1)[5]{}]{} ]{} (47,13.5)[(-1,1)[10.3]{}]{} (47,12.33333)[(-3,1)[17.3]{}]{} (48,5.88888)[(-9,5)[20.3]{}]{} (47,3.66666)[(-3,1)[25.3]{}]{} (12.5,22.5) (9.5,26.5) (14.5,36.5) (23.5,24.5) (19.5,46.5) (34.5,26.5) (22,14)(8,6)[3]{}[ (0,0)(-4,-3)[2]{}[(-3,4)[18]{}]{} (0,0)(-3,4)[7]{}[(-4,-3)[4]{}]{} ]{} (17.2,49.45)[(-4,9)[7.2]{}]{} (10.4,44.2)[(-1,10)[2.3]{}]{} (9,44)[(-2,9)[6.2]{}]{} (1.2,37.3)[(0,1)[35.3]{}]{} (14.5,73.5) (26.5,89.5) (9.5,83.5) (15.5,91.5) (30,90)(-8,6)[2]{}[ (0,0)(-4,3)[2]{}[(-3,-4)[18]{}]{} (0,0)(-3,-4)[7]{}[(-4,3)[4]{}]{} ]{} (22.5,97.5)[(5,-1)[25.3]{}]{} (29.5,91.5)[(9,5)[18.3]{}]{} (22.5,99.33333)[(6,1)[25.3]{}]{} (28.5,93.5)[(1,1)[19.3]{}]{}
[Current fluctuations]{} \[subsection current\] Equal time observables, such as the density profile of particles in the system or the number of clusters of consecutive particles can be extracted directly from the knowledge of $|P_{t}\rangle$. Other observables, however, require the $P_{t}(\mathcal{C})$ for several values of the time. A much studied example is the current of particles and especially the fluctuations around its mean value.\
We define the observable $Y_{t}$, which counts the total displacement of particles between time $0$ and time $t$. Starting with $Y_{0}=0$, it is then updated by $Y_{t}\to Y_{t}+1$ each time a particle hops anywhere in the system.\
The joint probability $P_{t}(\mathcal{C},Y)$ to observe the system in the configuration $\mathcal{C}$ with $Y_{t}=Y$ obeys the master equation $$\label{master eq Y}
\frac{dP_{t}(\mathcal{C},Y)}{dt}=\sum_{\mathcal{C}'\neq\mathcal{C}}\Big[w(\mathcal{C}\leftarrow\mathcal{C}')P_{t}(\mathcal{C}',Y-1)-w(\mathcal{C}'\leftarrow\mathcal{C})P_{t}(\mathcal{C},Y)\Big]\;.$$ It is convenient to introduce the quantity $F_{t}(\mathcal{C})=\sum_{Y=-\infty}^{\infty}\rme^{\gamma Y}P_{t}(\mathcal{C},Y)$. It verifies the deformed master equation [@DL98.1] $$\label{master eq gamma}
\frac{dF_{t}(\mathcal{C})}{dt}=\sum_{\mathcal{C}'\neq\mathcal{C}}\Big[\rme^{\gamma}w(\mathcal{C}\leftarrow\mathcal{C}')F_{t}(\mathcal{C}')-w(\mathcal{C}'\leftarrow\mathcal{C})F_{t}(\mathcal{C})\Big]\;.$$ Introducing the vector $|F_{t}\rangle=\sum_{\mathcal{C}\in\Omega}F_{t}(\mathcal{C})|\mathcal{C}\rangle$ and a deformation $M(\gamma)$ of the Markov matrix, one has $$\label{master eq matrix}
\frac{d}{dt}|F_{t}\rangle=M(\gamma)|F_{t}\rangle\;.$$ For $\gamma=0$, $F_{t}$ reduces to $P_{t}$ and $M(0)=M$. In the following, we will be interested in the spectrum of the operators $M$ and $M(\gamma)$.
[Mapping to a height model and continuous spectrum]{} It is well known that TASEP can be mapped to a model of growing interface [@HHZ95.1]: for each occupied site $i$ of the system, we draw a portion of interface decreasing from height $h_{i}$ to $h_{i+1}=h_{i}-(1-\rho)$, and for each empty site $i$, we draw a portion of interface increasing from height $h_{i}$ to $h_{i+1}=h_{i}+\rho$. The interface obtained is continuous and periodic, see fig. \[fig growth model\]. The dynamics of TASEP then implies that parallelograms (squares at half-filling $N=L/2$) deposit on local minima of the interface with rate 1.\
Unlike TASEP, the set of microstates of the growth model is not finite since the total height is not bounded: there always exists a local minimum from which the interface can grow. It is possible to identify any microstate of the growth model by the corresponding configuration of the exclusion process and the total current $Y$ defined in section \[subsection current\]. Then, (\[master eq Y\]) can be interpreted as the master equation for the growth model, to which is associated the infinite dimensional Markov matrix $$\mathcal{M}=M^{(+)}\otimes S+M^{(0)}\otimes{\mbox{{\small 1}$\!\!$1}}\;,$$ where $M^{(0)}$ and $M^{(+)}$ are respectively the diagonal and non-diagonal part of $M$, and $S$ is the translation operator in $Y$ space, $S=\sum_{Y=-\infty}^{\infty}|Y\rangle\langle Y-1|$.\
We would like to diagonalize $\mathcal{M}$ in order to study its spectrum. We note that $\mathcal{M}$ commutes with $S$. This implies that the eigenvectors $|\Psi\rangle$ of $\mathcal{M}$ must be of the form $$|\Psi\rangle=|\mathcal{\psi}\rangle\otimes|\phi(\theta)\rangle\;,$$ where $|\psi\rangle$ is a vector in configuration space and $$|\phi(\theta)\rangle=\frac{1}{\sqrt{2\pi}}\sum_{Y=-\infty}^{\infty}\rme^{-\rmi\,\theta\,Y}|Y\rangle\;$$ the right eigenvector of $S$ with eigenvalue $\rme^{\rmi\,\theta}$. The eigenvalue equation for $|\Psi\rangle$ can be written $$E|\Psi\rangle=[(\rme^{\rmi\,\theta}M^{(+)}+M^{(0)})|\psi\rangle]\otimes|\phi(\theta)\rangle=[M(\rmi\,\theta)|\psi\rangle]\otimes|\phi(\theta)\rangle\;,$$ where $M(\rmi\,\theta)$ is the deformation of the Markov matrix introduced in section \[subsection current\] and $|\psi\rangle$ and $E$ an eigenvector and an eigenvalue of $M(\rmi\,\theta)$. We finally find that the spectrum of $\mathcal{M}$ is the reunion of the spectra of the $M(\gamma)$ with $|\rme^{\gamma}|=1$. This spectrum is represented in fig. \[fig continuous spectrum\] for $L=8$, $N=4$.\
The spectrum of $\mathcal{M}$ can also be constructed from the finite spectra obtained by counting the total current $Y$ modulo $KL$ with $K$ a positive integer, and taking the limit $K\to\infty$. The case $K=1$ is the usual TASEP because of the cyclic structure of the graph of allowed transitions discussed at the end of section \[subsection master equation\]. The case $K=N$ corresponds to TASEP with distinguishable particles, restricted to a subspace with a given cyclic order of the particles since the particles cannot overtake each other.\
We call $M^{(K)}(\gamma)$ the corresponding deformed Markov matrix. The $K|\Omega|$ configurations arrange themselves in $KL$ sectors according to the value of $Y$. Calling $P_{r}$ the projector on the $r$-th sector and $M_{r,r+1}^{(K)}(\gamma)=\rme^{\gamma}P_{r+1}M^{(K)}P_{r}+P_{r}M^{(K)}P_{r}$, one has $$M^{(K)}(\gamma)=\sum_{r=1}^{KL}M_{r,r+1}^{(K)}(\gamma)\;.$$ Introducing $U=\sum_{r=1}^{KL}\rme^{r\gamma}P_{r}$, one finds $$U^{-1}M^{(K)}(\gamma)U=\sum_{r=1}^{KL-1}M_{r,r+1}^{(K)}+\rme^{KL\gamma}P_{1}M_{KL,1}^{(K)}P_{KL}+P_{KL}M_{KL,1}^{(K)}P_{KL}\;,$$ with $M_{r,r+1}^{(K)}=M_{r,r+1}^{(K)}(0)$. This implies that the spectrum of $M^{(K)}(\gamma)$ is invariant under the transformation $\gamma\to\gamma+2\rmi\pi/(KL)$. Starting from a given eigenvalue of $M^{(K)}(\gamma)$ and following the eigenvalue during the continuous change from $\gamma$ to $\gamma+2\rmi\pi/(KL)$, one does not in general come back to the initial eigenvalue: we observe by numerical diagonalization that one goes from one eigenvalue to the next one anticlockwise on the same continuous curve in fig. \[fig continuous spectrum\].
(140,70) (15,3) (35,3) (45,3) (85,3) (95,3) (105,3) (45,5)(50,10)(55,5) (55,5)[(1,-1)[0.2]{}]{} (50,10)[$1$]{} (10,0)[(1,0)[120]{}]{} (10,0)[(0,1)[5]{}]{} (20,0)[(0,1)[5]{}]{} (30,0)[(0,1)[5]{}]{} (40,0)[(0,1)[5]{}]{} (50,0)[(0,1)[5]{}]{} (60,0)[(0,1)[5]{}]{} (70,0)[(0,1)[5]{}]{} (80,0)[(0,1)[5]{}]{} (90,0)[(0,1)[5]{}]{} (100,0)[(0,1)[5]{}]{} (110,0)[(0,1)[5]{}]{} (120,0)[(0,1)[5]{}]{} (130,0)[(0,1)[5]{}]{} [(10,20)(1,1)[10]{}[(0,0)[$.$]{}]{}]{} [(20,30)(1,-1)[10]{}[(0,-1)[$.$]{}]{}]{} [(30,20)(1,1)[10]{}[(0,0)[$.$]{}]{}]{} [(40,30)[(1,-1)[10]{}]{}]{} [(50,20)[(1,1)[10]{}]{}]{} [(60,30)(1,-1)[10]{}[(0,-1)[$.$]{}]{}]{} [(70,20)(1,1)[10]{}[(0,0)[$.$]{}]{}]{} [(80,30)(1,-1)[10]{}[(0,-1)[$.$]{}]{}]{} [(90,20)(1,1)[10]{}[(0,0)[$.$]{}]{}]{} [(100,30)[(1,-1)[10]{}]{}]{} [(110,20)[(1,1)[10]{}]{}]{} [(120,30)(1,-1)[10]{}[(0,-1)[$.$]{}]{}]{} [(10,40)[(1,-1)[10]{}]{}]{} [(20,30)[(1,1)[10]{}]{}]{} [(30,40)[(1,-1)[10]{}]{}]{} [(60,30)[(1,1)[10]{}]{}]{} [(70,40)[(1,-1)[10]{}]{}]{} [(80,30)[(1,1)[10]{}]{}]{} [(90,40)[(1,-1)[10]{}]{}]{} [(120,30)[(1,1)[10]{}]{}]{} (50,37)[(0,-1)[14]{}]{} (51,30)[$1$]{} [(40,50)[(1,-1)[10]{}]{}]{} [(50,40)[(1,1)[10]{}]{}]{} [(40,50)[(1,1)[10]{}]{}]{} [(50,60)[(1,-1)[10]{}]{}]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Spectrum of the Markov matrix of TASEP with $4$ particles on $8$ sites. The black dots are the eigenvalues for undistinguishable particles, while the black $+$ red (gray in printed version) dots are the eigenvalues for distinguishable particles. The curves are the eigenvalues of the corresponding interface growth model.[]{data-label="fig continuous spectrum"}](ContinuousSpectrum.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Large scale description of the spectrum]{} Typical eigenvalues of the Markov matrix $M$ scale proportionally with $L$. Dividing all the eigenvalues by $L$, the rescaled spectrum fills a region of the complex plane in the thermodynamic limit. We call $e(\mu)$ a parametrization of the curve at the edge of this region. An exact representation of this curve, (\[e\[d\]\]), (\[d’\[d\]\]), is obtained in section \[section envelope\]. We observe that $e(\mu)$ is singular near $e=0$, with the scaling $$\label{Ree[Ime]}
{\mathrm{Re}}\,e\simeq-\frac{2^{2/3}3^{7/6}\pi^{2/3}}{10}|{\mathrm{Im}}\,e|^{5/3}\;.$$ The density of eigenvalues $D(e)$ in the rescaled spectrum is also studied in section \[section density\]. It grows for large $L$ as $$\label{D[s]}
D(e)\sim\rme^{L s(e)}\;.$$ The quantity $s(e)$ is computed exactly. For eigenvalues close to $0$, but far from the edge of the spectrum (*i.e.* with $|{\mathrm{Im}}\,e|^{5/3}\ll|{\mathrm{Re}}\,e|$), one finds in particular $$\label{s(e)}
\displaystyle s(e)\simeq\xi(-{\mathrm{Re}}\,e)^{2/5}
\quad
{\mathrm{with}}
\quad
\xi\simeq1.58600
\;.$$ The exponent $2/5$ is different from the exponent $1/3$ obtained in \[appendix free particles\] for undistinguishable non-interacting particles hopping unidirectionally on a periodic one-dimensional lattice.
[Trace of the time evolution operator and cumulants of the eigenvalues]{} \[subsection Q\] We consider the quantity $$\label{Q(t)}
Q(t)=\frac{1}{|\Omega|}\tr\rme^{tM}=\frac{1}{|\Omega|}\sum_{\mathcal{C}\in\Omega}\langle\mathcal{C}|\rme^{tM}|\mathcal{C}\rangle\;.$$ Here $\langle\mathcal{C}|\rme^{tM}|\mathcal{C}\rangle$ is the probability that the system is in the microstate $\mathcal{C}$ at time $t$ conditioned on the fact it was already in the microstate $\mathcal{C}$ at time $0$. Since all configurations are equally probable in the stationary state of periodic TASEP [@D98.1], $Q(t)$ is simply the stationary probability that the system is in the same microstate at both times $0$ and $t$.
[Perturbative expansion]{}\
We consider more generally (\[Q(t)\]) with $M$ replaced by the deformation $M(\gamma)$, and define $$\label{f[M]}
f(t)=\frac{1}{L}\log\tr\rme^{tM(\gamma)}\;.$$ The quantity $f(t)$ can be seen as the generating function of the cumulants of the eigenvalues of $M(\gamma)$ (or more precisely the cumulants of the uniform probability distribution on the set of the eigenvalues). The moments $\mu_{k}=|\Omega|^{-1}L^{-k}\tr M(\gamma)^{k}$ and the cumulants $c_{k}=L^{1-k}f^{(k)}(0)$ of the eigenvalues are indeed related from (\[f\[M\]\]) by $$\log\Big(1+\sum_{k=1}^{\infty}\frac{\mu_{k}t^{k}}{k!}\Big)=\sum_{k=1}^{\infty}\frac{c_{k}t^{k}}{k!}\;.$$ The first cumulants are $c_{1}=\mu_{1}=f'(0)$, $c_{2}=\mu_{2}-\mu_{1}^{2}=f''(0)/L$ and $c_{3}=\mu_{3}-3\mu_{1}\mu_{2}+2\mu_{1}^{3}=f'''(0)/L^{2}$\
The moments and cumulants of the eigenvalues are independent of the deformation $\gamma$. Indeed, one can write $$\label{tr(Mk)}
\tr M(\gamma)^{k}=\sum_{\mathcal{C}_{1},\ldots,\mathcal{C}_{k}}\rme^{\gamma\sum_{j=1}^{k}{\mbox{{\small 1}$\!\!$1}}_{\{\mathcal{C}_{j}\neq\mathcal{C}_{j+1}\}}}\langle\mathcal{C}_{1}|M|\mathcal{C}_{2}\rangle\langle\mathcal{C}_{2}|M|\mathcal{C}_{3}\rangle\ldots\langle\mathcal{C}_{k}|M|\mathcal{C}_{1}\rangle\;.$$ Because of the cyclic structure of allowed transitions explained at the end of section \[subsection master equation\], only $k$-tuples of configurations such that $\sum_{j=1}^{k}{\mbox{{\small 1}$\!\!$1}}_{\{\mathcal{C}_{j}\neq\mathcal{C}_{j+1}\}}$ is divisible by L contribute to (\[tr(Mk)\]). For $k<L$, it implies that $\tr M(\gamma)^{k}$ cannot depend on $\gamma$. One has then $$f(t)=\Big(\frac{1}{L}\log\tr\rme^{tM}+\mathcal{O}(t^{L})\Big)\;.$$ It is possible to calculate directly the coefficients of the expansion near $t=0$ of $f(t)$ by considering the case $\gamma\to-\infty$, for which the matrix $M(\gamma)$ becomes diagonal in configuration basis: $\langle\mathcal{C}|M(\gamma)|\mathcal{C}\rangle$ is equal to minus the number $m(\mathcal{C})$ of clusters of consecutive particles in the system. It implies $$\label{f[M] gamma=-infinity}
f(t)=\frac{1}{L}\sum_{\mathcal{C}\in\Omega}\rme^{-t m(\mathcal{C})}=\frac{1}{L}\sum_{m=1}^{N}|\Omega(m)|\rme^{-tm}\;.$$ The total number $|\Omega(m)|$ of configurations with $m$ clusters can be calculated in the following way: a configuration with $m$ clusters for which the last site is occupied can be described as $a(0)\geq0$ particles followed by $b(1)>0$ empty sites, $a(1)>0$ particles, …, $b(m)>0$ empty sites and $a(m)>0$ particles. The total number of such configurations is $A_{m+1}(N+1)A_{m}(L-N)$ with $$A_{m}(r)=\sum_{b(1),\ldots,b(m)=1}^{\infty}{\mbox{{\small 1}$\!\!$1}}_{\{b(1)+\ldots+b(m)=r\}}=\oint\frac{\rmd z}{2\rmi\pi}\,\frac{z^{m-r-1}}{(1-z)^{m}}={{r-1 \choose m-1}}\;.$$ From particle-hole symmetry, the number of configurations with $m$ clusters for which the last site is empty is $A_{m+1}(L-N+1)A_{m}(N)$ This implies $$|\Omega(m)|=\frac{mL}{N(L-N)}\,{{N \choose m}}{{L-N \choose m}}\;.$$ A saddle point approximation of the sum over $m$ in (\[f\[M\] gamma=-infinity\]) finally gives $$\begin{aligned}
\label{f(t)}
&& f(t)=\rho\log\Big(\frac{(1-2\rho)+2\rho\rme^{t}+\sqrt{1+4\rho(1-\rho)(\rme^{t}-1)}}{2\rho\rme^{t}}\Big)\nonumber\\
&& +(1-\rho)\log\Big(\frac{-(1-2\rho)+2(1-\rho)\rme^{t}+\sqrt{1+4\rho(1-\rho)(\rme^{t}-1)}}{2(1-\rho)\rme^{t}}\Big)\;,\end{aligned}$$ which simplifies at half-filling to $$\label{f(t) rho=1/2}
f(t)=\log(1+\rme^{-t/2})\;.$$ The expressions (\[f(t)\]) and (\[f(t) rho=1/2\]) must be understood as an equality between Taylor series.\
In section \[section trace\], we consider again the quantity $f(t)$ as a testing ground for the formulas derived from Bethe ansatz in section \[section eigenvalues\] for the eigenvalues of $M$ in the thermodynamic limit. We write the summation over the eigenvalues as an integral over a function $\eta$ that index the eigenstates. After a saddle point calculation in the functional integral, we recover (\[f(t)\]).
[Finite t]{}\
The total number of particles hopping during a finite time $t$ is roughly proportional to the average number of clusters of consecutive particles in the system. For typical configurations, this number scales proportionally with the system size in the thermodynamic limit at a finite density of particles. Because of the cyclic structure with period $L$ in the graph of allowed transitions described at the end of section \[subsection master equation\], the quantity $f(t)$ should have oscillations for finite times. The same reasoning also works for undistinguishable non-interacting particles. For distinguishable particles, on the other hand, a similar argument shows that $f(t)$ should show oscillations on the scale $t\sim L$, since the cyclic structure of the graph of allowed transitions has then period $NL$: all the particles need to come back to their initial state.\
The oscillations of $f(t)$ are observed for TASEP from numerical diagonalization, see fig. \[fig f\]. For non-interacting particles, they are confirmed by a direct calculation in \[appendix free particles\], see fig. \[fig f free\]. In both cases, we observe that the oscillations of $f(t)$ are not smooth: the function $f(t)$ is defined piecewise. There exists in particular a time $t(\gamma)$ such that $f(t)$ is analytic (and independent of $\gamma$) for $t$ between $0$ and $t(\gamma)$. We find $t(\gamma)\simeq1.7085\,\rme^{-\gamma}$ for undistinguishable free particles. For TASEP at half-filling, fig. \[fig f\] seems to indicate that $t(0)$ is slightly larger than $1$.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Plots of $L^{-1}\log\tr\rme^{tM(\gamma)}$ as a function of $t$ for a system at half-filling with $L=18$ sites, obtained from numerical diagonalization of the (deformed) Markov matrix. The different curves correspond to $\gamma=-\infty,0,0.1,0.2,0.3,0.4,0.5$, from bottom to top.[]{data-label="fig f"}](f.eps "fig:"){width="100mm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Parametric formulas for the eigenvalues]{} \[section eigenvalues\] In this section, we derive an exact parametric expression, (\[e\[k\]\]), (\[b\[k\]\]), for all the eigenvalues of the Markov matrix of TASEP.
[Bethe ansatz]{} The deformed Markov matrix $M(\gamma)$ of TASEP is equivalent by similarity transformation to minus the Hamiltonian of a ferromagnetic XXZ spin chain with anisotropy $\Delta=\infty$ and twisted boundary conditions [@GM06.1]. The integrability of TASEP is a consequence of this. The eigenfunctions of $M(\gamma)$ are given by the Bethe ansatz as [@D98.1; @S01.1; @GM06.1] $$\label{eigenvector}
\psi(x_{1},\ldots,x_{N})=\det\Big[\Big(\frac{y_{k}}{y_{j}}\Big)^{N-j}\rme^{\gamma x_{j}}(1-y_{k})^{x_{j}}\Big]_{j,k=1,\ldots,N}\;,$$ provided the quantities $y_{j}$, called Bethe roots, verify the Bethe equations $$\label{Bethe eq}
\frac{(1-y_{j})^{L}}{y_{j}^{N}}=(-1)^{N-1}\rme^{-L\gamma}\prod_{k=1}^{N}\frac{1}{y_{k}}\;.$$ The Bethe equations have many different solutions, corresponding to the various eigenstates of $M(\gamma)$. The eigenvalue of $M(\gamma)$ corresponding to (\[eigenvector\]) is $$\label{E[y]}
E=\sum_{j=1}^{N}\frac{y_{j}}{1-y_{j}}\;.$$ *Remark*: the Bethe ansatz is usually written in terms of the variables $z_{j}=\rme^{\rmi q_{j}}=\rme^{\gamma}(1-y_{j})$ instead of the $y_{j}$’s. The eigenvectors are then linear combinations of plane waves with momenta $q_{j}$.\
*Remark 2*: proving the completeness of the Bethe ansatz for periodic TASEP is still an open problem, although one observes numerically for small systems that it does give all the eigenstates. The main difficulties consist in proving that the Bethe equations (\[Bethe eq\]) have $|\Omega|$ solutions, and that all the eigenvectors generated form a basis of the configuration space of the model. Alternatively, the completeness would follow from a direct proof of the resolution of the identity, ${\mbox{{\small 1}$\!\!$1}}=\sum_{k=1}^{|\Omega|}|\psi_{k}\rangle\langle\psi_{k}|$. For periodic TASEP in a discrete time setting with parallel update, such a proof was given by Povolotsky and Priezzhev in [@PP07.1].
[Parametric solution of the Bethe equations]{} The Bethe equations (\[Bethe eq\]) of TASEP have the particularity that the rhs does not depend specifically on $y_{j}$, but is instead a symmetric function of all the $y_{k}$’s. We write this rhs $(-1)^{N-1}/B^{L}$. This allows to solve the Bethe equations in a parametric way, by first solving for each $y_{j}$ the polynomial equation $$\label{polynomial[y,B]=0}
(1-y_{j})^{L}/y_{j}^{N}=(-1)^{N-1}/B^{L}\;$$ as a function of $B$, and then solving a self-consistency equation for $B$. This “decoupling property” was already used in [@GS92.1; @GS92.2; @GM04.1; @GM05.1] for the calculation of the gap, and in [@DL98.1; @DA99.1] for the calculation of the eigenvalue of $M(\gamma)$ with largest real part. The same property is also true for periodic TASEP in a discrete time setting with parallel update [@PP07.1].\
Taking the power $1/L$ of the Bethe equations, there must exist numbers $k_{j}$, $j=1,\ldots,N$, integers if $N$ is odd, half-integers if $N$ is even, such that $$\label{BE 1/L}
\frac{1-y_{j}}{y_{j}^{\rho}}=\frac{\rme^{2\rmi\pi k_{j}/L}}{B}\;,$$ with $B$ a solution of $$\label{B[y]}
\log B=\gamma+\frac{1}{L}\sum_{j=1}^{N}\log y_{j}\;.$$ We also define $$\label{l[rho]}
\ell=-\rho\log\rho-(1-\rho)\log(1-\rho)\;.$$ The Bethe equations (\[BE 1/L\]) involve a function $g$, defined as $$\begin{aligned}
\label{g}
&& g:\mathbb{C}\backslash\mathbb{R^{-}}\to\mathbb{C}\backslash(\rme^{\rmi\pi\rho}[\rme^{\ell},\infty[\;\cup\;\rme^{-\rmi\pi\rho}[\rme^{\ell},\infty[)\nonumber\\
&&\hspace{14mm} y\mapsto\frac{1-y}{y^{\rho}}\;.\end{aligned}$$ It turns out that this function is a *bijection* for $0<\rho<1$, see \[appendix g\]. This is a key point, as it allows to formally solve the Bethe equations as $$\label{y[g,k]}
y_{j}=g^{-1}(\rme^{2\rmi\pi k_{j}/L}/B)\;,$$ Then, the corresponding eigenvalue can be computed from (\[E\[y\]\]), and $B$ is fixed by solving (\[B\[y\]\]). We assumed that $\rme^{2\rmi\pi k_{j}/L}/B$ does not belong to the cut of $g^{-1}$. If this is not the case for some eigenstate, a continuity argument in the parameter $\gamma$ should still allow to use (\[y\[g,k\]\]).\
*Remark*: there are exactly $|\Omega|$ ways to choose the $k_{j}$’s with the constraint $0<k_{1}<\ldots<k_{N}\leq L$. We observe numerically on small systems that all $|\Omega|$ eigenstates are recovered with this constraint. A similar argument was given in [@GM04.1], using instead a rewriting of the Bethe equations as a polynomial equation of degree $L$ for the $z_{j}$’s: the number of ways to choose $N$ roots from this polynomial equation is also equal to $|\Omega|$.
[Functions phi and psi]{} We define the rescaled eigenvalue $e=E/L$ and the parameter $b=\log B$. We introduce the functions $$\label{phipsi[g]}
\varphi(z)=\frac{g^{-1}(z)}{1-g^{-1}(z)}
\quad{\mathrm{and}}\quad
\psi(z)=\log g^{-1}(z)\;.$$ In terms of $\varphi$ and $\psi$, the parametric expression (\[E\[y\]\]), (\[B\[y\]\]) rewrites as $$\label{e[k]}
\displaystyle e=\frac{1}{L}\sum_{j=1}^{n}\varphi\Big(\rme^{\frac{2\rmi\pi k_{j}}{L}-b}\Big)\;$$ $$\label{b[k]}
\displaystyle b=\gamma+\frac{1}{L}\sum_{j=1}^{n}\psi\Big(\rme^{\frac{2\rmi\pi k_{j}}{L}-b}\Big)\;.$$ From (\[phipsi\[g\]\]) and (\[g\]), the functions $\varphi$ and $\psi$ verify the relations $$\begin{aligned}
\label{eq phi}
&& \log z+\rho\log\varphi(z)+(1-\rho)\log(1+\varphi(z))=0\\
\label{eq psi}
&& \rme^{\psi(z)}+z\,\rme^{\rho\,\psi(z)}=1\;.\end{aligned}$$ They are also related by the two equations $$\begin{aligned}
\label{relation phi psi 1}
&& \varphi(z)=\frac{\rme^{\psi(z)}}{1-\rme^{\psi(z)}}\\
\label{relation phi psi 2}
&& z\,\varphi(z)=\rme^{(1-\rho)\psi(z)}\;.\end{aligned}$$ The expansion near $z=0$ of $\varphi(z)$ and $\psi(z)$ at arbitrary filling can be computed explicitly from (\[phipsi\[g\]\]) and the observation [@DL98.1] that for any meromorphic function $h$, $h(g^{-1}(z))$ can be written as a contour integral. One has $$\label{h(g-1)}
h(g^{-1}(z))=\oint_{g(\Gamma)}\frac{\rmd w}{2\rmi\pi}\frac{h(g^{-1}(w))}{w-z}=\oint_{\Gamma}\frac{\rmd y}{2\rmi\pi}\frac{g'(y)h(y)}{g(y)-z}\;,$$ where the contour $\Gamma$ encloses $g^{-1}(z)$ but none of the poles of $h$, and does not cross the cut $\mathbb{R^{-}}$ of $g$. We will need to expand $h(g^{-1}(z))$ for small $z$. In the previous expression, it is possible to expand the integrand near $z=0$ as long as $|g(y)|>|z|$ for all $y\in\Gamma$. As shown in fig. \[fig curves g\], it is possible to find such a contour $\Gamma$ only if $|z|<\rme^{\ell}$, otherwise the contour would have to go through the cut.\
We use (\[h(g-1)\]) for $h(y)=\log y$ and expand for small $z$ inside the integral. After computing the residues at $y=1$ (which is always inside the contour, see fig. \[fig curves g\]), we obtain $$\label{psi(z)}
\psi(z)=\sum_{r=1}^{\infty}{{\rho\,r \choose r}}\frac{(-1)^{r}z^{r}}{\rho\,r}\;.$$ Using (\[h(g-1)\]) again for $h(y)=y/(1-y)$, one also finds $$\label{phi(z)}
\varphi(z)=z^{-1}-(1-\rho)\sum_{r=0}^{\infty}{{\rho\,r \choose r}}\frac{(-1)^{r}z^{r}}{r+1}\;.$$ The first term takes into account the pole at $y=1$ of $y/(1-y)$, so that the contour $\Gamma$ encloses both poles of the integrand $y=1$ and $y=g^{-1}(z)$.\
At half filling, the summation over $r$ in (\[phi(z)\]) and (\[psi(z)\]) can be done explicitly. One finds $$\begin{aligned}
\label{phi(z) rho=1/2}
&& \varphi(z)=-\frac{1}{2}+\frac{\sqrt{4+z^{2}}}{2z}\\
\label{psi(z) rho=1/2}
&& \psi(z)=-2\,{\mathrm{arcsinh}}(z/2)=-2\log\Big(\frac{z}{2}+\sqrt{1+\frac{z^{2}}{4}}\Big)\;.\end{aligned}$$ The expressions (\[psi(z)\]) and (\[phi(z)\]) rely on the assumption that the solution $b$ of (\[b\[k\]\]) is such that ${\mathrm{Re}}\,b>-\ell$ with $\ell$ defined in (\[l\[rho\]\]), otherwise the expansion of $g^{-1}$ for small argument would not be convergent. The condition is satisfied if $\gamma$ is a large enough real positive number, in which case $b\simeq\gamma$. Besides, when $\gamma=0$, numerical checks on small system seem to indicate that (\[b\[k\]\]) always has a solution inside the radius of convergence of the series in $\rme^{-b}$ for all the eigenstates, except the one with eigenvalue $0$. For the latter, it seems that (\[b\[k\]\]) does not have a solution $b\in\mathbb{C}$, although formally $b=-\infty$ is a solution of (\[b\[k\]\]) for which (\[e\[k\]\]) gives $e=0$. We emphasize that for *all* other solutions of (\[b\[k\]\]), even the ones for which the eigenvalue is close to $0$ such as the gap, equation (\[b\[k\]\]) seems to have a solution.
[ccc]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Curves of the points $y$ such that $|g(y)|=|z|$ for $\rho=1/3$ with $z=0.99\,\rme^{\ell}$ (left) and $z=1.01\,\rme^{\ell}$ (right). The function $g$ is defined in eq. (\[g\]) and $\ell$ is given by (\[l\[rho\]\]). The gray area corresponds to $|g(y)|<|z|$ and the white area to $|g(y)|>|z|$. The red half line is the cut $\mathbb{R^{-}}$ of the function $g$.[]{data-label="fig curves g"}](Curveg0.99.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
& &
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Curves of the points $y$ such that $|g(y)|=|z|$ for $\rho=1/3$ with $z=0.99\,\rme^{\ell}$ (left) and $z=1.01\,\rme^{\ell}$ (right). The function $g$ is defined in eq. (\[g\]) and $\ell$ is given by (\[l\[rho\]\]). The gray area corresponds to $|g(y)|<|z|$ and the white area to $|g(y)|>|z|$. The red half line is the cut $\mathbb{R^{-}}$ of the function $g$.[]{data-label="fig curves g"}](Curveg1.01.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Thermodynamic limit]{} \[subsection eigenvalues thermodynamic limit\] For large $L$, both (\[e\[k\]\]) and (\[b\[k\]\]) become independent of the detailed structure of the $k_{j}$’s, and only retain information about the density profile of the $k_{j}$’s. For each eigenstate, we introduce a function $\eta$ such that $L\,\eta(u)\rmd u$ is the number of $k_{j}$ in the interval $[L\,u,L(u+\rmd u)]$. The function $\eta$ then obeys the normalization $$\label{rho[eta]}
\rho=\int_{0}^{1}\rmd u\,\eta(u)\;,$$ while the eigenvalue (\[e\[k\]\]) becomes $$\label{e[eta]}
e=\int_{0}^{1}\rmd u\,\eta(u)\varphi\big(\rme^{2\rmi\pi u-b}\big)\;,$$ and the equation for the parameter $b$ (\[b\[k\]\]) rewrites $$\label{b[eta]}
b=\gamma+\int_{0}^{1}\rmd u\,\eta(u)\psi\big(\rme^{2\rmi\pi u-b}\big)\;.$$ In sections \[section density\] and \[section trace\], we will need to count the number of eigenstates corresponding to a given function $\eta$ in order to sum over the eigenvalues. Since the number of ways to place $L\,\eta(u)\rmd u$ $k_{j}$’s in any interval $L\,\rmd u$ is ${{L\,\rmd u \choose L\,\eta(u)\rmd u}}$, Stirling’s formula implies that the total number of eigenstates corresponding to $\eta$ is $\Omega[\eta]\sim\rme^{Ls}$, where we defined an “entropy per site” $$\label{s[eta]}
s=-\int_{0}^{1}\rmd u\,[\eta(u)\log\eta(u)+(1-\eta(u))\log(1-\eta(u))]\;.$$
[Edge of the spectrum]{} \[section envelope\] In the thermodynamic limit, the rescaled eigenvalues $e=E/L$ fill a bounded domain in the complex plane, see fig. \[fig spectrum\]. We study in this section the boundary of this domain, called in the following edge of the spectrum.\
We observe numerically that the eigenvalues located at the edge of the spectrum correspond to eigenstates for which the $k_{j}$’s are consecutive numbers. There are $L$ such possibilities, that we index with an integer $m$ between $0$ and $L-1$. We write $k_{j}=m+j-(N+1)/2$. The eigenvalue with largest real part corresponds to $m=0$ for $\gamma>0$. For $\gamma=0$, however, we remind that there is no solution to (\[b\[k\]\]) when $m=0$. We will see that it corresponds to a singular point for the curve at the edge of the spectrum.\
In the limit $L\to\infty$, the corresponding density profile $\eta(u)$ of the $k_{j}$’s is the function with period $1$ such that $$\eta(u)=\Big|
\begin{array}{lll}
1 && {\mathrm{for}}\;\mu-\frac{\rho}{2}<u<\mu+\frac{\rho}{2}\\
0 && {\mathrm{otherwise}}
\end{array}\;,$$ where we defined $\mu=m/L$. With this choice of $\eta$, writing explicitly the dependency in $\mu$ of the eigenvalue and of the parameter $b$, one finds $$e(\mu)=\int_{-\rho/2}^{\rho/2}\rmd u\,\varphi\big(\rme^{2\rmi\pi(u+\mu)-b(\mu)}\big)\;$$ and $$b(\mu)=\int_{-\rho/2}^{\rho/2}\rmd u\,\psi\big(\rme^{2\rmi\pi(u+\mu)-b(\mu)}\big)\;.$$ At half-filling, a nice parametric representation of the curve $e(\mu)$ can be written by replacing $b(\mu)$ by a new variable $d(\mu)$ defined by $$\label{d[b]}
d(\mu)=-\rmi\arccos\Big(\frac{\rme^{2\rmi\pi\mu-b(\mu)}}{2}\Big)\;.$$ Taking the derivative with respect to $\mu$ of the equation for $b(\mu)$ and calculating explicitly the integrals, one obtains $$\begin{aligned}
\label{e[d]}
&& e(\mu)=-\frac{\tanh d(\mu)-d(\mu)}{2\rmi\pi}\\
\label{d'[d]}
&& d'(\mu)=\frac{\pi^{2}}{d(\mu)\tanh d(\mu)}\;.\end{aligned}$$ The initial condition for the differential equation (\[d’\[d\]\]) depends on the value of $\gamma$. For $\gamma=0$, since $e(0)=0$ (largest eigenvalue of a Markov matrix), then one must have $d(0)=0$, which corresponds to $b(0)=-\log2$. This is a singular point for the differential equation (\[d’\[d\]\]), which is related to the fact that $b=-\log2$ corresponds to the border of the region where the expansion of $\varphi(z)$ and $\psi(z)$ for small $z$ in section \[section eigenvalues\] is convergent.\
Expanding the differential equation at second order near $\mu=0$ leads to $3$ solutions. Inserting them in the equation for $e(\mu)$, one finds that only the solution $$\label{d(mu) 0}
d(\mu)=\rme^{-2\rmi\pi/3}(3\pi^{2}\mu)^{1/3}+\frac{\pi^{2}\mu}{5}+{\mathcal{O}\left(\mu^{5/3}\right)}\;$$ gives ${\mathrm{Re}}\,e(\mu)<0$ for small $\mu$ positive or negative. One has $$\label{e(mu) 0}
e(\mu)=-\frac{\rmi\pi\mu}{2}+\frac{\rmi\rme^{2\rmi\pi/3}3^{2/3}\pi^{7/3}\mu^{5/3}}{10}+{\mathcal{O}\left(\mu^{7/3}\right)}\;.$$ Studying the stability of the solutions of the differential equation near $d=0$, one observes that taking as initial condition $d(0)=\epsilon\,\rme^{\rmi\,\theta}$ with $0<\epsilon\ll1$ and $-\pi<\theta<-\pi/3$ leads to the correct solution, see fig. \[fig stability d(0)\].\
We observe from (\[e(mu) 0\]) that the curve $e(\mu)$ is singular near the origin with a power $5/3$ as announced in (\[Ree\[Ime\]\]). In fig. \[fig spectrum\], the curve $e(\mu)$ obtained from (\[e\[d\]\]) and (\[d’\[d\]\]) is plotted along with the full spectrum of TASEP for $L=18$, $N=9$. The agreement is already very good with the large $L$ limit.
![Solutions of the differential equation (\[d’\[d\]\]) for small $\mu$ with initial condition $d(0)=10^{-10}\rme^{\rmi\,\theta}$, for $\theta=\pi/30,3\pi/30,\ldots,59\pi/30$. The initial values $d(0)$ are represented on the circle. The arrows indicate the direction of the increment given by the derivative at $\mu=0$, $d'(\mu)\simeq\pi^{2}/d(0)^{2}$. The curves starting on the circle represent the solution of the differential equation (\[d’\[d\]\]) solved numerically, which is essentially undistinguishable from the solution of $d'(\mu)=\pi^{2}/d(\mu)^{2}$ for the values of $d$ shown. The dotted lines represent the $3$ solutions of (\[d’\[d\]\]) unstable under a perturbation near $d=0$.[]{data-label="fig stability d(0)"}](StabilityInitConditionODEd.eps){width="70mm"}
[Dilogarithm]{} An alternative expression to (\[d’\[d\]\]) can be written by solving explicitly the differential equation for $d$ in terms of a dilogarithm function, as $$\label{Li2[d]}
\fl\hspace{5mm} \frac{d(\mu)^{2}}{2}-\frac{\pi^{2}}{24}+d(\mu)\log\big(1+\rme^{-2d(\mu)}\big)-\frac{1}{2}{\mathrm{Li}}_{2}\big(-\rme^{-2d(\mu)}\big)=\pi^{2}\mu-\pi^{2}\Big(1+\Big\lfloor\mu-\frac{1}{5}\Big\rfloor\Big)\;.$$ For $\mu$ in a neighbourhood of $0$, the rhs of the previous equation is equal to $\pi^{2}\mu$. The second term in the rhs is however needed since $\rme^{-2d(\mu)}$ crosses the cut $[1,\infty)$ of the dilogarithm at $\mu=1/5$, see fig. \[fig exp(-2d)\]. Indeed, inserting $d(1/5)=-\log((1+\sqrt{5})/2)-\rmi\pi/2$ and using $${\mathrm{Li}}_{2}\Big(\frac{3+\sqrt{5}}{2}\Big)=-\frac{11\pi^{2}}{15}-\log^{2}\Big(-\frac{1+\sqrt{5}}{2}\Big)\;,$$ one observes that the equation (\[Li2\[d\]\]) is verified, with $\rme^{-2d(1/5)}=(3+\sqrt{5})/2>1$. For $\mu=-1/5$, the solution is also explicit: using $${\mathrm{Li}}_{2}\Big(\frac{3-\sqrt{5}}{2}\Big)=\frac{\pi^{2}}{15}-\log^{2}\Big(\frac{1+\sqrt{5}}{2}\Big)\;,$$ one finds $d(-1/5)=\log((1+\sqrt{5})/2)-\rmi\pi/2$. Here however, $\rme^{-2d(-1/5)}=(3-\sqrt{5})/2<1$ does not cross the cut of the dilogarithm.
[ccc]{}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Curve in the complex plane of $d(\mu)$ (left) and $-\rme^{-2d(\mu)}$ (right), from a numerical resolution of the differential equation (\[d’\[d\]\]). In both graphs, the two dots are the explicit values $-\rme^{-2d(\pm1/5)}=(3\pm\sqrt{5})/2$. In the graph on the left, the outer, red curve encloses the domain for which ${\mathrm{Re}}\,b>-\log2$, for which the expansions (\[psi(z)\]) and (\[phi(z)\]) hold. In the graph on the right, the thick, red line correspond to the cut of the dilogarithm $[1,\infty)$.[]{data-label="fig exp(-2d)"}](d.eps "fig:"){width="50mm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
& &
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Curve in the complex plane of $d(\mu)$ (left) and $-\rme^{-2d(\mu)}$ (right), from a numerical resolution of the differential equation (\[d’\[d\]\]). In both graphs, the two dots are the explicit values $-\rme^{-2d(\pm1/5)}=(3\pm\sqrt{5})/2$. In the graph on the left, the outer, red curve encloses the domain for which ${\mathrm{Re}}\,b>-\log2$, for which the expansions (\[psi(z)\]) and (\[phi(z)\]) hold. In the graph on the right, the thick, red line correspond to the cut of the dilogarithm $[1,\infty)$.[]{data-label="fig exp(-2d)"}](Exp-2d.eps "fig:"){width="70mm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Spectrum of TASEP with $N=9$ particles on $L=18$ sites. The small black dots are the eigenvalues $E$ of the Markov matrix divided by $L$. The black curve is the edge of the spectrum in the thermodynamic limit at half-filling. The red (gray in printed version) curve corresponds to the asymptotic expression (\[Ree\[Ime\]\]) of the edge of the spectrum near the origin. The two big blue dots correspond to the explicit values $E/L=-\frac{1}{4}\pm\frac{1}{2\rmi\pi}\big(\sqrt{5}-\log\frac{1+\sqrt{5}}{2}\big)$. The first order correction to the edge from eq. (\[deltae\[d\]\]) is plotted in gray.[]{data-label="fig spectrum"}](Envelope.eps "fig:"){width="100mm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Edge of the spectrum, scale L\^0 (half-filling)]{} We observe in fig. \[fig spectrum\] that the edge of the spectrum shows $L$ small peaks. These peaks are a $1/L$ correction to the leading behaviour (\[e\[d\]\]), (\[d’\[d\]\]). They are a consequence of the constraint that all the integers $k_{j}$ are different. This phenomenon does not happen for non-interacting particles, see fig \[fig s(e) free U\].\
In order to study this correction, one must go back to the exact expressions (\[e\[k\]\]) and (\[b\[k\]\]). Numerically, one observes that the $k_{j}$ that contribute to the edge are such that all the $k_{j}$’s are consecutive except at most one of them. We will write $k_{j}=\mu L+j-(N+1)/2$ for $j=1,\ldots,N-1$ and $k_{N}=(\mu+\nu)L$. On the scale studied here, $\mu$ can only take the values $1/L$, $2/L$, …, $1$. The parameter $\nu$ verifies $0\leq\nu\leq1-\rho$. We focus again on the half-filled case $\rho=1/2$.\
At leading order in $L$, one recovers (\[e\[d\]\]), (\[d’\[d\]\]) using again the change of variable (\[d\[b\]\]). Writing $e=e(\mu+1/(2L))+\delta e(\mu,\nu)$, one finds at the end of the calculation $$\label{deltae[d]}
\hspace{-20mm}
\delta e(\mu,\nu)=\frac{\tanh d(\mu)}{2d(\mu)L}\Bigg(\frac{d(\mu)\sqrt{1-\rme^{4\rmi\pi\nu}\cosh^{2}d(\mu)}}{\rmi\,\rme^{2\rmi\pi\nu}\sinh d(\mu)}-{\mathrm{arcsinh}}\Big[\rmi\,\rme^{2\rmi\pi\nu}\cosh d(\mu)\Big]\Bigg)\;,$$ where $d(\mu)$ is the solution of (\[d’\[d\]\]).\
From (\[Li2\[d\]\]) and the relation ${\mathrm{Li}}_{2}(-\rme^{z})+{\mathrm{Li}}_{2}(-\rme^{-z})=-\pi^{2}/6-z^{2}/2$, the function $d(\mu)$ verifies the symmetry relation $d(-\mu)=-\overline{d(\mu)}$, where $\overline{\,\cdot\,}$ denotes complex conjugation. This implies $\delta e(-\mu,1/2-\nu)=\overline{\delta e(\mu,\nu)}$. The latter symmetry is a consequence of the term $+1/(2L)$ in the definition of $\delta e$. The first-order correction (\[deltae\[d\]\]) is plotted in fig. \[fig spectrum\] along with the exact spectrum for $L=18$, $N=9$.
[Density of eigenstates]{} \[section density\] The total number of eigenstates for TASEP is $|\Omega|\sim\rme^{\ell L}$, with $\ell$ defined in terms of the density of particles $\rho$ in (\[l\[rho\]\]). In the bulk of the spectrum, the number of eigenstates with a rescaled eigenvalue $E/L$ close to a given $e$ is expected to be of the form $\rme^{Ls(e)}$. The function $s(e)$ is studied in this section.
[Optimal function eta]{} The density of eigenstates near the rescaled eigenvalue $e$ can be formally defined by the functional integral $$\label{D[path integral]}
D(e)=\int\mathcal{D}\eta\,{\mbox{{\small 1}$\!\!$1}}_{\{e[\eta]=e\}}{\mbox{{\small 1}$\!\!$1}}_{\{\int_{0}^{1}\rmd u\,\eta(u)=\rho\}}\rme^{L s[\eta]}\;.$$ We write explicitly the dependency in $\eta$ of $e$, $b$ and $s$ in this section. We want to maximize for $\eta$ $$\label{s + Lagrange multipliers}
s[\eta]+\lambda\Big(\int_{0}^{1}\rmd u\,\eta(u)-\rho\Big)+{\mathrm{Re}}(2\omega(e[\eta]-e))\;.$$ It gives an optimal function $\eta^{*}$, which depends on the two Lagrange multipliers $\lambda\in\mathbb{R}$ and $\omega\in\mathbb{C}$. Those must then be set such that the constraints $\int_{0}^{1}\rmd u\,\eta(u)=\rho$ and $e[\eta^{*}]=e$ are satisfied, and we can finally write (\[D\[s\]\]) with $s(e)=s[\eta^{*}]$.\
For given values of the Lagrange multipliers, writing the variation of $s[\eta]$, $b[\eta]$ and $e[\eta]$ for a small variation $\delta\eta$ of $\eta$ and using (\[rho\[eta\]\]), (\[e\[eta\]\]) and (\[b\[eta\]\]), we find $$\label{deltas[deltaeta]}
\delta s=-\int_{0}^{1}\rmd u\,\delta\eta(u)\log\frac{\eta(u)}{1-\eta(u)}\;,$$ $$\label{deltab[deltaeta]}
\delta b=\frac{\int_{0}^{1}\rmd u\,\delta\eta(u)\psi\big(\rme^{2\rmi\pi u-b[\eta]}\big)}{1+\frac{1}{2\rmi\pi}\int_{0}^{1}\rmd u\,\eta(u)\partial_{u}\psi\big(\rme^{2\rmi\pi u-b[\eta]}\big)}\;,$$ and $$\label{deltae[deltaeta]}
\delta e=\int_{0}^{1}\rmd u\,\delta\eta(u)\Big[\varphi\big(\rme^{2\rmi\pi u-b[\eta]}\big)+a[\eta]\psi\big(\rme^{2\rmi\pi u-b[\eta]}\big)\Big]\;.$$ We have defined $$\label{a[eta]}
a[\eta]=-\frac{\frac{1}{2\rmi\pi}\int_{0}^{1}\rmd u\,\eta(u)\partial_{u}\varphi\big(\rme^{2\rmi\pi u-b[\eta]}\big)}{1+\frac{1}{2\rmi\pi}\int_{0}^{1}\rmd u\,\eta(u)\partial_{u}\psi\big(\rme^{2\rmi\pi u-b[\eta]}\big)}\;.$$ It implies that the optimal function $\eta^{*}$ verifies $$-\log\frac{\eta^{*}(u)}{1-\eta^{*}(u)}+\lambda+{\mathrm{Re}}[2\,\omega\,\varphi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)+2\,a[\eta^{*}]\,\omega\,\psi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)]\;.$$ The optimal function is then equal to $$\label{eta* density}
\eta^{*}(u)=\Big(1+\rme^{-\lambda-2\,{\mathrm{Re}}[\omega\,\varphi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)+a[\eta^{*}]\,\omega\,\psi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)]}\Big)^{-1}\;.$$ This is a real function that satisfies $0<\eta^{*}(u)<1$ for all $u$. The expression (\[eta\* density\]) is reminiscent of a Fermi-Dirac distribution. On the other hand, the corresponding expression for undistinguishable non-interacting particles, (\[eta\* free U\]), resembles a Bose-Einstein distributions: allowing some momenta to be equal gives a term $-1$ in the denominator of (\[eta\* free U\]), while forbidding equal momenta gives the term $+1$ in the denominator of (\[eta\* density\]).
[Contour integrals]{} For given values of the Lagrange multipliers $\lambda$ and $\omega$, the expression (\[eta\* density\]) is completely explicit except for the two unknown complex quantities $a[\eta^{*}]$ and $b[\eta^{*}]$, that must be determined self-consistently from (\[a\[eta\]\]) and (\[b\[eta\]\]). It is possible to simplify the problem a little by noticing that we can replace $a[\eta^{*}]$ and $b[\eta^{*}]$ by two real quantities $\alpha$ and $\beta$. Indeed, the definitions (\[a\[eta\]\]) and (\[eta\* density\]) imply $$\begin{aligned}
&& {\mathrm{Im}}(a[\eta^{*}]\omega)={\mathrm{Im}}\Big[-\frac{\omega}{2\rmi\pi}\int_{0}^{1}\rmd u\,\eta^{*}(u)\partial_{u}\Big(\varphi\big(\rme^{2\rmi\pi u-b}\big)+a\psi\big(\rme^{2\rmi\pi u-b}\big)\Big)\Big]\nonumber\\
&&\hspace{19mm} =\frac{1}{4\pi}\int_{0}^{1}\rmd u\,\frac{(\eta^{*})'(u)}{1-\eta^{*}(u)}=0\;.\end{aligned}$$ We define $\alpha=a[\eta^{*}]\omega\in\mathbb{R}$. The imaginary part of $b[\eta^{*}]$ can be eliminated by a shift of $u$ and a redefinition of $\eta$: we introduce $\sigma$ such that $$\sigma(\rme^{2\rmi\pi u})=\eta^{*}\Big(u+\frac{{\mathrm{Im}}(b)}{2\pi}\Big)\;.$$ Defining $\beta={\mathrm{Re}}(b[\eta^{*}])$, one has $$\label{sigma(z) density}
\sigma(z)=\Big(1+\rme^{-\lambda\,-2\,{\mathrm{Re}}[\omega\varphi(\rme^{-\beta}z)]\,-2\alpha\,{\mathrm{Re}}[\psi(\rme^{-\beta}z)]}\Big)^{-1}\;.$$ It is not possible to eliminate the quantity $\beta$ by changing the contour of integration, since *e.g.* ${\mathrm{Re}}[\psi(\rme^{-\beta}z)]$ is not an analytic function of $z$. It can also be seen by writing $2\,{\mathrm{Re}}[\psi(\rme^{-\beta}z)]=\psi(\rme^{-\beta}z)+\psi(\rme^{-\beta}z^{-1})$ for $|z|=1$.\
The quantities $s=s[\eta^{*}]$, $\beta$, $\alpha$ can be rewritten in terms of $\sigma$ as $$\label{s[sigma]}
s=-\frac{1}{2\rmi\pi}\oint\frac{\rmd z}{z}\,\sigma(z)\log\sigma(z)+(1-\sigma(z))\log(1-\sigma(z))\;,$$ $$\label{beta[sigma] density}
\beta={\mathrm{Re}}\,\gamma+{\mathrm{Re}}\Big[\frac{1}{2\rmi\pi}\oint\frac{\rmd z}{z}\,\sigma(z)\psi\big(\rme^{-\beta}z\big)\Big]\;,$$ and $$\label{alpha[sigma] density}
\alpha=-\frac{1}{2\rmi\pi}\oint\rmd z\,\sigma(z)\partial_{z}\Big(\omega\varphi\big(\rme^{-\beta}z\big)+\alpha\psi\big(\rme^{-\beta}z\big)\Big)\;.$$ All the contour integrals are over the circle of radius $1$ and center $0$ in the complex plane. Similarly, the constraints for $\rho$ and $e[\eta^{*}]$ give $$\label{rho[sigma]}
\rho=\frac{1}{2\rmi\pi}\oint\frac{\rmd z}{z}\,\sigma(z)\;,$$ and $$\label{e[sigma] density}
e=\frac{1}{2\rmi\pi}\oint\frac{\rmd z}{z}\,\sigma(z)\varphi\big(\rme^{-\beta}z\big)\;.$$ For given $\rho$ and $e$, one has to solve (\[rho\[sigma\]\]) and (\[e\[sigma\] density\]) in order to obtain $\lambda$ and $\omega$ in terms of them. Like for (\[alpha\[sigma\] density\]) and (\[beta\[sigma\] density\]), it does not seem that these equations can be solved analytically in general. At half-filling, however, it is possible to show that $$\label{lambda(omega)}
\lambda={\mathrm{Re}}\,\omega\;,$$ leaving only the equations for $\alpha$, $\beta$ and $\omega$ to be solved numerically. Indeed, for $\rho=1/2$, one has the identities $\varphi(-z)=-1-\varphi(z)$ and $\psi(-z)=-\psi(z)$. Setting $\lambda={\mathrm{Re}}\,\omega$ in (\[sigma(z) density\]) then implies $\sigma(z)+\sigma(-z)=1$, from which (\[rho\[sigma\]\]) follows at half-filling.\
We note that if $\omega\in\mathbb{R}$ then $e\in\mathbb{R}$. This is a consequence of (\[e\[sigma\] density\]), $\sigma(-z)=\sigma(z)$ and ${\mathrm{Im}}[\varphi(-z)]=-{\mathrm{Im}}[\varphi(z)]$. By the definition (\[s + Lagrange multipliers\]) of the Lagrange multiplier $\omega$, it implies that $s$ is also the density of eigenvalues with a given real part when $\omega$ is real.\
The maximum of $s$ is located at $e=-\rho(1-\rho)$, $s=\ell$. It corresponds to $\omega=0$, $\sigma(z)=\rho$, $\lambda=\log[\rho/(1-\rho)]$, $\alpha=0$, $\beta={\mathrm{Re}}\,\gamma$. Unlike free particles, the spectrum is not symmetric with respect to the maximum of $s$.\
In fig. \[fig e(omega)\], $e$ is plotted for various values of $\omega$ at half-filling, along with the optimal function $\sigma(\rme^{2\rmi\pi u})$. In fig. \[fig s(e)\], $s$ is plotted as a function of $e\in\mathbb{R}$ and compared with the density of real part of eigenvalues obtained from numerical diagonalization of the Markov matrix $M$ for $N=9$, $L=18$. The agreement is not very good for eigenvalues close to the edges. This is caused by the “arches” at distance $\sim1/L$ of the edge of the spectrum, which still contribute much for $N=9$, $L=18$, see fig. \[fig spectrum\].
[ccc]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the left, graph of $e(\omega)$ for fixed values of $|\omega|$ (black), and fixed values of $\arg\omega$ (gray), obtained from (\[e\[sigma\] density\]) after solving numerically the system (\[beta\[sigma\] density\]), (\[alpha\[sigma\] density\]) at half-filling (left). The different curves correspond to $|\omega|=0.25,0.5,1,2,4,8$ (from the center to the edge) and to $\arg\omega=0,\pi/10,2\pi/10,\ldots,19\pi/20$. The outer, red curve is the edge of the spectrum, computed numerically from (\[e\[d\]\]), which is recovered from (\[e\[sigma\] density\]) in the limit $|\omega|\to\infty$. On the right, optimal function $\sigma(\rme^{2\rmi\pi u})$ plotted as a function of $u$ for $\omega=8\,\rme^{\rmi\pi/10}$.[]{data-label="fig e(omega)"}](e_omega.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
&&
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the left, graph of $e(\omega)$ for fixed values of $|\omega|$ (black), and fixed values of $\arg\omega$ (gray), obtained from (\[e\[sigma\] density\]) after solving numerically the system (\[beta\[sigma\] density\]), (\[alpha\[sigma\] density\]) at half-filling (left). The different curves correspond to $|\omega|=0.25,0.5,1,2,4,8$ (from the center to the edge) and to $\arg\omega=0,\pi/10,2\pi/10,\ldots,19\pi/20$. The outer, red curve is the edge of the spectrum, computed numerically from (\[e\[d\]\]), which is recovered from (\[e\[sigma\] density\]) in the limit $|\omega|\to\infty$. On the right, optimal function $\sigma(\rme^{2\rmi\pi u})$ plotted as a function of $u$ for $\omega=8\,\rme^{\rmi\pi/10}$.[]{data-label="fig e(omega)"}](OptimalFunctionLargeomega.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Number of eigenvalues with real part $L\,e$ of the Markov matrix $M$ of TASEP at half-filling, plotted as a function of $e$. The thick black curve corresponds to the expressions (\[s\[sigma\]\]), (\[e\[sigma\] density\]) parametrized by $\omega$ ranging from $-50$ to $60$, where the quantities $\alpha$ and $\beta$ are solved numerically using equations (\[alpha\[sigma\] density\]) and (\[beta\[sigma\] density\]) for each value of the parametrization $\omega$. The thick red (gray in printed version) curve is the asymptotics (\[s(e)\]). The histograms correspond to the density of real part of eigenvalues obtained from numerical diagonalization of $M$ for the finite system with $N=9$ particles on $L=18$ sites. Because of the logarithmic corrections in $L$ of $L^{-1}\log|\Omega|\simeq\log2$, where $|\Omega|={{L \choose L/2}}$ is the total number of microstates, we shift the height of the histograms so that their maximum is $\log2$.[]{data-label="fig s(e)"}](DensityRealPartEigenvalues.eps "fig:"){width="70mm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Density of eigenvalues close to the origin (rho=1/2)]{} \[subsection density e=0\] We consider the limit $e\to0$ of $s(e)$ at half-filling for the Markov matrix (deformation $\gamma=0$). It corresponds to $|\omega|\to\infty$ with $\arg\omega\to0$.\
In the limit $|\omega|\to\infty$, the optimal function $\sigma(\rme^{2\rmi\pi u})$ approaches $1$ if $u_{1}<u<u_{2}$ and $0$ if $u_{2}<u<u_{1}+1$. The relation $\sigma(-z)=1-\sigma(z)$ at half-filling and the normalization condition (\[rho\[sigma\]\]) imply that $u_{2}=u_{1}+1/2$. Furthermore, if we consider a scaling such that $\arg\omega\to0$ when $|\omega|\to\infty$, one has $u_{1}=-1/4$ and $u_{2}=1/4$, which is the same as what we had in section \[section envelope\] for the edge of the spectrum near the eigenvalue $0$.\
In \[appendix density\], we compute explicitly the large $\omega$ limit of the integrals (\[s\[sigma\]\]), (\[beta\[sigma\] density\]), (\[alpha\[sigma\] density\]) and (\[e\[sigma\] density\]) with $\lambda$ given by (\[lambda(omega)\]). We find two different regimes, depending on the respective scaling between the real and imaginary part of $e$.\
The first regime, $e\to0$ with $|{\mathrm{Im}}\,e|^{5/3}/{\mathrm{Re}}\,e\to0$, corresponds to the central part of the spectrum, far from the edge. We find $\beta\simeq-\log2+{\delta\!\beta}|\omega|^{-2/3}$, where ${\delta\!\beta}$ is the solution of $$\label{deltabeta c=0}
\frac{1}{6\pi}=\int_{0}^{\infty}\rmd x\,\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{1/2}]}{1+\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}}\;.$$ The real part of the eigenvalue $e$ and the “entropy” $s$ are equal to $$\label{Re(e) c=0}
{\mathrm{Re}}\,e=\frac{4\sqrt{2}\,{\delta\!\beta}^{5/2}}{3|\omega|^{5/3}}\Big[\frac{1}{10\pi}-\int_{0}^{\infty}\rmd x\,\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}{1+\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}}\Big]\;$$ and $$\label{s c=0}
s=-5\,|\omega|\,{\mathrm{Re}}\,e\;.$$ Solving (\[deltabeta c=0\]) numerically, one finds ${\delta\!\beta}\simeq0.706532$, which implies ${\mathrm{Re}}\,e\simeq-0.147533|\omega|^{-5/3}$, $s\simeq0.737667|\omega|^{-2/3}$ and (\[s(e)\]).\
The second regime, $e\to0$ with ${\mathrm{Re}}\,e$ and ${\mathrm{Im}}\,e$ related by (\[Ree\[Ime\]\]), corresponds to the edge of the spectrum. In this regime, one finds $$s(e)\simeq\frac{2^{4/5}\pi^{2/5}}{3^{3/10}5^{1/10}}(-{\mathrm{Re}}\,e)^{2/5}\sqrt{1+\frac{2^{2/3}3^{7/6}\pi^{2/3}}{10}\,\frac{|{\mathrm{Im}}\,e|^{5/3}}{{\mathrm{Re}}\,e}}\;.$$ The crossover between the two regimes corresponds to $|{\mathrm{Im}}\,e|^{5/3}/(-{\mathrm{Re}}\,e)$ converging to a constant different from the $2^{2/3}3^{7/6}\pi^{2/3}/10$ characteristic of the edge. Explicit expressions are given in \[appendix density\].
[Trace of the time evolution operator]{} \[section trace\] In this section, we study the quantity $f(t)$ defined in eq. (\[f\[M\]\]). This is another application of the formulas (\[e\[eta\]\]), (\[b\[eta\]\]) derived in section \[section eigenvalues\] for the eigenvalues.
[Optimal function eta]{} As in section \[section density\] for the density of eigenvalues, one can write the summation over all eigenvalues as $$\label{tr[path integral]}
\tr\rme^{tM}=\sum_{\{k_{1},\ldots,k_{N}\}}\rme^{tE(k_{1},\ldots,k_{N})}\simeq\int\mathcal{D}\eta\,{\mbox{{\small 1}$\!\!$1}}_{\{\int_{0}^{1}\rmd u\,\eta(u)=\rho\}}\rme^{L(s[\eta]+te[\eta])}\;.$$ If the functional integral is dominated by the contribution of an optimal function $\eta^{*}$, one finds from (\[f\[M\]\]) $f(t)=s[\eta^{*}]+te[\eta^{*}]$. The function $\eta^{*}$ generally depends on $t$. The normalization (\[rho\[eta\]\]) of $\eta$ is enforced by the Lagrange multiplier $$\lambda\Big(-\rho+\int_{0}^{1}\rmd u\,\eta(u)\Big)\;.$$ The change in $s$, $b$ and $e$ from a variation $\delta\eta$ of $\eta$ is still given by (\[deltas\[deltaeta\]\]), (\[deltab\[deltaeta\]\]) and (\[deltae\[deltaeta\]\]). The optimal function then verifies $$-\log\frac{\eta^{*}(u)}{1-\eta^{*}(u)}+\lambda+t\Big(\varphi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)+a[\eta^{*}]\psi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)\Big)\;,$$ with $a[\eta]$ still given by (\[a\[eta\]\]). We obtain $$\label{eta* tr}
\eta^{*}(u)=\Big(1+\rme^{-\lambda-t\big(\varphi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)+a[\eta^{*}]\psi\big(\rme^{2\rmi\pi u-b[\eta^{*}]}\big)\big)}\Big)^{-1}\;.$$ *Remark*: unlike section \[section density\], the optimal function $\eta^{*}(u)$ is not a real function. This means that the saddle point of the functional integral (\[tr\[path integral\]\]) lies in the complex plane. The function $\eta^{*}$ of (\[eta\* tr\]) can be recovered from the function $\eta^{*}$ of (\[eta\* density\]) by the choice $\omega=t$ and $\overline{\omega}=0$, where $\overline{\,\cdot\,}$ denotes complex conjugation, and $\omega$ and $\overline{\omega}$ must be thought of as independent variables.
[Contour integrals]{} The optimal function $\eta^{*}(u)$ is an analytic function of $z=\rme^{2\rmi\pi u-b[\eta^{*}]}$. One defines $$\label{sigma(z) trace 0}
\sigma(z)=\eta^{*}(u)=\Big(1+\rme^{-\lambda-t(\varphi(z)+a[\eta^{*}]\psi(z))}\Big)^{-1}\;.$$ Under the assumption that the contours of integration can be freely deformed from $|z|=\rme^{-{\mathrm{Re}}\,b[\eta^{*}]}$ to something independent of $b[\eta^{*}]$, we recover the equations (\[s\[sigma\]\]) and (\[rho\[sigma\]\]) for $s\equiv s[\eta^{*}]$ and $\rho$, but with $\sigma$ now given by (\[sigma(z) trace 0\]) instead of (\[sigma(z) density\]). The equations for the quantities $e\equiv e[\eta^{*}]$ and $a=a[\eta^{*}]$ become $$\label{e[sigma] tr}
e=\frac{1}{2\rmi\pi}\oint\frac{\rmd z}{z}\,\sigma(z)\varphi\big(z\big)\;$$ and $$\label{a[sigma] tr}
a=-\oint\frac{\rmd z}{2\rmi\pi}\,\sigma(z)\,\partial_{z}\big(\varphi(z)+a\,\psi(z)\big)\;.$$ Eq. (\[sigma(z) trace 0\]) implies $$\varphi(z)+a\psi(z)=-\frac{\lambda}{t}-\frac{1}{t}\log\frac{1-\sigma(z)}{\sigma(z)}\;.$$ Combining this with (\[a\[sigma\] tr\]) gives $$a=\oint\frac{\rmd z}{2\rmi\pi t}\,\partial_{z}\log(1-\sigma(z))=\frac{w}{t}\;,$$ where $w\in\mathbb{Z}$ is the winding number of $1-\sigma(z)$ around the origin. We assume in the following that $w=0$, hence $a=0$ and $$\label{sigma(z) trace}
\sigma(z)=\Big(1+\rme^{-\lambda-t\varphi(z)}\Big)^{-1}\;.$$ The property $a=0$ is compatible, at least for small times, with numerical solutions of (\[rho\[sigma\]\]) with $a=0$, see fig. \[fig sigma\].\
Unlike section \[section density\], the expression (\[sigma(z) trace\]) for $\sigma(z)$ allows to calculate explicitly the Lagrange multiplier $\lambda$, the eigenvalue $e$, the “entropy” $s$ and the function $f(t)=s+t\,e$. Details are given in \[appendix trace\]. In the end, we recover (\[f(t)\]).
[Singularities of sigma(z)]{}
[ccc]{}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the left, the singularities of the function $\sigma$ for $t=5$ (poles and cuts) are drawn in red (gray in printed version). The circle of radius $r_{t}$ (\[r\[t\]\]) represents a possible contour of integration. On the right, the image by $\sigma$ of this contour is drawn for $t$ between $1$ (inner curve) to $5$ (outer curve).[]{data-label="fig sigma"}](Poles.eps "fig:"){width="60mm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
&&
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the left, the singularities of the function $\sigma$ for $t=5$ (poles and cuts) are drawn in red (gray in printed version). The circle of radius $r_{t}$ (\[r\[t\]\]) represents a possible contour of integration. On the right, the image by $\sigma$ of this contour is drawn for $t$ between $1$ (inner curve) to $5$ (outer curve).[]{data-label="fig sigma"}](sigma.eps "fig:"){width="50mm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In the previous subsection, we used a deformation of the contour of integration in order to eliminate completely the parameter $b$. We implicitly assumed that this was possible without crossing singularities of $\sigma(z)$. We come back to this issue here, with the expression (\[sigma(z) trace\]) for $\sigma(z)$ resulting from the assumption $a=0$.\
We focus on the half-filled case. From (\[phi(z) rho=1/2\]) and (\[lambda(t,rho)\]), the singularities of the function $\sigma$ consist in an essential singularity at $z=0$, two cuts starting at $\pm2\rmi$, and an infinity of poles located at $\pm z_{k}$, $k\in\mathbb{N}$, with $$z_{k}=\frac{2\rmi t}{\sqrt{t^{2}+4(2k+1)^{2}\pi^{2}}}\;.$$ For $t=0$, we observe that all the poles are located at the origin. For times $t<t(\gamma)$, where $t(\gamma)$ is the first non-analyticity of $f$ discussed at the end of section \[subsection Q\], the contour of integration in (\[s\[sigma\]\]) and (\[e\[sigma\] tr\]) then has to enclose all the poles, but must not cross the cuts. The circle of center $0$ and radius $$\label{r[t]}
r_{t}=1+t/\sqrt{t^{2}+4\pi^{2}}\;$$ is a possible contour.\
We observed in section \[subsection Q\] that the function $f(t)$ is defined piecewise. The non-analyticity of $f(t)$ should be the sign of the presence of several saddle points competing in the functional integral (\[tr\[path integral\]\]). This is similar to what happens in the direct calculations of $f(t)$ for free particles in \[appendix free particles\], with a simple integral instead of a functional integral. It is not completely clear, however, how several saddle points emerge from the functional integral (\[tr\[path integral\]\]) for TASEP. It might be due to a change in the contour of integration at $t=t(\gamma)$, with a new contour that does not enclose all the poles of $\sigma$. There could also be a transition in eq. (\[a\[sigma\] tr\]) from the solution $a=0$ to another value due to a change in the winding number of $1-\sigma(z)$ around $0$.
[Conclusion]{} Parametric expressions can be derived for all the eigenvalues of TASEP using Bethe ansatz. These expressions allow a study of large scale properties of the spectrum in the thermodynamic limit, in particular the curve marking the edge of the spectrum, the density of eigenvalues in the bulk of the spectrum and the generating function of the cumulants of the eigenvalues.\
A natural extension of the present work would be to analyse the structure of the eigenvalues closer to the origin. Of particular interest are eigenvalues with a real part scaling as $L^{-3/2}$, which control the relaxation to the stationary state. Another goal would be to obtain asymptotic expressions for the scalar product between an eigenstate characterized by a density $\eta$ of $k_{j}$’s as in sections \[section density\] and \[section trace\] and a microstate characterized by a density profile of particles. This would allow to study physical quantities more interesting than the trace of the time evolution operator.\
Another very interesting extension would be the case of the asymmetric simple exclusion process with partial asymmetry, where particles are allowed to hop in both directions, with an asymmetry parameter controlling the bias. It would be nice if it were possible to derive parametric expressions for the eigenvalues analogous to (\[e\[k\]\]) and (\[b\[k\]\]). A good starting point seems to be the quantum Wronskian formulation of the Bethe equations [@PS99.1; @P10.1], where a parameter analogous to the parameter $B$ we used here exists. It would allow to study how the spectrum changes at the transitions between equilibrium and non-equilibrium.\
Finally, it would be nice to understand how the approach used here to study the spectrum of TASEP relates to thermodynamic Bethe ansatz [@YY69.1]. The latter follows from the observation that, in the thermodynamic limit, Bethe roots accumulate on a curve in the complex plane. The density of Bethe roots along this curve can usually be shown to be the solution of a nonlinear integral equation. We would like to understand whether the absence of non-linear integral equations in our calculations is only due to the special decoupling (\[polynomial\[y,B\]=0\]) of the Bethe equations for TASEP.
\*[Acknowledgements]{} I thank B. Derrida for several very helpful discussions. I also thank D. Mukamel for his warm welcome at the Weizmann Institute of Science, where early stages of this work were done.
[Bijection g]{} \[appendix g\] The function $g$ is defined in (\[g\]) on the whole complex plane minus the negative real axis $(-\infty,0]$, if one chooses the usual cut of the logarithm. It is convenient to use polar coordinates, writing $y=r\,\rme^{\rmi\,\theta}$ with $r>0$ and $-\pi<\theta<\pi$. Then $g(y)$ is divergent in the limit of small and large values of $r$. One has $$\begin{aligned}
&& g(y)\simeq r^{-\rho}\rme^{-\rmi\,\rho\,\theta}\quad{\mathrm{for}}\;r\to0\\
&& g(y)\simeq-r^{1-\rho}\rme^{\rmi(1-\rho)\theta}\quad{\mathrm{for}}\;r\to\infty\;.\end{aligned}$$ This implies $\arg g(y)\in(-\rho\pi,\rho\pi)$ for small $r$ and $\arg g(y)\in(-\pi,-\rho\pi)\cup(\rho\pi,\pi]$ for large $r$. The image of a point on the cut of $g$ is $$g(-r\pm\rmi\,0^{\pm})=\rme^{\pm\rmi\pi\rho}\,\frac{1+r}{r^{\rho}}\;.$$ The derivative of $g(y)$ with respect to $\theta$ at this point verifies $$\frac{\partial_{\theta}g}{g}(-r\pm\rmi\,0^{\pm})=\frac{\rmi(1-\rho)}{r+1}\Big[r-\frac{\rho}{1-\rho}\Big]\;.$$ This implies that the curves $\{g(r\rme^{\rmi\,\theta}),\theta\in(-\pi,\pi)\}$ join the cuts in the image space orthogonally (which already follows from the local holomorphicity of $g$, as the image of the orthogonality of any circle of center $0$ with the negative real axis), from one side or the other depending on whether $r$ is smaller or larger than $\rho/(1-\rho)$.\
When $y$ spans $(-\infty,0]\pm0\,\rmi$, the image of the cut spans $\rme^{\rmi\pi\rho}[\rme^{\ell},\infty)\,\cup\,\rme^{-\rmi\pi\rho}[\rme^{\ell},\infty)$, with $\ell$ defined in equation (\[l\[rho\]\]). The bijective nature of $g$ is clearly seen in fig. \[fig grid g\] and fig. \[fig grid g-1\] where the functions $g$ and its inverse $g^{-1}$ are represented.
[ccc]{}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Deformation of a grid by the function $g$ (\[g\]), with $\rho=1/3$. On the left is a grid in polar coordinates, with angles $\theta$ regularly spaced of $2\pi/10$ and radii $0.0005$, $0.002$, $0.005$, $0.02$, $0.05$, $0.2$, $0.5$, $2$, $5$, $10$, $20$, $40$ coloured from blue for small radius to red for large radius. The thick red line corresponds to the cut of the function $g$. On the right, the image by the function $g$ of the previous grid is drawn, using the same colors for a curve in the initial grid and its image by $g$. The almost semi-circular curves on the left part of the complex plane are red, the ones on the right are blue. The two thick red lines correspond to the image of the cut by $g$.[]{data-label="fig grid g"}](Gridy.eps "fig:"){width="70mm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
&&
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Deformation of a grid by the function $g$ (\[g\]), with $\rho=1/3$. On the left is a grid in polar coordinates, with angles $\theta$ regularly spaced of $2\pi/10$ and radii $0.0005$, $0.002$, $0.005$, $0.02$, $0.05$, $0.2$, $0.5$, $2$, $5$, $10$, $20$, $40$ coloured from blue for small radius to red for large radius. The thick red line corresponds to the cut of the function $g$. On the right, the image by the function $g$ of the previous grid is drawn, using the same colors for a curve in the initial grid and its image by $g$. The almost semi-circular curves on the left part of the complex plane are red, the ones on the right are blue. The two thick red lines correspond to the image of the cut by $g$.[]{data-label="fig grid g"}](Gridg.eps "fig:"){width="70mm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[ccc]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Deformation of a grid by the inverse function $g^{-1}$ (\[g\]), with $\rho=1/3$. On the left is a grid in polar coordinates, with angles $\theta$ regularly spaced of $2\pi/15$ and radii $0.1\,\rme^{\ell}$, $0.25\,\rme^{\ell}$, $0.5\,\rme^{\ell}$, $0.75\,\rme^{\ell}$, $\rme^{\ell}$, $1.1\,\rme^{\ell}$, $1.5\,\rme^{\ell}$ coloured from blue for small radius to red for large radius, with $\ell$ defined in (\[l\[rho\]\]). The two thick red lines correspond to the cuts of the function $g^{-1}$. On the right is represented the image by the function $g^{-1}$ of the previous grid, using the same colors for a curve in the initial grid and its image by $g^{-1}$. The almost circular shapes of large radius and small radius around $0$ are red. The small almost circular shapes around $1$ are blue.[]{data-label="fig grid g-1"}](Gridy2.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
&&
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Deformation of a grid by the inverse function $g^{-1}$ (\[g\]), with $\rho=1/3$. On the left is a grid in polar coordinates, with angles $\theta$ regularly spaced of $2\pi/15$ and radii $0.1\,\rme^{\ell}$, $0.25\,\rme^{\ell}$, $0.5\,\rme^{\ell}$, $0.75\,\rme^{\ell}$, $\rme^{\ell}$, $1.1\,\rme^{\ell}$, $1.5\,\rme^{\ell}$ coloured from blue for small radius to red for large radius, with $\ell$ defined in (\[l\[rho\]\]). The two thick red lines correspond to the cuts of the function $g^{-1}$. On the right is represented the image by the function $g^{-1}$ of the previous grid, using the same colors for a curve in the initial grid and its image by $g^{-1}$. The almost circular shapes of large radius and small radius around $0$ are red. The small almost circular shapes around $1$ are blue.[]{data-label="fig grid g-1"}](Gridg-1.eps "fig:"){width="70mm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[Density of eigenvalues close to the origin]{} \[appendix density\] In this appendix, we perform the calculations related to the limit $e\to0$ of the density of eigenvalues of the Markov matrix at half-filling. We compute the limit $|\omega|\to\infty$, $\arg\omega\to0$ of the integrals for the quantities $\alpha$ (\[alpha\[sigma\] density\]), $\beta$ (\[beta\[sigma\] density\]), $e$ (\[e\[sigma\] density\]) and $s$ (\[s\[sigma\]\]). The integrals must be decomposed as a sum of $4$ terms. Up to terms exponentially small in $\epsilon^{-1}\gg1$, one has $$\begin{aligned}
&& \int_{0}^{1}\rmd u\,F[\sigma(\rme^{2\rmi\pi u}),\rme^{2\rmi\pi u}]\simeq\int_{-\frac{1}{4}+\epsilon}^{\frac{1}{4}-\epsilon}\rmd u\,F[1,\rme^{2\rmi\pi u}]+\int_{\frac{1}{4}+\epsilon}^{\frac{3}{4}-\epsilon}\rmd u\,F[0,\rme^{2\rmi\pi u}]\\
&&\hspace{25mm} +\int_{-\frac{1}{4}+\epsilon}^{-\frac{1}{4}-\epsilon}\rmd u\,F[\sigma(\rme^{2\rmi\pi u}),\rme^{2\rmi\pi u}]+\int_{\frac{1}{4}+\epsilon}^{\frac{1}{4}-\epsilon}\rmd u\,F[\sigma(\rme^{2\rmi\pi u}),\rme^{2\rmi\pi u}]\;.\nonumber\end{aligned}$$ After a little rewriting, one obtains $$\begin{aligned}
\label{intF}
&&\fl\hspace{5mm} \int_{0}^{1}\rmd u\,F[\sigma(\rme^{2\rmi\pi u}),\rme^{2\rmi\pi u}]\simeq\int_{-\frac{1}{4}}^{\frac{1}{4}}\rmd u\,F[1,\rme^{2\rmi\pi u}]+\int_{\frac{1}{4}}^{\frac{3}{4}}\rmd u\,F[0,\rme^{2\rmi\pi u}]\\
&&\fl\hspace{4mm} +\int_{-\epsilon}^{0}\rmd u\,\Big(F[\sigma(\rmi\,\rme^{2\rmi\pi u}),\rmi\,\rme^{2\rmi\pi u}]+F[\sigma(-\rmi\,\rme^{2\rmi\pi u}),-\rmi\,\rme^{2\rmi\pi u}]-F[1,\rmi\,\rme^{2\rmi\pi u}]-F[0,-\rmi\,\rme^{2\rmi\pi u}]\Big)\nonumber\\
&&\fl\hspace{4mm} +\int_{0}^{\epsilon}\rmd u\,\Big(F[\sigma(\rmi\,\rme^{2\rmi\pi u}),\rmi\,\rme^{2\rmi\pi u}]+F[\sigma(-\rmi\,\rme^{2\rmi\pi u}),-\rmi\,\rme^{2\rmi\pi u}]-F[0,\rmi\,\rme^{2\rmi\pi u}]-F[1,-\rmi\,\rme^{2\rmi\pi u}]\Big)\;.\nonumber\end{aligned}$$ For the four quantities $\alpha$, $\beta$, $e$ and $s$, the terms with $F[0,\rme^{2\rmi\pi u}]$ vanishes, as well as the terms with $F[1,\rme^{2\rmi\pi u}]$ for $s$.\
In order to continue the calculations, several scalings need to be considered for ${\mathrm{Im}}\,\omega$ when $|\omega|\to\infty$. We write $\omega=r+\rmi\chi r^{c}$ with $r>0$, $\chi>0$, $c<1$, and will take the limit $r\to\infty$. The case $\chi<0$ then follows from the invariance of the spectrum by complex conjugation. A summary of the different scalings obtained is given in table \[table scalings\]. The scalings $c=-2$, $-1$, $-1/2$, $0$, $1/6$, $1/5$, $1/4$, $1/3$, $1/2$, $2/3$, $3/4$, $4/5$ were checked by solving numerically (\[alpha\[sigma\] density\]), (\[beta\[sigma\] density\]) for $r=100,200,\ldots,1000$. We used the BST algorithm [@HS88.1] in order to improve the convergence to $r\to\infty$. For all the scalings studied, the relative errors in the numerical coefficients of $\alpha$, $\beta$, ${\mathrm{Re}}\,e$, ${\mathrm{Im}}\,e$ and $s$ obtained from the BST algorithm were smaller than $10^{-3}$ compared to the exact values.
$\displaystyle
\begin{array}{cccccc}
& \alpha-r/4 & \beta+\log2 & {\mathrm{Re}}\,e & {\mathrm{Im}}\,e & s\\
&&&&&\\
c<0 & -\frac{0.164473}{r^{1/3}} & \frac{0.706532}{r^{2/3}} & -\frac{0.147533}{r^{5/3}} & \frac{0.224251\,\chi}{r^{-c+4/3}} & \frac{0.737667}{r^{2/3}}\\
& & \rotatebox{90}{$=$} & \rotatebox{90}{$=$} & & \rotatebox{90}{$=$}\\
0<c<\frac{1}{3} & \frac{0.107018\,\chi^{2}}{r^{-2c+1/3}} & \frac{0.706532}{r^{2/3}} & -\frac{0.147533}{r^{5/3}} & \frac{0.224251\,\chi}{r^{-c+4/3}} & \frac{0.737667}{r^{2/3}}\\
&&&&&\\
\frac{1}{3}<c<1 & \frac{\sqrt{3}\chi r^{c}}{12} & \frac{\sqrt{3}\,\chi}{2r^{1-c}} & -\frac{2^{1/2}3^{3/4}\,\chi^{5/2}}{5\pi r^{5(1-c)/2}} & -\frac{2^{1/2}\,\chi^{3/2}}{3^{1/4}\pi r^{3(1-c)/2}} & \frac{\pi}{2^{1/2}3^{3/4}\chi^{1/2}r^{(1+c)/2}}
\end{array}$
[Scaling c=0.]{} \[subsection scaling c=0\] Writing $\alpha=r/4+{\delta\!\alpha}/r^{1/3}$, $\beta=-\log2+{\delta\!\beta}/r^{2/3}$ and expanding the equations (\[beta\[sigma\] density\]) and (\[alpha\[sigma\] density\]) respectively up to order $r^{-1}$ and $r^{-2/3}$ with $\epsilon\sim r^{-2/3}$ in (\[intF\]) give after straightforward, but rather tedious calculations (\[deltabeta c=0\]) and $$\begin{aligned}
\label{deltaalpha c=0}
&& {\delta\!\alpha}\Big[\frac{1}{2\pi}-\int_{0}^{\infty}\rmd x\,\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{-1/2}]}{1+\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}}\Big]\\
&& =\frac{{\delta\!\beta}^{2}}{3}\Big[\frac{1}{10\pi}-\int_{0}^{\infty}\rmd x\,\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}{1+\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}}\Big]\nonumber\\
&&\hspace{-7mm} +\frac{\sqrt{2{\delta\!\beta}}\,\chi^{2}}{4}\,\int_{0}^{\infty}\rmd x\,\frac{{\mathrm{Re}}[(1+2\rmi\pi x)^{1/2}]\,{\mathrm{Re}}[(1+2\rmi\pi x)^{-1/2}]\,\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}}{\Big(1+\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}\Big)^{2}}\;.\nonumber\end{aligned}$$ We used the expansion $$\int_{-1/4}^{1/4}\!\rmd u\,\psi(2\,\rme^{2\rmi\pi u-\epsilon})=-\log2+\epsilon-\frac{4\sqrt{2}}{3\pi}\,\epsilon^{3/2}+\frac{2\sqrt{2}}{15\pi}\,\epsilon^{5/2}+{\mathcal{O}\left(\epsilon^{7/2}\right)}\;.$$ We observe that the equations (\[deltabeta c=0\]) and (\[deltaalpha c=0\]) for ${\delta\!\beta}$ and ${\delta\!\alpha}$ decouple in the scaling $c=0$, unlike the original equations (\[beta\[sigma\] density\]) and (\[alpha\[sigma\] density\]) for $\beta$ and $\alpha$.\
Expanding the equation (\[e\[sigma\] density\]) for $e$, we find at leading order in $r$ (\[Re(e) c=0\]) and $$\label{Im(e) c=0}
{\mathrm{Im}}\,e=-\frac{4{\delta\!\beta}^{2}\chi}{r^{4/3}}\,\int_{0}^{\infty}\rmd x\,\frac{[{\mathrm{Re}}[(1+2\rmi\pi x)^{1/2}]]^{2}\,\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}}{\Big(1+\rme^{\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}\Big)^{2}}\;,$$ while the equation (\[s\[sigma\]\]) for $s$ gives (\[s c=0\]). Numerically, one has ${\delta\!\alpha}\simeq-0.164473+0.107018\,\chi^{2}$ and ${\mathrm{Im}}\,e\simeq-0.224251\,\chi\,r^{-4/3}$.
[Scaling c=1/3.]{} Writing $\alpha=r/4+{\delta\!\alpha}\,r^{1/3}$, $\beta=-\log2+{\delta\!\beta}\,r^{-2/3}$ and expanding the equations for $\beta$ (\[beta\[sigma\] density\]) and $\alpha$ (\[alpha\[sigma\] density\]) respectively up to order $r^{-1}$ and $r^{0}$ with $\epsilon\sim r^{-2/3}$ in (\[intF\]) give after again long but straightforward calculations $$\label{deltabeta c=1/3}
\frac{1}{3\pi}=\int_{0}^{\infty}\rmd x\,\Big[\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{1/2}]}{1+\rme^{\Phi_{+}(x)}}+\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{1/2}]}{1+\rme^{\Phi_{-}(x)}}\Big]\;$$ and $$\begin{aligned}
\label{deltaalpha c=1/3}
&& {\delta\!\alpha}\Big(\frac{1}{\pi}-\int_{0}^{\infty}\rmd x\,\Big[\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{-1/2}]}{1+\rme^{\Phi_{+}(x)}}+\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{-1/2}]}{1+\rme^{\Phi_{-}(x)}}\Big]\Big)\nonumber\\
&&\hspace{10mm} =-\frac{\chi}{4}\int_{0}^{\infty}\rmd x\,\Big[\frac{{\mathrm{Re}}[(1+2\rmi\pi x)^{-1/2}]}{1+\rme^{\Phi_{+}(x)}}-\frac{{\mathrm{Re}}[(1+2\rmi\pi x)^{-1/2}]}{1+\rme^{\Phi_{-}(x)}}\Big]\;,\end{aligned}$$ with the definition $$\begin{aligned}
&& \Phi_{\pm}(x)=\frac{(2{\delta\!\beta})^{3/2}}{3}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]-4\sqrt{2{\delta\!\beta}}\,{\delta\!\alpha}\,{\mathrm{Im}}[(1+2\rmi\pi x)^{1/2}]\nonumber\\
&&\hspace{20mm} \pm\sqrt{2{\delta\!\beta}}\,\chi\,{\mathrm{Re}}[(1+2\rmi\pi x)^{1/2}]\;.\end{aligned}$$ We observe that the equations for ${\delta\!\alpha}$ and ${\delta\!\beta}$ are now coupled, unlike in the scaling $c=0$. Similar calculations with the equations for $e$ (\[e\[sigma\] density\]) and $s$ (\[s\[sigma\]\]) give at leading order in $r$ $$\fl\hspace{15mm} {\mathrm{Re}}\,e=\frac{2\sqrt{2}\,{\delta\!\beta}^{5/2}}{3r^{5/3}}\Big(\frac{1}{5\pi}-\int_{0}^{\infty}\rmd x\,\Big[\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}{1+\rme^{\Phi_{+}(x)}}+\frac{{\mathrm{Im}}[(1+2\rmi\pi x)^{3/2}]}{1+\rme^{\Phi_{-}(x)}}\Big]\Big)\;,$$ $$\fl\hspace{15mm} {\mathrm{Im}}\,e=\frac{\sqrt{2}\,{\delta\!\beta}^{3/2}}{r}\int_{0}^{\infty}\rmd x\,\Big[\frac{{\mathrm{Re}}[({\delta\!\beta}+2\rmi\pi x)^{1/2}]}{1+\rme^{\Phi_{+}(x)}}-\frac{{\mathrm{Re}}[(1+2\rmi\pi x)^{1/2}]}{1+\rme^{\Phi_{-}(x)}}\Big]\;,$$ and $$\label{s c=1/3}
s=-5\,r\,{\mathrm{Re}}\,e+3\chi\,r^{1/3}\,{\mathrm{Im}}\,e\;.$$ In the limit $\chi\to0$, we see that ${\delta\!\beta}$, ${\mathrm{Re}}\,e$ and $s$ converge to their value in the scaling $c=0$, while ${\delta\!\alpha}\simeq0.107018\,\chi^{2}$ and ${\mathrm{Im}}\,e\simeq-0.224251\,\chi/r$, with the same numerical constants as in the scaling $c=0$.\
The limit $\chi\to\infty$ is a bit more complicated. We check that ${\delta\!\alpha}\simeq\sqrt{3}\chi/12$ and ${\delta\!\beta}\simeq\sqrt{3}\chi/2$ in this limit. Indeed, with these values for ${\delta\!\alpha}$ and ${\delta\!\beta}$, we observe that $\Phi_{+}(\chi u)>0$ for all $u>0$, while $\Phi_{-}(\chi u)<0$ when $u<3/(4\pi)$ and $\Phi_{-}(\chi u)>0$ otherwise. In the integrals of (\[deltabeta c=1/3\]) and (\[deltaalpha c=1/3\]), the first term of the integrands gives a contribution exponentially small in $\chi$ to the integral, while the second term of the integrands contributes only for $v/\chi<3/(4\pi)$, with $\rme^{\Phi_{-}(v)}\to0$. Performing explicitly the integrals then shows that (\[deltabeta c=1/3\]) and (\[deltaalpha c=1/3\]) are indeed verified. We also checked numerically that the solutions of (\[deltabeta c=1/3\]) and (\[deltaalpha c=1/3\]) for finite $\chi$ seem to grow as ${\delta\!\alpha}\simeq\sqrt{3}\chi/12$ and ${\delta\!\beta}\simeq\sqrt{3}\chi/2$ for large $\chi$.\
Similar calculations lead to ${\mathrm{Re}}\,e\simeq-2^{1/2}3^{3/4}\chi^{5/2}/(5\pi r^{5/3})$ and ${\mathrm{Im}}\,e\simeq-2^{1/2}\chi^{3/2}/(3^{1/4}\pi r)$ in the limit $\chi\to\infty$. Using (\[s c=1/3\]), it implies $s=0$ on the scale $\chi^{5/2}$. Going beyond that requires some more work: one has to take into account the contribution of the integrals near $v=3\chi/(4\pi)$, making the change of variables $v=3\chi/(4\pi)+\chi^{-1/2}u$. Writing ${\delta\!\alpha}=\sqrt{3}\chi/12+\chi^{-2}{\delta\!\alpha}_{2}$ and ${\delta\!\beta}=\sqrt{3}\chi/2+\chi^{-2}{\delta\!\beta}_{2}$, one finds at the end of the calculation ${\delta\!\alpha}_{2}\simeq-\pi^{2}/144$, ${\delta\!\beta}_{2}\to0$, ${\mathrm{Re}}\,e\simeq-2^{1/2}3^{3/4}\chi^{5/2}/(5\pi r^{5/3})-\pi/(2^{7/2}3^{3/4}\chi^{1/2}r^{5/3})$ and ${\mathrm{Im}}\,e\simeq-2^{1/2}\chi^{3/2}/(3^{1/4}\pi r)+\pi/(2^{7/2}3^{3/4}\chi^{3/2}r)$. It finally leads to $s\simeq\pi/(2^{1/2}3^{3/4}\chi^{1/2}r^{2/3})$.
[Scalings c<0 and 0<c<1/3.]{} The crossover scaling $c=0$ is surrounded by the $2$ regimes $c<0$ and $0<c<1/3$. Similar calculations to the ones performed for the scalings $c=0$ allow to compute the quantities $\alpha$, $\beta$, ${\mathrm{Re}}\,e$, ${\mathrm{Im}}\,e$ and $s$ at leading order in $r$.\
We observe that the results found in the regime $c<0$ are identical to the limit $\chi\to0$ in the scaling $c=0$. Similarly, the results found in the regime $0<c<1/3$ are identical to the limit $\chi\to\infty$ in the scaling $c=0$.\
Since the regimes $c<0$ and $0<c<1/3$ differ only by the value of the parameter ${\delta\!\alpha}$, which is just an intermediate quantity needed for the calculations, one can for all purpose consider this as a unique regime, $c<1/3$: the quantities of interest $e$ and $s$ then depend in a simple way on $r$, $\chi$ and $c$ in the whole regime.
[Scaling 1/3<c<1.]{} The regime $1/3<c<1$ is much more complicated. There, one is lead to make a change of variable of the form $u=\pm1/4+d_{1}/r^{1-c}+d_{2}/r^{2(1-c)}+\ldots+d_{m}/r^{m(1-c)}+v/r^{(c+1)/2}$ in the integrals, where the constants $d_{j}$ must be such that the argument of the exponential in $\sigma(u)$ is of order $r^{0}$. Using a similar rewriting to (\[intF\]), but with $\pm1/4$ replaced by $\pm1/4+d_{1}/r^{1-c}+d_{2}/r^{2(1-c)}+\ldots+d_{m}/r^{m(1-c)}$, the same kind of calculations as in the other scalings can in principle be done.\
We checked only the specific case $c=2/3$. There, making the change of variables $u=\pm1/4+d_{1}/r^{1/3}+d_{2}/r^{2/3}+v/r^{5/6}$ in the integrals, we find that $d_{1}$ must be solution of the equation $$(3\chi-4\pi d_{1}){\mathrm{Re}}[({\delta\!\beta}+2\rmi\pi d_{1})^{1/2}]=2({\delta\!\beta}-6{\delta\!\alpha}){\mathrm{Im}}[({\delta\!\beta}+2\rmi\pi d_{1})^{1/2}]\;,$$ while $d_{2}$ has a rather complicated (but completely explicit) expression in terms of $\chi$, ${\delta\!\alpha}$, ${\delta\!\beta}$ and $d_{1}$. The equation for $\beta$ at order $r^{-1/2}$ then gives $${\mathrm{Re}}[({\delta\!\beta}+2\rmi\pi d_{1})^{3/2}]=0\;,$$ while the equation for $\alpha$ at order $r^{1/2}$ leads to $$3\chi-4\pi d_{1}=2\rmi({\delta\!\beta}-6{\delta\!\alpha})\;.$$ Gathering the last $3$ equations, one finds $d_{1}=3\chi/(4\pi)$, ${\delta\!\alpha}=\sqrt{3}\chi/12$ and ${\delta\!\beta}=\sqrt{3}\chi/2$. The equation for $d_{2}$ then gives $d_{2}=-7\chi^{2}/(10\sqrt{3}\pi)$. From the equation for $e$, one obtains ${\mathrm{Re}}\,e=-2^{1/2}3^{3/4}\chi^{5/2}/(5\pi r^{5/6})$ and ${\mathrm{Im}}\,e=-2^{1/2}\chi^{3/2}/(3^{1/4}\pi r^{1/2})$, while the equation for $s$ leads to $s=\pi/(2^{1/2}3^{3/4}\chi^{1/2}r^{5/6})$. We observe that the numerical constants are the same as in the limit $\chi\to\infty$ of the scaling $c=1/3$. We conjecture that this is the case for all the scaling $1/3<c<1$.
[Explicit calculations for the generating function f(t)]{} \[appendix trace\] In this appendix, we calculate explicitly the contour integrals in (\[rho\[sigma\]\]), (\[e\[sigma\] tr\]) and (\[s\[sigma\]\]), with $\sigma(z)$ given by (\[sigma(z) trace\]). In order to do this, we first prove two useful formulas, (\[exp(psi)\]) and (\[exp(phi)\]) for the exponential of the functions $\psi$ (\[psi(z)\]) and $\varphi$ (\[phi(z)\]).
[Formula for exp(x psi(z))]{} From (\[phipsi\[g\]\]), one has $\rme^{x\,\psi(z)}=(g^{-1}(z))^{x}$. Using (\[h(g-1)\]) with $h(y)=y^{x}$, the expansion near $z=0$ leads to $$\label{exp(psi)}
\rme^{x\,\psi(z)}=x\,\sum_{r=0}^{\infty}{{\rho\,r+x \choose r}}\frac{(-1)^{r}z^{r}}{\rho\,r+x}\;.$$
[Formula for exp(-x phi(z))]{} Expanding the exponential in $\rme^{-x\varphi(z)}$ and using (\[relation phi psi 2\]), one finds $$\rme^{-x\varphi(z)}=1+\sum_{k=1}^{\infty}\frac{(-z)^{-k}x^{k}}{k!}\rme^{k(1-\rho)\psi(z)}\;.$$ Eq. (\[exp(psi)\]) then leads to $$\label{exp(phi)}
\hspace{-5mm}
\rme^{-x\varphi(z)}=1+(1-\rho)\sum_{k=1}^{\infty}\frac{x^{k}}{(k-1)!}\sum_{r=0}^{\infty}{{\rho\,r+(1-\rho)k \choose r}}\frac{(-1)^{r-k}z^{r-k}}{\rho\,r+(1-\rho)k}\;.$$ In particular, one has $$\label{residue(exp(phi))}
\oint\frac{\rmd z}{2\rmi\pi}\,\frac{\rme^{-x\varphi(z)}}{z}=\rho+(1-\rho)\rme^{x}\;.$$
[Exact expression for the Lagrange multiplier lambda]{} The Lagrange multiplier $\lambda$ is fixed by the normalization condition (\[rho\[sigma\]\]) with $\sigma$ given by (\[sigma(z) trace\]). One has $$\rho=\sum_{r=0}^{\infty}(-1)^{r}(\rme^{-\lambda})^{r}\oint\frac{\rmd z}{2\rmi\pi}\,\frac{\rme^{-rt\varphi(z)}}{z}\;.$$ Using (\[residue(exp(phi))\]) with $x=rt$ implies $$\rho=\frac{\rho}{1+\rme^{-\lambda}}+\frac{1-\rho}{1+\rme^{t-\lambda}}\;.$$ Solving for $\lambda$ finally gives $$\label{lambda(t,rho)}
\lambda=\log\frac{2\rho\,\rme^{t}}{1-2\rho+\sqrt{1+4\rho(1-\rho)(\rme^{t}-1)}}\;,$$ which simplifies at half filling to $\lambda=t/2$. The sign $+$ is chosen in front of the square root for continuity at $t=0$, for which one has $\lambda=\log[\rho/(1-\rho)]$.
[Exact expression for the eigenvalue e]{} The expression (\[e\[sigma\] tr\]) for the eigenvalue can be made completely explicit. From (\[sigma(z) trace\]), one has $$\begin{aligned}
&& e=\sum_{r=0}^{\infty}(-1)^{r}(\rme^{-\lambda})^{r}\oint\frac{\rmd z}{2\rmi\pi}\,\frac{\rme^{-rt\varphi(z)}\varphi(z)}{z}\nonumber\\
&&\hspace{2mm} =\oint\frac{\rmd z}{2\rmi\pi}\,\frac{\varphi(z)}{z}+\partial_{t}\Big[\sum_{r=1}^{\infty}\frac{(-1)^{r-1}(\rme^{-\lambda})^{r}}{r}\oint\frac{\rmd z}{2\rmi\pi}\,\frac{\rme^{-rt\varphi(z)}}{z}\Big]\;.\end{aligned}$$ Using (\[residue(exp(phi))\]) to compute the residue, one finds $$e=-\frac{1-\rho}{1+\rme^{t-\lambda}}=\frac{1-\sqrt{1+4\rho(1-\rho)(\rme^{t}-1)}}{2(\rme^{t}-1)}\;.$$
[Exact expression for s]{} After a little rewriting, the expression (\[s\[sigma\]\]) for $s$ becomes $$s=\oint\frac{\rmd z}{2\rmi\pi z}\,\Big[(1-\sigma(z))\big(\lambda+t\varphi(z)\big)+\log\big(1+\rme^{-\lambda-t\varphi(z)}\big)\Big]\;.$$ Using (\[rho\[sigma\]\]), (\[e\[sigma\] tr\]) and (\[sigma(z) trace\]), one has $$s=\lambda(1-\rho)-t(1-\rho)-te+\sum_{r=1}^{\infty}\frac{(-1)^{r-1}\rme^{r\lambda}}{r}\oint\frac{\rmd z}{2\rmi\pi}\,\frac{\rme^{-rt\varphi(z)}}{z}\;.$$ Using (\[residue(exp(phi))\]) to compute the residue, one finds $$s=-te+\rho\log(1+\rme^{-\lambda})+(1-\rho)\log(1+\rme^{\lambda-t})\;.$$
[Free particles]{} \[appendix free particles\] In this appendix, we study a system of $N=\rho L$ non-interacting particles hopping to the nearest site on the right with rate $1$ on a periodic lattice of $L$ sites. Unlike TASEP, there is no exclusion constraint. We will consider both the case of distinguishable particles and the case of undistinguishable particles.\
Similarly to TASEP, we call $M(\gamma)$ the deformation of the Markov matrix which counts the current of particles. The action of $M(\gamma)$ on a microstate with particles at positions $x_{1},\ldots,x_{N}$ is $$M(\gamma)|x_{1},\ldots,x_{N}\rangle=\sum_{j=1}^{N}\big(\rme^{\gamma}|\ldots,x_{j}+1,\ldots\rangle-|\ldots,x_{j},\ldots\rangle\big)\;.$$ The $x_{j}$’s need not be distinct. For distinguishable particles, the $j$-th element of $|x_{1},\ldots,x_{N}\rangle$ is the position of the $j$-th particle, and the total number of microstates is $|\Omega_{{\mathrm{free}}}^{{\mathrm{d}}}|=L^{N}$. For undistinguishable particles, the positions are kept ordered as $x_{1}\leq\ldots\leq x_{N}$, and the number of configurations is then $|\Omega_{{\mathrm{free}}}^{{\mathrm{u}}}|={{L+N-1 \choose N}}$. In both cases, the eigenvectors of $M(\gamma)$ are products of plane waves, and the eigenvalues are of the form $$\label{E free}
E=\sum_{j=1}^{N}\big(\rme^{\gamma-2\rmi\pi k_{j}/L}-1\big)\;,$$ where each $k_{j}$ is an integer between $1$ and $L$. For distinguishable particles, there is no further restriction on the $k_{j}$’s. For undistinguishable particles the $k_{j}$’s must be ordered, $k_{1}\leq\ldots\leq k_{N}$.\
As in the case of TASEP, we define a density of eigenvalues $D(e)$ as in (\[D\[path integral\]\]) and a quantity $f(t)$ as in (\[f\[M\]\]). We calculate in this appendix $D(e)$ in the thermodynamic limit in the case of undistinguishable particles, and $f(t)$ for both distinguishable and undistinguishable particles.
[Density of eigenvalues for undistinguishable particles]{}
[ccc]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the left, spectrum of the Markov matrix $M$ for a system of $N=6$ non-interacting undistinguishable particles on $L=12$ sites. The black dots are the eigenvalues rescaled by a factor $1/L$. The black circle is the edge of the spectrum in the thermodynamic limit with $\rho=1/2$. The red (gray in printed version) curve is the parabolic approximation of the circle near the origin. On the right, number of rescaled eigenvalues with a given real part $e$ for non-interacting undistinguishable particles at half-filling, as a function of $e$. The black curve corresponds to the expressions (\[s\[eta\] free U\]), (\[e\[eta\] free U\]) parametrized by $\omega$ ranging from $-1000$ to $1000$. The red (gray in printed version) curve is the asymptotics (\[s(e) free U\]). The histograms correspond to the density of real part of eigenvalues obtained from numerical diagonalization for the finite system with $N=9$ particles on $L=18$ sites. The histograms are shifted so that their maximum is $(\log2+3\log3)/2\simeq L^{-1}\log|\Omega_{{\mathrm{free}}}^{{\mathrm{u}}}|$, with $|\Omega_{{\mathrm{free}}}^{{\mathrm{u}}}|$ the total number of microstates.[]{data-label="fig s(e) free U"}](EnvelopeFreeU.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
&&
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![On the left, spectrum of the Markov matrix $M$ for a system of $N=6$ non-interacting undistinguishable particles on $L=12$ sites. The black dots are the eigenvalues rescaled by a factor $1/L$. The black circle is the edge of the spectrum in the thermodynamic limit with $\rho=1/2$. The red (gray in printed version) curve is the parabolic approximation of the circle near the origin. On the right, number of rescaled eigenvalues with a given real part $e$ for non-interacting undistinguishable particles at half-filling, as a function of $e$. The black curve corresponds to the expressions (\[s\[eta\] free U\]), (\[e\[eta\] free U\]) parametrized by $\omega$ ranging from $-1000$ to $1000$. The red (gray in printed version) curve is the asymptotics (\[s(e) free U\]). The histograms correspond to the density of real part of eigenvalues obtained from numerical diagonalization for the finite system with $N=9$ particles on $L=18$ sites. The histograms are shifted so that their maximum is $(\log2+3\log3)/2\simeq L^{-1}\log|\Omega_{{\mathrm{free}}}^{{\mathrm{u}}}|$, with $|\Omega_{{\mathrm{free}}}^{{\mathrm{u}}}|$ the total number of microstates.[]{data-label="fig s(e) free U"}](DensityRealPartEigenvaluesFreeU.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Defining the density profile $\eta(u)$ of the $k_{j}$’s as in section \[subsection eigenvalues thermodynamic limit\], the rescaled eigenvalue $e=E/L$ is expressed in terms of $\eta$ as $$\label{e[eta] free U}
e[\eta]=\int_{0}^{1}\rmd u\,\eta(u)(\rme^{\gamma-2\rmi\pi u}-1)\;.$$ The number $\Omega[\eta]$ of ways to place $L\,\eta(u)\,du$ $k_{j}$’s in an interval of length $L\,du$ is ${{L\,du+L\,\eta(u)\,du-1 \choose L\,\eta(u)\,du}}$. Stirling’s formula implies $\Omega[\eta]\sim\rme^{Ls}$, where the “entropy” $s[\eta]$ is $$\label{s[eta] free U}
s[\eta]=\int_{0}^{1}\rmd u\,[-\eta(u)\log\eta(u)+(1+\eta(u))\log(1+\eta(u))]\;.$$ The difference with eq. (\[s\[eta\]\]) for TASEP comes from the fact that several $k_{j}$’s can be equal for non-interacting particles.\
We introduce the two Lagrange multipliers $\lambda\in\mathbb{R}$ and $\omega\in\mathbb{C}$ as in (\[s + Lagrange multipliers\]). The optimal function $\eta^{*}(u)$ that maximizes (\[s + Lagrange multipliers\]) with $s[\eta]$ given by (\[s\[eta\] free U\]) is $$\label{eta* free U}
\eta^{*}(u)=\Big(-1+\rme^{-\lambda-2{\mathrm{Re}}[\omega(\rme^{\gamma-2\rmi\pi u}-1)]}\Big)^{-1}\;.$$ Solving numerically (\[rho\[eta\]\]) for several values of $\omega$ allows to plot $s(e)$, see fig. \[fig s(e) free U\]. We are interested in the limit $e\to0$, which corresponds to $\omega\to\infty$, $\gamma=0$. We will only treat the case $e\in\mathbb{R}$, for which $\omega\in\mathbb{R}$. We first expand the denominator and the exponential in the expression (\[eta\* density\]) of $\eta^{*}$, as $$\eta^{*}(u)=\sum_{k=1}^{\infty}\sum_{j=0}^{\infty}\frac{k^{j}}{j!}\rme^{k(\lambda-2\omega)}(\omega\rme^{-2\rmi\pi u}+\overline{\omega}\rme^{2\rmi\pi u})^{j}\;,$$ where $\overline{\,\cdot\,}$ denotes complex conjugation. The integral over $u$ can then be performed in (\[rho\[eta\]\]), (\[e\[eta\] free U\]) and (\[s\[eta\] free U\]). After summing over $j$, we find $$\rho=\sum_{k=1}^{\infty}\rme^{k(\lambda-2\omega)}{\mathrm{I}}_{0}(2k\omega)\;,$$ $$e=-\rho+\sum_{k=1}^{\infty}\rme^{k(\lambda-2\omega)}{\mathrm{I}}_{1}(2k\omega)\;,$$ and $$s=-\rho\lambda-2\omega e+\sum_{k=1}^{\infty}\frac{\rme^{k(\lambda-2\omega)}}{k}\,{\mathrm{I}}_{0}(2k\omega)\;.$$ In the previous expressions, ${\mathrm{I}}_{0}$ and ${\mathrm{I}}_{1}$ are modified Bessel functions of the first kind. Taking the asymptotics of the Bessel functions for large argument and summing over $j$ gives $$\sqrt{2\pi}\rho\simeq\frac{{\mathrm{Li}}_{1/2}(\rme^{\lambda})}{\sqrt{2\omega}}+\frac{{\mathrm{Li}}_{3/2}(\rme^{\lambda})}{8(2\omega)^{3/2}}+\frac{9\,{\mathrm{Li}}_{5/2}(\rme^{\lambda})}{128(2\omega)^{5/2}}\;,$$ $$\sqrt{2\pi}(e+\rho)\simeq\frac{{\mathrm{Li}}_{1/2}(\rme^{\lambda})}{\sqrt{2\omega}}-\frac{3{\mathrm{Li}}_{3/2}(\rme^{\lambda})}{8(2\omega)^{3/2}}-\frac{15\,{\mathrm{Li}}_{5/2}(\rme^{\lambda})}{128(2\omega)^{5/2}}\;,$$ and $$\sqrt{2\pi}(s+\rho\lambda+2\omega e)\simeq\frac{{\mathrm{Li}}_{3/2}(\rme^{\lambda})}{\sqrt{2\omega}}+\frac{{\mathrm{Li}}_{5/2}(\rme^{\lambda})}{8(2\omega)^{3/2}}+\frac{9\,{\mathrm{Li}}_{7/2}(\rme^{\lambda})}{128(2\omega)^{5/2}}\;.$$ In the limit $\omega\to\infty$, one has $\lambda\to0^{-}$. After expanding the polylogarithms, we finally obtain $$\begin{aligned}
\label{s(e) free U}
&& s(e)\simeq\frac{3\,\zeta(3/2)^{2/3}(-e)^{1/3}}{2\pi^{1/3}}-\frac{\pi^{1/3}(-e)^{2/3}}{\rho\,\zeta(3/2)^{2/3}}\nonumber\\
&&\hspace{15mm} +\Big(\frac{2\pi}{3\rho^{2}\zeta(3/2)^{2}}+\frac{\zeta(1/2)}{\rho^{2}\zeta(3/2)}-\frac{\zeta(5/2)}{4\,\zeta(3/2)}\Big)e\;,\end{aligned}$$ where $\zeta$ is Riemann zeta function. We observe that $s(e)$ vanishes for $e=0$ with an exponent $1/3$. This exponent should not be confused with the exponent $1/2$ obtained from ${\mathrm{Re}}\,E\simeq-2\pi^{2}\sum_{j=1}^{N}k_{j}^{2}/L^{2}$ for eigenvalues that do not scale proportionally with $L$.
[Function f(t) for distinguishable particles]{}
[ccc]{}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Logarithm of the trace of the time evolution operator for non-interacting particles. On the left, graph of $f(\tau L)/L$ as a function of $\tau$ for distinguishable particles, with $f$ defined in (\[f\[M\]\]). In black are exact computations for $\rho=1/2$, $\gamma=0$ with $L=20$ (upper curve) and $L=100$ (lower curve). The thick, red curve corresponds to the large $L$ limit (\[f free d asymptotic L\]). On the right, graph of $f(t)$ as a function of $t$ for undistinguishable particles, with $f$ defined in (\[f\[M\]\]). In black are exact computations for $\rho=1/2$, $\gamma=0$ with $L=20$ (lower curve) and $L=80$ (upper curve). The thick, red curve corresponds to the large $L$ limit (\[f free u asymptotic L\]).[]{data-label="fig f free"}](ffreeDistinguishable.eps "fig:"){width="70mm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
&&
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Logarithm of the trace of the time evolution operator for non-interacting particles. On the left, graph of $f(\tau L)/L$ as a function of $\tau$ for distinguishable particles, with $f$ defined in (\[f\[M\]\]). In black are exact computations for $\rho=1/2$, $\gamma=0$ with $L=20$ (upper curve) and $L=100$ (lower curve). The thick, red curve corresponds to the large $L$ limit (\[f free d asymptotic L\]). On the right, graph of $f(t)$ as a function of $t$ for undistinguishable particles, with $f$ defined in (\[f\[M\]\]). In black are exact computations for $\rho=1/2$, $\gamma=0$ with $L=20$ (lower curve) and $L=80$ (upper curve). The thick, red curve corresponds to the large $L$ limit (\[f free u asymptotic L\]).[]{data-label="fig f free"}](ffreeUndistinguishable.eps "fig:"){width="70mm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
From the definition (\[f\[M\]\]) and the expression (\[E free\]) for the eigenvalues, one has $$\label{f free d}
f(t)=\rho\log\Big(\sum_{k=1}^{L}\rme^{t(\rme^{\gamma-2\rmi\pi k/L}-1)}\Big)\;.$$ For finite times, the sum becomes an integral in the large $L$ limit. After a rewriting as a contour integral, one finds $$f(t)-\rho\log L=\rho\log\Big(\oint\frac{\rmd z}{z}\,\rme^{t(\rme^{\gamma}z-1)}\Big)=-\rho\,t\;.$$ We are also interested in $f(t)$ for times $t$ of order $L$. Expanding the exponential in (\[f free d\]) leads to $$f(\tau L)=\rho\log\Big(\sum_{k=1}^{L}\sum_{j=0}^{\infty}\sum_{m=0}^{j}{{j \choose m}}\frac{(-1)^{j-m}L^{j}\tau^{j}\rme^{m\gamma}\rme^{-2\rmi\pi km/L}}{j!}\Big)\;.$$ Exchanging the order of the summations over $j$ and $m$ allows to perform the summation over $j$. One finds $$f(\tau L)=\rho\log\Big(\sum_{k=1}^{L}\sum_{m=0}^{\infty}\frac{L^{m}\tau^{m}\rme^{-\tau L}\rme^{m\gamma}\rme^{-2\rmi\pi km/L}}{m!}\Big)\;.$$ The summation over $k$ is then done with the help of $$\label{sum k}
\sum_{k=1}^{L}\rme^{-2\rmi\pi km/L}=L\sum_{r=0}^{\infty}\delta_{m,rL}\;,$$ which leads to $$f(\tau L)-\rho\log L=\rho\log\Big(\sum_{r=0}^{\infty}\frac{L^{rL}\tau^{rL}\rme^{-\tau L}\rme^{r\gamma L}}{(rL)!}\Big)\;.$$ Using Stirling’s formula for $(rL)!$ (except for the term $r=0$) and extracting the leading term of the sum, one has $$\begin{aligned}
\label{f free asymptotic L + corrections}
&& \frac{f(\tau L)}{L}\simeq\max_{r\in\mathbb{N}}\rho(-\tau+r(1+\log(\rme^{\gamma}\tau/r)))\nonumber\\
&&\hspace{20mm} -{\mbox{{\small 1}$\!\!$1}}_{\{r(\tau)\geq1\}}\frac{\rho\log(2\pi r(\tau)L)}{2L}-{\mbox{{\small 1}$\!\!$1}}_{\{r(\tau)\geq1\}}\frac{1}{12r(\tau)L^{2}}\;,\end{aligned}$$ with the convention $r\log r=0$ for $r=0$. In the second and third terms, $r(\tau)$ is the $r$ corresponding to the maximum in the first term. Defining $\tau_{0}=0$ and for $r\in\mathbb{N}^{*}$ $$\rme^{\gamma}\tau_{r}=\frac{\rme^{-1}r^{r}}{(r-1)^{r-1}}\;,$$ one finally finds for $\tau_{r}\leq\tau\leq\tau_{r+1}$ $$\label{f free d asymptotic L}
\lim_{L\to\infty}\frac{f(\tau L)}{L}=-\rho\,\tau+\rho\,r+\rho\,r\log\frac{\rme^{\gamma}\tau}{r}\;.$$ For large $r$, one has $\rme^{\gamma}\tau_{r}\simeq r-1/2$, hence for large $\tau$ $$\lim_{L\to\infty}\frac{f(\tau L)}{L}\simeq\rho(\rme^{\gamma}-1)\tau-\frac{\rho(\rme^{\gamma}\tau-[\rme^{\gamma}\tau])^{2}}{2\rme^{\gamma}\tau}\;,$$ where $[\rme^{\gamma}\tau]$ is the integer closest to $\rme^{\gamma}\tau$.
[Function f(t) for undistinguishable particles]{} From the definition (\[f\[M\]\]) and the expression (\[E free\]) for the eigenvalues, one has $$f(t)=\frac{1}{L}\log\Big(\sum_{1\leq k_{1}\leq\ldots\leq k_{N}\leq L}\prod_{i=1}^{N}\rme^{t(\rme^{\gamma-2\rmi\pi k_{i}/L}-1)}\Big)\;.$$ Expanding the exponential leads to $$\fl\hspace{5mm}
f(t)=\frac{1}{L}\log\Big(\sum_{1\leq k_{1}\leq\ldots\leq k_{N}\leq L}\prod_{i=1}^{N}\sum_{j=0}^{\infty}\sum_{m=0}^{j}{{j \choose m}}\frac{(-1)^{j-m}t^{j}\rme^{m\gamma}\rme^{-2\rmi\pi k_{i}m/L}}{j!}\Big)\;.$$ Exchanging the order of the summations over $j$ and $m$ allows to perform the summation over $j$. One finds $$f(t)=\frac{1}{L}\log\Big(\sum_{1\leq k_{1}\leq\ldots\leq k_{N}\leq L}\prod_{i=1}^{N}\sum_{m=0}^{\infty}\frac{t^{m}\rme^{-t}\rme^{m\gamma}\rme^{-2\rmi\pi k_{i}m/L}}{m!}\Big)\;.$$ So far, the calculation parallels the one for distinguishable particles in the scaling $t\sim L$. In order to perform the summation over the $k_{i}$, we first use the relation $$\label{S(N,L)}
\hspace{-5mm}
S_{N}(L)=\!\!\!\sum_{1\leq k_{1}\leq\ldots\leq k_{N}\leq L}\prod_{i=1}^{N}f(k_{i})=\oint\frac{\rmd z}{2\rmi\pi z^{N+1}}\exp\Big[\sum_{a=1}^{\infty}\sum_{k=1}^{L}\frac{z^{a}f(k)^{a}}{a}\Big]\;,$$ The contour integral is over a contour enclosing $0$. Eq. (\[S(N,L)\]) can be proved by considering the formal series in $z$ $$\sum_{N=0}^{\infty}z^{N}S_{N}(L)=\prod_{k=1}^{L}\frac{1}{1-z f(k)}\;.$$ Eq. (\[S(N,L)\]) gives $$\hspace{-20mm}
f(t)=\frac{1}{L}\log\Big(\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}}{z^{N+1}}\exp\Big[\sum_{a=1}^{\infty}\frac{z^{a}}{a}\sum_{k=1}^{L}\sum_{m_{1},\ldots,m_{a}=0}^{\infty}\prod_{i=1}^{a}\frac{t^{m_{i}}\rme^{m_{i}\gamma}\rme^{-2\rmi\pi km_{i}/L}}{m_{i}!}\Big]\Big)\;.$$ Using (\[sum k\]) to sum over $k$ leads to $$\hspace{-20mm}
f(t)=\frac{1}{L}\log\Big(\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}}{z^{N+1}}\exp\Big[L\sum_{a=1}^{\infty}\frac{z^{a}}{a}\sum_{r=0}^{\infty}\sum_{m_{1},\ldots,m_{a}=0}^{\infty}\delta_{rL,\sum\limits_{i=1}^{a}m_{i}}\prod_{i=1}^{a}\frac{t^{m_{i}}\rme^{m_{i}\gamma}}{m_{i}!}\Big]\Big)\;.$$ The multinomial sum over the $m_{i}$’s can be performed. One has $$f(t)=\frac{1}{L}\log\Big(\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}}{z^{N+1}}\exp\Big[L\sum_{a=1}^{\infty}\frac{z^{a}}{a}\sum_{r=0}^{\infty}\frac{(a\,\rme^{\gamma}t)^{rL}}{(rL)!}\Big]\Big)\;.$$ The summation over $a$ can be done explicitly. For $r\geq1$, it gives a polylogarithm. One finds $$f(t)=\frac{1}{L}\log\Big(\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}}{z^{N+1}(1-z)^{L}}\exp\Big[L\sum_{r=1}^{\infty}\frac{(\rme^{\gamma}t)^{rL}}{(rL)!}{\mathrm{Li}}_{1-rL}(z)\Big]\Big)\;.$$ We deform the contour of integration so that it encloses the negative real axis. In the thermodynamic limit, it is then possible to use the asymptotics $${\mathrm{Li}}_{-n}(z)\simeq\Gamma(n+1)(-\log z)^{-n-1}\;$$ for large $n$. It leads to $$f(t)\simeq\frac{1}{L}\log\Big(\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}}{z^{N+1}(1-z)^{L}}\exp\Big[\sum_{r=1}^{\infty}\frac{1}{r}\Big(-\frac{\rme^{\gamma}t}{\log z}\Big)^{rL}\Big]\Big)\;.$$ Summing explicitly over $r$ gives $$f(t)\simeq\frac{1}{L}\log\Big(\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}}{z^{N+1}(1-z)^{L}\Big(1-\Big(-\frac{\rme^{\gamma}t}{\log z}\Big)^{L}\Big)}\Big)\;.$$ Expanding the last factor of the denominator, we finally obtain $$f(t)\simeq\frac{1}{L}\log\Big(\sum_{r=0}^{\infty}\oint\frac{\rmd z}{2\rmi\pi}\frac{\rme^{-Nt}\Big(-\frac{\rme^{\gamma}t}{\log z}\Big)^{rL}}{z^{N+1}(1-z)^{L}}\Big)\;.$$ The thermodynamic limit of $f(t)$ is extracted by calculating the saddle point $z_{r}$ of the contour integral. One finds $$\label{f free u asymptotic L}
f(t)\simeq\max_{r\in\mathbb{N}}\Big(-\rho\,t-\rho\log z_{r}-\log(1-z_{r})+r\log\Big(-\frac{\rme^{\gamma}t}{\log z_{r}}\Big)\Big)\;,$$ where $z_{r}$ verifies the equation $$\frac{r}{\log z_{r}}=\frac{z_{r}}{1-z_{r}}-\rho\;.$$ One has $z_{0}=\rho/(1+\rho)$. For $\rho=1/2$, we find numerically $z_{1}\simeq0.085$, $z_{2}\simeq0.016$, $z_{3}\simeq0.0024$. We introduce times $t_{r}$, $r\in\mathbb{N}$ such that when $t_{r}<t<t_{r+1}$, the saddle point that dominates (\[f free u asymptotic L\]) is $z_{r}$. The value of $t_{r}$ is determined by continuity of $f(t)$. One has $t_{0}=0$, and for $\rho=1/2$, $\rme^{\gamma}t_{1}=1.7085$, $\rme^{\gamma}t_{2}=3.236$, $\rme^{\gamma}t_{3}=5.040$.\
For large $r$, $z_{r}$ decreases to $0$ as $z_{r}\simeq\rme^{-r/\rho}$. This implies, for $t_{r}<t<t_{r+1}$, $$f(t)\simeq-\rho t+r+r\log\frac{\rho\,\rme^{\gamma}t}{r}\;,$$ with $t_{r}$ given for large $r$ by $$\label{t(r) free u asymptotic t}
\rho\,\rme^{\gamma}t_{r}\simeq\frac{\rme^{-1}r^{r}}{(r-1)^{r-1}}\simeq r-\frac{1}{2}\;.$$ For large times, we finally obtain $$f(t)\simeq\rho(\rme^{\gamma}-1)t-\frac{(\rho\,\rme^{\gamma}t-[\rho\,\rme^{\gamma}t])^{2}}{2\rho\,\rme^{\gamma}t}\;,$$ where $[\rho\,\rme^{\gamma}t]$ is the integer closest to $\rho\,\rme^{\gamma}t$. This expression is very similar to the one found for distinguishable particles on the scale $t\sim L$.
References {#references .unnumbered}
==========
[10]{}
F. Spitzer. Interaction of [M]{}arkov processes. , 5:246–290, 1970.
B. Derrida. An exactly soluble non-equilibrium system: The asymmetric simple exclusion process. , 301:65–83, 1998.
G.M. Schütz. In [*Exactly Solvable Models for Many-Body Systems Far from Equilibrium*]{}, volume 19 of [*Phase Transitions and Critical Phenomena*]{}. San Diego: Academic, 2001.
O. Golinelli and K. Mallick. The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. , 39:12679–12705, 2006.
B. Derrida. Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. , 2007:P07023.
T. Sasamoto. Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques. , 2007:P07007.
P.L. Ferrari and H. Spohn. Random growth models. In G. Akemann, J. Baik, and P. Di Francesco, editors, [*The Oxford Handbook of Random Matrix Theory*]{}. Oxford University Press, 2011.
K. Mallick. Some exact results for the exclusion process. , 2011:P01024.
T. Chou, K. Mallick, and R.K.P. Zia. Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport. , 74:116601, 2011.
L.-H. Gwa and H. Spohn. Six-vertex model, roughened surfaces, and an asymmetric spin [H]{}amiltonian. , 68:725–728, 1992.
L.-H. Gwa and H. Spohn. ethe solution for the dynamical-scaling exponent of the noisy [B]{}urgers equation. , 46:844–854, 1992.
D. Kim. ethe ansatz solution for crossover scaling functions of the asymmetric [XXZ]{} chain and the [K]{}ardar-[P]{}arisi-[Z]{}hang-type growth model. , 52:3512–3524, 1995.
O. Golinelli and K. Mallick. Bethe ansatz calculation of the spectral gap of the asymmetric exclusion process. , 37:3321–3331, 2004.
O. Golinelli and K. Mallick. Spectral gap of the totally asymmetric exclusion process at arbitrary filling. , 38:1419–1425, 2005.
K. Johansson. Shape fluctuations and random matrices. , 209:437–476, 2000.
M. Prähofer and H. Spohn. Current fluctuations for the totally asymmetric simple exclusion process. In [*In and Out of Equilibrium: Probability with a Physics Flavor*]{}, volume 51 of [*Progress in Probability*]{}, pages 185–204. Boston: Birkhäuser, 2002.
A. Borodin, P.L. Ferrari, M. Prähofer, and T. Sasamoto. Fluctuation properties of the [TASEP]{} with periodic initial configuration. , 129:1055–1080, 2007.
C.A. Tracy and H. Widom. Total current fluctuations in the asymmetric simple exclusion process. , 50:095204, 2009.
T. Sasamoto and H. Spohn. The crossover regime for the weakly asymmetric simple exclusion process. , 140:209–231, 2010.
G. Amir, I. Corwin, and J. Quastel. Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. , 64:466–537, 2011.
B. Derrida and J.L. Lebowitz. Exact large deviation function in the asymmetric exclusion process. , 80:209–213, 1998.
B. Derrida and C. Appert. Universal large-deviation function of the [K]{}ardar-[P]{}arisi-[Z]{}hang equation in one dimension. , 94:1–30, 1999.
S. Prolhac and K. Mallick. Current fluctuations in the exclusion process and [B]{}ethe ansatz. , 41:175002, 2008.
S. Prolhac. Fluctuations and skewness of the current in the partially asymmetric exclusion process. , 41:365003, 2008.
S. Prolhac and K. Mallick. Cumulants of the current in a weakly asymmetric exclusion process. , 42:175001, 2009.
S. Prolhac. Tree structures for the current fluctuations in the exclusion process. , 43:105002, 2010.
A. Lazarescu and K. Mallick. An exact formula for the statistics of the current in the [TASEP]{} with open boundaries. , 44:315001, 2011.
M. Gorissen, A. Lazarescu, K. Mallick, and C. Vanderzande. Exact current statistics of the asymmetric simple exclusion process with open boundaries. , 109:170601, 2012.
A. Lazarescu. Matrix ansatz for the fluctuations of the current in the [ASEP]{} with open boundaries. , 46:145003, 2013.
S. Katz, J.L. Lebowitz, and H. Spohn. Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors. , 34:497–537, 1984.
B. Schmittmann and R.K.P. Zia. Driven diffusive systems. [A]{}n introduction and recent developments. , 301:45–64, 1998.
A.L. Barabási and H.E. Stanley. . Cambridge university press, 1995.
P. Meakin. . Cambridge University Press, 1998.
T. Halpin-Healy and Y.-C. Zhang. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. [A]{}spects of multidisciplinary statistical mechanics. , 254:215–414, 1995.
E. Brunet and B. Derrida. Probability distribution of the free energy of a directed polymer in a random medium. , 61:6789–6801, 2000.
V. Dotsenko. ethe ansatz derivation of the [T]{}racy-[W]{}idom distribution for one-dimensional directed polymers. , 90:20003, 2010.
P. Calabrese, P. Le Doussal, and A. Rosso. Free-energy distribution of the directed polymer at high temperature. , 90:20002, 2010.
M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic scaling of growing interfaces. , 56:889–892, 1986.
T. Kriecherbauer and J. Krug. A pedestrian’s view on interacting particle systems, [KPZ]{} universality and random matrices. , 43:403001, 2010.
T. Sasamoto and H. Spohn. The 1 + 1-dimensional [K]{}ardar-[P]{}arisi-[Z]{}hang equation and its universality class. , 2010:P11013.
A.M. Povolotsky and V.B. Priezzhev. Determinant solution for the totally asymmetric exclusion process with parallel update: [II]{}. ring geometry. , 2007:P08018.
G.P. Pronko and Y.G. Stroganov. Bethe equations ‘on the wrong side of the equator’. , 32:2333–2340, 1999.
C.N. Yang and C.P. Yang. Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction. , 10:1115–1122, 1969.
M. Henkel and G.M. Schütz. Finite-lattice extrapolation algorithms. , 21:2617–2633, 1988.
|
---
abstract: |
We present a path algorithm for the generalized lasso problem. This problem penalizes the $\ell_1$ norm of a matrix $D$ times the coefficient vector, and has a wide range of applications, dictated by the choice of $D$. Our algorithm is based on solving the dual of the generalized lasso, which facilitates computation and conceptual understanding of the path. For $D=I$ (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary $D$, we derive an unbiased estimate of the degrees of freedom of the generalized lasso fit. This estimate turns out to be quite intuitive in many applications.\
Keywords: [*lasso; path algorithm; Lagrange dual; LARS; degrees of freedom*]{}
author:
- '[Ryan J. Tibshirani]{} [^1]'
- '[Jonathan Taylor]{} [^2]'
bibliography:
- 'ryantibs.bib'
title: The Solution Path of the Generalized Lasso
---
Introduction {#sec:intro}
============
Regularization with the $\ell_1$ norm seems to be ubiquitous throughout many fields of mathematics and engineering. In statistics, the best-known example is the [*lasso*]{}, the application of an $\ell_1$ penalty to linear regression [@lasso; @bp]. Let $y \in
\R^n$ be a response vector and $X \in \R^{n \times p}$ be a matrix of predictors. If the response and the predictors have been centered, we can omit an intercept term from the model, and then the lasso problem is commonly written as $$\label{eq:lasso}
{\mathop{\mathrm{minimize}}}_{\beta \in \R^p} \; \half\|y-X\beta\|_\ltwo^2 +
\lambda\|\beta\|_\lone,$$ where $\lambda \geq 0$ is a tuning parameter. There are many fast algorithms for solving the lasso at a single value of the parameter $\lambda$, or over a discrete set of parameter values. The [*least angle regression*]{} (LARS) algorithm, on the other hand, is unique in that it solves for all $\lambda \in [0,\infty]$ [@lars] (see also the earlier [ *homotopy*]{} method of [@homotopy], and the even earlier work of [@parqp]). This is possible because the lasso solution is piecewise linear with respect to $\lambda$.
The LARS path algorithm may provide a computational advantage when the solution is desired at many values of the tuning parameter. For large problems, this is less likely to be the case because the number of knots (changes in slope) in the solution path tends to be very large, and this renders the path intractable. Computational efficiency aside, the LARS method fully characterizes the tradeoff between goodness-of-fit and sparsity in the lasso solution (this is controlled by $\lambda$), and hence yields interesting statistical insights into the problem. Most notably, the LARS paper established a result on the degrees of freedom of the lasso fit, which was further developed by [@lassodf].
The first of its kind, LARS inspired the development of path algorithms for various other optimization problems that appear in statistics [@svmpath; @rosset; @holger; @dasso], and our case is no exception. In this paper, we derive a path algorithm for problems that use the $\ell_1$ norm to enforce certain structural constraints—instead of pure sparsity—on the coefficients in a linear regression. These problems are nicely encapsulated by the formulation: $$\label{eq:dlasso}
{\mathop{\mathrm{minimize}}}_{\beta \in \R^p} \; \half \|y-X\beta\|_\ltwo^2 +
\lambda\|D\beta\|_\lone,$$ where $D \in \R^{m \times p}$ is a specified penalty matrix. We refer to problem as the [*generalized lasso*]{}. Depending on the application, we choose $D$ so that sparsity of $D\beta$ corresponds to some other desired behavior for $\beta$, typically one that is structural or geometric in nature. In fact, various choices of $D$ in give problems that are already well-known in the literature: the fused lasso, trend filtering, wavelet smoothing, and a method for outlier detection. We derive a simple path algorithm for the minimization that applies to a general matrix $D$, hence this entire class of problems. Like the lasso, the generalized lasso solution is piecewise linear as a function of $\lambda$. We also prove a result on the degrees of freedom of the fit for a general $D$. It is worth noting that problem has been considered by other authors, for example [@tisp]. This last work establishes some asymptotic properties of the solution, and proposes a computational technique that relates to simulated annealing.
The paper is organized as follows. We begin in Section \[sec:apps\] by motivating the use of a penalty matrix $D$, offering several examples of problems that fit into this framework. Section \[sec:lasso\] explains that some instances of the generalized lasso can be transformed into a regular lasso problem, but many instances cannot, which emphasizes the need for a new path approach. In Section \[sec:dual\] we derive the Lagrange dual of , which serves as the jumping point for our algorithm and all of the work that follows. For the sake of clarity, we build up the algorithm over the next 3 sections. Sections \[sec:1d\] and \[sec:d\] consider the case $X=I$. In Section \[sec:1d\] we assume that $D$ is the 1-dimensional fused lasso matrix, in which case our path algorithm takes an especially simple (and intuitive) form. In Section \[sec:d\] we give the path algorithm for a general penalty matrix $D$, which requires adding only one step in the iterative loop. Section \[sec:x\] extends the algorithm to the case of a general design matrix $X$. Provided that $X$ has full column rank, we show that our path algorithm still applies, by rewriting the dual problem in a more familiar form. We also outline a path approach for the case when $X$ has rank less than its number of columns. Practical considerations for the path’s computation are given in Section \[sec:comp\].
In Section \[sec:lars\] we focus on the lasso case, $D=I$, and compare our method to LARS. Above, we described LARS as an algorithm for computing the solution path of . This actually refers to LARS in its “lasso” state, and although this is probably the best-known version of LARS, it is not the only one. In its original (unmodified) state, LARS does not necessarily optimize the lasso criterion, but instead performs a more “democratic” form of forward variable selection. It turns out that with an easy modification, our algorithm gives this selection procedure exactly. In Section \[sec:df\] we derive an unbiased estimate of the degrees of freedom of the fit for a general matrix $D$. The proof is quite straightforward because it utilizes the dual fit, which is simply the projection onto a convex set. As we vary $D$, this result yields interpretable estimates of the degrees of freedom of the fused lasso, trend filtering, and more. Finally, Section \[sec:discuss\] contains some discussion.
To save space (and improve readability), many of the technical details in the paper are deferred to a supplementary document, available online at <http://www-stat.stanford.edu/~ryantibs/>.
Applications {#sec:apps}
============
There are a wide variety of interesting applications of problem . What we present below is not meant to be an exhaustive list, but rather a set of illustrative examples that motivated our work on this problem in the first place. This section is split into two main parts: the case when $X=I$ (often called the “signal approximation” case), and the case when $X$ is a general design matrix.
The signal approximation case, $X=I$ {#sec:appsi}
------------------------------------
When $X=I$, the solution of the lasso problem is given by soft-thresholding the coordinates of $y$. Therefore one might think that an equally simple formula exists for the generalized lasso solution when the design matrix is the identity—but this is not true. Taking $X=I$ in the generalized lasso gives an interesting and highly nontrivial class of problems. In this setup, we observe data $y \in \R^n$ which is a noisy realization of an underlying signal, and the rows of $D \in \R^{m \times n}$ reflect some believed structure or geometry in the signal. The solution of problem fits adaptively to the data while exhibiting some of these structural properties. We begin by looking at piecewise constant signals, and then address more complex features.
### The fused lasso
Suppose that $y$ follows a 1-dimensional structure, that is, the coordinates of $y$ correspond to successive positions on a straight line. If $D$ is the $(n-1)\times n$ matrix $$\label{eq:d1d}
D_\mathrm{1d} = \left[\begin{array}{rrrrrr}
-1 & 1 & 0 & \ldots & 0 & 0 \\
0 & -1 & 1 & \ldots & 0 & 0 \\
& & & \ldots & & \\
0 & 0 & 0 & \ldots & -1 & 1
\end{array}\right],$$ then problem penalizes the absolute differences in adjacent coordinates of $\beta$, and is known as the [*1d fused lasso*]{} [@fuse]. This gives a piecewise constant fit, and is used in settings where coordinates in the true model are closely related to their neighbors. A common application area is comparative genomic hybridization (CGH) data: here $y$ measures the number of copies of each gene ordered linearly along the genome (actually $y$ is the log ratio of the number of copies relative to a normal sample), and we believe for biological reasons that nearby genes will exhibit a similar copy number. Identifying abnormalities in copy number has become a valuable means of understanding the development of many human cancers. See Figure \[fig:cgh\] for an example of the 1d fused lasso applied to some CGH data on glioblastoma multiformes, a particular type of malignant brain tumor, taken from [@gbm].
![*The 1d fused lasso applied to some glioblastoma multiforme data. The red line represents the inferred copy number from the 1d fused lasso solution (for $\lambda=3$).*[]{data-label="fig:cgh"}](cgh.pdf){width="5in"}
A natural extension of this idea penalizes the differences between neighboring pixels in an image. Suppose that $y$ represents a noisy image that has been unraveled into a vector, and each row of $D$ again has a $1$ and $-1$, but this time arranged to give both the horizontal and vertical differences between pixels. Then problem is called the [*2d fused lasso*]{} [@fuse], and is used to denoise images that we believe should obey a piecewise constant structure. This technique is a special type of [*total variation denoising*]{}, a well-studied problem that carries a vast literature spanning the fields of statistics, computer science, electrical engineering, and others (for example, see [@tv]). Figure \[fig:s\] shows the 2d fused lasso applied to a toy example.
We can further extend this idea by defining adjacency according to an arbitrary graph structure, with $n$ nodes and $m$ edges. Now the coordinates of $y \in \R^n$ correspond to nodes in the graph, and we penalize the difference between each pair of nodes joined by an edge. Hence $D$ is $m \times n$, with each row having a $-1$ and $1$ in the appropriate spots, corresponding to an edge in the graph. In this case, we simply call problem the [*fused lasso*]{}. Note that both the 1d and 2d fused lasso problems are special cases of this, with the underlying graph a chain and a 2d grid, respectively. But the fused lasso is a very general problem, as it can be applied to any graph structure that exhibits a piecewise constant signal across adjacent nodes. See Figure \[fig:us\] for application in which the underlying graph has US states as nodes, with two states joined by an edge if they share a border. This graph has 48 nodes (we only include the mainland US states) and 105 edges.
The observant reader may notice a discrepancy between the usual fused lasso definition and ours, as the fused lasso penalty typically includes an additional term $\|\beta\|_1$, the $\ell_1$ norm of the coefficients themselves. We refer to this as the [*sparse fused lasso*]{}, and to represent this penalty we just append the $n\times n$ identity matrix to the rows of $D$. Actually, this carries over to all of the applications yet to be discussed—if we desire pure sparsity in addition to the structural behavior that is being encouraged by $D$, we append the identity matrix to the rows of $D$.
### Linear and polynomial trend filtering
Suppose again that $y$ follows a 1-dimensional structure, but now $D$ is the $(n-2)\times n$ matrix $$\label{eq:dtf}
D_{\mathrm{tf},1} = \left[\begin{array}{rrrrrrr}
-1 & 2 & -1 & \ldots & 0 & 0 & 0 \\
0 & -1 & 2 & \ldots & 0 & 0 & 0 \\
\ldots & & & & & & \\
0 & 0 & 0 & \ldots & -1 & 2 & -1
\end{array}\right].$$ Then problem is equivalent to [*linear trend filtering*]{} (also called [*$\ell_1$ trend filtering*]{}) [@l1tf]. Just as the 1d fused lasso penalizes the discrete first derivative, this technique penalizes the discrete second derivative, and so it gives a piecewise linear fit. This has many applications, namely, any settings in which the underlying trend is believed to be linear with (unknown) changepoints. Moreover, by recursively defining $$\label{eq:Dtfk}
D_{\mathrm{tf},k} = D_\mathrm{1d} \cdot D_{\mathrm{tf},k-1} \;\;\;
\text{for}\;\;k=2,3,\ldots,$$ (here $D_\mathrm{1d}$ is the $(n-k-1)\times (n-k)$ version of ) we can fit a piecewise polynomial of any order $k$, further extending the realm of applications. We call this [*polynomial trend filtering of order $k$*]{}. Figure \[fig:tf\] shows examples of linear, quadratic, and cubic fits.
The polynomial trend filtering fits (especially for $k=3$) are similar to those that one could obtain using regression splines and smoothing splines. However, the knots (changes in $k$th derivative) in the trend filtering fits are selected adaptively based on the data, jointly with the inter-knot polynomial estimation. This phenomenon of simultaneous selection and estimation—analogous to that concerning the nonzero coefficients in the lasso fit, and the jumps in the piecewise constant fused lasso fit—does not occur in regression and smoothing splines. Regression splines operate on a fixed set of knots, and there is a substantial literature on knot placement for this problem (see Chapter 9.3 of [@gam], for example). Smoothing splines place a knot at each data point, and implement smoothness via a generalized ridge regression on the coefficients in a natural spline basis. As a result (of this $\ell_2$ shrinkage), they cannot represent both global smoothness and local wiggliness in a signal. On the other hand, trend filtering has the potential to represent both such features, a property called “time and frequency localization” in the signal processing field, though this idea has been largely unexplored. The classic example of a procedure that allows time and frequency localization is wavelet smoothing, discussed next.
### Wavelet smoothing
This is a quite a popular method in signal processing and compression. The main idea is to model the data as a sparse linear combination of wavelet functions. Perhaps the most common formulation for wavelet smoothing is [*SURE shrinkage*]{} [@sure], which solves the lasso optimization problem $$\label{eq:trans}
{\mathop{\mathrm{minimize}}}_{\theta \in \R^n} \; \half \|y-W\theta\|_\ltwo^2 +
\lambda\|\theta\|_\lone,$$ where $W \in \R^{n\times n}$ has an orthogonal wavelet basis along its columns. By orthogonality, we can change variables to $\beta=W\theta$ and then becomes a generalized lasso problem with $D=W\T$.
In many applications it is desirable to use an overcomplete wavelet set, so that $W \in \R^{n\times m}$ with $n < m$. Now problem and the generalized lasso with $D=W\T$ (and $X=I$) are no longer equivalent, and in fact give quite different answers. In signal processing, the former is called the [*synthesis*]{} approach, and the latter the [*analysis*]{} approach, to wavelet smoothing. Though attention has traditionally been centered around synthesis, a recent paper by [@avs] suggests that synthesis may be too sensitive, and shows that it can be outperformed by its analysis counterpart.
A general design matrix $X$ {#sec:appsx}
---------------------------
For any of the fused lasso, trend filtering, or wavelet smoothing penalties discussed above, the addition of a general matrix $X$ of covariates significantly extends the domain of applications. For a fused lasso example, suppose that each row of $X$ represents a $k_1 \times k_2 \times k_3$ MRI image of a patient’s brain, unraveled into a vector (so that $p=k_1 \cdot k_3 \cdot k_3$). Suppose that $y$ contains some continuous outcome on the patients, and we model these as a linear function of the MRIs, $\E(y_i|X_i)=\beta\T X_i$. Now $\beta$ also has the structure of a $k_1 \times k_2 \times k_3$ image, and by choosing the matrix $D$ to give the sparse 3d fused lasso penalty (that is, the fused lasso on a 3d grid with an additional $\ell_1$ penalty of the coefficients), the solution of attempts to explain the outcome with a small number of contiguous regions in the brain.
As another example, the inclusion of a design matrix $X$ in the trend filtering setup provides an alternative way of fitting [*varying-coefficient models*]{} [@vcm; @lrm]. We consider a data set from [@vcm], which examines $n=88$ observations on the exhaust from an engine fueled by ethanol. The response $y$ is the concentration of nitrogen dioxide, and the two predictors are a measure of the fuel-air ratio $E$, and the compression ratio of the engine $C$. Studying the interactions between $E$ and $C$ leads the authors of [@vcm] to consider the model $$\label{eq:vcm}
\E(y_i|E_i,C_i) = \beta_0(E_i) + \beta_1(E_i)\cdot C_i.$$ This is a linear model with a different intercept and slope for each $E_i$, subject to the (implicit) constraint that the intercept and slope should vary smoothly along the $E_i$’s. We can fit this using , in the following way: first we discretize the continuous observations $E_1,\ldots E_n$ so that they fall into, say, 25 bins. Our design matrix $X$ is $88 \times 50$, with the first 25 columns modeling the intercept $\beta_0$ and the last 25 modeling the slope $\beta_1$. The $i$th row of $X$ is $$X_{ij} = \begin{cases}
1 & \text{if $E_i$ lies in the $j$th bin} \\
C_i & \text{if $E_i$ lies in the $(j+25)$th bin} \\
0 & \text{otherwise.} \\
\end{cases}$$ Finally, we choose $$D = \left[\begin{array}{cc}
D_{\mathrm{tf},3} & 0 \\
0 & D_{\mathrm{tf},3}
\end{array}\right],$$ where $D_{\mathrm{tf},3}$ is the cubic trend filtering matrix (the choice $D_{\mathrm{tf},3}$ is not crucial and of course can be replaced by a higher or lower order trend filtering matrix.) The matrix $D$ is structured in this way so that we penalize the smoothness of the first 25 and last 25 components of $\beta=(\beta_0,\beta_1)\T$ individually. With $X$ and $D$ as described, solving the optimization problem gives the coefficients shown in Figure \[fig:vcm\], which appear quite similar to those plotted in [@vcm].
![*The intercept and slope of the varying-coefficient model for the engine data of [@vcm], fit using with a cubic trend filtering penalty matrix (and $\lambda=3$). The dashed lines show 85% bootstrap confidence intervals from 500 bootstrap samples.*[]{data-label="fig:vcm"}](beta0.pdf "fig:"){width="2.5in"} ![*The intercept and slope of the varying-coefficient model for the engine data of [@vcm], fit using with a cubic trend filtering penalty matrix (and $\lambda=3$). The dashed lines show 85% bootstrap confidence intervals from 500 bootstrap samples.*[]{data-label="fig:vcm"}](beta1.pdf "fig:"){width="2.5in"}
We conclude this section with a generalized lasso application of [@owenoutlie], in which the penalty is not structurally-based, unlike the examples discussed previously. Suppose that we observe $y_1,\ldots y_n$, and we believe the majority of these points follow a linear model $\E(y_i|X_i) = \beta\T X_i$ for some covariates $X_i=(X_{i1},\ldots X_{ip})\T$, except that a small number of the $y_i$ are outliers and do not come from this model. To determine which points are outliers, one might consider the problem $$\label{eq:outlie}
{\mathop{\mathrm{minimize}}}_{z \in \R^n, \, \beta \in \R^p} \; \half \|z-X\beta\|_2^2 \;\;
\text{subject to} \; \|z-y\|_0 \leq k$$ for a fixed integer $k$. Here $\|x\|_0 = \sum_i 1(x_i \not= 0)$. Thus by setting $k=3$, for example, the solution $\hat{z}$ of would indicate which 3 points should be considered outliers, in that $\hat{z}_i \not= y_i$ for exactly 3 coordinates. A natural convex relaxation of problem is $$\label{eq:outlie1}
{\mathop{\mathrm{minimize}}}_{z \in \R^n, \, \beta \in \R^p} \; \half \|z-X\beta\|_2^2 +
\lambda \|z-y\|_1,$$ where we have also transformed the problem from bound form to Lagrangian form. Letting $\alpha=y-z$, this can be rewritten as $$\label{eq:outlie2}
{\mathop{\mathrm{minimize}}}_{\alpha \in \R^n, \, \beta \in \R^p} \; \half
\|y-\alpha-X\beta\|_2^2 +
\lambda \|\alpha\|_1,$$ which fits into the form of problem , with design matrix $\widetilde{X}=[\, I \;\, X \,]$, coefficient vector $\tbeta=(\alpha,\beta)\T$, and penalty matrix $D=[\, I \;\, 0 \,]$. Figure \[fig:outlie\] shows a simple example with $p=1$.
![*A simple example of using problem to perform outlier detection. Written in the form , the blue line denotes the fitted slope $\hbeta$, while the red circles indicate the outliers, as determined by the coordinates of $\hat\alpha$ that are nonzero (for $\lambda=8$).*[]{data-label="fig:outlie"}](outlie.pdf){width="3in"}
After reading the examples in this section, a natural question is: when can a generalized lasso problem be transformed into a regular lasso problem ? (Recall, for example, that this is possible for an orthogonal $D$, as we discussed in the wavelet smoothing example.) We discuss this in the next section.
When does a generalized lasso problem reduce to a lasso problem? {#sec:lasso}
================================================================
If $D$ is $p \times p$ and invertible, we can transform variables in problem by $\theta=D\beta$, yielding the lasso problem $$\label{eq:dlasso2}
{\mathop{\mathrm{minimize}}}_{\theta \in \R^p} \; \half\|y-XD^{-1}\theta\|^2_2 +
\lambda\|\theta\|_1.$$ More generally, if $D$ is $m \times p$ and $\rank(D)=m$ (note that this necessarily means $m\leq p$), then we can still transform variables and get a lasso problem. First we construct a $p \times p$ matrix $\tD=\left[\begin{array}{c} D \\ A \end{array}\right]$ with $\rank(\tD)=p$, by finding a $(p-m)\times p$ matrix $A$ whose rows are orthogonal to those in $D$. Then we change variables to $\theta=(\theta_1,\theta_2)\T
= \tD\beta$, so that the generalized lasso becomes $$\label{eq:dlasso3}
{\mathop{\mathrm{minimize}}}_{\theta \in \R^p} \; \half\|y-X\tD^{-1}\theta\|^2_2 +
\lambda\|\theta_1\|_1.$$ This is almost a regular lasso, except that the $\ell_1$ penalty only covers part of the coefficient vector. First write $X\tD^{-1}\theta = X_1\theta_1 + X_2\theta_2$; then, it is clear that at the solution the second block of the coefficients is given by a linear regression: $$\hat{\theta}_2 = (X_2\T X_2)^{-1} X_2\T (y-X_1\hat{\theta}_1).$$ Therefore we can rewrite problem as $$\label{eq:dlasso4}
{\mathop{\mathrm{minimize}}}_{\theta_1 \in \R^m} \; \half\|(I-P)y-(I-P)X_1\theta_1\|^2_2
+ \lambda\|\theta_1\|_1,$$ where $P=X_2\T(X_2\T X_2)^{-1}X_2\T$, the projection onto the column space of $X_2$. The LARS algorithm provides the solution path of such a lasso problem , from which we can back-transform to get the generalized lasso solution: $\hbeta = \tD^{-1} \hat{\theta}$.
However, if $D$ is $m \times p$ and $\rank(D)<m$, then such a transformation is not possible, and LARS cannot be used to find the solution path of the generalized lasso problem . Further, in this case, the authors of [@avs] establish what they call an “unbridgeable” gap between problems and , based on the geometric properties of their solutions.
While several of the examples from Section \[sec:apps\] satisfy $\rank(D)=m$, and hence admit a lasso transformation, a good number also fall into the case $\rank(D)<m$, and suggest the need for a novel path algorithm. These are summarized in Table \[table:rank\]. Therefore, in the next section, we derive the Lagrange dual of problem , which leads to a nice algorithm to compute the solution path of for an arbitrary penalty matrix $D$.
The Lagrange dual problem {#sec:dual}
=========================
Roughly speaking, the generalized lasso problem is difficult to analyze directly because the nondifferentiable $\ell_1$ penalty is composed with a linear transformation of $\beta$. We hence turn to the corresponding Lagrange dual problem where the story is conceptually clearer. First we consider the generalized lasso in the signal approximation case, $X=I$: $$\label{eq:dlassoi}
{\mathop{\mathrm{minimize}}}_{\beta \in \R^n} \; \half \|y-\beta\|_\ltwo^2 +
\lambda\|D\beta\|_\lone.$$ Following an argument of [@l1tf], we rewrite this problem as $${\mathop{\mathrm{minimize}}}_{\beta \in \R^n, \, z \in \R^m} \; \half \|y-\beta\|_\ltwo^2
+ \lambda\|z\|_\lone \;\;\; \text{subject to} \;\; z=D\beta.$$ The Lagrangian is hence $$\cL(\beta,z,u) = \half\|y-\beta\|_\ltwo^2 + \lambda\|z\|_\lone + u\T
(D\beta-z),$$ and to derive the dual problem, we minimize this over $\beta,z$. The terms involving $\beta$ are just a quadratic, and up to some constants (not depending on $u$) $$\min_\beta \bigg(\half\|y-\beta\|_\ltwo^2 + u\T D\beta\bigg) =
-\half\|y-D\T u\|_\ltwo^2,$$ while $$\min_z \Big(\lambda\|z\|_\lone - u\T z\Big) =
\begin{cases}
0 & \text{if} \;\; \|u\|_\linf \leq \lambda \\
-\infty & \text{otherwise}.
\end{cases}$$ Therefore the dual problem of is $$\label{eq:dual}
{\mathop{\mathrm{minimize}}}_{u \in \R^m} \; \half\|y-D\T u\|_\ltwo^2 \;\;\;
\text{subject to} \;\; \|u\|_\linf \leq \lambda.$$ Immediately we can see that has a “nice” constraint set, $\{u : \|u\|_\linf \leq \lambda\}$, which is simply a box, free of any linear transformation. It is also important to note the difference in dimension: the dual problem has a variable $u
\in \R^m$, whereas the original problem , called the primal problem, has a variable $\beta \in \R^n$.
When $\rank(D)<m$, the dual problem is not strictly convex, and so it can have many solutions. On the other hand, the primal problem is always strictly convex and always has a unique solution. The primal problem is also strictly feasible (it has no constraints), and so strong duality holds (see Section 5.2 of [@convex]). Let us denote the solutions of the primal problem and dual problem by $\hbeta_\lambda$ and $\hu_\lambda$, respectively, to emphasize their dependence on $\lambda$. By taking the gradient of the Lagrangian $\cL(\beta,z,u)$ with respect to $\beta$ and setting this equal to zero, we get the primal-dual relationship $$\label{eq:primaldual}
\hbeta_\lambda = y - D\T \hu_\lambda.$$ Taking the gradient of $\cL(\beta,z,u)$ with respect to $z$ and setting this equal to zero, we find that each coordinate $i=1,\ldots
m$ of the dual solution satisfies $$\label{eq:dualsign}
\hu_{\lambda,i} \in \begin{cases}
\{+\lambda\} & \text{if} \;\; (D\hbeta_\lambda)_i > 0 \\
\{-\lambda\} & \text{if} \;\; (D\hbeta_\lambda)_i < 0 \\
[-\lambda,\lambda] & \text{if} \;\; (D\hbeta_\lambda)_i = 0.
\end{cases}$$ This last equation tells us that the dual coordinates that are equal to $\lambda$ in absolute value, $$\label{eq:bset}
\cB=\{ i : |\hu_{\lambda,i}| = \lambda\},$$ are the coordinates of $D\hbeta_\lambda$ that are “allowed” to be nonzero. But this does necessarily mean that $(D\hbeta_\lambda)_i \not= 0$ for all $i \in \cB$.
For a general design matrix $X$, we can apply a similar argument to derive the dual of : $$\begin{aligned}
\label{eq:xdual}
{\mathop{\mathrm{minimize}}}_{u \in \R^m} \;\;
&\half (X\T y-D\T u)\T (X\T X)^+ (X\T y - D\T u) \\
\nonumber
\text{subject to} \;\;
&\|u\|_\linf \leq \lambda, \; D\T u \in \row(X), \end{aligned}$$ This looks complicated, certainly in comparison to problem . However, the inequality constraint on $u$ is still a simple (un-transformed) box. Moreover, we can make look like by changing the response $y$ and penalty matrix $D$. This will be discussed later in Section \[sec:x\].
In the next two sections, Sections \[sec:1d\] and \[sec:d\], we restrict our attention to the case $X=I$ and derive an algorithm to find a solution path of the dual . This gives the desired primal solution path, using the relationship . Since our focus is on solving the dual problem, we write simply “solution” or “solution path” to refer to the dual versions. Though we will eventually consider an arbitrary matrix $D$ in Section \[sec:d\], we begin by studying the 1d fused lasso in Section \[sec:1d\]. This case is especially simple, and we use it to build the framework for the path algorithm in the general $D$ case.
The 1d fused lasso {#sec:1d}
==================
In this setting we have $D=D_\mathrm{1d}$, the $(n-1)\times n$ matrix given in . Now the dual problem is strictly convex (since $D_\mathrm{1d}$ has rank equal to its number of rows), and therefore it has a unique solution. In order to efficiently compute the solution path, we use a lemma that allows us, at different stages, to reduce the dimension of the problem by one.
The boundary lemma {#sec:blem}
------------------
Consider the constraint set $\{u: \|u\|_\linf \leq \lambda\} \subseteq
\R^{n-1}$: this is a box centered around the origin with side length $2\lambda$. We say that coordinate $i$ of $u$ is “on the boundary” (of this box) if $|u_i|=\lambda$. For the 1d fused lasso, it turns out that coordinates of the solution that are on the boundary will remain on the boundary indefinitely as $\lambda$ decreases. This idea can be stated more precisely as follows:
\[lemma:boundary\] Suppose that $D=D_\mathrm{1d}$, the 1d fused lasso matrix in . For any coordinate $i$, the solution $\hu_\lambda$ of satisfies $$\hu_{\lambda_0,i} = \lambda_0 \;\; \Rightarrow \;\;
\hu_{\lambda,i} = \lambda \;\;\;
\text{for all} \;\; 0 \leq \lambda \leq \lambda_0,$$ and $$\hu_{\lambda_0,i} = -\lambda_0 \;\; \Rightarrow \;\;
\hu_{\lambda,i} = -\lambda \;\;\;
\text{for all} \;\; 0 \leq \lambda \leq \lambda_0.$$
The proof is given in the online supplement. It is interesting to note a connection between the boundary lemma and a lemma of [@pco], which states that $$\label{eq:fuselem}
\hbeta_{\lambda_0,i} = \hbeta_{\lambda_0,i+1} \;\; \Rightarrow \;\;
\hbeta_{\lambda,i} = \hbeta_{\lambda,i+1} \;\;\; \text{for all} \;\;
\lambda\geq\lambda_0$$ for this same problem. In other words, this lemma says that no two equal primal coordinates can become unequal with increasing $\lambda$. In general $|\hu_{\lambda,i}|=\lambda$ is not equivalent to $(D\hbeta_\lambda)_i\not=0$, but these two statements are equivalent for the 1d fused lasso problem (see the primal-dual correspondence in Section \[sec:1dprops\]), and therefore the boundary lemma is equivalent to .
Path algorithm {#sec:1dpath}
--------------
This section is intended to explain the path algorithm from a conceptual point of view, and no rigorous arguments for its correctness are made here. We defer these until Section \[sec:dpath\], when we revisit the problem in the context of a general matrix $D$.
The boundary lemma describes the behavior of the solution as $\lambda$ decreases, and therefore it is natural to construct the solution path by moving the parameter from $\lambda=\infty$ to $\lambda=0$. As will be made apparent from the details of the algorithm, the solution path is a piecewise linear function of $\lambda$, with a change in slope occurring whenever one of its coordinate paths hits the boundary. The key observation is that, by the boundary lemma, if a coordinate hits the boundary it will stay on the boundary for the rest of the path down to $\lambda=0$. Hence when it hits the boundary we can essentially eliminate this coordinate from consideration (since we know its value at each smaller $\lambda$), recompute the slopes of the other coordinate paths, and move until another coordinate hits the boundary.
As we construct the path, we maintain two lists: $\cB=\cB(\lambda)$, which contains the coordinates that are currently on the boundary; and $s=s(\lambda)$, which contains their signs. For example, if we have $\cB(\lambda)=(5,2)$ and $s(\lambda)=(-1,1)$, then this means that $\hu_{\lambda,5}=-\lambda$ and $\hu_{\lambda,2}=\lambda$. We call the coordinates in $\cB$ the “boundary coordinates”, and the rest the “interior coordinates”. Now we can describe the algorithm:
\[alg:1d\]
- Start with $\lambda_0=\infty$, $\cB=\emptyset$, and $s=\emptyset$.
- For $k=0,\ldots n-2$:
1. Compute the solution at $\lambda_k$ by least squares, as in .
2. Continuing in a linear direction from the solution, compute $\lambda_{k+1}$, when an interior coordinate will next hit the boundary, as in and .
3. Add this coordinate to $\cB$ and its sign to $s$.
The algorithm’s details appear slightly more complicated, but this is only because of notation. If $\cB=(i_1,\ldots i_k)$, then we define for a matrix $A$ and a vector $x$ $$A_\cB = \left[\begin{array}{c}A_{i_1} \\ \vdots \\ A_{i_k}
\end{array}\right] \;\; \text{and} \;\;
x_\cB = (x_{i_1},\ldots x_{i_k})\T,$$ where $A_i$ is the $i$th row of $A$. In words: $A_\cB$ gives the rows of $A$ that are in $\cB$, and $x_\cB$ gives the coordinates of $x$ in $\cB$. We use the subscript $-\cB$, as in $A_{-\cB}$ or $x_{-\cB}$, to index over the rows or coordinates except those in $\cB$. Note that $\cB$ as defined above (in the paragraph preceding the algorithm) is consistent with our previous definition , except that here we treat $\cB$ as an ordered list instead of a set (its ordering only needs to be consistent with that of $s$). Also, we treat $s$ as a vector when convenient.
When $\lambda=\infty$, the problem is unconstrained, and so clearly $\cB=\emptyset$ and $s=\emptyset$. But more generally, suppose that we are at the $k$th iteration, with boundary set $\cB=\cB(\lambda_k)$ and signs $s=s(\lambda_k)$. By the boundary lemma, the solution satisfies $$\hu_{\lambda,\cB} = \lambda s \;\;\;
\text{for all} \;\; \lambda \in [0,\lambda_k].$$ Therefore, for $\lambda\leq\lambda_k$, we can reduce the optimization problem to $$\label{eq:dualrewrite}
{\mathop{\mathrm{minimize}}}_{u_{-\cB}} \; \half \|y - \lambda (D_\cB)\T s
- (D_{-\cB})\T u_{-\cB} \|_\ltwo^2 \;\;\; \text{subject to} \;\;
\|u_{-\cB}\|_\linf \leq \lambda,$$ which involves solving for just the interior coordinates. By construction, $\hu_{\lambda_k,-\cB}$ lies strictly between $-\lambda_k$ and $\lambda_k$ in every coordinate. Therefore it is found by simply minimizing the objective function in , which gives the least squares estimate $$\label{eq:ls}
\hu_{\lambda_k,-\cB} = \big( D_{-\cB}(D_{-\cB})\T \big)^{-1} D_{-\cB}
\big(y-\lambda_k (D_\cB)\T s \big).$$ Now let $a-\lambda_k b$ denote the right-hand side above. For $\lambda\leq\lambda_k$, the interior solution will continue to be $\hu_{\lambda,-\cB}=a-\lambda b$ until one of its coordinates hits the boundary. This critical value is determined by solving, for each $i$, the equation $a_i-\lambda b_i = \pm\lambda$; a simple calculation shows that the solution is $$\label{eq:hittime}
t_i = \frac{a_i}{b_i \pm 1} =
\frac{\Big[\big(D_{-\cB}(D_{-\cB})\T \big)^{-1}D_{-\cB} y\Big]_i}
{\Big[\big(D_{-\cB}(D_{-\cB})\T \big)^{-1}D_{-\cB}(D_\cB)\T s\Big]_i
\pm 1}$$ (only one of $+1$ or $-1$ will yield a value $t_i \in
[0,\lambda_k]$), which we call the “hitting time” of coordinate $i$. We take $\lambda_{k+1}$ to be maximum of these hitting times $$\label{eq:nextlam}
\lambda_{k+1} = \max_i \, t_i.$$ Then we compute $$i_{k+1} = {\mathop{\mathrm{argmax}}}_i \, t_i \;\; \text{and} \;\;
s_{k+1} = \sign\big(\hu_{\lambda_{k+1},i_{k+1}}\big),$$ and append $i_{k+1}$ and $s_{k+1}$ to $\cB$ and $s$, respectively.
Properties of the solution path {#sec:1dprops}
-------------------------------
Here we study some of the path’s basic properties. Again we defer any rigorous arguments until Section \[sec:dprops\], when we consider a general penalty matrix $D$. Instead we demonstrate them by way of a simple example.
Consider Figure \[fig:dualpaths\], which shows the coordinate paths $\hu_{\lambda,i}$ for an example with $n=8$. Recall that it is natural to interpret the paths from right to left ($\lambda=\infty$ to $\lambda=0$). Initially all of the slopes are zero, because when $\lambda=\infty$ the solution is just the least squares estimate $(DD\T )^{-1}Dy$, which has no dependence on $\lambda$. When a coordinate path first hits the boundary (the topmost path, drawn in red) the slopes of the other paths change, and they don’t change again until another coordinate hits the boundary (the bottommost path, drawn in green), and so on, until all coordinates are on the boundary.
The picture suggests that the path $\hu_\lambda$ is continuous and piecewise linear with respect to $\lambda$, with changes in slope or “kinks” at the values $\lambda_1, \ldots \lambda_{n-1}$ visited by the algorithm. (Piecewise linearity is obvious from the algorithm’s construction of the path, but continuity is not.) This is also true in the general $D$ case, although the solution path can have more than $m$ kinks for an $m\times n$ matrix $D$.
On the other hand, Figure \[fig:primalpaths\] shows the corresponding primal coordinate paths $$\hbeta_{\lambda,i} = (y-D\T \hu_\lambda)_i.$$ As $\hu_\lambda$ is a continuous piecewise linear function of $\lambda$, so is $\hbeta_\lambda$, again with kinks at $\lambda_1,
\ldots \lambda_{n-1}$. In contrast to the dual versions, it is natural to interpret the primal coordinate paths from left to right, because in this direction the coordinate paths become adjoined, or “fused”, at some values of $\lambda$. The primal picture suggests that these fusion values are the same as the kinks $\lambda_1,\ldots
\lambda_{n-1}$, that is:
- [*Primal-dual correspondence for the 1d fused lasso.*]{} The values of $\lambda$ at which two primal coordinates fuse are exactly the values of $\lambda$ at which a dual coordinate hits the boundary.
A similar property holds for the fused lasso on an arbitrary graph, though the primal-dual correspondence is a little more complicated for this case.
Note that as $\lambda$ decreases in Figure \[fig:dualpaths\], no dual coordinate paths leave the boundary. This is prescribed by the boundary lemma. As $\lambda$ increases in Figure \[fig:primalpaths\], no primal two coordinates split apart, or “un-fuse”. This is prescribed by a lemma of [@pco] that we paraphrased in , and the two lemmas are equivalent.
A general penalty matrix $D$ {#sec:d}
============================
Now we consider for general $m\times n$ matrix $D$. The first question that comes to mind is: does the boundary lemma still hold? If $DD\T $ is diagonally dominant, that is $$\label{eq:dd}
(DD\T )_{ii} \geq \sum_{j\not=i} |(DD\T )_{ij}|
\;\;\; \text{for} \;\; i=1,\ldots m,$$ then indeed the boundary lemma is still true. (See the online supplement.) Therefore the path algorithm for such a $D$ is the same as that presented in the previous section, Algorithm \[alg:1d\].
It is easy to check the 1d fused lasso matrix is diagonally dominant, as both the left- and right-hand sides of the inequality in are equal to 2 when $D=D_\mathrm{1d}$. Unfortunately, neither the 2d fused lasso matrix nor any of the trend filtering matrices satisfy condition . In fact, examples show that the boundary lemma does not hold for these cases. However, inspired by the 1d fused lasso, we can develop a similar strategy to compute the full solution path for an arbitrary matrix $D$. The difference is: in addition to checking when coordinates will hit the boundary, we have to check when coordinates will leave the boundary as well.
Path algorithm {#sec:dpath}
--------------
Recall that we defined, at a particular $\lambda_k$, the “hitting time” of an interior coordinate path to the value of $\lambda\leq\lambda_k$ at which this path hits the boundary. Similarly, let us define the “leaving time” of a boundary coordinate path to be the value of $\lambda\leq\lambda_k$ at which this path leaves the boundary (we will make this idea more precise shortly). We call the coordinate with the largest hitting time the “hitting coordinate”, and the one with the largest leaving time the “leaving coordinate”. As before, we maintain a list $\cB$ of boundary coordinates, and $s$ contains their signs. The algorithm for a general matrix $D$ is:
\[alg:d\]
- Start with $k=0$, $\lambda_0=\infty$, $\cB=\emptyset$, and $s=\emptyset$.
- While $\lambda_k>0$:
1. Compute a solution at $\lambda_k$ by least squares, as in .
2. Compute the next hitting time $h_{k+1}$, as in and .
3. Compute the next leaving time $l_{k+1}$, as in , , and .
4. Set $\lambda_{k+1}=\max\{h_{k+1},l_{k+1}\}$. If $h_{k+1} > l_{k+1}$ then add the hitting coordinate to $\cB$ and its sign to $s$, otherwise remove the leaving coordinate from $\cB$ and its sign from $s$. Set $k=k+1$.
Although the intuition for this algorithm comes from the 1d fused lasso problem, its details are derived from a more technical point of view, via the Karush-Kuhn-Tucker (KKT) optimality conditions. For our problem , the KKT conditions are $$\label{eq:kkt}
DD\T \hu_\lambda - Dy + \alpha \gamma = 0,$$ where $\hu_\lambda,\alpha,\gamma$ are subject to the constraints $$\begin{aligned}
\stepcounter{equation}
\label{eq:a}
\tag{\theequation a}
\|\hu_\lambda\|_\linf &\leq \lambda \\
\label{eq:b}
\tag{\theequation b}
\alpha &\geq 0 \\
\label{eq:c}
\tag{\theequation c}
\alpha\cdot(\|\hu_\lambda\|_\linf-\lambda) &= 0 \\
\label{eq:d}
\tag{\theequation d}
\|\gamma\|_\lone &\leq 1 \\
\label{eq:e}
\tag{\theequation e}
\gamma\T \hu_\lambda &= \|\hu_\lambda\|_\linf.\end{aligned}$$ Constraints , say that $\gamma$ is a subgradient of the function $x \mapsto \|x\|_\linf$ evaluated at $x=\hu_\lambda$. Subgradients are a generalization of gradients to the case of nondifferentiable functions—for an overview, see [@bert].
A necessary and sufficient condition for $\hu_\lambda$ to be a solution to is that $\hu_\lambda,\alpha,\gamma$ satisfy and – for some $\alpha$ and $\gamma$. The basic idea is that hitting times are events in which is violated, and leaving times are events in which – are violated. We now describe what happens at the $k$th iteration. At $\lambda=\lambda_k$, the path’s solution is $\hu_{\lambda_k,\cB}=\lambda_k s$ for the boundary coordinates, and the least squares estimate $$\label{eq:dls}
\hu_{\lambda_k,-\cB} = \big( D_{-\cB}(D_{-\cB})\T \big)^+ D_{-\cB}
\big(y-\lambda_k (D_\cB)\T s \big)$$ for the interior coordinates. Here $A^+$ denotes the (Moore-Penrose) pseudoinverse of a matrix $A$, which is needed as $D$ may not have full row rank. Write $\hu_{\lambda_k,-\cB} = a-\lambda_k
b$. Like the 1d fused lasso case, we decrease $\lambda$ and continue in a linear direction from the interior solution at $\lambda_k$, proposing $\hu_{\lambda,-\cB}=a-\lambda b$. We first determine when a coordinate of $a-\lambda b$ will hit the boundary. Setting $a_i -
> \lambda b_i = \pm \lambda$ and solving for $\lambda$ gives the hitting time of the $i$th interior coordinate, $$\label{eq:dhittime}
t^{(\mathrm{hit})}_i = \frac{a_i}{b_i \pm 1} =
\frac{\Big[\big(D_{-\cB}(D_{-\cB})\T \big)^+D_{-\cB} y\Big]_i}
{\Big[\big(D_{-\cB}(D_{-\cB})\T \big)^+D_{-\cB}(D_\cB)\T s \Big]_i
\pm 1}$$ (only one of $+1$ or $-1$ will yield a value in $[0,\lambda_k]$.) Hence the next hitting time is $$\label{eq:dnextht}
h_{k+1} = \max_i \, t^{(\mathrm{hit})}_i.$$
The new step is to determine when a boundary coordinate will next leave the boundary. After examining the constraints –, we can express the leaving time of the $i$th boundary coordinate by first defining $$\begin{aligned}
\label{eq:dpreleave}
c_i &= s_i \cdot \bigg[ D_\cB\Big[I-(D_{-\cB})\T
\big(D_{-\cB}(D_{-\cB})\T \big)^+ D_{-\cB}\Big]y \bigg]_i \\
\nonumber
d_i &= s_i \cdot \bigg[ D_\cB\Big[I-(D_{-\cB})\T
\big(D_{-\cB}(D_{-\cB})\T \big)^+ D_{-\cB}\Big] (D_\cB)\T s \bigg]_i, \end{aligned}$$ and then the leaving time is $$\label{eq:dleavetime}
t^{(\mathrm{leave})}_i = \begin{cases}
c_i/d_i & \text{if} \;\; c_i<0 \;\; \text{and} \;\; d_i<0 \\
0 & \text{otherwise.}
\end{cases}$$ Therefore the next leaving time is $$\label{eq:dnextlt}
l_{k+1} = \max_i \, t^{(\mathrm{leave})}_i.$$
The last step of the iteration moves until the next critical event—hitting time or leaving time, whichever happens first. We can verify that the path visited by the algorithm satisfies the KKT conditions and – at each $\lambda$, and hence is indeed a solution path of the dual problem . This argument, as well a derivation of the leaving times given in and , can be found in the online supplement.
Properties of the solution path {#sec:dprops}
-------------------------------
Suppose that the algorithm terminates after $T$ iterations. By construction, the returned solution path $\hu_\lambda$ is piecewise linear with respect to $\lambda$, with kinks at $\lambda_1 \geq \ldots \geq \lambda_T$. Continuity, on the other hand, is a little more subtle: because of the specific choice of the pseudoinverse solution in , the path $\hu_\lambda$ is continuous over $\lambda$. (When $A$ does not have full column rank, there are many minimizers of $\|z-Ax\|_\ltwo$, and $x=(A\T A)^+ A\T z$ is only one of them.) The proof of continuity appears in the online supplement.
Since the primal solution path $\hbeta_\lambda$ can be recovered from $\hu_\lambda$ by the linear transformation , the path $\hbeta_\lambda$ is also continuous and piecewise linear in $\lambda$. The kinks in this path are necessarily a subset of $\{\lambda_1,\ldots \lambda_T\}$. However, this could be a strict inclusion as $\rank(D)$ could be $<m$, that is, $D\T$ could have a nontrivial null space. So when does the primal solution path change slope? To answer this question, it helps to write the solutions in a more explicit form.
For any given $\lambda$, let $\cB=\cB(\lambda)$ and $s=s(\lambda)$ be the current boundary coordinates and their signs. Then we know that the dual solution can be written as $$\begin{aligned}
\hu_{\lambda,\cB} &= \lambda s \\
\hu_{\lambda,-\cB} &= \big( D_{-\cB}(D_{-\cB})\T \big)^+
D_{-\cB}\big(y-\lambda(D_\cB)\T s \big).\end{aligned}$$ This means that the dual fit $D\T\hu_\lambda$ is just $$\label{eq:dualfit}
D\T\hu_\lambda \;=\; (D_\cB)\T\hu_{\lambda,\cB} +
(D_{-\cB})\T\hu_{\lambda,-\cB} \;=\;
\lambda (D_\cB)\T s +
P_{\row(D_{-\cB})}\big(y-\lambda(D_\cB)\T s \big),$$ where $P_M$ denotes the projection operator onto a linear subspace $M$ (here the row space of $D_{-\cB}$). Therefore, applying , the primal solution is given by $$\label{eq:dlassosol2}
\hbeta_\lambda \;=\; \big(I -
P_{\row(D_{-\cB})}\big)\big(y-\lambda(D_\cB)\T s \big) \;=\;
P_{\nul(D_{-\cB})} \big(y-\lambda(D_\cB)\T s \big).$$
Equation is useful for several reasons. Later, in Section \[sec:df\], we use it along with a geometric argument to prove a result on the degrees of freedom of $\hbeta_\lambda$. But first, equation can be used to answer our immediate question about the primal path’s changes in slope: it turns out that $\hbeta_\lambda$ changes slope at $\lambda_{k+1}$ if $\nul(D_{-\cB(\lambda_k)}) \not=
\nul(D_{-\cB(\lambda_{k+1})})$, that is, the null space of $D_{-\cB}$ changes from iterations $k$ to $k+1$. (The proof is left to the online supplement.) Thus we have achieved a generalization of the primal-dual correspondence of Section \[sec:1dprops\]:
- [*Primal-dual correspondence for a general $D$.*]{} The values of $\lambda$ at which at which the primal coordinates changes slope are the values of $\lambda$ at which the null space of $D_{-\cB(\lambda)}$ changes.
For various applications, the null space of $D_{-\cB}$ can have a nice interpretation. We present the case for the fused lasso on an arbitrary graph $\cG$, with $m$ edges and $n$ nodes. We assume without a loss of generality that $\cG$ is connected (otherwise the problem decouples into smaller fused lasso problems). Recall that in this setting each row of $D$ gives the difference between two nodes connected by an edge. Hence the null space of $D$ is spanned by the vector of all ones $$\mathds{1} = (1,1,\ldots 1)\T \in \R^n.$$ Furthermore, removing a subset of the rows, as in $D_{-\cB}$, is like removing the corresponding subset of edges, yielding a subgraph $\cG_{-\cB}$. It is not hard to see that the dimension of the null space of $D_{-\cB}$ is equal to the number of connected components in $\cG_{-\cB}$. In fact, if $\cG_{-\cB}$ has connected components $A_1,\ldots A_k$, then the null space of $D_{-\cB}$ is spanned by $\mathds{1}_{A_1},\ldots \mathds{1}_{A_k} \in \R^m$, the indicator vectors on these components, $$(\mathds{1}_{A_i})_j = 1(\text{node } j \in A_i) \;\;\; \text{for}
\;\; j=1,\ldots n.$$
When $\cG_{-\cB}$ has connected components $A_1,\ldots A_k$, the projection $P_{\nul(D_{-\cB})}$ performs a coordinate-wise average within each group $A_i$: $$P_{\nul(D_{-\cB})}(x) = \sum_{i=1}^k
\bigg(\frac{(\mathds{1}_{A_i}) \T x}{|A_i|}\bigg) \cdot
\mathds{1}_{A_i}.$$ Therefore, recalling , we see that coordinates of the primal solution $\hbeta_\lambda$ are constant (or in other words, fused) on each group $A_i$.
As $\lambda$ decreases, the boundary set $\cB$ can both grow and shrink in size; this corresponds to adding an edge to and removing an edge from the graph $\cG_{-\cB}$, respectively. Since the null space of $D_{-\cB}$ can only change when $\cG_{-\cB}$ undergoes a change in connectivity, the general primal-dual correspondence stated above becomes:
- [*Primal-dual correspondence for the fused lasso on a graph.*]{} In two parts:
- the values of $\lambda$ at which two primal coordinate groups fuse are the values of $\lambda$ at which a dual coordinate hits the boundary and disconnects the graph $\cG_{-\cB(\lambda)}$;
- the values of $\lambda$ at which two primal coordinate groups un-fuse are the values of $\lambda$ at which a dual coordinate leaves the boundary and reconnects the graph $\cG_{-\cB(\lambda)}$.
Figure \[fig:Dpaths\] illustrates this correspondence for a graph with $n=6$ nodes and $m=9$ edges. Note that the primal-dual correspondence for the fused lasso on a graph, as stated above, is consistent with that given in Section \[sec:1dprops\]. This is because the 1d fused lasso corresponds to a chain graph, so removing an edge always disconnects the graph, and furthermore, no dual coordinates ever leave the boundary by the boundary lemma.
A general design matrix $X$ {#sec:x}
===========================
In the last two sections, we focused on the signal approximation case $X=I$. In this section we consider the problem when $X$ is a general $n\times p$ matrix of predictors (and $D$ is a general $m \times p$ penalty matrix). Our strategy is to again solve the equivalent dual problem . At first glance this problem looks much more difficult than the dual when $X=I$. Moreover, the relationship between the primal and dual solutions is now $$\label{eq:xprimaldual}
\hbeta_\lambda = (X\T X)^+ (X\T y - D\T \hu_\lambda) + b,$$ where $b \in \nul(X)$, which is also more complicated.
It turns out that we can make look more familiar—that is, more like —by defining $$\label{eq:yd}
\ty = XX^+ y \;\;\text{and}\;\; \tD = DX^+.$$ (Here the pseudoinverse of a rectangular matrix $A$ is $A^+=(A\T
A)^+A\T$.) Abbreviating $P=P_{\col(X)}=XX^+$, the objective function in becomes $$\begin{aligned}
(X\T y-D\T u)\T (X\T X)^+ (X\T y - D\T u)
&= y\T Py - 2y\T \tD\T u + u\T \tD \tD\T u \\
&= (y-\tD\T u)\T P (y -\tD\T u) \\
&= (y-\tD\T u)\T P^2 (y -\tD\T u) \\
&= (\ty - \tD\T u)\T (\ty - \tD\T u).\end{aligned}$$ The first equality above is by the definition of $\tD$; the second holds because $P\tD\T=\tD\T$; the third is because $P$ is idempotent; and the fourth is again due to the identity $P\tD\T=\tD\T$. Therefore we can rewrite the dual problem in terms of our transformed data and penalty matrix: $$\begin{aligned}
\label{eq:xdual2}
{\mathop{\mathrm{minimize}}}_{u \in \R^m} \;\;
&\half \|\ty-\tD\T u\|_\ltwo^2 \\
\nonumber
\text{subject to} \;\;
&\|u\|_\linf \leq \lambda, \; D\T u \in \row(X), \end{aligned}$$ It is also helpful to rewrite the relationship in terms of our new data and penalty matrix: $$\label{eq:xprimaldual2}
\hbeta_\lambda = X^+(\ty - \tD\T \hu_\lambda) + b,$$ where $b \in \nul(X)$, which implies that the fit is simply $$\label{eq:xprimaldual3}
X\hbeta_\lambda = \ty - \tD\T \hu_\lambda.$$
Modulo the row space constraint, $D\T u \in \row(X)$, problem has exactly the same form as the dual studied in Section \[sec:d\]. In the case that $X$ has full column rank, this extra constraint has no effect, so we can treat the problem just as before. We discuss this next.
The case $\rank(X)=p$ {#sec:xfull}
---------------------
Suppose that $\rank(X)=p$, so $\row(X)=\R^p$ (note that this necessarily means $p \leq n$). Hence we can remove the constraint $D\T u \in
\row(X)$ from problem , so that it is really just the same as problem that we solved in Section \[sec:d\], except with $y,D$ replaced by $\ty,\tD$, respectively. Therefore we can apply Algorithm \[alg:d\] to find a dual solution path $\hu_\lambda$. This gives the (unique) primal solution path using with $b=0$ (since $\nul(X)=\{0\}$), or the fit using .
Fortunately, all of the properties in Section \[sec:dprops\] apply to the current setting as well. First, we know that the constructed dual path $\hu_\lambda$ is continuous and piecewise linear, because we are using the same algorithm as before, just with a different data vector and penalty matrix. This means that $\hbeta_\lambda$ is also continuous and piecewise linear, since it is given by the linear transformation . Next, we can follow the same logic in writing out the dual fit $\tD\T\hu_\lambda$ to conclude that $$\label{eq:dlassosol3}
\hbeta_\lambda = X^+ P_{\nul(\tD_{-\cB})}
\big(\ty - \lambda(\tD_{-\cB})\T s\big),$$ or $$\label{eq:dlassosol4}
X\hbeta_\lambda = P_{\nul(\tD_{-\cB})}
\big(\ty - \lambda(\tD_{-\cB})\T s\big).$$ Hence $0=\tD_{-\cB}X\hbeta_\lambda=D_{-\cB}\hbeta_\lambda$, which means that $\hbeta_\lambda \in \nul(D_{-\cB})$, as before.
Though working with equations and may seem complicated (as one would need to expand the newly defined variables $\ty,\tD$ in terms of $y,D$), it is straightforward to show that the general primal-dual correspondence still holds here. This is given in the online supplement. That is: the primal path $\hbeta_\lambda$ changes slope at the values of $\lambda$ at which the null space of $D_{-\cB(\lambda)}$ changes. For the fused lasso on a graph $\cG$, we indeed still get fused groups of coordinates in the primal solution, since $\hbeta_\lambda \in \nul(D_{-\cB})$ implies that $\hbeta_\lambda$ is fused on the connected components of $\cG_{-\cB}$. Therefore, fusions still correspond to dual coordinates hitting the boundary and disconnecting the graph, and un-fusions still correspond to dual coordinates leaving the boundary and reconnecting the graph.
The case $\rank(X)<p$ {#the-case-rankxp}
---------------------
If $\rank(X)<p$, then $\row(X)$ is a strict subspace of $\R^p$. One easy way to avoid dealing with the constraint $D\T u \in \row(X)$ of is to add an $\ell_2$ penalty to our original problem, giving $$\label{eq:dlassoridge}
{\mathop{\mathrm{minimize}}}_{\beta \in \R^p} \; \half\|y-X\beta\|_\ltwo^2 +
\lambda\|D\beta\|_\lone + \epsilon\|\beta\|_\ltwo^2,$$ where $\epsilon>0$ is fixed. This is analogous to the [*elastic net*]{}, which adds an $\ell_2$ penalty to the lasso criterion [@enet]. Problem can be rewritten as $${\mathop{\mathrm{minimize}}}_{\beta\in\R^p} \; \half\|y^*-(X^*)\beta\|_\ltwo^2 +
\lambda\|D\beta\|_\lone,$$ where $y^*=(y,0)\T$ and $X^*=\left[\begin{array}{c} X \\ \epsilon \cdot I
\end{array}\right]$. Since $\rank(X^*)=p$, we can now use the strategy discussed in the last section, which is just applying Algorithm \[alg:d\] to a transformed problem, to find the solution path of . Putting aside computational concerns, it may still be preferable to study problem instead of the original generalized lasso problem . Some reasons are:
- the (unique) solution path of may be more stable than the (non-unique) generalized lasso path, analogous to the behavior of the elastic net and lasso paths;
- the fit of may actually outperform the generalized lasso fit in terms prediction error, again like the comparison between the elastic net and lasso prediction errors.
Though adding an $\ell_2$ penalty is easier and, as we suggested, perhaps even desirable, we can still solve the unmodified problem in the $\rank(X)<p$ case, by looking at its dual . We only give a rough sketch of the path algorithm because in the present setting the solution and its computation are more complicated.
We can rewrite the row space constraint in as $D\T u \perp \nul(X)$. Write $P=P_{\nul(X)}$, and then problem is $$\begin{aligned}
\label{eq:xdual3}
{\mathop{\mathrm{minimize}}}_{u \in \R^m} \;\;
&\half \|\ty-\tD\T u\|_\ltwo^2 \\
\nonumber
\text{subject to} \;\;
&\|u\|_\linf \leq \lambda, \; (DP)\T u=0. \end{aligned}$$ To find a solution path of , the KKT conditions and - need to be modified to incorporate the new equality constraint. These are now $$\tD\tD\T \hu_\lambda - \tD\ty + \alpha \gamma + DP\delta = 0,$$ where $\hu_\lambda,\alpha,\gamma,\delta$ are subject to the same constraints as before, –, and additionally $(DP)\T \hu_\lambda = 0$. Instead of simply using the appropriate least squares estimate at each iteration, we now need to solve for $\hu_\lambda$ and $\delta$ jointly. When $\lambda=\infty$, this can be done by solving the block system $$\label{eq:block}
\left[\begin{array}{cc}
\tD\tD\T & DP \\
(DP)\T & 0
\end{array}\right]
\left[\begin{array}{c}
\hu_\lambda \\ \delta
\end{array}\right]
=
\left[\begin{array}{c}
\tD\ty \\ 0
\end{array}\right],$$ and in future iterations the expressions are similar. Having done this, satisfying the rest of the constraints – can be done by finding the hitting and leaving times just as we did previously.
Computational considerations {#sec:comp}
============================
Here we discuss an efficient implementation of Algorithm \[alg:d\], which gives the solution path of the signal approximation problem , after applying the transformation from dual to primal variables. For a design with $\rank(X)=p$, we can modify $y$ and $X$ as in , and then the same algorithm gives the solution path of , this time relying on the transformation (with $b=0$) for the primal
At each iteration of the algorithm, the dominant work is in computing expressions of the form $$\big(D_{-\cB}(D_{-\cB})\T\big)^+
D_{-\cB}x$$ for a vector $x \in \R^n$, where $\cB$ is the current boundary set (see equations and ). Equivalently, the complexity of each iteration is based on finding $$\label{eq:form}
{\mathop{\mathrm{argmin}}}_v \bigg\{ \|v\|_\ltwo \;:\; v \in {\mathop{\mathrm{argmin}}}_w
\|x-(D_{-\cB})\T w\|_\ltwo \bigg\},$$ the least squares solution with the smallest $\ell_2$ norm. In the next iteration, $D_{-\cB}$ has either one less or one more row (depending on whether a coordinate hit or left the boundary).
We can exploit the fact that the problems are highly related from one iteration to the next (our strategy that is similar to that in the LARS implementation). Suppose that when $\cB=\emptyset$, we solve the problem by using a matrix factorization (for example, a QR decomposition). In future iterations, this factorization can be efficiently updated after a row has been deleted from or added to $D_{-\cB}$. This allows us to compute the new solution of with much less work than it would take to solve the problem from “scratch”.
Recall that $D$ is $m\times n$, and the dual variable $u$ is $m$-dimensional. Let $T$ denote the number of iterations taken by the algorithm (note that $T \geq m$, and can be strictly greater if dual coordinates leave the boundary). When $m \leq n$, we can use a QR decomposition of $D\T$ to compute the full dual solution path in $$O(mn^2 + Tm^2)$$ operations. When $m>n$, using a QR decomposition of $D$ allows us to compute the full dual solution path in $$O(m^2 n + Tn^2)$$ operations. The main idea behind this implementation is fairly straightforward. However, the details are somewhat complicated because we require the minimum $\ell_2$ norm solution , instead of a generic solution, to the appropriate least squares problem at each iteration. See Chapters 5 and 12 of [@golub] for an extensive coverage of the QR decomposition.
We mention two simple points to improve practical efficiency:
- The algorithm starts at the fully regularized end of the path ($\lambda=\infty$) and works towards the un-regularized solution ($\lambda=0$). Therefore, for problems in which the highly or moderately regularized solutions are the only ones of interest, the algorithm can compute only part of the path and terminate early. This could end up being a large savings in practice.
- One can obtain an approximate solution path by not permitting dual coordinates to leave the boundary (achieved by setting $l_{k+1}=0$ in Step 3 of Algorithm \[alg:d\]). This makes $T=m$, and so computing this approximate path only requires $O(mn^2)$ or $O(m^2n)$ operations when $m \leq n$ or $m > n$, respectively. This approximation can be quite accurate if the number times a dual coordinate leaves the boundary is (relatively) small. Furthermore, its legitimacy is supported by the following fact: for $D=I$ (the lasso problem) and $\rank(X)=p$, this approximate path is exactly the LARS path when LARS is run in its original (unmodified) state. We discuss this in the next section.
Finally, it is important to note that if one’s goal is to find the solution of or over a discrete set of $\lambda$ values, and the problem size is very large, then it is unlikely that our path algorithm is the most efficient approach. A reason here is the same reason that LARS is not generally used to solve large-scale lasso problems: the set of critical points (changes in slope) in the piecewise linear solution path $\hbeta_\lambda$ becomes very dense as the problem size increases. But there is a further disadvantage is that is specific to the generalized lasso (and not the lasso). At the regularized end of the path (which is typically the region of interest), the generalized lasso solution is orthogonal to many of the rows of $D$; therefore we must solve large least squares problems, as in , in order to compute the path. On the other hand, at the regularized end of the lasso path, the solution lies in the span of a small number of columns of $X$; therefore we only need to solve small least squares problems to compute the path.
For solving a large generalized lasso (or lasso) problem at a fixed $\lambda$, it is preferable to use a convex optimization technique that was specifically developed for the purposes of computational efficiency. First-order methods, for example, can efficiently solve large-scale instances of or for $\lambda$ in a discrete set (see [@nesta] as an example). Another optimization method of recent interest is coordinate descent [@cd], which is quite efficient in solving the lasso at discrete values of $\lambda$ [@pco], and is favored for its simplicity. But coordinate descent cannot be used for the minimizations and , because the penalty term $\|D\beta\|_\lone$ is not separable in $\beta$, and therefore coordinate descent does not necessarily converge. In the important signal approximation case , however, the dual problem is separable, so coordinate descent will indeed converge if applied to the dual. Moreover, for various applications, the matrix $D$ is sparse and structured, which means that the coordinate-wise updates for are very fast. This makes coordinate descent on the dual a promising method for solving many of the signal approximation problems from Section \[sec:appsi\].
Connection to LARS {#sec:lars}
==================
In this section, we return to the LARS algorithm, described in the introduction as a point of motivation for our work. Here we assume that $D=I$, so that is the same as the standard lasso problem ; furthermore we assume that $\rank(X)=p$. Our algorithm computes the lasso path $\hbeta_\lambda$, via the dual path $\hu_\lambda$ (using Algorithm \[alg:d\] applied to $\ty,\tD$ as defined in ). Another way of finding the lasso path is to use the LARS algorithm in its “lasso” mode. Since the problem is strictly convex ($X$ has full column rank), there is only one solution at each $\lambda$, so of course these two algorithms must give the same result.
In its original or unmodified state, LARS returns a different path, obtained by selecting variables in order to continuously decrease the maximal absolute correlation with the residual (technically these are inner products with the residual, but they become correlations if we think of standardized data). We refer to this as the “LARS path”. Interestingly, the LARS path can be viewed as an approximation to the lasso path (see [@lars] for an elegant interpretation and discussion of this). In our framework, we can obtain an approximate dual solution path if we never check for dual coordinates leaving the boundary, which can be achieved by dropping Step 3 from Algorithm \[alg:d\] (or more precisely, by setting $l_{k+1}=0$ for each $k$). If we denote the resulting dual path by $\tu_\lambda$, then this suggests a primal path $$\label{eq:tbeta}
\tbeta_\lambda=(X\T X)^{-1}(X\T y - \tu_\lambda),$$ based on the transformation (noting that $b=0$ as $\nul(X)=\{0\}$). The question is: how does this approximate solution path $\tbeta_\lambda$ compare to the LARS path?
Figure \[fig:lar\] shows the two paths in question. On the left is the familiar plot of [@lars], showing the LARS path for the “diabetes data”. The colored dots on the x-axis mark when variables enter the model. The right plot shows our approximate solution path on this same data set, with vertical dashed lines marking when variables (coordinates) hit the boundary. The paths look identical, and this is not a coincidence: we show that our approximate path, which is given by ignoring dual coordinates leaving the boundary, is equal to the LARS path in general.
![ *Comparing the LARS path and our approximate lasso path, on the diabetes data. For this data set $n=442$ and $p=10$. The paths by parametrized by the $\ell_1$ norm of their (respective) coefficient vectors.*[]{data-label="fig:lar"}](lars.pdf){width="5.5in"}
\[lemma:lars\] Suppose that $\rank(X)=p$ and consider using Algorithm \[alg:d\] to compute an approximate lasso path in the following way: we use $\ty=XX^+y, \; \tD=X^+$ in place of $y,D$, and we ignore Step 3 (that is, set $l_{k+1}=0$). Let $\tu_\lambda$ denote the corresponding dual path, and define a primal path $\tbeta_\lambda$ according to . Then $\tbeta_\lambda$ is exactly the LARS path.
First define the residual $r_\lambda=y-X\tbeta_\lambda$. Notice that by rearranging , we get $\tu_\lambda = X\T r_\lambda$. Therefore, the coordinates of the dual path are equal to the correlations (inner products) of the columns of $X$ with the current residual. Hence we have a procedure that:
- moves in a direction so that the absolute correlation with the current residual is constant within $\cB$ (and maximal among all variables) for all $\lambda$;
- adds variables to $\cB$ once their absolute correlation with the residual matches that realized in $\cB$.
This almost proves that $\tbeta_\lambda$ is the LARS path, with $\cB$ being the “active set” in LARS terminology. What remains to be shown is that the variables not in $\cB$ are all assigned zero coefficients. But, recalling that $D=I$, the same arguments given in Section \[sec:dprops\] and Section \[sec:xfull\] apply here to give that $\tbeta_\lambda \in \nul(I_{-\cB})$ (really, $\tu_\lambda$ still solves a sequence of least squares problems, and the only difference between $\tu_\lambda$ and $\hu_\lambda$ is in how we construct $\cB$). This means that $\tbeta_{\lambda,-\cB}=0$, as desired.
Degrees of freedom {#sec:df}
==================
In applied statistics, degrees of freedom describes the effective number of parameters used by a fitting procedure. This is typically easy to compute for linear procedures (linear in the data $y$) but difficult for nonlinear, adaptive procedures. In this section, we derive the degrees of freedom of the fit of the generalized lasso problem, when $\rank(X)=p$ and $D$ is an arbitrary penalty matrix. This produces corollaries on degrees of freedom for various problems presented in Section \[sec:apps\]. We then briefly discuss model selection using these degrees of freedom results, and lastly we discuss the role of shrinkage, a fundamental property of $\ell_1$ regularization.
Degrees of freedom results
--------------------------
We assume that the data vector $y \in \R^n$ is drawn from the normal model $$y \sim N(\mu,\sigma^2 I),$$ and the design matrix $X$ is fixed (nonrandom). For a function $g:
\R^n \rightarrow \R^n$, with $i$th coordinate function $g_i: \R^n
\rightarrow \R$, the degrees of freedom of $g$ is defined as $$\df(g) = \frac{1}{\sigma^2} \sum_{i=1}^n
\Cov\big(g_i(y),y_i\big).$$ For our problem, the function of interest is $g(y)=X\hbeta_\lambda(y)$, for fixed $\lambda$.
An alternative and convenient formula for degrees of freedom comes from Stein’s unbiased risk estimate [@stein]. If $g$ is continuous and almost differentiable, then Stein’s formula states that $$\label{eq:stein}
\frac{1}{\sigma^2} \sum_{i=1}^n
\Cov\big(g_i(y),y_i\big) =
\E[(\nabla \cdot g)(y)].$$ Here $\nabla \cdot g = \sum_{i=1}^n
\partial g_i/\partial y_i$ is called the divergence of $\theta$. This is useful because typically the right-hand side of is easier to calculate; for our problem this is the case. But using Stein’s formula requires checking that the function is continuous and almost differentiable. In addition to checking these regularity conditions for $g(y)=X\hbeta_\lambda(y)$, we establish below that for almost every $y$ the fit $X\hbeta_\lambda(y)$ is a locally affine projection. Essentially, this allows us to take the divergence in when $X=I$, or for the general $X$ case, and treat $\cB$ and $s$ as constants.
As in our development of the path algorithm in Sections \[sec:1d\], \[sec:d\], and \[sec:x\], we first consider the case $X=I$, because it is easier to understand. Notice that we can express the dual fit as $D\T \hu_\lambda (y) = P_{C_\lambda}(y)$, the projection of $y$ onto the convex polytope $$C_\lambda = \{D\T u: \|u\|_\linf \leq \lambda\} \subseteq \R^n.$$ From , the primal solution is just the residual from this projection, $\hbeta_\lambda(y) =
(I-P_{C_\lambda})(y)$. The projection map onto a convex set is always nonexpansive, that is, Lipschitz continuous with constant $\leq
1$. In fact, so is the residual from projecting onto a convex set (for example, see the proof of Theorem 1.2.2 in [@schneider]). Therefore $\hbeta_\lambda(y)$ is nonexpansive in $y$, and hence continuous and almost differentiable (this follows from the standard proof a result called “Rademacher’s theorem”; for example, see Theorem 2 in Section 3.2 of [@evans]).
Furthermore, thinking geometrically about the projection map onto $C_\lambda$ yields a crucial insight. Examine Figure \[fig:geom\]—as drawn, it is clear that we can move the point $y$ slightly and it still projects to the same face of $C_\lambda$. In fact, it seems that the only points $y$ for which this property does not hold necessarily lie on rays that emanate orthogonally from the corners of $C_\lambda$ (two such rays are drawn leaving the bottom right corner). In other words, we are lead to believe that for almost every $y$, the projection map onto $C_\lambda$ is a locally constant affine projection. This is indeed true:
\[lemma:locconst\] For fixed $\lambda$, there exists a set $\cN_\lambda$ such that:
- $\cN_\lambda$ has Hausdorff dimension $n-1$, hence Lebesgue measure zero;
- for any $y \notin \cN_\lambda$, there exists a neighborhood $U$ of $y$ such that $P_{C_\lambda} : U \rightarrow \R^n$ is simply the projection onto an affine subspace. In particular, the affine subspace is $$\label{eq:affspace}
\lambda (D_\cB)\T s + \row(D_{-\cB}),$$ where $\cB$ and $s$ are the boundary set and signs for a solution $\hu_\lambda(y)$ of the dual problem , $$\cB = \{i:|\hu_{\lambda,i}(y)| = \lambda\} \;\; \text{and} \;\;
s = \sign\big(\hu_{\lambda,\cB}(y)\big).$$ The quantity is well-defined in the sense that it is invariant under different choices of $\cB$ and $s$ (as the dual solution may not be unique).
The proof, which follows the intuition described above, is given in the online supplement.
![ *An illustration of the geometry surrounding $\hu_\lambda$ and $\hbeta_\lambda$, for the case $X=I$. Recall that $\hbeta_\lambda(y)
=y-D\T\hu_\lambda(y)$, where $D\T\hu_\lambda(y)$ is the projection of $y$ onto the convex polytope $C_\lambda=\{D\T u: \|u\|_\linf \leq
\lambda\}$. Almost everywhere, small perturbations of $y$ don’t change the face on which its projection lies. The exceptional set $\cN_\lambda$ of points for which this property does not hold has dimension $n-1$, and is a union of rays like the two drawn as dotted lines in the bottom right of the figure.*[]{data-label="fig:geom"}](geom.pdf){width="4in"}
Hence we have the following result:
\[theorem:df\] For fixed $\lambda$, the solution $\hbeta_\lambda$ of the signal approximation problem has degrees of freedom $$\df(\hbeta_\lambda) =
\E[\mathrm{nullity}(D_{-\cB(y)})],$$ where the nullity of a matrix is the dimension of its null space. The expectation here is taken over $\cB(y)$, the boundary set of a dual solution $\hu_\lambda(y)$.
Note: above, we can choose any dual solution at $y$ to construct the boundary set $\cB(y)$, because by Lemma \[lemma:locconst\], all dual solutions give rise to the same linear subspace $\nul(D_{-\cB(y)})$ (almost everywhere in $y$).
Consider $y \notin \cN_\lambda$, and let $\cB$ and $s$ be the boundary set and signs of a dual solution $\hu_\lambda(y)$. By Lemma \[lemma:locconst\], there is a neighborhood $U$ of $y$ such that $$\hbeta_\lambda(y') = (I - D\T \hu_\lambda)(y') =
P_{\nul(D_{-\cB})}\big(y'-\lambda(D_\cB)^T s\big)$$ for all $y' \in U$. Taking the divergence at $y$ we get $$(\nabla \cdot \hbeta_\lambda)(y) = \tr(P_{\nul(D_{-\cB})}) =
\mathrm{nullity}(D_{-\cB}),$$ since the trace of a projection matrix is just its rank. This holds for almost every $y$ because $\cN_\lambda$ has measure zero, and we can use Stein’s formula to conclude that $\df(\hbeta_\lambda) = \E[\mathrm{nullity}(D_{-\cB(y)})]$.
We now consider problem , with the predictor matrix satisfying $\rank(X)=p$, and it turns out that the same degrees of freedom formula holds for the fit $X\hbeta_\lambda$. This is relatively straightforward to show, but requires sorting out the details of how to turn statements involving $\ty,\tD$ into those involving $y,D$. First, by the same arguments as before, we know that $X\hbeta_\lambda(\ty)$ is nonexpansive as a function of $\ty$. But $\ty=P_{\col(X)}(y)$ is nonexpansive in $y$, so $X\hbeta_\lambda(y)$ is indeed nonexpansive, hence continuous and almost differentiable, as a function of $y$.
Next we must establish that $\tD\T\hu_\lambda(y)$ is a locally affine projection for almost every $y$. Well, by Lemma \[lemma:locconst\], this is true of $\tD\T\hu_\lambda(\ty)$ for $\ty \notin \cN_\lambda$, so we have the desired result except on $\cM_\lambda=(P_{\col(X)})^{-1}(\cN_\lambda)$. Following the arguments in the proof of Lemma \[lemma:locconst\], it is not hard to see that $\cN_\lambda$ now has dimension $p-1$, so $\cM_\lambda$ has measure zero.
With these properties satisfied, we have the general result:
\[theorem:xdf\] Suppose that $\rank(X)=p$. For fixed $\lambda$, the fit $X\hbeta_\lambda$ of the generalized lasso has degrees of freedom $$\df(X\hbeta_\lambda) =
\E[\mathrm{nullity}(D_{-\cB(y)})],$$ where $\cB(y)$ is the boundary set of a dual solution $\hu_\lambda(y)$.
Note: as before, we can construct the boundary set $\cB(y)$ from any dual solution at $y$, because by Lemma \[lemma:locconst\] the quantity $\nul(D_{-\cB(y)})$ is invariant (almost everywhere in $y$).
Let $y \notin \cM_\lambda$. We show that $(\nabla \cdot X\hbeta_\lambda)(y)=
\mathrm{nullity}(D_{-\cB(y)})$, and then applying Stein’s formula (along with the fact that $\cM_\lambda$ has measure zero) gives the result.
Let $\cB$ denote the boundary set of a dual solution $\hu_\lambda(y)$. Then the fit is $$X\hbeta_\lambda(y) =
P_{\nul(\tD_{-\cB})}
P_{\col(X)} y + c,$$ where $c$ denotes the terms that have zero derivative with respect to $y$. Using the fact $\nul(X^+)=\nul(X\T)$, and that $\nul(\tD_{-\cB}) \supseteq \nul(X^+)$, $$\begin{aligned}
P_{\nul(\tD_{-\cB})}
P_{\col(X)}
&= P_{\nul(\tD_{-\cB})} -
P_{\nul(\tD_{-\cB})}
P_{\nul(X^+)} \\
&= P_{\nul(\tD_{-\cB})} -
P_{\nul(X^+)}.\end{aligned}$$ Therefore, computing the divergence: $$\begin{aligned}
\big(\nabla\cdot X\hbeta_\lambda\big)(y)
&= \mathrm{nullity}(D_{-\cB}X^+) -
\mathrm{nullity}(X^+) \\
&= \mathrm{nullity}(D_{-\cB}),\end{aligned}$$ where the last equality follows because $X$ has full column rank. This completes the proof.
We saw in Section \[sec:dprops\] that the null space of $D$ has a nice interpretation for the fused lasso problem. In this case, the theorem also becomes easier to interpret:
\[cor:fuse\] Suppose that $\rank(X)=p$ and that $D$ corresponds to the fused lasso penalty on an arbitrary graph. Then for fixed $\lambda$, the fit $X\hbeta_\lambda$ of has degrees of freedom $$\df(X\hbeta_\lambda)=
\E[\,\mathrm{number}\;\mathrm{of}\;\mathrm{fused}\;\mathrm{groups}
\;\mathrm{in}\;\hbeta_\lambda(y)\,].$$
If $\cG$ denotes the underlying graph, we showed in Section \[sec:dprops\] that $\mathrm{nullity}(D_{-\cB(y)})$ is the number of connected components in $\cG_{-\cB(y)}$. We also showed (see Section \[sec:xfull\] for the extension to a general design $X$) that the coordinates of $\hbeta_\lambda(y)$ are fused on the connected components of $\cG_{-\cB(y)}$, giving the result.
By slightly modifying the penalty matrix, we can also derive the degrees of freedom of the sparse fused lasso:
\[cor:sfuse\] Suppose that $\rank(X)=p$ and write $X_i \in \R^p$ for the $i$th row of $X$. Consider the sparse fused lasso problem: $$\label{eq:sfuse}
{\mathop{\mathrm{minimize}}}_{\beta \in \R^p} \;
\sum_{i=1}^n (y_i - X_i\T \beta)^2 + \lambda_1 \sum_{i=1}^p |\beta_i|
+ \lambda_2 \hspace{-0.08in} \sum_{(i,j) \in E} |\beta_i-\beta_j|,$$ where $E$ is an arbitrary set of edges between nodes $\beta_1,\ldots \beta_p$. Then for fixed $\lambda_1,\lambda_2$, the fit $X\hbeta_{\lambda_1,\lambda_2}$ of has degrees of freedom $$\df(X\hbeta_{\lambda_1,\lambda_2})=
\E[\,\mathrm{number}\;\mathrm{of}\;\mathrm{nonzero}\;
\mathrm{fused}\;\mathrm{groups}\;\mathrm{in}\;
\hbeta_{\lambda_1,\lambda_2}(y)\,].$$
We can write in the generalized lasso framework by taking $\lambda=\lambda_2$ and $$D = \left[\begin{array}{c}
D_\mathrm{fuse} \smallskip \\
\frac{\lambda_1}{\lambda_2} I \end{array}\right],$$ where $D_\mathrm{fuse}$ is the fused lasso matrix corresponding to the underlying graph, with each row giving the difference between two nodes connected by an edge.
In Section \[sec:dprops\], we analyzed the null space of $D_\mathrm{fuse}$ to interpret the primal-dual correspondence for the fused lasso. A similar interpretation can be achieved with $D$ as defined above. Let $\cG$ denote the underlying graph and suppose that it has $m$ edges (and $p$ nodes), so that $D_\mathrm{fuse}$ is $m\times p$ and $D$ is $(m+p)\times p$. Also, suppose that we decompose the boundary set as $\cB= \cB_1\cup \cB_2$, where $\cB_1$ contains the dual coordinates in $\{1,\ldots m\}$ and $\cB_2$ contains those in $\{m+1,\ldots m+p\}$. We can associate the first $m$ coordinates with the $m$ edges, and the last $p$ coordinates with the $p$ nodes. Then the matrix $D_{-\cB}$ defines a subgraph $\cG_{-\cB}$ that can be constructed as follows:
1. delete the edges of $\cG$ that correspond to coordinates in $\cB_1$, yielding $\cG_{-\cB_1}$;
2. keep only the nodes of $\cG_{-\cB_1}$ that correspond to coordinates in $\cB_2$, yielding $\cG_{-\cB}$.
It is straightforward to show that the nullity of $D_{-\cB}$ is the number of connected components in $\cG_{-\cB}$. Furthermore, the solution $\hbeta_{\lambda_1,\lambda_2}(y)$ is fused on each connected component of $\cG_{-\cB}$ and zero in all other coordinates. Applying Theorem \[theorem:xdf\] gives the result.
The above corollary proves a conjecture of [@fuse], in which the authors hypothesize that the degrees of freedom of the sparse 1d fused lasso fit is equal to the number of nonzero fused coordinate groups, in expectation. But Corollary \[cor:sfuse\] covers any underlying graph, which makes it a much more general result.
By examining the null space of $D_{-\cB}$ for other applications, and applying Theorem \[theorem:xdf\], one can obtain more corollaries on degrees of freedom. We omit the details for the sake of brevity, but list some such results in Table \[table:df\], along with those on the fused lasso for the sake of completeness. The table’s first result, on the degrees of freedom of the lasso, was already established in [@lassodf]. The results on trend filtering and outlier detection can actually be derived from this lasso result, because these problems correspond to the case $\rank(D)=m$, and can be transformed into a regular lasso problem . For the outlier detection problem, we actually need to make a modification in order for the design matrix to have full column rank. Recall the problem formulation , where the coefficient vector is $(\alpha,\beta)\T$, the first block concerning the outliers, and the second the regression coefficients. We set $\alpha_1=\ldots=\alpha_p=0$, the interpretation being that we know a priori $p$ points $y_1,\ldots y_p$ come from the true model, and only rest of the points $y_{p+1},\ldots y_n$ can possibly be outliers (this is quite reasonable for a method that simultaneous performs a $p$-dimensional linear regression and detects outliers).
**Problem** **Unbiased estimate of** $\df(X\hbeta_\lambda)$
--------------------------------------- -------------------------------------------------
Lasso Number of nonzero coordinates
Fused lasso Number of fused groups
Sparse fused lasso Number of nonzero fused groups
Polynomial trend filtering, order $k$ Number of knots $+\;k+1$
Outlier detection Number of outliers $+\;p$
: *Corollaries of Theorem \[theorem:xdf\], giving unbiased estimates of $\df(X\hbeta_\lambda)$ for various problems discussed in Section \[sec:apps\]. These assume that $\rank(X)=p$.*[]{data-label="table:df"}
Model selection
---------------
Note that the estimates Table \[table:df\] are all easily computable from the solution vector $\hbeta_\lambda$. The estimates for the lasso, (sparse) fused lasso, and outlier detection problems can be obtained by simply counting the appropriate quantities in $\hbeta_\lambda$. The estimate for trend filtering may be difficult to determine visually, as it may be difficult to identify the knots in a piecewise polynomial by eye, but the knots can be counted from the nonzeros of $D\hbeta_\lambda$. All of this is important because it means that we can readily use model selection criteria like $C_p$ or BIC for these problems, which employ degrees of freedom to assess risk. For example, for the estimate $X\hbeta_\lambda$ of the underlying mean $\mu$, the $C_p$ statistic is $$C_p (\lambda) = \|y-X\hbeta_\lambda\|_\ltwo^2 -n\sigma^2 +
2\sigma^2\df(X\hbeta_\lambda),$$ (see [@mallows] for the classic linear regression case), and is an unbiased estimate of the true risk $\E\big[\|\mu-X\hbeta_\lambda\|_\ltwo^2\big]$. Hence we can define $$\widehat{C}_p (\lambda) = \|y-X\hbeta_\lambda\|_\ltwo^2 -n\sigma^2 +
2\sigma^2\mathrm{nullity}(D_{-\cB}),$$ replacing $\df(X\hbeta_\lambda)$ by its own unbiased estimate $\mathrm{nullity}(D_{-\cB})$. This modified statistic $\widehat{C}_p(\lambda)$ is still unbiased as an estimate of the true risk, and this suggests choosing $\lambda$ to minimize $\widehat{C}_p(\lambda)$. For this task, it turns out that $\widehat{C}_p(\lambda)$ obtains its minimum at one of the critical points $\{\lambda_1,\ldots \lambda_T\}$ in the solution path of $\hbeta_\lambda$. This is true because $\mathrm{nullity}(D_{-\cB})$ is a step function over these critical points, and the residual sum of squares $\|y-X\hbeta_\lambda\|_\ltwo^2$ is monotone nondecreasing for $\lambda$ in between critical points (this can be checked using ). Therefore Algorithm \[alg:d\] can be used to simultaneously compute the solution path and select a model, by simply computing $\widehat{C}_p(\lambda_k)$ at each iteration $k$.
Shrinkage and the $\ell_1$ norm
-------------------------------
At first glance, the results in Table \[table:df\] seem both intuitive and unbelievable. For the fused lasso, for example, we are told that on average we spend a single degree of freedom on each group of coordinates in the solution. But these groups are being adaptively selected based on the data, so aren’t we using more degrees of freedom in the end? As another example, consider the trend filtering result: for a cubic fit, the degrees of freedom is the number of knots $+\;4$, in expectation. A cubic regression spline also has degrees of freedom equal to the number of knots $+\;4$; however, in this case we fix the knot locations ahead of time, and for cubic trend filtering the knots are selected automatically. How can this be?
This seemingly remarkable property—that searching for the nonzero coordinates, fused groups, knots, or outliers doesn’t cost us anything in terms of degrees of freedom—is explained by the shrinking nature of the $\ell_1$ penalty. Looking back at the criterion in , it is not hard to see that the nonzero entries in $D\hbeta_\lambda$ are shrunken towards zero (imagine the problem in constrained form, instead of Lagrangian form). For the fused lasso, this means that once the groups are “chosen”, their coefficients are shrunken towards each other, which is less greedy than simply fitting the group coefficients to minimize the squared error term. Roughly speaking, this makes up for the fact that we chose the fused groups adaptively, and in expectation, the degrees of freedom turns out “just right”: it is simply the number of groups.
This leads us to think about the $\ell_0$-equivalent of problem , which is achieved by replacing the $\ell_1$ norm by an $\ell_0$ norm (giving best subset selection when $D=I$). Solving this problem requires a combinatorial optimization, and this makes it difficult to study the properties of its solution in general. However, we do know that the solution of the $\ell_0$ problem does not enjoy any shrinkage property like that of the lasso solution: if we fix which entries of $D\beta$ are nonzero, then the penalty term is constant and the problem reduces to an equality-constrained regression. Therefore, in light of our above discussion, it seems reasonable to conjecture that the $\ell_0$ fit has greater than $\E[\mathrm{nullity}(D_{-\cB})]$ degrees of freedom. When $D=I$, this would mean that the degrees of freedom of the best subset selection fit is more than the number of nonzero coefficients, in expectation.
Discussion {#sec:discuss}
==========
We have studied a generalization of the lasso problem, in which the penalty is $\|D\beta\|_1$ for a matrix $D$. Several important problems (such as the fused lasso and trend filtering) can be expressed as a special case of this, corresponding to a particular choice of $D$. We developed an algorithm to compute a solution path for this general problem, provided that the design matrix $X$ has full column rank. This is achieved by instead solving the (easier) Lagrange dual problem, which, using simple duality theory, yields a solution to the original problem after a linear transformation.
Both the dual solution path and the primal (original) solution path are continuous and piecewise linear with respect to $\lambda$. The primal solution $\hbeta_\lambda$ can be written explicitly in terms of the boundary set $\cB$, which contains the coordinates of the dual solution that are equal to $\pm \lambda$, and the signs of these coordinates $s$. Further, viewing the dual solution as a projection onto a convex set, we derived a simple formula for the degrees of freedom of the generalized lasso fit. For the fused lasso problem, this result reveals that the number of nonzero fused groups in the solution is an unbiased estimate of the degrees of freedom of the fit, and this holds true for any underlying graph structure. Other corollaries follow, as well.
An implementation of our path algorithm, following the ideas presented in Section \[sec:comp\], is a direction for future work, and will be made available as an R package “genlasso” on the CRAN website [@cran].
There are several other directions for future research. We describe three possibilities below.
- [*Specialized implementation for the fused lasso path algorithm.*]{} When $D$ is the fused lasso matrix corresponding to a graph $\cG$, projecting onto the null space of $D_{-\cB}$ is achieved by a simple coordinate-wise average on each connected component of $\cG_{-\cB}$. It may therefore be possible (when $X=I$) to compute the solution path $\hbeta_\lambda$ without having to use any linear algebra, but by instead tracking the connectivity of $\cG$. This could improve the computational efficiency of each iteration, and could also lead to a parallelized approach (in which we work on each connected component in parallel).
- [*Number of steps until termination.*]{} The number of steps $T$ taken by our path algorithm, for a general $D$, is determined by how many times dual coordinates leave the boundary. This is related to an interesting problem in geometry studied by [@donoho], and investigating this connection could lead to a more definitive statement about the algorithm’s computational complexity.
- [*Connection to forward stagewise regression.*]{} When $D=I$ (and $\rank(X)=p$), we proved that our path algorithm yields the LARS path (when LARS is run in its original, unmodified state) if we just ignore dual coordinates leaving the boundary. LARS can be modified to give forward stagewise regression, which is the limit of forward stepwise regression when the step size goes to zero (see [@lars]). A natural follow-up question is: can our algorithm be changed to give this path too?
In general, we believe that Lagrange duality deserves more attention in the study of convex optimization problems in statistics. The dual problem can have a complementary (and interpretable) structure, and can offer novel mathematical or statistical insights into the original problem.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Robert Tibshirani for his many interesting suggestions and great support. Nick Henderson and Michael Saunders provided valuable input with the computational considerations. We also thank thank Trevor Hastie for his help with the LARS algorithm. Finally, we thank the referees and especially the Editor for all of their help making this paper more readable.
[^1]: Dept. of Statistics, 390 Serra Mall, Stanford, CA 94305, U.S.A. email: [ryantibs@stanford.edu]{}. Supported by a National Science Foundation Vertical Integration of Graduate Research and Education fellowship.
[^2]: Dept. of Statistics, 390 Serra Mall, Stanford, CA 94305, U.S.A. email: [jtaylo@stanford.edu]{}. Partially supported by National Science Foundation grants DMS-0852227 and DMS-0906801.
|
---
abstract: 'We present an analysis of high precision radial velocity (RV) observations of stars hosting multi-planet systems with Jovian companions. We use dynamical stability constraints and quasi-global methods of optimization. As an illustration, we present new results derived for the RV data of the Sun-like dwarfs HD 155358 and $\tau^1$ Gruis.'
bibliography:
- 'biblio.bib'
---
Introduction
============
Extrasolar planetary systems have became a major challenge for contemporary astrophysics and dynamical astronomy. One of the most difficult problems in this field concerns the orbital stability of such systems, in particular when related to the observations and their interpretation. Usually, the investigations of long-term evolution are the domain of direct, numerical integrations. The stability of planetary systems is often understood in terms of the Lagrange definition implying that orbits remain well bounded over infinite time. Other definitions may be formulated as well, like the astronomical stability [@Lissauer1999] requiring that the system persists over a very long, Gyr time-scale or Hill stability [@Szebehely1984] that requires the constant ordering of the planets. In our studies, we prefer a more formal and stringent definition related to the fundamental Kolmogorov-Arnold-Moser Theorem (KAM), see [@Arnold1978]. Planetary $N$-body systems, involving a dominant mass of the parent star and significantly smaller planetary masses, are well modeled by close-to-integrable, Hamiltonian dynamical systems. According to the KAM Theorem, their evolution may be quasi-periodic (with a discrete number of fundamental frequencies, which are stable forever — and also stable with respect to the other notions of stability quoted above), periodic (or resonant; stable or unstable) or chaotic (with a continuous spectrum of frequencies, and unstable). In the last case, initially close phase trajectories diverge exponentially, i.e., their Maximum Lyapunov Characteristic Exponent (MLCE, denoted also by $\sigma$) is positive. This understanding of the global structure of the phase space is widely adopted, in particular with regard to the Solar system dynamics [e.g., @Wisdom1991; @Laskar1990; @Holman1996; @Malhotra1988; @Nesvorny1999; @Robutel2001; @Murray2001; @Michtchenko2001; @Lecar2001; @Morbidelli2002 and references therein.]. However, the distinction between regular and chaotic trajectories is a difficult task that, in practice, may be resolved only with numerical methods relying on efficient and accurate integrators of the equations of motion.
Stability indicators
====================
To detect chaotic motions in the phase space, many numerical tools are available. Concerning the dynamics of close to integrable Hamiltonian systems, these tools can be roughly divided onto two classes: spectral algorithms that resolve the fundamental frequencies and/or their diffusion rates [@Laskar1993; @Nesvorny1997; @Michtchenko2001], and methods based on the divergence rate of initially close phase trajectories, expressed in terms of the Lyapunov exponents [@Benettin1980; @Froeschle1984]. These fast indicators can be correlated with geometrical evolution of orbital osculating elements, like the maximal eccentricity ($\max e$), maximal amplitude of the critical angle of a resonance ($\max \theta$) or with event time $T_E$ (indicating a collision or an ejection of a body from the system). Unfortunately, no general relation between these indicators can be determined [@Lecar2001; @Michtchenko2001]. The event time $T_E$, relying on CPU-intensive long-term integrations of the equations of motion [@Lecar2001] can be considered as a direct measure of the astronomical stability.
In our work, among the the spectral tools, we often choose the method invented by @Michtchenko2001; its idea is very simple — to detect chaotic behavior one counts the number of frequencies in the FFT-spectrum of an appropriately chosen dynamical signal. We deal with conservative Hamiltonian systems; so in a regular case, the spectrum of fundamental frequencies is discrete and we obtain only a few dominant peaks in the FFT spectrum. Chaotic signals do not have well defined frequencies, and their FFT spectrum is very complex. The number of peaks in the spectrum above some noise level $p$ (typically, $p$ is set to a few percent of the dominant amplitude) tell us about the character (regular vs chaotic) of a given phase trajectory.
The basic tool to discover exponentially unstable bounded orbits, i.e. chaotic orbits, is the Maximum Lyapunov Characteristic Exponent (MLCE) $\sigma$. The direct computation of the MLCE is based on the analysis of the tangent vectors $\vec{\delta}$ which are solutions to the variational equations of the equations of planetary motions: $$\frac{\mbox{d}}{\mbox{d}t} \vec{x} = \vec{f}(\vec{x}),$$ where $\vec{x}$ denotes the state vector including coordinates and momenta, and $\vec{f}$ stands for the gravitational forces. For its solution $\vec{\phi}=\vec{\phi}(t)$ we define: $$\label{eq}
\frac{\mbox{d}}{\mbox{d}t} \vec{\delta} = \vec{A}(t) \vec{\delta},
\quad \vec{A}(t) := \frac{\partial \vec{f}}{\partial \vec{x}}[\vec{\phi}(t)],
\quad \delta = ||\vec{\delta}||.$$ Asymptotically, the MLCE value is given by [@Cincotta2000]: $$\label{sig}
\sigma = \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t
\frac{\dot{{\delta}}(s)}{{\delta}(s)} \mbox{d}s.$$ If $\sigma$ converges to some positive value, we conclude that the nominal orbit $\vec{\phi}$ and some initially close orbit diverge exponentially at the rate $\exp(\sigma t)$. Two practical difficulties arise when the direct definition (\[sig\]) is used: the convergence of $\sigma$ is often very slow, and it is difficult to tell how small the final value of $\sigma$ should be to consider it $\sigma=0$.
A large variety of methods has been proposed to overcome the problem of slowly convergent MLCE estimates. Recently, a new algorithm offering excellent convergence, called MEGNO (Mean Exponential Growth factor of Nearby Orbits), has been proposed by @Cincotta2000. The definition of MEGNO and its mean value is the following [@Cincotta2003]: $$\label{mfm2}
Y(t) = \frac{2}{t} \int_0^t
\frac{\dot{{\delta}}(s)}{{\delta}(s)} s \mbox{d}s , \qquad
\left<Y\right>(t) = \frac{1}{t} \int_0^t Y(s) \mbox{d}s.$$ It was shown that if $\vec{\phi}(t)$ is a regular solution with a linear divergence of nearby orbits then $\lim_{t\rightarrow\infty} \left<Y\right>(t) =
2$, and if $\vec{\phi}(t)$ is a chaotic solution then $\left<Y\right>(t) \sim
(\sigma/2) t$, as $t \rightarrow \infty$. Moreover, when $\left<Y\right>(t)$ tends towards a value different from 2, then it indicates that close trajectories diverge according to a certain power law. If $\vec{\phi}(t)$ is a periodic solution then $\left<Y\right>(t)$ tends to 0. The asymptotic behavior of $\left<Y\right>(t)$ is given by a uniform formula $ \left<Y\right>(t) \sim
a t + d, $ where $a \sim 0$ and $d\sim 2$ for a quasi-periodic solution, while $a \sim \sigma/2$ and $d\sim 0$ for an irregular and stochastic motion. Having $Y(t)$ we can indirectly estimate the MLCE on a finite time interval. The weight function $s$ in the definition of MEGNO reduces the contribution of the initial part of the tangent vector evolution, when the exponential divergence is too small to be observed relative to other linear and nonlinear effects [@Morbidelli2002]. Thus, fitting the straight line to the final part of $Y(t)$, we obtain good estimates of $\sigma$ from a relatively shorter piece of trajectory than in the direct MLCE evaluation.
Modeling the RV data – an overview
==================================
The radial velocity (RV) is still the most efficient technique for detecting extrasolar planets. To model the RV signal, the standard formulae by [@Smart1949] are commonly used. Each planet in the system contributes to the reflex motion of the star at time $t$ with: $$V_{\idm{r}}(t) = K [ \cos (\omega+\nu(t)) + e \cos \omega] + V_0,
\label{eq:eq1}$$ where $K$ is the semi-amplitude, $\omega$ is the argument of pericenter, $\nu(t)$ is the true anomaly (involving implicit dependence on the orbital period $P$ and time of periastron passage $T_{\idm{p}}$), $e$ is the eccentricity, $V_0$ is the velocity offset. We interpret the primary model parameters $(K,P,e,\omega,T_{\idm{p}})$ in terms of the Keplerian elements and minimal masses related to coordinates of Jacobi [@Lee2003] or Poincaré [@FerrazMello2006].
In our previous work, we tested and tried to optimize different tools helpful for exploring the multi-parameter space of ${{(\chi^2_\nu)^{1/2}}}$ for the model Eq. \[eq:eq1\] and its generalizations. In the case when ${{(\chi^2_\nu)^{1/2}}}$ may possess many local extrema, we found that good results can be obtained with hybrid optimization [@Gozdziewski2006b]. A single run of the hybrid code starts the quasi-global genetic algorithm [GA, @Charbonneau1995]. GA makes it easy to carry out a constrained optimization within prescribed parameter bounds or to add a penalty term to ${{(\chi^2_\nu)^{1/2}}}$. The best fits found with GAs are not very accurate in terms of ${{(\chi^2_\nu)^{1/2}}}$, so finally, a number of the best fit members of the “population” are refined using a relatively fast local method like the simplex of Melder and Nead [@Press1992]. The simplex is a matter of choice, so we could use other fast local methods. However, the code using non-gradient methods works with minimal requirements for user-supplied information. It is only required to define the model function \]the so called [*fitness function*]{}, usually equal to $1/{{(\chi^2_\nu)^{1/2}}}$\] — conveniently, this function is the same for the GAs and simplex — and to determine (even very roughly) the bounds of the parameters. The repeated runs provide an ensemble of the best-fits that helps us to detect local minima of ${{(\chi^2_\nu)^{1/2}}}$, even if they are distant in the parameter space. We can also obtain reliable approximation to the parameter errors [@Bevington2003] within the $1\sigma$, $2\sigma$ and $3\sigma$ confidence intervals of $\chi^2$ at selected 2-dim parameter planes.
While we prefer the GAs as the quasi-global optimization tool, other efficient and robust algorithms for identifying and characterizing multiple planetary orbits in precision RV data are known. In particular, the Bayesian Kepler periodogram [@Gregory2007] and Markov chain Monte Carlo (MCMC) technique [@Ford2005a] are proven to be robust tools for calculating the model marginal likelihood which is used to compare the probabilities of models with different numbers of planets and for investigating the uncertainty of parameters in the orbital solutions.
Due to the limited time-span of the observations, we often encounter a problem that the data only partially cover the longest orbital period. In this situation, it is possible either that ${{(\chi^2_\nu)^{1/2}}}$ does not have a well defined minimum, or that its shape is very “flat”, so the confidence levels may cover large ranges of the fit parameters. To illustrate [*the shape*]{} of ${{(\chi^2_\nu)^{1/2}}}$ in selected 2-dim parameter planes, we perform a systematic scanning of the space of initial conditions with the fast Levenberg-Marquardt (L-M) algorithm [@Press1992]. Usually, for representing such scans, we choose the semimajor-axis—eccentricity $(a,e)$ plane of the outermost planet. We fix $(a,e)$ and then search for the best fit, initiating the L-M algorithm with starting points selected randomly (but within reasonably wide parameter bounds). The L-M scheme ensures a rapid convergence. It is heavily CPU-consuming and may be effectively applied in low-dimensional problems. In reward it provides a clear picture of the parameter space.
In the case of resonant or mutually interacting planets, the problem is even more complex. How to interpret the RV measurements in that case often remains an open and difficult question. The $N$-planet configurations are parameterized by at least $5 N+1$ parameters, even assuming that the system is coplanar. The RV signal in terms of Eq. \[eq:eq1\] is degenerate — we have no information on the inclinations of orbits and the true masses of the companions. Moreover, we do not know [*a priori*]{} the number of planets in the system, so the resolved solutions are often not unique. It is also well known that the Keplerian (kinematic) model is usually not adequate to properly explain the RV variability. Instead, a self-consistent $N$-body Newtonian model should be applied [@Rivera2001; @Laughlin2001]. Basically, the effects of mutual interactions included in the Newtonian model could make it possible to determine or estimate the inclinations and masses provided that long enough time-series of precision data are available. Nevertheless, the measurements can be affected by many sources of error, like complex systematic instrumental effects, short time-series of the observations, irregular sampling due to observing conditions, and stellar noise. Little is known on the real statistical characteristics of the error distributions and the assumption that these are Gaussian distributions is not necessarily valid [@FerrazMello2005], see also [@Baluev2007].
All these factors, in particular the unspecified number of planets and undetermined or weakly constrained parameters, can (and often do) lead to best-fit solutions representing unrealistic, quickly disrupting configurations [@FerrazMello2005; @Lee2006; @Gozdziewski2006x]. But according to the Copernican principle, the detection of strongly unstable systems during two decades of RV observations is not likely, we would rather expect that the dynamical stability should be preserved over a significant part of the parent star life-time counted in Gyrs. Stability is therefore a natural requirement of a model solution consistent with observations. Many authors take it into account when analyzing the dynamics of the best-fit configurations with different constraints, e.g., to mention only a few examples in an endless list of references: through long-term integrations, requiring astronomical stability [@Lissauer2001; @Laughlin2002] or Lagrange and/or Hill stability [@Barnes2007], also through fast indicators like the diffusion of characteristic frequencies [@Correia2005], fast Lyapunov indicators [@Dvorak2005], maximal eccentricity [@Vogt2005; @Ford2005b], critical angles [@Ji2003; @Lee2006; @Beauge2006 see also paper by Beaugé et al. (2008) in this volume], or $T_E$ determined over short-time scale (the Systemic project initiated and led by Greg Laughlin, www.oklo.org). However, a common approach to search for stable solutions in a neighborhood of unstable best-fit configuration by trial and error does not necessarily provide stable fits that are simultaneously optimal, in terms of ${{(\chi^2_\nu)^{1/2}}}$ or an rms. What is even more important, the stability requirement imply non-continuous and complex structure of the space of initial conditions that depends on adopted definition of stability.
Hence, as a general way of modeling the observations we propose to eliminate unstable (for instance, strongly chaotic) solutions [*during*]{} the fitting procedure. The idea is very simple: we modify the hybrid algorithm by adding a suitable penalty term to the function determining the fit quality, e.g., ${{(\chi^2_\nu)^{1/2}}}$ or rms for unstable solutions. The penalty term must rely on some signature of the system stability. We found that MEGNO is particularly useful for that purpose thanks to its rapid convergence and great sensitivity to chaotic motions. We have shown that on many examples, this method (called Genetic Algorithm with MEGNO Penalty, GAMP) is very useful in modeling resonant or close-to-resonant planetary configurations when even small errors of orbital phases or other parameters may lead to quick self-destruction of the system. Unfortunately, the algorithm cannot give a definite answer when we want to resolve the ${{(\chi^2_\nu)^{1/2}}}$ shape in detail or find strictly stable solutions because, in practice, the penalty term can be calculated only a over relatively short period of time (due to CPU time requirements). This is particularly important for systems affected by long-term resonances, i.e., configurations with three and more planets, inclined orbits [@Gozdziewski2007b]. Hence, in an additional step, we need to examine the stability of individual best fits selected with GAMP over a period of time related to the time-scale of relevant unstable behaviors (resonances).
- a system with two planets
============================
To illustrate the algorithms and problems discussed above on a new planetary system, we consider the RV observations of by [@Cochran2007]. The data published in this paper consist of 71 observations and span $\sim 2100$ days. The single-planet Keplerian model does not fully explain the RV variability, so the discovery team studied a 2-planet Keplerian model of the RV. This yields three acceptable fits: a solution which is stable for 100 Myrs with $P_{\idm{c}}\sim 195$ days, $P_{\idm{c}}\sim 526$ days and minimizes ${{(\chi^2_\nu)^{1/2}}}$ as well as two unstable fits with $P_{\idm{c}} \sim 500$ and 1500 days, respectively. These fits have very large $e_{\idm{c}}$ leading to catastrophically unstable configurations [@Cochran2007].
We try to extend the analysis of the RV data by considering an $N$-body model of the observations and looking more closely at the phase-space structure of the putative 2-planet system with the help of the dynamical tools described above.
At first, we did two systematic scans of ${{(\chi^2_\nu)^{1/2}}}$ in the $(P_{\idm{c}},e_{\idm{c}})$-plane. The results are shown in Fig. \[fig:fig1\]. Three local minima of ${{(\chi^2_\nu)^{1/2}}}$ reported by [@Cochran2007] are evident. Two of them lie far over the collision line of orbits, defined in terms of semi-axes and eccentricities through $a_b (1+e_b) = a_c (1-e_c)$. This line marks the zone in which the mutual interactions of relatively massive companions can quickly destabilize the configuration. The solution with $P_{\idm{c}} \sim 300$ days yields ${{(\chi^2_\nu)^{1/2}}}$ similar to that one of the best fit with $P_{\idm{c}} \sim
500$ days.
Next, we refined the Keplerian fits with $N$-body model. The osculating parameters of the dominant solution (see Table 1, fit II) are slightly different from that one of the Kepler model, nevertheless it appears also stable. Its neighborhood is illustrated in dynamical maps shown in Fig. \[fig:fig2\]. Curiously, the best-fit configuration it located in the very edge of a stable zone between 5:2 and 3:1 mean motion resonances (MMRs) of two planets with masses $\sim 0.8$ m$_{\idm{J}}$ and $\sim 0.5$ m$_{\idm{J}}$, respectively. As we have observed in other cases, the $\max e$ and $\max
\theta$ indicators are in excellent correlation with the measure of formal stability (here, $\log SN$). We can also see a very complex border of the stable zone. The system would be located in dynamically active region of the phase space spanned by a few low-order MMRs. The proximity of the best fit configuration to the 5:2 MMR and moderate eccentricities indicate a dynamical similarity of the system to the Solar system.
To illustrate the hybrid optimization, we performed an independent search for 2-planet Keplerian fits assuming orbital periods in the range of \[100,3000\] days and eccentricities in the range of \[0,0.8\]. The ensemble of gathered fits is shown in the top-left panel in Fig.\[fig:fig3\]. To make the comparison with the results of systematic scanning more transparent, in this panel we also plot contour levels of $1\sigma, 2\sigma$ and $3\sigma$ confidence intervals seen in Fig. \[fig:fig1\]. Note that we plot only solutions in the range of $P_{\idm{c}}\in [300,700]$ days (compare with the right panel in Fig. \[fig:fig1\]). The hybrid algorithm also reveals the minima seen in Fig. \[fig:fig1\]. Besides, it detected one more minimum at $P_{\idm{c}}\sim
300$ days and moderate $e_{\idm{c}}\sim 0.1$, at the same depth. This justifies the efficiency and robustness of the hybrid code. The algorithm not only detects the best fits solutions but also helps to resolve to some extent the shape of ${{(\chi^2_\nu)^{1/2}}}$.
The best fits with $P_{\idm{c}}\sim 300$ days are both very unstable. It does not necessarily mean that in their neighborhood some stable solutions do not exist, so it is the case in which the application of GAMP can be helpful. Indeed, we can detect two clumps of stable fits (see the top-right panel of Fig. \[fig:fig3\]) in the regime of large $e_{\idm{c}}$, both lying far over the collision line. In one of these islands, we pick up a rigorously stable solution with ${{(\chi^2_\nu)^{1/2}}}\sim 1.14$ comparable to that one of the best fit II (Table 1) yielding only marginally worse rms $\sim 6.3$ m/s. The evolution of MEGNO and osculating elements in this quasi-periodic configuration is shown in Fig. \[fig:fig4\]. We notice extremely large variations of eccentricities up to $0.8$. The system would be involved in 5:3 MMR protecting companions from close encounter (see evolution of the critical argument of this resonance in the bottom-right panel in Fig. \[fig:fig4\]).
Yet the fit parameters are determined within some error ranges that should be interpreted with taking into account the structure of the phase space (see Fig. \[fig:fig2\]). To illustrate this problem we examined more closely the neighborhood of the best fit II. At this time, we performed two experiments. In the first search, we applied the hybrid code without stability constraints driven by the “usual” $N$-body model of the RV. The results are illustrated in the bottom-left panel in Fig. \[fig:fig3\]. The quality of fits within $1\sigma,2\sigma$ and $3\sigma$ confidence intervals of the best fit (marked with crossed circle; see Table 1, fit II) is color coded with blue, light-blue and gray, respectively. The best fits only marginally worse from the best one, are marked in red. Curiously, the plot reveals a subtle structure with three additional local minima of ${{(\chi^2_\nu)^{1/2}}}$, in relatively small range of $a_{\idm{c}} \in
[1.1,1.3]$ AU. Moreover, these minima are spread over wide range of $e_{\idm{c}}
\in [0,0.7]$. Simultaneously, this zone covers many low order resonances, between 5:2 and 3:1 MMR and the ${{(\chi^2_\nu)^{1/2}}}$ “valley” is crossed by the collision line. Close to this line, the stability could be preserved only if the planets are protected from close encounter through an MMR.
Now, we could examine the stability of every fit that we found but we choose a new search for stable solutions in a self-consistent manner with GAMP. The results are shown in the bottom-right panel in Fig. \[fig:fig3\]. In this panel, we overplot the stable solutions within $1\sigma,2\sigma$ and $3\sigma$ confidence interval of the best stable fit II over all solutions within $3\sigma$ level which are found in the previous search (i.e., without stability constraints). It now is evident that only a part of the ${{(\chi^2_\nu)^{1/2}}}$ valley can consist of dynamically stable solutions, nevertheless the acceptable fits are spread over significant range of $\Delta a_{\idm{c}} \sim 0.2$ AU. Apparently, this error is quite small but in fact it is large enough to cover a few low-order MMRs. We conclude that the current set of RV data cannot fully characterize the system state and new observations are required to constrain the elements of the outer planet.
Finally, both Keplerian and Newtonian 2-planet solutions lead to apparent excess of the residuals, in particular at the end parts of the RV curve. It may indicate that the 2-planet model does not fully explain the RV variability. In particular, the system may involve more than two planets. A heuristic argument supporting such a claim may be the proximity of the best fits to the collision line. We know similar cases, for instance $\mu$ Arae [@Gozdziewski2007a; @Pepe2007], or HD 37124 [@Vogt2005]. To check such hypothesis we looked first for 3-planet Keplerian solutions with the hybrid code. The best fit found yields ${{(\chi^2_\nu)^{1/2}}}\sim 0.87$ and significantly better rms $\sim 4.6$ m/s but is unstable. The GAMP search yields stable configurations with [*quasi-circular*]{} orbits of the outermost planets, yielding rms $\sim
5.5$ m/s. An example fit of this type, yielding ${{(\chi^2_\nu)^{1/2}}}\sim 1.07$ and an rms $\sim
5.6$ m/s, given in terms of osculating element at the epoch of the first observation in tuples of ($m$ \[m$_{\idm{J}}$\], $a$ \[AU\],$e$, $\omega$ \[deg\], ${\cal M}(t_0)$ \[deg\]) is the following (0.115, 0.383, 0.025, 281.3, 154.4), (0.770, 0.627, 0.040, 108.9, 173.7), (0.490, 1.187, 0.000, 359.1, 65.7), for planets $d,b,c$ respectively, the offset $V_0=10.21$ m/s. We note the small mass of the innermost planet. Its RV signal is at the level of noise, so additional observations would be required to confirm of withdrawn such a model.
Trojan planets in the $\tau^1$ Gruis system?
============================================
[@Laughlin2002a] predict that the reflex signal of a single planet in a quasi-circular orbit may be also interpreted by two Jovian Trojan planets, i.e., two objects sharing similar orbits (involved in 1:1 MMR). That possibility is intriguing because stable Trojan companions to the stars may be quite common. It can be indicated by a number of stable Trojan configurations in the Solar system. Some argue that they can be a frequent by-product of planet formation and and/or dynamical evolution [@Laughlin2002a]. However, the genesis of Trojan planets is not quite clear because on the contrary, there is some evidence that formation of such bodies could be difficult [@Beauge2007]. Still, many authors expect that Trojan planets can exist \[see, for instance, the work of Dvorak et al. in this volume and references therein, also [@Ford2006]\].
Recently, we found a similar kind of ambiguity of the RV models concerning 2:1 MMR configurations. At present, we know five extrasolar systems presumably involved in 2:1 MMR, i.e., Gliese 876 [@Marcy2001], HD 82943 [@Mayor2003], HD 128311 [@Vogt2005], [@Tinney2006], and $\mu$ Arae [@Jones2002; @Gozdziewski2007a; @Pepe2007]. However, the 2:1 MMR model of the radial velocity observations can also be non-unique. The periodogram of the 2:1 MMR RV signal is very similar to that one of the 1:1 MMR. Indeed, the RV variability of HD 128311 and HD 82943 can be explained by highly inclined systems in 1:1 MMR [@Gozdziewski2006x]. We also found that the RV of can be modeled with two highly inclined Jovian Trojans. The modeling of the 1:1 MMR is a challenging problem because Jupiter-like planets sharing eccentric orbits with similar semi-major axes interact heavily and the collisional configurations are generic. Hence, stability constraints are critical in the search for optimal and stable configurations. This seems to be one of the best applications of GAMP like algorithms.
Among a few cases we analyzed so far, the $\tau^1$ Gruis appears to be a particularly interesting example of the possible “Jupiter on circular orbit”–“two Trojans” ambiguity. A Jovian companion to the G0 dwarf $\tau^1$ Gruis in a wide and almost circular orbit has been announced in the work by [@Jones2002]. In our analysis, we use updated RV data comprising of 59 precision measurements [@Butler2006]. The best-fit single planet model to these data yields $P_{\idm{b}}\sim1300$ days, and $e_{\idm{b}}\sim0.1$. We re-analyse the data to look for possible Trojan planet solutions. Curiously, we quite easily found many [*stable, coplanar*]{} configurations involved in 1:1 MMR (Fig. \[fig:fig5\]) yielding similar or slightly better fit quality (rms $\in [5,6]$ m/s). The reflex signal of the 1:1 MMR (Fig. \[fig:fig6\]) can hardly be distinguished from that of a single-planet system. The osculating elements of the Trojans are given in Table 1 (fit III). In this case, both planets would move on quasi-circular orbits and these would be [*coplanar*]{}. The dynamical maps shown in Fig. \[fig:fig5\] (also accompanying other best-fits solutions with acceptable quality which we found in the search) illustrate the extreme variability of the 1:1 MMR islands. The map for the best-fit with elements in Table 1 is shown in the top-left panel of Fig. \[fig:fig5\]. Another peculiar solution is illustrated in the top-right map in Fig. \[fig:fig5\]. The initial eccentricities are moderate, and the system would be found in extremely large island of stable motions. It spans whole range of $e_{\idm{b}}$. The possibility of existence of such extended stable zones may strength the hypothesis of stable extrasolar Trojans.
![ *The RV data of $\tau^1$ Gruis and the synthetic signals. The red curve is for the the best, single-planet Keplerian fit yielding ${{(\chi^2_\nu)^{1/2}}}\sim1$ and an rms $\sim6.1$ m/s. The blue curve (darker one) is for coplanar, edge-on configuration (an rms $\sim5.9$ m/s) involved in 1b:1c MMR (see the top-left panel in Fig. \[fig:fig5\] for the dynamical map).* []{data-label="fig:fig6"}](fig6.png){width="80mm"}
![ *The RV data of $\tau^1$ Gruis and the synthetic signals. The red curve is for the the best, single-planet Keplerian fit yielding ${{(\chi^2_\nu)^{1/2}}}\sim1$ and an rms $\sim6.1$ m/s. The blue curve (darker one) is for coplanar, edge-on configuration (an rms $\sim5.9$ m/s) involved in 1b:1c MMR (see the top-left panel in Fig. \[fig:fig5\] for the dynamical map).* []{data-label="fig:fig6"}](fig6.eps){width="80mm"}
The ambiguity of the RV fits implies interesting issues concerning the models of creation and stability of Earth-like planets interior to the orbits of the putative Jovian Trojans. In the $\tau^1$ Gruis, the space interior to the Jovian planet is “empty” as no smaller planets have been yet detected. So we can try to predict in which regions of the habitable zone ($\sim 1$ AU) smaller planet could survive. For this purpose we computed dynamical maps for putative Earth-like masses with initial conditions varied in the $(a_0,e_0)$ plane, and initial orbital angles set to $0^{\circ}$. We considered two dynamical environments: the one with the best-fit Jovian companion in close to circular orbit and the second one with Trojans in quasi-circular orbits (their elements are given in Table 1, fit III). The results are shown in Fig.\[fig:fig7\]. For the first configuration, we detect an extended zone of stable motions. Additional experiments regarding creation of Earth-like planets through coagulation of Mars and Moon-size protopolanets (see Raymond, 2008 in this volume) performed with the Mercury code [@Chambers1999] assures us that such planets emerge easily in that zone. In the case of a configuration with Trojans, the stable zone shrinks significantly. Moreover, the creation of Earth-like planets is much more difficult. We found that they could form only in the zones of relatively stable motions, up to $\sim 0.9$ AU and in the “gap” between the 4:1 MMR and the border of global instability. Yet in that case, the simulations are very difficult to carry out due to frequent close encounters between planetesimals and the Jovian planets.
\[tab:tab1\]
------------------------------ ----------------------- -------------------- ----------------------- -------------------- ----------------------- --------------------
Parameter planet [**b**]{} planet [**c**]{} planet [**b**]{} planet [**c**]{} planet [**b**]{} planet [**c**]{}
$m \sin i$ \[m$_{\idm{J}}$\] 0.827 0.490 0.863 0.497 0.401 0.923
$a$ \[AU\] 0.623 0.875 0.628 1.212 2.471 2.565
$e$ 0.121 0.743 0.128 0.198 0.027 0.053
$\omega$ \[deg\] 131.6 86.11 161.9 272.0 99.9 163.8
${\cal M}(t_0)$ \[deg\] 106.32 313.7 130.3 198.8 3.5 3.6
${{(\chi^2_\nu)^{1/2}}}$
$\sigma_{\idm{j}}$ \[m/s\]
rms \[m s$^{-1}$\]
$V_0$ \[m s$^{-1}$\]
$M_{\star}$ \[$M_{\circ}$\]
------------------------------ ----------------------- -------------------- ----------------------- -------------------- ----------------------- --------------------
: *The best-fit astro-centric, osculating Keplerian elements of stable, coplanar and edge-on planetary configurations at the epoch of the respective first observation. Original errors of the data are rescaled by adding the “jitter” $\sigma_{\idm{j}}$ in quadrature.*
Conclusions
===========
In this work we consider some problems related to modeling observations of stars hosting multi-planet systems. It is well known that the phase space of such system has a non-continuous and complex structure with respect to any stability criterion. Hence, when searching for initial conditions, one has to take into account the dynamical character of putative planetary configurations. Due to narrow observational windows, significant measurement errors, stellar jitter and other uncertainties, the formal best-fits may appear very unstable. Searching for stable solutions in their neighborhood of the phase space by trial and error, we should not expect that the results could be statistically optimal. Thus, an intuitively natural approach is to eliminate unstable configurations during the fitting process, through penalizing unstable solutions with a suitably large value of the ${{(\chi^2_\nu)^{1/2}}}$ function, or of another measure of the fit quality. In that way, the stability plays a role of an additional, implicit observable. That method is suitable for multi-body systems with Jovian planets presumably involved in low-order mean motion resonances (MMRs). In particular, we considered two new examples in which the model of the RV may be non-unique. The RV of HD 155358 by [@Cochran2007] permit a few local minima of ${{(\chi^2_\nu)^{1/2}}}$ related to different orbital configurations. We also found an example illustrating the ambiguity of Keplerian, close to circular single-planet solutions. The RV data of $\tau^1$ Gruis could be equally well modeled with coplanar configurations of Jovian planets involved in 1:1 MMRs.
Moreover, our fitting method, used mainly for RV data, is quite general and may be applied to other types of observations as well. As the stability criterion, one can use the maximal Lyapunov exponent, the most stringent and formal characteristic of stable/unstable motion. Other suitable indicators like the maximal eccentricity, the spectral number, or the diffusion rate of fundamental frequencies may also be applied. These fast indicators help us to search for and find long-term stable solutions, but also make it possible to efficiently explore and to visualize the sophisticated and varying structure of the phase space. We can see the planetary system in its dynamical environment.
Many multi-planet systems are found on the edge of long-term dynamical stability. It is not clear yet whether this is a general property of multi-planet systems, the outcome of poor statistics or just the consequence of a bad choice of the RV model. Large eccentricities in multi-planet systems may “hide” other, unknown planets. Yet in that case, the dynamical modeling of the RV with stability constraints provides valuable information on the dynamical structure of the putative planetary configurations. Finding the best fits on the very edge of stable zones may provide good hints and motivation to look for alternate models of the RV.
Acknowledgments {#acknowledgments .unnumbered}
===============
K. G. thanks the organizers of the IAU 249 symposium for the invitation and great hospitality. We are grateful to the anonymous referee for comments that improved the manuscript. Many thanks to Boud Roukema for corrections of the text. This work is supported by the Polish Ministry of Science, Grant 1P03D 021 29.
|
---
abstract: 'We analyze $L^2$-normalized solutions of nonlinear Schrödinger systems of Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable standing waves. Such waves correspond to global minimizers of the associated energy in the $L^2$-subcritical and critical cases, and to local ones in the $L^2$-supercritical case. Notably, our study includes also the Sobolev-critical case.'
author:
- 'Benedetta Noris, Hugo Tavares, Gianmaria Verzini'
title: Normalized solutions for Nonlinear Schrödinger systems on bounded domains
---
[**AMS-Subject Classification**]{}. 35Q55, 35B33, 35B35, 35J50.\
[**Keywords**]{} Gross-Pitaevskii systems, constrained critical points, solitary waves, orbital stability, critical exponents.
Introduction
============
In this paper, we carry on the study of normalized solutions for Nonlinear Schrödinger (NLS) equations and systems, started in [@ntvAnPDE; @ntvDCDS].
Let $\Omega\subset{{\mathbb{R}}}^N$, $N\geq 1$, be a bounded smooth domain, $\mu_1,\mu_2>0$ and $\beta\in{{\mathbb{R}}}$. We consider the following system of coupled Gross-Pitaevskii equations $$\label{eq:system_schro}
\begin{cases}
{{\mathrm{i}}}\partial_t\Psi_1 + \Delta\Psi_1 + \Psi_1( \mu_1 |\Psi_1|^{p-1} +\beta |\Psi_1|^{(p-3)/2}|\Psi_2|^{(p+1)/2} )=0\\
{{\mathrm{i}}}\partial_t\Psi_2 + \Delta\Psi_2 + \Psi_2( \mu_2 |\Psi_2|^{p-1} +\beta |\Psi_2|^{(p-3)/2}|\Psi_1|^{(p+1)/2} )=0
\end{cases}$$ with $\Psi_i:{{\mathbb{R}}}^+\times\Omega\to{{\mathbb{C}}}$ and, for every $t>0$, $\Psi_i(t,\cdot)\in H^1_0(\Omega;{{\mathbb{C}}})$ ($i=1,2$). Throughout the paper we will distinguish several cases in the range $$\begin{cases}
p>1 & N=1,2,\\
1<p\le 2^*-1 & N\ge 3,
\end{cases}$$ where $2^*=2N/(N-2)$ denotes the Sobolev critical exponent.
NLS systems with power-type nonlinearities appear in several different physical models from quantum mechanics, in particular when $p = 3$ or $p = 5$. Such models include Bose–Einstein condensation in multiple hyperfine spin states [@PhysRevLett.81.5718; @MR2090357] and the propagation of mutually incoherent waves packets in nonlinear optics [@agrawal2000]. Moreover, both the cases $\Omega = {{\mathbb{R}}}^N$ and $\Omega$ bounded are of interest [@FibichMerle2001; @Fukuizumi2012], the latter one appearing also as a limiting case of the system on ${{\mathbb{R}}}^N$ with (confining) trapping potential.
System preserves, at least formally, both the masses $${{\mathcal{Q}}}(\Psi_i)=\int_\Omega |\Psi_i|^2 \qquad i=1,2,$$ and the energy $${{\mathcal{E}}}(\Psi_1,\Psi_2) := \frac12\int_{\Omega}|\nabla \Psi_1|^2+|\nabla \Psi_2|^2 -
\frac{1}{p+1}\int_\Omega \mu_1 |\Psi_1|^{p+1} + 2\beta |\Psi_1|^{(p+1)/2} |\Psi_2|^{(p+1)/2} + \mu_2 |\Psi_2|^{p+1}.$$ We look for standing wave solutions $(\Psi_1(t,x),\Psi_2(t,x))=(e^{{{\mathrm{i}}}\omega_1 t}u_1(x),e^{{{\mathrm{i}}}\omega_2 t}u_2(x))$ of such that $(u_1,u_2) \in H^1_0(\Omega;{{\mathbb{R}}}^2)$ and $$\label{eq:mass_constraint}
{{\mathcal{Q}}}(u_i)=\rho_i, \quad i=1,2,$$ for some $\rho_1,\rho_2\ge0$ prescribed a priori. Then, $(u_1,u_2)$ is a normalized solution of an elliptic system; namely, there exist $(\omega_1,\omega_2)\in {{\mathbb{R}}}^2$ such that $$\label{eq:system_elliptic}
\begin{cases}
-\Delta u_1+ \omega_1 u_1=\mu_1 u_1|u_1|^{p-1}+\beta u_1|u_1|^{(p-3)/2} |u_2|^{(p+1)/2}\\
-\Delta u_2+ \omega_2 u_2=\mu_2 u_2|u_2|^{p-1}+\beta u_2 |u_2|^{(p-3)/2} |u_1|^{(p+1)/2}\\
\int_\Omega u_i^2=\rho_i, \quad i=1,2,\\
(u_1,u_2) \in H^1_0(\Omega;{{\mathbb{R}}}^2).
\end{cases}$$ Solutions of can be seen as critical points of ${{\mathcal{E}}}$, constrained to the Hilbert manifold $$\label{eq:defM}
{{\mathcal{M}}}={{\mathcal{M}}}_{\rho_1,\rho_2} := \left\{(u_1,u_2)\in H^1_0(\Omega;{{\mathbb{R}}}^2):\int_\Omega u_i^2=\rho_i, \ i=1,2 \right\},$$ in which case the unknowns $\omega_i$ play the role of Lagrange multipliers. Our main aim is to provide conditions on $p$ and $(\rho_1,\rho_2)$ (and also on $\mu_1,\mu_2,\beta$) so that $\left.{{\mathcal{E}}}\right|_{{{\mathcal{M}}}}$ admits minima, either global or local. We call such solutions *least energy solutions*, or *ground states*. Secondly, we consider the stability properties of such ground states, with respect to the evolution system .
An alternative, non equivalent point of view —which we do not treat here— is that of considering the parameters $\omega_i$ in as fixed, without any normalizing condition on the functions $u_i$. This leads to an alternative definition of ground states, that of *least action solutions*: for a detailed discussion of this topic we refer the interested reader to the introduction of [@ntvAnPDE]. Starting from [@MR2135447; @MR2358296; @MR2263573; @MR2252973; @MR2302730; @Sirakov2007; @MR1939088; @MR2629888], the literature dealing with this approach is vast and we do not even make an attempt to summarize it here. As a matter of fact, the results for non-normalized solutions cannot be directly extended to the normalized ones: among the other reasons, because in the latter case the ambient space is the Hilbert manifold ${{\mathcal{M}}}$ (rather than a vector space).
Going back to normalized solutions, the simplest case one can face is that of a single NLS equation on ${{\mathbb{R}}}^N$, with a pure power nonlinearity. In such case, the problem can be completely solved by simple scaling arguments. This structure breaks down whenever one considers a system, as well as non-homogeneous nonlinearities, bounded domains or confining potentials. Apart when global minimization can be applied, see [@RoseWeinstein88], as far as we know the first result in the literature is due to Jeanjean [@MR1430506], for the superlinear, Sobolev-subcritical NLS single equation on ${{\mathbb{R}}}^N$ with a non-homogeneous nonlinearity. In recent years, other papers appeared, dealing with the NLS equation or system, always in the Sobolev subcritical regime, either on ${{\mathbb{R}}}^N$ [@MR3539467; @MR3534090; @MR3639521; @MR3638314; @MR3777573; @0951-7715-31-5-2319; @2017arXiv170302832B] or on a bounded domain [@MR2928850; @ntvAnPDE; @ntvDCDS; @MR3660463; @MR3689156; @BonheureJeanjeanNoris]. These two settings are rather different in nature: each one requires a specific approach, and the results are in general not comparable. A key difference is that ${{\mathbb{R}}}^N$ is invariant under translations and dilations, which has pros and cons: on the one hand, translations are responsible for a loss of compactness; on the other hand, in the Sobolev subcritical case, dilations can be used to produce variations and eventually construct natural constraints such as the so-called Pohozaev manifold. This tool is not available when working in bounded domains, and also the gain of compactness is lost when we face the Sobolev critical case.
However, a common key tool in the study of normalized solution is the Gagliardo-Nirenberg inequality (see below), which can be used to estimate the non-quadratic part in ${{\mathcal{E}}}$ in terms of the quadratic one. As a consequence, the exponent $p$ in can be classified according to the following four cases:
- superlinear, $L^2$–subcritical: $1<p<1+4/N$;
- $L^2$–critical: $p=1+{4/N}$;
- $L^2$-supercritical, Sobolev–subcritical: $1+4/N<p<1 + 4/(N-2)^+=2^*-1$;
- Sobolev–critical: $p=2^*-1$, for $N\geq 3$.
In the first three cases, the study of the single equation $$\label{eq:singleeq}
\begin{cases}
-\Delta u_1 + \omega_1 u=\mu_1 u_1|u_1|^{p-1}\\
\int_\Omega u_1^2=\rho_1,\quad u_1\in H^1_0(\Omega),
\end{cases}$$ has been carried on in [@ntvAnPDE; @MR3689156]. Notice that is a particular case of , when $\rho_2=0$, with associated energy $u_1\mapsto {{\mathcal{E}}}(u_1,0)$. Summarizing, it is known that
- (H1) implies that has a solution, which is a global minimizer, for $\rho_1\ge0$;
- (H2) implies that has a solution, which is a global minimizer, for $0\le\rho_1<\rho_*(\Omega,N,p,\mu_1)<+\infty$;
- (H3) implies that has a solution, which is a local minimizer, for $0\le\rho_1<\rho_*(\Omega,N,p,\mu_1)<+\infty$, and a second one of mountain pass type.
Moreover, all the minimizers above are associated to orbitally stable solitary waves of the corresponding evolutive equation.
Up to our knowledge, the only paper dealing with the NLS system (with both $\rho_i>0$) is [@ntvDCDS]. Among other things, in that paper we deal with the $L^2$-supercritical, Sobolev–subcritical case (H3), obtaining the existence of orbitally stable solitary waves, in case both $\rho_1$, $\rho_2$ are sufficiently small and $\rho_1/\rho_2$ is uniformly bounded away from $0$ and $+\infty$. This result is perturbative in nature, the existence following by a multi-parametric extension of a Ambrosetti-Prodi-type reduction [@AmbrosettiProdiBook] and the stability by the Grillakis-Shatah-Strauss stability theory [@GrillakisShatahStrauss]. The corresponding solutions are close to suitably normalized first eigenfunctions of the Dirichlet laplacian.
The aim of the present paper is twofold: on the one hand, in the cases (H1)-(H2)-(H3), we extend to systems the above described results obtained in [@ntvAnPDE; @MR3689156] for the single equation; on the other hand, we treat for the first time the Sobolev critical case (H4), obtaining results which are new also in the case of a single equation. Now we describe in details our results.
In what follows, we take $$\Omega\subset {{\mathbb{R}}}^N \text{ a Lipschitz bounded domain ($N\geq 1$),}\qquad \mu_1,\mu_2>0,\qquad \beta\in {{\mathbb{R}}},$$ and denote by $C_N$ the best constant appearing in the Gagliardo-Nirenberg inequality in the $L^2$-critical case (see ahead) while $S_N$ is the best constant appearing in the Sobolev inequality (see ).
To start with, as we already mentioned, both (H1) and (H2) can be treated in a quite standard way by using the Gagliardo-Nirenberg inequality. Even though this result is somewhat expected, we provide it here since we could not find a precise reference to cite.
\[prop:subcritical\] Suppose that one of the following cases occurs
- $1<p<1+{4/N}$ and $\rho_1,\rho_2\ge0$;
- $p=1+{4/N}$ and $\rho_1,\rho_2\ge0$ are such that (see Fig. \[fig:N=2\]) $$\label{N=2condition2}
\max\left\{\mu_1 \rho_1^{2/N},\ \mu_2 \rho_2^{2/N},\ \mu_1\rho_1^{2/N}+\mu_2 \rho_2^{2/N} +\frac{NC_N}{N+2} \left((\beta^+)^2-\mu_1 \mu_2\right)(\rho_1\rho_2)^{2/N}\right\}<\frac{N+2}{NC_{N}}$$
Then:
- the level $\inf_{{{\mathcal{M}}}}{{\mathcal{E}}}$ is achieved at some $(u_1,u_2)\in {{\mathcal{M}}}$, which is a non-negative solution of for some $(\omega_1,\omega_2)\in {{\mathbb{R}}}^2$ ($u_i>0$ when $\rho_i>0$);
- the set of ground states $$G=\left\{ (u_1,u_2) \in H^1_0(\Omega;{{\mathbb{C}}}^2): \ (|u_1|,|u_2|)\in{{\mathcal{M}}},\ {{\mathcal{E}}}(u_1,u_2)=
\inf_{{{\mathcal{M}}}}{{\mathcal{E}}}\right\}$$ is conditionally orbitally stable.
We recall the definition of orbital stability in Section \[sec:stab\]. Actually, notice that we prove *conditional* orbital stability, where the condition is that the solution of system , with Cauchy datum $(\psi_{1},\psi_{2})\in H^1_0(\Omega;{{\mathbb{C}}}^2)$, exists locally in time for a time interval which is uniform in $\|(\psi_{1},\psi_{2})\|_{H^1_0}$, and that ${{\mathcal{Q}}}$ and ${{\mathcal{E}}}$ are preserved along the solutions. This holds true under further assumptions on $\mu_i$, $\beta$ and $\Omega$, see also [@ntvDCDS] and references therein; however, being the field so vast, even a rough summary of well-posedness for Schrödinger systems on bounded domains is far beyond the scopes of this paper.
\[rem:uniform\_beta1\] Observe that, for $\beta\leq 0$, reduces to $$\max\left\{\mu_1 \rho_1^{2/N},\ \mu_2 \rho_2^{2/N}\right\}<\frac{N+2}{NC_{N}}.$$ which is independent from $\beta$.
(1.4,1.4) node\[below left, draw=black, fill=white\] [$\beta \le0$]{}; (0,0)–(0,1)–(1,1)–(1,0)–cycle;
(1.4,1.4) node\[below left, draw=black, fill=white\] [$0<\beta <\sqrt{\mu_1\mu_2}$]{}; ; (0,1)–(0,0)–(1,0); (0,1)–(1,1)–(1,0);
\
(1.4,1.4) node\[below left, draw=black, fill=white\] [$\beta =\sqrt{\mu_1\mu_2}$]{}; (0,0)–(0,1)–(1,0)–cycle; (0,1)–(1,1)–(1,0);
(1.4,1.4) node\[below left, draw=black, fill=white\] [$\beta >\sqrt{\mu_1\mu_2}$]{}; ; (0,1)–(0,0)–(1,0); (0,1)–(1,1)–(1,0);
Now we turn to the case in which either (H3) or (H4) hold true, i.e. when $1+4/N < p \le 2^*-1$ (no upper bound if $N=1,2$). Contrarily to the previous cases, in this one it is known that $\left.{{\mathcal{E}}}\right|_{{{\mathcal{M}}}}$ is not bounded below, see Lemma \[lemma:geometry\_mp\] ahead. Nonetheless we will show that, even though no global minima can exist, local ones do, in case $(\rho_1,\rho_2)$ belongs to some explicit set. To detect the existence of such minima we need to introduce some auxiliary problem and further notations. Let, for $\alpha\geq \lambda_1(\Omega)$ (the first Dirichlet eigenvalue of $-\Delta$ in $\Omega$), $$\label{eq:Balpha}
\begin{split}
{{\mathcal{B}}}_\alpha &:= \left\{(u_1,u_2)\in{{\mathcal{M}}}:\int_\Omega |\nabla u_1|^2+|\nabla u_2|^2 \leq
(\rho_1+\rho_2)\alpha \right\},\\
{{\mathcal{U}}}_\alpha &:= \left\{(u_1,u_2)\in{{\mathcal{M}}}:\int_\Omega |\nabla u_1|^2+
|\nabla u_2|^2=
(\rho_1+\rho_2)\alpha \right\}
\end{split}$$ Notice that ${{\mathcal{B}}}_\alpha$ is not empty, since it contains at least a pair of suitably normalized first eigenfunctions, and that ${{\mathcal{U}}}_\alpha$ is the topological boundary of $\mathcal{B}_\alpha$ in $\mathcal{M}$. Moreover, let us define $$\label{eq:calpha}
c_\alpha := \inf_{{{\mathcal{B}}}_\alpha}{{\mathcal{E}}},\qquad \hat c_\alpha := \inf_{{{\mathcal{U}}}_\alpha}{{\mathcal{E}}}.$$ Being ${{\mathcal{B}}}_\alpha$ weakly closed in ${{\mathcal{M}}}$, in the Sobolev subcritical case the level $c_\alpha$ is achieved for any $\alpha\geq \lambda_1(\Omega)$, possibly on ${{\mathcal{U}}}_\alpha\subset{{\mathcal{B}}}_\alpha$. Therefore, in order to find a solution of , it is sufficient to find $\alpha$ such that $c_\alpha < \hat c_\alpha$ (and this will be our strategy).
On the contrary, in the Sobolev critical case, it is also an issue to prove that $c_\alpha$ is achieved: indeed, since $H^1_0(\Omega)$ is not compactly embedded in $L^{2^*}(\Omega)$, ${{\mathcal{E}}}|_{{{\mathcal{M}}}}$ is no longer weakly lower semicontinuous. To overcome this difficulty, in the spirit of the celebrated paper by Brezis and Nirenberg [@MR709644], we are able to recover the compactness of the minimizing sequences associated to $c_\alpha$ by imposing a bound on the masses $\rho_1,\rho_2$ and on $\alpha$. More precisely, we have the following key result.
\[prop:compact\_intro\] Consider $N\ge3$ and $p=2^*-1$. Suppose that $\rho_1,\rho_2\geq 0$ and $\alpha\geq \lambda_1(\Omega)$ are such that $$\label{eq:compact_intro}
(\rho_1+\rho_2)(\alpha - \lambda_1(\Omega)) \le \frac{1}{{\Lambda}^{(N-2)/2}},$$ where $$\label{eq:def_Lambda}
\Lambda:=\frac{2S_{N}}{2^*} \max_{\{x^2+y^2 =1\}}
\left(\mu_1 |x|^{2^*} + \mu_2 |y|^{2^*}
+2\beta^+ |xy|^{2^*/2} \right).$$ Then any minimizing sequence associated to $c_\alpha$ is relatively compact in ${{\mathcal{B}}}_\alpha$. In particular, $c_\alpha$ is achieved.
Based on the previous proposition, we introduce the following set of admissible masses $$\label{eq:defA}
A := \left\{(\rho_1,\rho_2)\in [0,\infty)^2 : \begin{array}{l}
\text{$c_\alpha<\hat c_\alpha$ for some $\alpha\ge \lambda_1(\Omega)$,}\smallskip\\
\text{with $\alpha$ satisfying \eqref{eq:compact_intro} if $p=2^*-1$}
\end{array}
\right\}\cup\left\{(0,0)\right\}.$$ Notice that, as a matter of fact, $A$ depends on $\Omega$, $N$, $p$, $\mu_1$, $\mu_2$ and $\beta$. Moreover, if $(\rho_1,\rho_2)\in A$, then we can choose the local minimizer $(u_1,u_2)\in {{\mathcal{M}}}$ to be a non-negative solution of for some $(\omega_1,\omega_2)\in {{\mathbb{R}}}^2$. Finally,we introduce the exponents $a$ and $r$ as $$\label{eq:newexponent}
a = a(N,p):= \frac{N(p-1)}{4} ,\qquad r = r(N,p) := \frac{p+1}{4} - \frac{N(p-1)}{8}.$$ Notice that these two constants appear naturally in this context because (up to a suitable multiple) they enter in the Gagliardo-Nirenberg inequality. Observe also that $$\label{eq:sign_a-1}
0<a<1 \text{ if }1<p<1+4/N;
\quad a=1 \text{ if }p=1+4/N;
\quad a>1 \text{ if }p=1+4/N.$$
\[prop:supercritical\] Let $1+4/N < p \le 2^*-1$. If $A$ is defined as in , then $$\text{$A$ is star-shaped with respect to $(0,0)$.}$$ Moreover, there exists a positive constant $R=R(\Omega,N,p)$ such that if $\rho_1,\rho_2\ge0$ are such that $$\label{eq:assnice}
\left[\max\{ \mu_1 \rho_1^{2r},\mu_2 \rho_2^{2r}\} +
\beta^+ \rho_1^r\rho_2^{r} \right]\cdot (\rho_1+\rho_2)^{a-1} \le
R(\Omega,N,p),$$ then $(\rho_1,\rho_2)\in A$. Here $a$ and $r$ are defined as in and $R$ is explicit (see ahead).
When $N\ge3$ and $p=2^*-1$, explicit calculations show that $a=N/(N-2)$, $r=0$ and rewrites as $$\rho_1+\rho_2 \le \left[\frac{R(\Omega,N,2^*-1)}{\max\{ \mu_1 ,\mu_2 \} +
\beta^+}\right]^{(N-2)/2}.$$
\[rem:uniform\_beta2\] When $\beta\leq 0$, condition is independent from $\beta$ and reduces to: $$\left[\max\{ \mu_1 \rho_1^{2r},\mu_2 \rho_2^{2r}\} \right]\cdot (\rho_1+\rho_2)^{a-1} \le R(\Omega,N,p),$$
As we mentioned before, when $p>1+4/N$ the functional ${{\mathcal{E}}}$ is unbounded from below on ${{\mathcal{M}}}$. We deduce that, under the same assumptions of Theorem \[prop:supercritical\], ${{\mathcal{E}}}$ has a mountain pass geometry on ${{\mathcal{M}}}$ (see for instance [@Struwebook Thm. 4.2, Ch. II]). In a standard way, this implies that
> if $1+\dfrac{4}{N} < p < 2^*-1$ and $(\rho_1,\rho_2)$ satisfies , then $\left.{{\mathcal{E}}}\right|_{{{\mathcal{M}}}}$ has two critical points: one local minimum and one mountain pass.
We cannot obtain the same result for $p=2^*-1$, since our compactness result in Proposition \[prop:compact\_intro\] holds only for minimizing sequences.
It is natural to expect that the set $A$ is bounded in ${{\mathbb{R}}}^2_+$. Actually, we know from [@ntvAnPDE] that this is the case, for the single equation, at least in the Sobolev subcritical case. The proof of this fact should follow by a careful blow-up analysis based on suitable pointwise a priori controls, along the lines of [@ntvAnPDE Section 4], and will be the object of further investigation.
Since the solutions we found in the $L^2$-supercritical cases are local minima of the energy, it is natural to expect that they correspond to orbitally stable solitary waves. The proof of this fact requires some modification of the standard arguments used for global minimizers. Notably, the lack of compactness of the embedding $H^1_0 \hookrightarrow L^{2^*}$ is an issue here, too. We have the following result.
\[thm:stab\] Let $1+4/N < p \le 2^*-1$ and $(\rho_1,\rho_2)\in A$. Let $\bar\alpha\ge \lambda_1(\Omega)$ be such that $$c_{\bar\alpha}<\hat c_{\bar\alpha},
\qquad
\text{and $\bar\alpha$ satisfies \eqref{eq:compact_intro} if $p=2^*-1$}.$$ Then the set of *local* ground states $$\label{eq:GroundStates}
G_{\bar\alpha}:=\left\{ (u_1,u_2) \in H^1_0(\Omega;{{\mathbb{C}}}^2): \ (|u_1|,|u_2|)\in{{\mathcal{B}}}_{\bar \alpha},\ {{\mathcal{E}}}(u_1,u_2)=
c_{\bar\alpha} \right\},$$ is (conditionally) orbitally stable.
As we noticed, in the Sobolev critical case our results are new also for the single equation. In particular, choosing $\rho_2=0$, $\rho_1=\rho$, $\mu_1=\mu$, we have the following direct consequence.
Let $\mu>0$. If $$0<\rho\le \left[\frac{R(\Omega,N,2^*-1)}{\mu}\right]^{(N-2)/2}$$ then the problem $$\begin{cases}
-\Delta u + \omega u=\mu u|u|^{2^*-2}\\
\int_\Omega u^2=\rho,\quad u\in H^1_0(\Omega)
\end{cases}$$ admits a positive solution $u$, for a suitable $\omega\in (-\lambda_1(\Omega),0)$, which is a local minimizer of the associated energy. Moreover, the corresponding set of local ground states is (conditionally) orbitally stable.
To conclude, we remark that all the assumptions in our results involve $\beta^+$, the positive part of $\beta$. As a consequence, all our estimates are uniform in $\beta<0$. To summarize, recalling Theorems \[prop:subcritical\] and \[prop:supercritical\] (see also and Remarks \[rem:uniform\_beta1\] and \[rem:uniform\_beta2\]), we prove existence of solutions whenever $\beta<0$ and $(\rho_1,\rho_2)$ satisfy $$\label{eq:uniform_beta}
\begin{cases}
\rho_1,\rho_2>0 & \quad \text{ if }1<p<1+4/N, \\ \smallbreak
0<\mu_1\rho_1^\frac{2}{N},\mu_2\rho_2^\frac{2}{N}<\frac{N+2}{NC_{N}} &\quad \text{ if } p=1+\frac{4}{N}\\ \smallbreak
\max\{ \mu_1 \rho_1^{2r},\mu_2 \rho_2^{2r}\} \cdot (\rho_1+\rho_2)^{a-1} \le R(\Omega,N,p) &\quad \text{ if }1+\frac{4}{N}<p\leq 2^*-1.
\end{cases}$$ This allows to exploit results in [@MR2599456; @SZ15; @STTZ] in order to perform a segregation analysis as $\beta\to-\infty$.
\[thm\_beta-infty\] Let $\Omega\subset{{\mathbb{R}}}^N$ be a smooth bounded domain, $\mu_1,\mu_2>0$, $\beta< 0$ and $(\rho_1,\rho_2)$ be such that holds. Let also $(u_{1,\beta},u_{2,\beta})$ be a corresponding ground state of , with multipliers $(\omega_{1,\beta},\omega_{2,\beta})$ and such that $u_{1,\beta},u_{2,\beta}>0$ in $\Omega$. Then $\{(u_{1,\beta},u_{2,\beta})\}_{\beta<0}$ is uniformly bounded in $C^{0,\alpha}(\overline \Omega)$ and, up to subsequences, $(u_{1,\beta},u_{2,\beta})\to(w^+,w^-)$ as $\beta\to-\infty$, in $C^{0,\alpha}(\overline{\Omega})\cap H^1_0(\Omega)$, where $w\in C^{0,1}(\overline \Omega)$ solves $$\begin{cases}
-\Delta w + \omega_1 w^+-\omega_2 w^- = \mu_1 (w^+)^p-\mu_2 (w^-)^p & \text{in } \Omega,\\
\int_\Omega (w^+)^2=\rho_1,\ \int_\Omega (w^-)^2=\rho_2,\quad w\in H^1_0(\Omega),
\end{cases}$$ for $\omega_i:=\lim_{\beta\to -\infty}\omega_{i,\beta}$.
The paper is structured as follows. In the next subsection we make some preliminary remarks and definitions which will be used in the text; in particular, we recall some facts about the Gagliardo-Nirenberg inequality and deduce some direct consequences.
Section \[sec:subcrit\_crit\] is devoted to the existence results under (H1)-(H2), i.e., to the proof of Theorem \[prop:subcritical\]-a) as well as to a detailed explanation of condition (which leads to Fig. \[fig:N=2\]).
The existence results under (H3)-(H4) (Theorem \[prop:supercritical\]) are proved in Section \[sec:3\]. Therein, we provide lower estimates for $\hat c_\alpha$ (see Subsection \[sec:hat\_c\_below\]), we prove a slightly more general version of Proposition \[prop:compact\_intro\] (Subsection \[sec:calpha\_achiev\]), while in Subsection \[sec:abstractexist\] we introduce an abstract criterium that guarantees that $(\rho_1,\rho_2)\in A$. Finally, Subsections \[sec:star\] and \[sec:explicit\_est\] contain respectively the proofs of the qualitative properties of $A$ and the deduction of condition .
Section \[sec:stab\] is concerned with the proof of the stability results, namely the proof of Theorems \[prop:subcritical\]-b) and \[thm:stab\]. Finally, Theorem \[thm\_beta-infty\] is proved in Section \[sec:segregation\].
Notations and Preliminaries {#eq:preliminaries}
---------------------------
Throughout the paper we denote by $\lambda_1(\Omega)$ the first eigenvalue of the Dirichlet Laplacian in $\Omega$, and by $\varphi_1$ the corresponding first eigenfunction, which we assume normalized in $L^2(\Omega)$ and positive in $\Omega$.
We use the following $L^q(\Omega)$ ($1\le q<\infty$) and $H^1_0(\Omega)$–norms: $$\|u\|_{L^q(\Omega)}^q:=\int_\Omega |u|^q,\qquad \|u\|_{H^1_0(\Omega)}^2:= \int_\Omega |\nabla u|^2.$$ Where there is no risk of confusion, we will denote $\|\cdot\|_{L^q(\Omega)}$ simply by $\|\cdot \|_q$.
The Gagliardo-Nirenberg inequality asserts that there exists a constant $C_{N,p}$ such that $$\label{eq:GN}
\begin{split}
\|v\|_{L^{p+1}({{\mathbb{R}}}^N)}^{p+1}&\leq C_{N,p} \|\nabla v\|_{L^2({{\mathbb{R}}}^N)}^{N(p-1)/2}\|v\|_{L^2({{\mathbb{R}}}^N)}^{p+1-{N(p-1)/2}}\\
&= C_{N,p} \|\nabla v\|_{L^2({{\mathbb{R}}}^N)}^{2a}\|v\|_{L^2({{\mathbb{R}}}^N)}^{4r},\qquad \forall v\in H^1({{\mathbb{R}}}^N),
\end{split}$$ where the exponents $a$ and $r$ were defined in . We remark that this inequality holds also in $H^1_0(\Omega)$, for any bounded domain $\Omega$, with the same constant $C_{N,p}$. It is proved in [@Weinstein1983] that $$\label{eq:CN4}
C_{N,p}:=\inf_{v\in H^1({{\mathbb{R}}}^N)\setminus\{0\}} \frac{\|v\|_{L^{p+1}({{\mathbb{R}}}^N)}^{p+1}}{\|\nabla v\|_{L^2({{\mathbb{R}}}^N)}^{2a} \|v\|_{L^2({{\mathbb{R}}}^N)}^{4r} } = \frac{\|Z\|_{L^{p+1}({{\mathbb{R}}}^N)}^{p+1}}{\|\nabla Z\|_{L^2({{\mathbb{R}}}^N)}^{2a} \|Z\|_{L^2({{\mathbb{R}}}^N)}^{4r} },$$ where $Z$ is, up to translations, the unique (see [@Kwong1989]) positive solution of $$\label{eq:Kwong}
-\Delta Z+Z=Z^p, \qquad Z\in H^1({{\mathbb{R}}}^N).$$ In particular, the inequality on $H^1_0(\Omega)$ is strict unless $v$ is trivial. In the special case $p=1+{4/N}$ we denote $$\label{eq:def_CN}
C_N:=C_{N,1+{4/N}},$$ while for $p={(N+2)/(N-2)}$ and $N\geq 3$, $$\label{eq:Sobolev_constant}
S_N:=C_{N,(N+2)/(N-2)}.$$ Observe that $S_N$ is just the best Sobolev constant of the embedding $\mathcal{D}^{1,2}({{\mathbb{R}}}^N)\hookrightarrow L^{2N/(N-2)}({{\mathbb{R}}}^N)$: $$\|v\|_{L^{2N/(N-2)}({{\mathbb{R}}}^N)}^{2N/N-2}\leq S_N \|\nabla v\|_{L^2({{\mathbb{R}}}^N)}^{2N/(N-2)}, \qquad \forall v\in \mathcal{D}^{1,2}({{\mathbb{R}}}^N)$$ For $(u_1,u_2)\in {{\mathcal{M}}}$, defined as in , using the Hölder and Gagliardo-Nirenberg inequalities (on bounded domains) we have $$\label{eq:importantestimateG}
\begin{split}
\int_\Omega \mu_1 |u_1|^{p+1} + &2\beta |u_1|^{(p+1)/2} |u_2|^{(p+1)/2} + \mu_2 |u_2|^{p+1}\\
&\le \mu_1\|u_1\|_{p+1}^{p+1}+\mu_2\|u_2\|_{p+1}^{p+1}+2\beta^+ \|u_1\|_{p+1}^{(p+1)/2}\|u_2\|_{p+1}^{(p+1)/2}\\
&< C_{N,p}\left(\mu_1 \rho_1^{2r}\|\nabla u_1\|_2^{2a} +\mu_2 \rho_2^{2r}\|\nabla u_2\|_2^{2a} +2\beta^+ \rho_1^r\rho_2^{r} \|\nabla u_1\|_2^{a} \| \nabla u_2\|_2^{a}\right).
\end{split}$$ where the exponents $a$ and $r$ are defined in . As a consequence we have, for $(u_1,u_2)\in {{\mathcal{M}}}$, $$\begin{gathered}
\label{eq:importantestimate}
{{\mathcal{E}}}(u_1,u_2)
>
\frac{1}{2}(\|\nabla u_1\|_2^2 + \|\nabla u_2\|_2^2) \\
-\frac{C_{N,p}}{p+1}\left(\mu_1 \rho_1^{2r}\|\nabla u_1\|_2^{2a}
+\mu_2 \rho_2^{2r}\|\nabla u_2\|_2^{2a}
+2\beta^+ \rho_1^r\rho_2^{r} \|\nabla u_1\|_2^{a} \| \nabla u_2\|_2^{a}\right).\end{gathered}$$ According to and to the previous inequality, the $L^2$-critical value $p=1+4/N$ is the threshold for the coercivity of ${{\mathcal{E}}}$ over ${{\mathcal{M}}}$, as we shall see more in detail in the following.
The L–subcritical and L–critical cases {#sec:subcrit_crit}
======================================
In this section we deal with conditions (H1) and (H2), meaning that $$\label{H1}
1<p\leq 1+\frac{4}{N}.$$ Recall the definition of $C_N$ in .
Let us show in the two cases that ${{\mathcal{E}}}$ restricted to ${{\mathcal{M}}}$ is coercive. Then, by the direct method of the calculus of variations, $\inf_{{{\mathcal{M}}}}{{\mathcal{E}}}$ is achieved by a couple $(u_1,u_2)$ (which belongs to ${{\mathcal{M}}}$ because of the compact embedding $H^1_0(\Omega)\hookrightarrow L^2(\Omega)$). By the Lagrange multipliers rule, $(u_1,u_2)$ solves for some $(\omega_1,\omega_2)\in{{\mathbb{R}}}^2$. By possibly taking $|u_i|$, we can suppose $u_i\geq0$ and, if $\rho_1,\rho_2>0$, the maximum principle provides $u_i>0$ (indeed, since $\Omega$ is Lipschitz, each $u_i$ is continuous up to the boundary).
If (H1) holds, then $0<{N(p-1)/2}<2$, so that, in , $a<1$; we immediately deduce that ${{\mathcal{E}}}$ restricted to ${{\mathcal{M}}}$ is coercive for every $\rho_1,\rho_2>0$.
In case we have (H2), continuing from and since in this case $a=1$, $r={1/N}$, we have $$\begin{aligned}
{{\mathcal{E}}}(u_1,u_2) &> \frac{1}{2}(\|\nabla u_1\|_2^2 + \|\nabla u_2\|_2^2) \\
& \qquad -\frac{NC_N}{2(N+2)}\left(\mu_1 \rho_1^{2/N} \|\nabla u_1\|_2^2 +\mu_2 \rho_2^{2/N} \|\nabla u_2\|_2^2+2\beta^+ (\rho_1 \rho_2)^{1/N}\|\nabla u_1\|_2\| \nabla u_2\|_2\right) \\
&= \frac{1}{2}\|\nabla u_1\|_2^2\left(1-\frac{NC_N\mu_1\rho_1^{2/N}}{N+2}\right) + \frac{1}{2}\|\nabla u_2\|_2^2\left(1-\frac{NC_N\mu_2\rho_2^{2/N}}{N+2}\right)\\
& \qquad -\frac{\beta^+ N C_N(\rho_1\rho_2)^{1/N}}{N+2}\|\nabla u_1\|_2\|\nabla u_2\|_2\\
&= \frac{1}{2} \begin{bmatrix} \|\nabla u_1\|_2 & \|\nabla u_2\|_2 \end{bmatrix} \cdot A \cdot \begin{bmatrix} \|\nabla u_1\|_2 & \|\nabla u_2\|_2 \end{bmatrix}^T \label{eq:L^2crit_energyestimate},\end{aligned}$$ where $$A=\begin{bmatrix}
1-\frac{NC_N\mu_1\rho_1^{2/N}}{N+2} & -\frac{\beta^+ N C_N(\rho_1\rho_2)^{1/N}}{N+2} \\
-\frac{\beta^+ N C_N(\rho_1\rho_2)^{1/N}}{N+2} & 1-\frac{NC_N\mu_2\rho_2^{2/N}}{N+2}.
\end{bmatrix}$$ If $A$ is positive definite then our result follows. Now $A$ is positive definite if and only if the following inequalities are simultaneously satisfied $$\begin{aligned}
&1-\frac{NC_N\mu_1\rho_1^{2/N}}{N+2}>0, \qquad 1-\frac{NC_N\mu_2\rho_2^{2/N}}{N+2}>0, \\
&\left(1-\frac{NC_N\mu_1\rho_1^{2/N}}{N+2}\right)\left(1-\frac{NC_N\mu_2\rho_2^{2/N}}{N+2}\right)-\left(\frac{N C_N}{N+2}\right)^2(\beta^+)^2 (\rho_1 \rho_2)^{2/N}>0,\end{aligned}$$ that is to say holds.
\[rem:N=2\] In this remark we interpret in the $(\mu_1\rho_1^{2/N},\mu_2\rho_2^{2/N})$–plane the condition (see Fig. \[fig:N=2\] for a visualization of this remark). Let $\bar x=\mu_1\rho_1^{2/N}$, $\bar y=\mu_2\rho_2^{2/N}$ so that corresponds to $(\bar x,\bar y)\in C$, where $$C:=\left\{(x,y)\in {{\mathbb{R}}}^2:\ 0< x,y< \frac{N+2}{NC_{N}}\ \text{ and }\ x+y+\frac{NC_N}{N+2} \frac{(\beta^+)^2-\mu_1\mu_2}{\mu_1\mu_2}xy<\frac{N+2}{NC_{N}} \right\}$$ For $\beta\leq 0$, the condition reduces to $(\bar x,
\bar y)$ lying in the square $$Q:=\left\{ (x,y)\in {{\mathbb{R}}}^2:\ 0< x,y< \frac{N+2}{NC_{N}}\right\}$$ For $\beta=\sqrt{\mu_1\mu_2}$, we have a half-square: $$Q_1:=Q\cap \left\{(x,y)\in {{\mathbb{R}}}^2:\ x+y< \frac{N+2}{NC_{N}}\right\}.$$ For $\beta>0$, $\beta\neq \sqrt{\mu_1\mu_2}$, the curve $$\left\{(x,y)\in {{\mathbb{R}}}^2:\ x+y+\frac{NC_N}{N+2} \frac{(\beta^+)^2-\mu_1\mu_2}{\mu_1\mu_2}xy=\frac{N+2}{NC_{N}} \right\}$$ is an hyperbola which contains the points $(0,\frac{N+2}{NC_{N}})$, $(\frac{N+2}{NC_{N}},0)$. This hyperbola is the graph of $$y=\left(\frac{N+2}{NC_{N}}-x\right)\left(1+\frac{NC_N}{N+2}\frac{(\beta^+)^2-\mu_1\mu_2}{\mu_1\mu_2} x\right)^{-1}$$ or, equivalently, $$y=\frac{N+2}{NC_{N}}\left(-\frac{\mu_1\mu_2}{(\beta^+)^2-\mu_1\mu_2}+\frac{N+2}{NC_{N}} \frac{(\beta^+)^2}{(\beta^+)^2-\mu_1\mu_2}\left(\frac{N+2}{NC_{N}}+\frac{(\beta^+)^2-\mu_1\mu_2}{\mu_1\mu_2}x\right)^{-1} \right)$$ which has a vertical asymptote at $x= \frac{N+2}{NC_{N}}\frac{\mu_1\mu_2}{\mu_1\mu_2-(\beta^+)^2}$. Thus, the set $C$ always contains the sides of the square $Ox^+\cap \overline{Q}$ and $Oy^+\cap \overline{Q}$. When $0<\beta<\sqrt{\mu_1\mu_2}$ it contains $Q_1$, and when $\beta>\sqrt{\mu_1\mu_2}$ it is contained in $Q_1$.
When $\beta\leq 0$, the condition reads as $$\mu_1 \rho_1^{2/N}, \mu_2 \rho_2^{2/N}<\frac{N+2}{NC_{N}}.$$ Going to [@ntvAnPDE p. 1833] we see that, as a consequence of Pohozaev’s identity: $$\frac{N+2}{NC_{N}}=\|Z\|_2^{2/N},$$ with $Z$ defined in . Therefore the condition is equivalent to $$\rho_1<\|Z\|_2 \mu_1^{-N/2},\qquad \rho_2<\|Z\|_2 \mu_2^{-N/2}.$$ This is consistent with the results in [@FibichMerle2001; @ntvAnPDE], which correspond to the case $\beta=0$ in .
The L–supercritical and Sobolev–subcritical case. The Sobolev–critical case {#sec:3}
===========================================================================
Preliminaries
-------------
Assume from now on that $p$ satisfies either (H3) or (H4), that is $p>1+4/N$, with $p\leq (N+2)/(N-2)$ if $N\geq 3$. Along this section we do not make any distinction between the Sobolev-critical and the Sobolev-subcritical cases, unless otherwise specified.
In Proposition \[prop:subcritical\] we proved that ${{\mathcal{E}}}$ restricted to ${{\mathcal{M}}}$ is coercive for any $\rho_1,\rho_2>0$ under (H1) or for $(\rho_1,\rho_2)$ satisfying under (H2). Thus, solutions were found as global minimizers of ${{\mathcal{E}}}|_{{\mathcal{M}}}$. In the $L^2$-supercritical case $p>1+4/N$ the previous approach cannot work, since ${{\mathcal{E}}}$ restricted to ${{\mathcal{M}}}$ is not coercive for every value of $(\rho_1,\rho_2)$, as we show in the following lemma. Notice that this was already suggested by equation , since now $a=N(p-1)/4>1$.
\[lemma:geometry\_mp\] Let $p>1+4/N$. Then there exists $(U_{1,k},U_{2,k})\in {{\mathcal{M}}}$, with nonnegative components, such that, as $k\to \infty$, $$\|(U_{1,k},U_{2,k})\|_{H^1_0(\Omega)}\to +\infty \quad \text{ and }\qquad {{\mathcal{E}}}(U_{1,k},U_{2,k})\to -\infty.$$
Let $\phi\in C^\infty_c(B_1)$ with $\phi>0$ in $B_1$ and $\int_{B_1} \phi^2=1$, and $x_1,x_2\in\Omega$ such that $x_1\neq x_2$. For $k\in {{\mathbb{N}}}$ and $i=1,2$, we define $$U_{i,k}(x)=\rho_i^{1/2} k^{N/2} \phi(k(x-x_i)), \qquad x\in \Omega.$$ For $k$ sufficiently large we have $\textrm{supp}(U_{i,k})\subset B_{1/k}(x_i)\subset\Omega$, $i=1,2$, and $\textrm{supp}(U_{1,k})\cap \textrm{supp}(U_{2,k})=\emptyset$. Furthermore, $$\int_\Omega U_{i,k}^2=\rho_i \int_{B_1} \phi^2 =\rho_i,$$ so that $(U_{1,k},U_{2,k}) \in {{\mathcal{M}}}$ for $k$ sufficiently large. We compute $$\|(U_{1,k},U_{2,k})\|_{H^1_0(\Omega)} = k \sqrt{\rho_1+\rho_2} \|\nabla\phi\|_{L^2(B_1)}
\to+\infty$$ as $k\to+\infty$ and $${{\mathcal{E}}}(U_{1,n},U_{2,n})=
k^2 \frac{\rho_1+\rho_2}{2}\|\nabla \phi\|_{L^2(B_1)}^2
- k^{2a} \frac{\mu_1\rho_1^{(p+1)/2}+\mu_2\rho_2^{(p+1)/2}}{p+1} \|\phi\|_{L^{p+1}(B_1)}^{p+1}
\to -\infty$$ as $k\to+\infty$, since $a>1$.
A basic estimate on hat c alpha {#sec:hat_c_below}
-------------------------------
In order to prove the existence of a solution of under (H3) or (H4) for certain values of $\rho_1,\rho_2$, we use a different approach than the one used in Section \[sec:subcrit\_crit\]. Recall that, for $\alpha\geq \lambda_1(\Omega)$, ${{\mathcal{B}}}_\alpha$ and ${{\mathcal{U}}}_\alpha$ are defined in , while $c_\alpha$ and $\hat c_\alpha$ are as in . Observe that ${{\mathcal{B}}}_\alpha\neq \emptyset$, since it contains at least $(\sqrt{\rho_1}\varphi_1,\sqrt{\rho_2}\varphi_1)$. Moreover $$c_{\lambda_1(\Omega)}=\hat c_{\lambda_1(\Omega)}={{\mathcal{E}}}(\sqrt{\rho_1}\varphi_1,\sqrt{\rho_2}\varphi_1).$$
Recalling and using the identification $x=\|\nabla u_1\|_2$, $y=\|\nabla u_2\|_2$, we end up studying the function $\varphi:{{\mathbb{R}}}^2_+\to {{\mathbb{R}}}$ defined by $$\label{eq:vphi_def}
\Phi(x,y)=\frac{1}{2}(x^2+y^2)-\frac{C_{N,p}}{p+1}\left(\mu_1 \rho_1^{2r}x^{2a}
+\mu_2 \rho_2^{2r}y^{2a}
+2\beta^+ \rho_1^r\rho_2^{r} x^{a} y^{a}\right)$$ where now $a>1$. Indeed, by we obtain that $$\label{eq:Phi_ineq}
{{\mathcal{E}}}(u_1,u_2)\geq \Phi \left(\|\nabla u_1\|_2,\|\nabla u_1\|_2\right)\qquad\text{ for every }(u_1,u_2)\in {{\mathcal{M}}}.$$ In particular, this allows to estimate $\hat c_\alpha$ from below. To do that, let us define the following subsets of ${{\mathbb{R}}}^2$: $$U_\alpha=\left\{(x,y)\in {{\mathbb{R}}}^2_+:\ x^2+y^2= (\rho_1+\rho_2)\alpha\right\},\quad V_\alpha=U_\alpha\cap \left\{x\geq \sqrt{\rho_1\lambda_1(\Omega)},\ y\geq \sqrt{\rho_2\lambda_1(\Omega)}\right\}.$$ The set $U_\alpha$ is obtained from ${{\mathcal{U}}}_\alpha$ through the identification $x=\|\nabla u_1\|_2$, $y=\|\nabla u_2\|_2$. The set $V_\alpha$ is motivated by the fact that, for $(u_1,u_2)\in {{\mathcal{M}}}$, $\|\nabla u_1\|^2\geq \rho_1\lambda_1(\Omega)$ and $\|\nabla u_2\|^2\geq \rho_2\lambda_1(\Omega)$. With this notation, using , we obtain $$\label{eq:iniziostima}
\hat c_\alpha \ge \min_{(x,y)\in V_\alpha} \Phi(x,y) = \frac12 \alpha
-\frac{C_{N,p}}{p+1} \max_{(x,y)\in V_\alpha} \left(\mu_1 \rho_1^{2r}x^{2a}
+\mu_2 \rho_2^{2r}y^{2a}
+2\beta^+ \rho_1^r\rho_2^{r} x^{a} y^{a}\right).$$ Now, due to the limitations in the definition of $V_\alpha$, the last maximum can not be written explicitly in terms of $\alpha$ (except for a few particular cases). For this reason, we prefer the more rough estimate in which $V_\alpha$ is replaced with $U_\alpha$. This allows more readable results, without modifying the qualitative structure of the estimates.
\[lem:hatc\_from\_below\] Let $$\label{eq:defdd}
{\Lambda}={\Lambda}(\rho_1,\rho_2):= \frac{2C_{N,p}}{p+1} \max_{t\in [0,\pi/2]}
\left(\mu_1 \rho_1^{2r}\cos^{2a}t
+\mu_2 \rho_2^{2r}\sin^{2a}t
+2\beta^+ \rho_1^r\rho_2^{r} \cos^{a}t \sin^{a}t\right).$$ Then, for every $\alpha > \lambda_1(\Omega)$, $$\hat c_\alpha > \frac12\left((\rho_1+\rho_2)\alpha - {\Lambda}(\rho_1,\rho_2)(\rho_1+\rho_2)^a \alpha^a \right).$$
Since $$V_\alpha\subset U_\alpha = \left\{(\cos t,\sin t)\sqrt{(\rho_1+\rho_2) \alpha}:t\in[0,\pi/2]\right\},$$ the lemma follows by continuing the estimate in .
Notice that ${\Lambda}$ depends on $\mu_1,\mu_2,\beta$, and also on $p$ and $N$ (via $a$ and $r$). On the other hand, in case $N\ge 3$ and $p=2^*-1$, we have that $a=2^*/2$, $r=0$ and ${\Lambda}$ does not depend on $\rho_1,\rho_2$, and actually its definition coincides with that given in . Then we have, for any $(v_1,v_2)\in H^1_0(\Omega;{{\mathbb{R}}}^2)$, $$\label{eq:lambda_max_def}
\frac{2 S_N}{2^*} \left(\mu_1\|\nabla v_1\|_2^{2^*}+2\beta^+ \|\nabla v_1\|_2^{2^*/2}\|\nabla v_{2}
\|_2^{2^*/2}+\mu_2\|\nabla v_{2}\|_2^{2^*}\right) \le {\Lambda}\left(\|\nabla v_1\|_2^{2}+
\|\nabla v_{2}\|_2^2\right)^{2^*/2}$$ (recall the definition of $S_N = C_{N,2^*-1}$ given in ). To see this, we notice that for any $(v_1,v_2)$ one can find $t\in[0,\pi/2]$ such that $$\|\nabla v_1\|_2 = \left(\|\nabla v_1\|_2^{2}+
\|\nabla v_{2}\|_2^2\right)^{1/2}\cos t,\qquad
\|\nabla v_2\|_2 = \left(\|\nabla v_1\|_2^{2}+
\|\nabla v_{2}\|_2^2\right)^{1/2}\sin t,$$ and we substitute in .
The level c alpha is achieved {#sec:calpha_achiev}
-----------------------------
As we mentioned, we will look for local minimizers of ${{\mathcal{E}}}$ on ${{\mathcal{B}}}_\alpha$, hence at level $c_\alpha$, for suitable values of $\alpha$. A first necessary step is to prove that $c_\alpha$ is achieved (possibly on ${{\mathcal{U}}}_\alpha$, the topological boundary of ${{\mathcal{B}}}_\alpha$). This is easily obtained, for every $\alpha\ge \lambda_1(\Omega)$, in the Sobolev subcritical case: indeed, in such situation, ${{\mathcal{B}}}_\alpha$ is weakly compact and ${{\mathcal{E}}}$ weakly lower semicontinuous. On the other hand, if $N\ge3$ and $p=2^*-1$, ${{\mathcal{E}}}$ is no longer weakly lower semicontinuous. In this situation, inspired by the celebrated paper by Brezis and Nirenberg [@MR709644], we can recover compactness of the minimizing sequences by imposing a smallness condition on the masses $\rho_1,\rho_2$ and on $\alpha$, as stated in Proposition \[prop:compact\_intro\]. Actually, here we will prove a slightly more general result, considering sequences in which also the masses are not fixed; this will be useful when dealing with stability issues.
In the following, recall that ${\Lambda}$ has been introduced in (or, equivalently, in ) and that, in the Sobolev critical case, it does not depend on $\rho_1,\rho_2$.
\[prop:Sobcritconv\] Let $\alpha>\lambda_1(\Omega)$, $\rho_1,\rho_2>0$ satisfy $$(\rho_1+\rho_2)(\alpha - \lambda_1(\Omega)) <
\frac{1}{{\Lambda}^{(N-2)/2}},$$ and let $(u_{1,n},u_{2,n})_n$ be such that $$\label{eq:minimizing_seq}
\begin{cases}
\|u_{i,n}\|_2^2=\rho_i + o(1) \qquad & \text{for } i=1,2, \\
\|\nabla u_{1,n}\|_2^2+\|\nabla u_{2,n}\|_2^2 \leq \alpha(\rho_1+\rho_2) + o(1)
\qquad & \\
c_\alpha \leq \mathcal{E}(u_{1,n},u_{2,n}) \leq c_\alpha +o(1) \quad &
\end{cases}$$ as $n\to \infty$. Then, up to subsequences, $$(u_{1,n},u_{2,n}) \to (\bar u_{1},\bar u_{2}), \qquad\text{strongly in }H^1_0(\Omega).$$ In particular, $c_\alpha$ is achieved.
The proposition follows as a particular case of Proposition \[prop:Sobcritconv\], when in both $\|u_{i,n}\|_2^2=\rho_i$, $i=1,2$, and $\|\nabla u_{1,n}\|_2^2+\|\nabla u_{2,n}\|_2^2 \leq \alpha(\rho_1+\rho_2)$.
By assumption there exists $(\bar u_1,\bar u_2)\in H^1_0(\Omega;{{\mathbb{R}}}^2)$ such that, up to subsequences, $$\begin{cases}
\|\bar u_i\|_2^2=\rho_i \qquad & \text{for } i=1,2 \\
u_{i,n} \rightharpoonup \bar u_i \qquad & H^1_0(\Omega)\text{-weak } \text{for } i=1,2\\
\|\nabla \bar u_i\|_2^2 \leq \liminf_{n\to\infty} \|\nabla u_{i,n}\|_2^2 \qquad & \text{for } i=1,2.
\end{cases}$$ Notice that $(\bar u_1,\bar u_2)$ is admissible for the minimization problem $c_\alpha$, whence $$\label{eq:bar_u_i_admissible}
\mathcal{E}(\bar u_1,\bar u_2)\geq c_\alpha.$$ Let $v_{i,n}=u_{i,n}-\bar u_i$ and notice that, for $i=1,2$, $$\label{eq:v_i_n}
v_{i,n}\rightharpoonup 0 \quad\text{weakly both } H^1_0(\Omega) \text{ and } L^{2^*}(\Omega),
\qquad v_{i,n}\to0 \quad \text{strongly } L^2(\Omega),$$ where $2^*=2N/(N-2)$.
Notice that the strong convergence of a subsequence of $(u_{1,n},u_{2,n})$ is equivalent to the statement: $$\label{eq:BN_easy_case}
\text{ there exists a subsequence } (v_{1,n_k},v_{2,n_k}) \text{ such that } \|\nabla v_{1,n_k}\|_2^2+\|\nabla v_{2,n_k}\|_2^2 \to 0.$$ In such a case, by continuity of the Sobolev embeddings, we have that $c_\alpha=\mathcal{E}(\bar u_1,\bar u_2)$. Since a minimizing sequence for $c_\alpha$ exists, and it satisfies , we deduce that $c_\alpha$ is achieved.
To conclude the proof, suppose by contradiction that does not hold, so that $$\label{eq:BN_difficult_case}
\|\nabla v_{1,n}\|_2^2+\|\nabla v_{2,n}\|_2^2 \geq K>0 \quad \text{eventually.}$$ We can write $$\begin{gathered}
\label{eq:energy_rewritten}
\mathcal{E}(u_{1,n},u_{2,n}) =
\frac{1}{2} \left(\|\nabla(\bar u_1+v_{1,n})\|_2^2+\|\nabla(\bar u_2+v_{2,n})\|_2^2\right) \\
-\frac{1}{2^*} \left(\mu_1 \|\bar u_1+v_{1,n}\|_{2^*}^{2^*}
+2\beta \|(\bar u_1+v_{1,n})(\bar u_2+v_{2,n})\|_{2^*/2}^{2^*/2}
+\mu_2 \|\bar u_2+v_{2,n}\|_{2^*}^{2^*} \right).\end{gathered}$$ Notice that, by weak convergence, we have, for $i=1,2$, $$\label{eq:weak_conv}
\|\nabla (\bar u_i+v_{i,n})\|_2^2=\|\nabla \bar u_i\|_2^2+\|\nabla v_{i,n}\|_2^2+o(1)
\qquad \text{as }n\to\infty.$$ In order to estimate the remaining terms of , we recall the following Lemma by Brezis and Lieb [@BL83]: given $1\leq q<\infty$, if $\{f_n\}_n\subset L^q(\Omega)$ is a sequence bounded in $L^q(\Omega)$, such that $f_n\to f$ almost everywhere, then $$\label{eq:BL}
\|f_n\|_q^q=\|f\|_q^q+\|f_n-f\|_q^q+o(1) \qquad \text{as }n\to\infty.$$
We apply first with $f_n=u_{i,n}=\bar u_i+v_{i,n}$ and $q=2^*$ to get $$\label{eq:L4conv}
\|\bar u_i+v_{i,n}\|_{2^*}^{2^*}=\|\bar u_i\|_{2^*}^{2^*}+\|v_{i,n}\|_{2^*}^{2^*}+o(1)
\qquad \text{as }n\to\infty,$$ then we apply it with $f_n=(\bar u_1+v_{1,n})(\bar u_2+v_{2,n})$ and $q=2^*/2$ to obtain $$\label{eq:mixed_term_conv1}
\|(\bar u_1+v_{1,n})(\bar u_2+v_{2,n})\|_{2^*/2}^{2^*/2}
=\|\bar u_1\bar u_2\|_{2^*/2}^{2^*/2}+\|\bar u_1 v_{2,n}+\bar u_2 v_{1,n} +v_{1,n}v_{2,n}\|_{2^*/2}^{2^*/2}+o(1)$$ as $n\to\infty$. In order to estimate the second term in the right hand side of , we shall need two inequalities. For every $q>1$ and for every $a,b\in{{\mathbb{R}}}$ it holds $$\label{eq:ineq1}
|a+b|^q\leq 2^{q-1} (|a|^q+|b|^q);$$ $$\label{eq:ineq2}
||a+b|^q-|a|^q| \leq C(|a|^{q-1}|b|+|b|^q),$$ for a constant $C$ not depending on $a$ and $b$. By , we have $$\label{eq:ineq3}
\|\bar u_1 v_{2,n}+\bar u_2 v_{1,n}\|_{2^*/2}^{2^*/2} \leq 2^{(2^*-2)/2}
\left( \|\bar u_1v_{2,n}\|_{2^*/2}^{2^*/2} +\|\bar u_2v_{1,n}\|_{2^*/2}^{2^*/2} \right)
=o(1) \qquad \text{as }n\to\infty,$$ because $|v_{i,n}|^{2^*/2}\rightharpoonup 0$ in $L^2(\Omega)$-weak as $n\to+\infty$, for $i=1,2$. Then using , the Hölder inequality and , we compute $$\begin{gathered}
\left| \|\bar u_1 v_{2,n}+\bar u_2 v_{1,n} +v_{1,n}v_{2,n}\|_{2^*/2}^{2^*/2} -
\|v_{1,n}v_{2,n}\|_{2^*/2}^{2^*/2}\right| \\
\leq C \int_\Omega \left( |v_{1,n}v_{2,n}|^{(2^*-2)/2}|\bar u_1 v_{2,n}+\bar u_2 v_{1,n}|+|\bar u_1 v_{2,n}+\bar u_2 v_{1,n}|^{2^*/2} \right)\,dx\\
\leq C \|v_{1,n}\|_{2^*}^{(2^*-2)/2}\|v_{2,n}\|_{2^*}^{(2^*-2)/2}\|\bar u_1 v_{2,n}+\bar u_2 v_{1,n}\|_{2^*/2}+\|\bar u_1 v_{2,n}+\bar u_2 v_{1,n}\|_{2^*/2}^{2^*/2} =o(1)\end{gathered}$$ as $n\to+\infty$. This last estimate, replaced into , provides $$\label{eq:mixed_term_conv}
\|(\bar u_1+v_{1,n})(\bar u_2+v_{2,n})\|_{2^*/2}^{2^*/2}
=\|\bar u_1\bar u_2\|_{2^*/2}^{2^*/2}+\|v_{1,n}v_{2,n}\|_{2^*/2}^{2^*/2}+o(1)$$ as $n\to+\infty$.
By replacing , and into , we see that $$\mathcal{E}(u_{1,n},u_{2,n})=\mathcal{E}(\bar u_1,\bar u_2)+\mathcal{E}(v_{1,n},v_{2,n})+o(1) \qquad \text{as }n\to\infty.$$ The last expression, together with and , implies $$\mathcal{E}(v_{1,n},v_{2,n})\leq o(1) \qquad \text{as }n\to\infty,$$ whence, using (with $r=0$, $a=2^*/2$ and $C_{N,2^*-1}=S_N$) and , $$\begin{split}
\|\nabla v_{1,n}\|_2^2+\|\nabla v_{2,n}\|_2^2 &\leq
\frac{2 S_N}{2^*} \left(\mu_1\|\nabla v_{1,n}\|_2^{2^*}+2\beta^+ \|\nabla v_{1,n}\|_2^{2^*/2}\|\nabla v_{2,n}\|_2^{2^*/2}+\mu_2\|\nabla v_{2,n}\|_2^{2^*}\right)+o(1)\\
&\leq{\Lambda}\left(\|\nabla v_1\|_2^{2}+
\|\nabla v_{2}\|_2^2\right)^{2^*/2}
+o(1).
\end{split}$$ Now, we can use to rewrite the last inequality as $$\left(\|\nabla v_{1,n}\|_2^2+\|\nabla v_{2,n}\|_2^2\right)^{(2^*-2)/2}\ge \frac{1}{{{\Lambda}}} +o(1).$$ We combine the previous inequality with to obtain $$\begin{split}
\left( \frac{1}{{{\Lambda}}} + o(1) \right)^{(N-2)/2} &\leq
\|\nabla v_{1,n}\|_2^2+\|\nabla v_{2,n}\|_2^2
=\|\nabla u_{1,n}\|_2^2+\|\nabla u_{2,n}\|_2^2 - (\|\nabla \bar u_1 \|_2^2+\|\nabla \bar u_2\|_2^2)+o(1)\\
&\le (\rho_1+\rho_2)\alpha - \lambda_1(\Omega)(\rho_1+\rho_2) +o(1),
\end{split}$$ as $n\to+\infty$, which contradicts the assumption.
Existence of ground states {#sec:abstractexist}
--------------------------
This section is devoted to prove the following result.
\[thm:existence\_bad\_cond\] Let $\rho_1,\rho_2\ge0$ be such that $$\label{eq:mainassL}
{\Lambda}(\rho_1,\rho_2) \cdot (\rho_1+\rho_2)^{a-1} \le \frac{(a-1)^{a-1}}{a^a}
\lambda_j(\Omega)^{-(a-1)},$$ where $j=1$ if $\beta\ge-\sqrt{\mu_1\mu_2}$, $j=2$ otherwise. Let $\bar \alpha= \frac{a}{a-1}\lambda_i(\Omega)$.
Then $c_{\bar\alpha}$ is achieved by $(\bar u_1,\bar u_2)\in {{\mathcal{B}}}_{\bar \alpha}\setminus {{\mathcal{U}}}_{\bar \alpha}$ such that ${{\mathcal{E}}}(\bar u_1, \bar u_2)= c_{\bar \alpha}$, which implies that $(\bar u_1, \bar u_2)$ is a local minimum of ${{\mathcal{E}}}|_{{{\mathcal{M}}}}$, corresponding to a positive solution of for some $(\omega_1,\omega_2)\in {{\mathbb{R}}}^2$. Equivalently, $(\rho_1,\rho_2)\in A$ as defined in .
First of all, we state a sufficient condition for the above theorem to hold, in terms of $\hat c_\alpha$.
\[lem:c<hat\_c\] Let us assume that $\rho_1,\rho_2>0$ are such that, for some $\alpha_1,\alpha_2$, $$\lambda_1(\Omega)\le \alpha_1 < \alpha_2
\qquad \text{ and } \qquad
\hat c_{\alpha_1} < \hat c_{\alpha_2};$$ furthermore, in the Sobolev critical case $N\ge3$, $p=2^*-1$, let us also assume that $$\alpha_2 < \lambda_1(\Omega) +
\frac{{{\Lambda}}^{-(N-2)/2}}{\rho_1+\rho_2}.$$ Then $c_{\alpha_2} < \hat c_{\alpha_2}$, and $c_{\alpha_2}$ is achieved by a positive solution of .
Firstly, $c_{\alpha_2}$ is achieved by some $(\bar u_1,\bar u_2)\in{{\mathcal{B}}}_{\alpha_2}$: as we already observed, this is trivial in the Sobolev subcritical case, while in the critical one it follows by Proposition \[prop:Sobcritconv\]. Next we observe that $$c_{\alpha_2} = \min\left\{\hat c_\alpha:\lambda_1(\Omega)\le \alpha \le \alpha_2\right\}\le
\hat c_{\alpha_1} < \hat c_{\alpha_2}.$$ We deduce that $(\bar u_1,\bar u_2)\in{{\mathcal{B}}}_{\alpha_2}\setminus{{\mathcal{U}}}_{\alpha_2}$, and the lemma follows.
Let us denote by $\lambda_2(\Omega)$ the second eigenvalue of $-\Delta$ in $H^1_0(\Omega)$, and by $\varphi_2$ a corresponding eigenfunction.
\[lem:estimate\_c\] We have $$(\sqrt{\rho_1}\varphi_1,\sqrt{\rho_2} \varphi_1)\in {{\mathcal{U}}}_{\lambda_1(\Omega)},\qquad \left(\sqrt{\rho_1}\frac{\varphi_2^+}{\|\varphi_2^+\|_2},\sqrt{\rho_2} \frac{\varphi_2^-}{\|\varphi_2^-\|_2}\right)\in {{\mathcal{U}}}_{\lambda_2(\Omega)}.$$ In particular $$\hat c_{\lambda_j(\Omega)} \leq \frac{\rho_1+\rho_2}{2} \lambda_j(\Omega), \qquad
\text{for }j=
\begin{cases}
1 & \text{if }\beta\geq -\sqrt{\mu_1\mu_2},\\
2 & \text{if }\beta< -\sqrt{\mu_1\mu_2}.
\end{cases}$$
The first assertion is direct. Then $$\begin{split}
\hat c_{\lambda_1(\Omega)}& \leq {{\mathcal{E}}}(\sqrt{\rho_1}\varphi_1,\sqrt{\rho_2}\varphi_1) \\&=
\frac{\rho_1+\rho_2}{2}
\lambda_1(\Omega)-\frac{\mu_1 \rho_1^{p+1}+2\beta (\rho_1\rho_2)^\frac{p+1}{2} +\mu_2 \rho_2^{p+1}}{p+1}\int_\Omega \varphi_1^4 \leq \frac{\rho_1+\rho_2}{2}
\lambda_1(\Omega),
\end{split}$$ since $\beta\ge -\sqrt{\mu_1\mu_2}$ implies that $\mu_1 \rho_1^{p+1}+2\beta (\rho_1\rho_2)^\frac{p+1}{2} +\mu_2 \rho_2^{p+1} \geq 0$ for every $\rho_1,\rho_2>0$.
On the other hand, $$\begin{aligned}
\hat c_{\lambda_2(\Omega)} &\leq {{\mathcal{E}}}\left(\sqrt{\rho_1}\frac{\varphi_2^+}{\|\varphi_2^+\|_{L^2(\Omega)}},\sqrt{\rho_2}\frac{\varphi_2^-}{\|\varphi_2^-\|_{L^2(\Omega)}}\right) \\
& \leq \frac{\rho_1+\rho_2}{2} \lambda_2(\Omega)
-\frac{\mu_1\rho_1^{p+1}}{(p+1)\|\varphi_2^+\|_{L^2(\Omega)}^{p+1}}\int_\Omega (\varphi_2^+)^{p+1}\,dx
-\frac{\mu_2\rho_2^{p+1}}{(p+1)\|\varphi_2^-\|_{L^2(\Omega)}^{p+1}}\int_\Omega (\varphi_2^-)^{p+1}\,dx\\
&\leq \frac{\rho_1+\rho_2}{2} \lambda_2(\Omega). \qedhere\end{aligned}$$
In the following let $j=1$ if $\beta\ge -\sqrt{\mu_1\mu_2}$ and $j=2$ otherwise. In view of the application of Lemma \[lem:c<hat\_c\], our aim is to find $\bar\alpha>\lambda_j(\Omega)$ such that $$\label{eq:c<hat_c_app}
\hat c_{\lambda_j(\Omega)} < \hat c_{\bar\alpha}.$$ To start with, we look for a sufficient condition implying . Using Lemmas \[lem:hatc\_from\_below\] and \[lem:estimate\_c\], it is sufficient to find $\bar\alpha>\lambda_j(\Omega)$ such that $$\frac{\rho_1+\rho_2}{2} \lambda_j(\Omega) \le \frac12\left(\bar\alpha(\rho_1+\rho_2) - {\Lambda}(\rho_1,\rho_2) (\rho_1+\rho_2)^a\bar\alpha^a \right)$$ (by Lemma \[lem:hatc\_from\_below\], the right hand side is strictly less than $\hat c_{\bar\alpha}$). Equivalently, $$\label{eq:passa}
{\Lambda}(\rho_1,\rho_2) (\rho_1+\rho_2)^{a-1} \le \frac{\bar \alpha - \lambda_j(\Omega) }{\bar\alpha^a}.$$ By a direct computation, recalling that $a>1$, the best possible choice for the right hand side is $$\max_{\alpha\ge\lambda_j(\Omega)}\frac{\alpha - \lambda_j(\Omega) }{\alpha^a} =
\frac{(a-1)^{a-1}}{a^a}
\lambda_j(\Omega)^{-(a-1)}, \qquad\text{achieved by }
\bar \alpha= \frac{a}{a-1}\lambda_j(\Omega).$$ This choice of $\bar \alpha$ is possible, since it makes equivalent to , the assumption of the theorem. Furthermore, it is clear that $\bar \alpha>\lambda_j(\Omega)$. Then, in order to apply Lemma \[lem:c<hat\_c\] and conclude the proof, we only need to check that, in case $N\ge3$ and $p=2^*-1$, the additional assumption $$\label{eq:passa2}
\bar\alpha < \lambda_1(\Omega) +
\frac{1}{{{\Lambda}}^{(N-2)/2}(\rho_1+\rho_2)}$$ holds true. This is straightforward since, being $a=N/(N-2)$, relation provides $${\Lambda}(\rho_1+\rho_2)^{2/(N-2)} \le \frac{\bar \alpha - \lambda_j(\Omega) }{\bar\alpha^{N/(N-2)}}
< \frac{\bar \alpha - \lambda_1(\Omega) }{(\bar \alpha - \lambda_1(\Omega))^{N/(N-2)}} = \frac{1}{(\bar \alpha - \lambda_1(\Omega))^{2/(N-2)}},$$ which is equivalent to .
The solution $(\bar u_1,\bar u_2)$ does *not* coincide with $(\sqrt{\rho_1} \varphi_1,\sqrt{\rho_2} \varphi_1)$, unless $\beta=-\mu_1=-\mu_2$. Indeed, this last pair solves if and only if, for every $i=1,2$, $$\begin{gathered}
(\lambda_1(\Omega)+\omega_i)\sqrt{\rho_i}\varphi_1=(\mu_i+\beta)\rho_i^\frac{p}{2}\varphi_1^{p} \iff
\lambda_1(\Omega)+\omega_i=(\mu_i+\beta)\rho_i^\frac{p-1}{2}\varphi_1^{p-1} \\
\iff \lambda_1(\Omega)=-\omega_i,\ \beta=-\mu_1=-\mu_2.\end{gathered}$$
The set A is star-shaped {#sec:star}
------------------------
This section is devoted to the proof of the following result.
\[prop\_starshaped\] Let $A$ be defined as in . Then $A$ is star-shaped with respect to $(0,0)$.
We follow a strategy inspired by [@ntvDCDS] but, since such paper does not extend directly to the Sobolev critical case, we provide here a self-contained argument. In this section it is convenient to make explicit the dependence of some quantities with respect to $\rho_1,\rho_2$: in view of this, we write $c_\alpha(\rho_1,\rho_2)$, $\hat c_\alpha(\rho_1,\rho_2)$, ${{\mathcal{B}}}_\alpha(\rho_1,\rho_2)$, ${{\mathcal{U}}}_\alpha(\rho_1,\rho_2)$. For shorter notation, we define $$F(u_1,u_2):=\int_\Omega \mu_1 |u_1|^{p+1} + 2\beta |u_1|^{(p+1)/2} |u_2|^{(p+1)/2} + \mu_2 |u_2|^{p+1}$$ and we introduce the optimization problem $$M_\alpha(\rho_1,\rho_2):= \sup _{{{\mathcal{U}}}_\alpha(\rho_1,\rho_2)} F$$ (a quantity thoroughly investigated in [@ntvDCDS]). Notice that $$\hat c_\alpha(\rho_1,\rho_2)=\frac{1}{2}\alpha(\rho_1+\rho_2)-\frac{1}{p+1} M_\alpha(\rho_1,\rho_2),$$ and that $\hat c_\alpha(\rho_1,\rho_2)$ is achieved at $(u_1,u_2)\in {{\mathcal{U}}}_\alpha(\rho_1,\rho_2)$ if, and only if, $M_\alpha(\rho_1,\rho_2)$ is achieved at the same pair.
Fix, if any, $(\rho_1,\rho_2)\in A\setminus\{(0,0)\}$. By definition of $A$, there exist $\alpha>\lambda_1(\Omega)$ and $(\bar u_1,\bar u_2)\in {{\mathcal{B}}}_\alpha$, a solution of , such that ${{\mathcal{E}}}(\bar u_1,\bar u_2)=c_{\alpha}< \hat c_{\alpha}$, and $\alpha$ satisfies in case $N\ge3$, $p=2^*-1$. Notice that the assumption $c_{\alpha}< \hat c_{\alpha}$ implies $\int_\Omega |\nabla \bar u_1|^2+|\nabla \bar u_2|^2<(\rho_1+\rho_2)\alpha$, so that $$\bar \alpha:=\frac{1}{\rho_1+\rho_2}\int_\Omega |\nabla \bar u_1|^2+|\nabla \bar u_2|^2<\alpha.$$ As a consequence, $(\bar u_1,\bar u_2)\in {{\mathcal{U}}}_{\bar \alpha}(\rho_1,\rho_2)$ achieves $\hat c_{\bar \alpha}
= c_\alpha$.
\[lem:starshap1\] If $s>0$ then $(s \bar u_1,s \bar u_2)\in {{\mathcal{U}}}_{\bar \alpha}(s^2\rho_1,s^2\rho_2)$ achieves $$\hat c_{\bar \alpha}(s^2 \rho_1,s^2\rho_2)=\frac{s^2}2 \bar \alpha (\rho_1+\rho_2) -
\frac{s^{p+1}}{p+1} F(\bar u_1,\bar u_2).$$
This follows by noticing that $$(u_1,u_2)\in{{\mathcal{U}}}_{\bar\alpha}(\rho_1,\rho_2)
\qquad\iff\qquad
(su_1,su_2)\in{{\mathcal{U}}}_{\bar\alpha}(s^2\rho_1,s^2\rho_2),$$ with $$F(su_1,su_2) = s^{p+1} F(u_1,u_2).$$ Then $M_\alpha(s^2\rho_1,s^2\rho_2) = s^{p+1}M_\alpha(\rho_1,\rho_2)$ and the lemma follows.
\[lem:curvetta\] Let $s\in (0,1)$ and $(v_1,v_2)\in H^1_0(\Omega,{{\mathbb{R}}}^2)$ be such that $$\int_\Omega \bar u_1v_1=\int_\Omega \bar u_2 v_2=0,\qquad \int_\Omega \nabla \bar u_1\cdot \nabla v_1+\nabla \bar u_2 \cdot \nabla v_2<0.$$ Let, for $|t|$ small, $$(U_1(t),U_2(t)):=\left( s\sqrt{\rho_1} \frac{\bar u_1+tv_1}{\|\bar u_1+tv_1\|_2},s\sqrt{\rho_2} \frac{\bar u_2+tv_2}{\|\bar u_2+tv_2\|_2}\right).$$ Then $(U_1(t),U_2(t))\in {{\mathcal{M}}}_{s^2\rho_1,s^2\rho_2}$ for every $t$ and $$\left.\frac{d}{dt} \|(U_1(t),U_2(t)) \|^2_{H^1_0(\Omega)} \right|_{t=0}<0,\qquad \left.\frac{d}{dt} {{\mathcal{E}}}(U_1(t),U_2(t)) \right|_{t=0}<0.$$
By direct inspection we have that $(U_1(t),U_2(t))\in {{\mathcal{M}}}_{s^2\rho_1,s^2\rho_2}$ for every $t$, and that $$\left.\frac{d}{dt} (U_1(t),U_2(t)) \right|_{t=0} = (sv_1,sv_2).$$ Then $$\left.\frac{d}{dt} \|(U_1(t),U_2(t)) \|^2_{H^1_0(\Omega)} \right|_{t=0} =
2s^2\int_\Omega \nabla \bar u_1\cdot \nabla v_1+\nabla \bar u_2 \cdot \nabla v_2<0$$ by assumption. On the other hand, recalling that $(\bar u_1,\bar u_2)$ solves , we have that $$\left.\frac{d}{dt} F(U_1(t),U_2(t)) \right|_{t=0} = s^{p+1}
F'(\bar u_1,\bar u_2)[v_1,v_2] = s^{p+1}
\int_\Omega \nabla \bar u_1\cdot \nabla v_1+\nabla \bar u_2 \cdot \nabla v_2$$ and $$\left.\frac{d}{dt} {{\mathcal{E}}}(U_1(t),U_2(t)) \right|_{t=0} =
(s^2- s^{p+1}) \int_\Omega \nabla \bar u_1\cdot \nabla v_1+\nabla \bar u_2 \cdot \nabla v_2<0,$$ as $0<s<1$ and $p>1$.
With the notation of Lemma \[lem:curvetta\], being $(U_1(0),U_2(0)) = (s\bar u_1,s\bar u_2)\in{{\mathcal{U}}}_{\bar\alpha}(s^2\rho_1,s^2\rho_2)$, there exist positive and small constants $\varepsilon,\tau$ such that $$(U_1(\tau),U_2(\tau)) \in{{\mathcal{U}}}_{\bar\alpha-\varepsilon}(s^2\rho_1,s^2\rho_2)$$ and $$\hat c_{\bar\alpha-\varepsilon}(s^2\rho_1,s^2\rho_2) \le {{\mathcal{E}}}(U_1(\tau),U_2(\tau))
< {{\mathcal{E}}}(U_1(0),U_2(0)) = \hat c_{\bar\alpha}(s^2\rho_1,s^2\rho_2)$$ (the last equality following by Lemma \[lem:starshap1\]). Then we can apply Lemma \[lem:c<hat\_c\], with $\alpha_1 = \bar\alpha - \varepsilon$ and $\alpha_2 = \bar\alpha$, obtaining that $(s^2\rho_1,s^2\rho_2)\in A$. Since this holds true for any $s\in(0,1)$, the proposition follows.
In [@ntvDCDS Theorem 1.1], we show that, in the Sobolev subcritical case, $M_{\alpha}(\rho_1,\rho_2)$ is achieved and that an associated maximum point $(u_1,u_2)$ satisfies $$\begin{cases}
-\Delta u_1+ \omega_1 u_1=\gamma(\mu_1 u_1|u_1|^{p-1}+\beta u_1|u_1|^{(p-3)/2} |u_2|^{(p+1)/2})\\
-\Delta u_2+ \omega_2 u_2=\gamma(\mu_2 u_2|u_2|^{p-1}+\beta u_2 |u_2|^{(p-3)/2} |u_1|^{p+1/2})\\
\int_\Omega u_i^2=\rho_i, \quad i=1,2,\qquad (u_1,u_2) \in H^1_0(\Omega;{{\mathbb{R}}}^2).
\end{cases}$$ for a suitable Lagrange multiplier $\gamma>0$. Then, repeating the above arguments, we obtain that $$\gamma>1 \text{ for some $\alpha$} \implies \hat c_{\alpha-{\varepsilon}}<\hat c_\alpha,$$ so that Theorem \[thm:existence\_bad\_cond\] applies.
Explicit estimates for Lambda {#sec:explicit_est}
-----------------------------
At this point, the main assumption in Theorem \[thm:existence\_bad\_cond\] is written in terms of the function $\Lambda(\rho_1,\rho_2)$ defined in , which we recall here for the reader’s convenience $${\Lambda}(\rho_1,\rho_2)=\frac{2C_{N,p}}{p+1} \max_{t\in [0,\pi/2]}
\left(\mu_1 \rho_1^{2r}\cos^{2a}t
+\mu_2 \rho_2^{2r}\sin^{2a}t
+2\beta^+ \rho_1^r\rho_2^{r} \cos^{a}t \sin^{a}t\right),$$ where $$a = \frac{N(p-1)}{4} \in \left(1,\frac{N}{N-2}\right],\qquad r = \frac{p+1}{4} - \frac{N(p-1)}{8}\in \left[0,\frac{1}{N}\right).$$ It is clear that ${\Lambda}$ is a $r$-homogeneous polynomial of $(\rho_1,\rho_2)$, but its explicit expression can be derived only in few particular cases. The aim of this subsection is to prove Theorem \[prop:supercritical\] by showing that condition in Theorem \[prop:supercritical\], with $R=R(\Omega,N,p)$ defined as $$\label{eq:def_R}
R(\Omega,N,p)=\frac{p+1}{2C_{N,p}}\frac{(a-1)^{a-1}}{a^a} \lambda_j(\Omega)^{-(a-1)},$$ implies assumption in Theorem \[thm:existence\_bad\_cond\]. Here, as usual, $j=1$ if $\beta\geq -\sqrt{\mu_1\mu_2}$ and $j=2$ otherwise (or simply $j=2$ for any $\beta$, in case one wants to avoid this weak dependence of $R$ on $\beta$). The advantage of with respect to is that of being more explicit; furthermore, the two conditions coincide in the case $\beta\leq0$, as proved in Remark \[rem:beta<0\] below.
We proved that $A$ is star-shaped with respect to the origin in Proposition \[prop\_starshaped\]. We estimate ${\Lambda}$ from above noticing that, as $a>1$, we have $$\begin{split}
{\Lambda}(\rho_1,\rho_2)\le{\Lambda}'&(\rho_1,\rho_2):=\frac{2C_{N,p}}{p+1} \max_{t\in [0,\pi/2]}
\left(\mu_1 \rho_1^{2r}\cos^{2}t
+\mu_2 \rho_2^{2r}\sin^{2}t
+2\beta^+ \rho_1^r\rho_2^{r} \cos t \sin t\right)\\
&=\frac{C_{N,p}}{p+1} \max_{t\in [0,\pi/2]}
\left[\left(\mu_1 \rho_1^{2r}+\mu_2 \rho_2^{2r}\right) +
\left(\mu_1 \rho_1^{2r}-\mu_2 \rho_2^{2r}\right)\cos 2t
+2\beta^+ \rho_1^r\rho_2^{r} \sin 2t\right]\\
&=\frac{C_{N,p}}{p+1} \max_{x^2+y^2=1}
\left[\left(\mu_1 \rho_1^{2r}+\mu_2 \rho_2^{2r}\right) +
\left(\mu_1 \rho_1^{2r}-\mu_2 \rho_2^{2r}\right)x
+2\beta^+ \rho_1^r\rho_2^{r} y\right].
\end{split}$$ Next, explicit computations show that $$\begin{split}
{\Lambda}'(\rho_1,\rho_2) &= \frac{C_{N,p}}{p+1}\left(\mu_1 \rho_1^{2r}+\mu_2 \rho_2^{2r} +
\sqrt{(\mu_1 \rho_1^{2r}-\mu_2 \rho_2^{2r})^2 + 4(\beta^+ \rho_1^r\rho_2^{r})^2 } \right)\\
&\le \frac{C_{N,p}}{p+1}\left(\mu_1 \rho_1^{2r}+\mu_2 \rho_2^{2r} +
|\mu_1 \rho_1^{2r}-\mu_2 \rho_2^{2r}| + 2\beta^+ \rho_1^r\rho_2^{r} \right)\\
&= \frac{2C_{N,p}}{p+1}\left[\max\{\mu_1 \rho_1^{2r},\mu_2 \rho_2^{2r}\} +
\beta^+ \rho_1^r\rho_2^{r} \right].
\end{split}$$ Therefore assumption , with $R$ as in , implies , so that we can apply Theorem \[thm:existence\_bad\_cond\] to conclude.
\[rem:beta<0\] Relations and coincide for $\beta\leq 0$. Indeed, in such case, the maximum in the definition of $\Lambda'$ is achieved when either $t=0$ or $t=\pi/2$, and for such values the estimate is an equality: $${\Lambda}(\rho_1,\rho_2)={\Lambda}'(\rho_1,\rho_2) = \frac{2C_{N,p}}{p+1} \max\{\mu_1 \rho_1^{2r},\mu_2 \rho_2^{2r}\} \qquad \text{ for } \beta\leq 0.$$
Orbital stability of the set of ground states {#sec:stab}
=============================================
This section is devoted to the proof of the stability statements, namely of Theorems \[prop:subcritical\]-b) and \[thm:stab\]. Our aim is to prove the stability of the sets $G$ and $G_{\bar \alpha}$ defined in the statements. Actually, in the case of global minimizers (i.e. $1<p\le 1+4/N$), the stability follows from the conservation of the energy and masses, and from the compactness of any minimizing sequence, see e.g. [@Cazenave2003 Remark 8.3.9]. The case of local minimizers (i.e. $1+4/N < p \le 2^*-1$), however, requires an adaptation of such arguments, in particular in the Sobolev critical case.
In order to provide a unified proof for all the cases, we first observe that global minimizers are also local ones. Recall the definitions of $B_\alpha, {{\mathcal{U}}}_\alpha$ in , and those of $c_\alpha,\hat{c}_\alpha$ in . For $p\leq 1+4/N$ and $(\rho_1, \rho_2)$ satisfying the assumptions of Theorem \[prop:subcritical\], by the inequalities and , we readily infer the existence of $\bar \alpha>\lambda_1(\Omega)$ such that $\{(|u_1|,|u_2|)\in H^1_0(\Omega;{{\mathbb{R}}}^2):\ (u_1,u_2)\in G\}\subseteq {{\mathcal{B}}}_{\bar \alpha}$ and $\inf_{{{\mathcal{M}}}}{{\mathcal{E}}}=c_{\bar \alpha}< \hat c_{\bar \alpha}$; in particular, $G=G_{\bar \alpha}$. For $p>1+4/N$ and $(\rho_1, \rho_2)\in A$, take $\bar\alpha\ge \lambda_1(\Omega)$ such that $c_{\bar\alpha}<\hat c_{\bar\alpha}$, as in the statement of Theorem \[thm:stab\], and satisfying moreover in the case $p=2^*-1$. Therefore, for the previous choice of $\bar \alpha$, we are reduced in all cases to prove the stability of the set $G_{\bar \alpha}$.
To this aim, we recall that a set ${{\mathcal{G}}}\subset H^1_0(\Omega;{{\mathbb{C}}}^2)$ is orbitally stable if for every ${\varepsilon}>0$ there exists $\delta>0$ such that, whenever $(\psi_1,\psi_2)\in H^1_0(\Omega;{{\mathbb{C}}}^2)$ satisfies $\operatorname{dist}_{H^1_0}((\psi_1,\psi_2),{{\mathcal{G}}})<\delta$, $\operatorname{dist}_{H^1_0}$ denoting the $H^1_0$–distance, then the solution $(\Psi_{1},\Psi_{2})$ of $$\begin{cases}
{{\mathrm{i}}}\partial_t\Psi_1 + \Delta\Psi_1 + \Psi_1( \mu_1 |\Psi_1|^{p-1} +\beta |\Psi_1|^{(p-3)/2}|\Psi_2|^{(p+1)/2} )=0\\
{{\mathrm{i}}}\partial_t\Psi_2 + \Delta\Psi_2 + \Psi_2( \mu_2 |\Psi_2|^{p-1} +\beta |\Psi_2|^{(p-3)/2}|\Psi_1|^{(p+1)/2} )=0\\
\Psi_i(0,\cdot) = \psi_i(\cdot),\qquad \Psi_i(t,\cdot)\in H^1_0(\Omega;{{\mathbb{C}}}^2).
\end{cases}$$ is such that $$\label{eq:os1}
(\Psi_1(t,\cdot),\Psi_2(t,\cdot))\text{ can be continued to a solution in } 0\leq t <+\infty$$ and $$\label{eq:os2}
\sup_{t>0} \operatorname{dist}_{H^1_0}((\Psi_1(t,\cdot),(\Psi_2(t,\cdot)),{{\mathcal{G}}}) <{\varepsilon},$$
As we mentioned, we prove the orbital stability of $G_{\bar\alpha}$ under the condition that for every $M>0$ there exists $T_0=T_0(M)$ such that if $\|(\psi_{1},\psi_{2})\|_{H^1_0(\Omega;{{\mathbb{C}}}^2)}\le M$ then the above Cauchy problem admits an unique solution on $[0,T_0)$, and that both ${{\mathcal{Q}}}$ and ${{\mathcal{E}}}$ are preserved along the solutions. Notice that, under these conditions, the failure of implies the failure of . Indeed, if does not hold, then since $T_0$ depends on the norm of the initial data we necessarily have $\|(\Psi_{1}(t,\cdot),\Psi_{2}(t,\cdot)\|_{H^1_0(\Omega;{{\mathbb{C}}}^2)} \to +\infty$ as $t$ approaches a finite endpoint of the maximal existence time-interval. Since $G_{\bar\alpha}$ is bounded in $H^1_0(\Omega;{{\mathbb{C}}}^2)$, cannot hold.
We start with the following preliminary considerations.
\[lemma:c\_c’\] Let $(u_1,u_2)\in G_{\bar \alpha}$. Then there exist $\theta_1,\theta_2\in {{\mathbb{R}}}$ such that $(u_1,u_2)=(e^{{{\mathrm{i}}}\theta_1} |u_1|,e^{{{\mathrm{i}}}\theta_2}|u_2|)$. In particular, $$\inf \left\{ {{\mathcal{E}}}(v_1,v_2): (v_1,v_2) \in H^1_0(\Omega;{{\mathbb{C}}}^2), (|v_1|,|v_2|)
\in{{\mathcal{B}}}_{\bar\alpha}\right\} = c_{\bar \alpha},$$ while $$\inf \left\{ {{\mathcal{E}}}(v_1,v_2): (v_1,v_2) \in H^1_0(\Omega;{{\mathbb{C}}}^2), (|v_1|,|v_2|)
\in{{\mathcal{U}}}_{\bar\alpha}\right\} =: \tilde c_{\bar \alpha}\le \hat c_{\bar \alpha}.$$
Given $(v_1,v_2)\in {{\mathcal{U}}}_{\bar\alpha}$, we have clearly that $(v_1,v_2) \in H^1_0(\Omega;{{\mathbb{C}}}^2)$ and that $(|v_1|,|v_2|)
\in{{\mathcal{U}}}_{\bar\alpha}$, so that $\tilde c_{\bar\alpha}\le \hat c_{\bar\alpha}$.
Now let $(u_1,u_2)\in G_{\bar \alpha}$. By the diamagnetic inequality [@LiebLoss Theorem 7.21], we have $\int_\Omega |\nabla |u_i||^2 \le \int_\Omega |\nabla u_i|^2$, for $i=1,2$, so that $c_{\bar \alpha}\le {{\mathcal{E}}}(|u_1|,|u_2|) \le {{\mathcal{E}}}(u_1,u_2)=c_{\bar \alpha}$. As a consequence, $\int_\Omega |\nabla |u_i||^2 = \int_\Omega |\nabla u_i|^2$, for $i=1,2$ so that equality holds in the diamagnetic inequality, whence $u_i$ is a complex multiple of $|u_i|$, that is to say $u_i=e^{{{\mathrm{i}}}\theta_i} |u_i|$ for some $\theta_i\in {{\mathbb{R}}}$, and the rest of the lemma follows.
Our general criterion for stability is the following.
\[prop:stability\] Let $\bar \alpha$ be as above. If $c_{\bar \alpha}<\tilde c_{\bar \alpha}$, then $G_{\bar \alpha}$ is (conditionally) orbitally stable.
The proof is presented after the following lemma.
\[lem:stability\_compact\] Let $\bar \alpha$ be as above. Let $\{(\psi_{1,n},\psi_{2,n})\}\subset H^1_0(\Omega;{{\mathbb{C}}}^2)$ satisfy, as $n\to\infty$, $$\label{eq:stability_compact1}
\int_\Omega |\psi_{i,n}|^2 \to\rho_i \quad\text{for }i=1,2, \qquad {{\mathcal{E}}}(\psi_{1,n},\psi_{2,n})\to c_{\bar \alpha}$$ and, for every $n$ sufficiently large, $$\label{eq:stability_compact2}
\int_\Omega |\nabla \psi_{1,n}|^2 + |\nabla \psi_{2,n}|^2 \leq (\rho_1+\rho_2)\bar\alpha + \text{o}(1).$$ Then there exists $(u_1,u_2)\in G_{\bar \alpha}$ such that, up to a subsequence, $(\psi_{1,n},\psi_{2,n})\to (u_1,u_2)$, strongly in $H^1_0(\Omega;{{\mathbb{C}}}^2)$.
By there exists $(\bar\psi_1,\bar\psi_2)\in H^1_0(\Omega;{{\mathbb{C}}}^2)$ such that, up to a subsequence, $\psi_{i,n}\rightharpoonup u_i$ weakly in $H^1_0(\Omega;{{\mathbb{C}}})$ and $\psi_{i,n}\to u_i$ in $L^2(\Omega;{{\mathbb{C}}})$ for $i=1,2$, as $n\to+\infty$. Then , and Lemma \[lemma:c\_c’\] provide, for $i=1,2$, $$\int_\Omega |u_i|^2 =\rho_i, \qquad
\int_\Omega |\nabla u_1|^2 + |\nabla u_2|^2 \leq (\rho_1+\rho_2)\bar\alpha, \qquad
{{\mathcal{E}}}(u_{1},u_{2})\ge c_{\bar \alpha}.$$
Now, in case $p<2^*-1$, we have that $(\psi_{1,n},\psi_{2,n})\to (u_1,u_2)$ also in $L^{p+1}(\Omega;{{\mathbb{C}}}^2)$. Then $$c_{\bar \alpha}\le {{\mathcal{E}}}(u_{1},u_{2}) \le \liminf_{n\to+\infty} {{\mathcal{E}}}(\psi_{1,n},\psi_{2,n}) = c_{\bar \alpha},$$ and the strong $H^1_0$ convergence follows, together with the fact that $(u_1,u_2)\in G_{\bar \alpha}$.
On the other hand, in case $p=2^*-1$, the result follows by Proposition \[prop:Sobcritconv\]: actually, such proposition is stated for real valued functions, but after Lemma \[lemma:c\_c’\] it is straightforward to check that its proof holds also for complex valued ones.
Suppose by contradiction that $\{(\psi_{1,n},\psi_{2,n})\}\subset H^1_0(\Omega;{{\mathbb{C}}}^2)$, $(u_{1,n},u_{2,n})\in G_{\bar \alpha}$ and $\bar{\varepsilon}>0$ are such that $$\label{eq:psi_to_u}
\lim_{n\to\infty} \|(\psi_{1,n},\psi_{2,n})-(u_{1,n},u_{2,n})\|_{H^1_0(\Omega;{{\mathbb{C}}}^2)}=0$$ and $$\sup_{t>0} \operatorname{dist}_{H^1_0}((\Psi_{1,n}(t,\cdot),\Psi_{2,n}(t,\cdot)),G_{\bar \alpha}) \geq 2\bar{\varepsilon},$$ where $(\Psi_{1,n},\Psi_{2,n})$ is the solution of with initial condition $(\psi_{1,n},\psi_{2,n})$. Then there exists $\{t_n\}$ such that, letting $\phi_{i,n}(x):=\Psi_{i,n}(t_n,x)$, $i=1,2$, $$\label{eq:stability_phi}
\operatorname{dist}_{H^1_0}((\phi_{1,n},\phi_{2,n}),G_{\bar \alpha}) \geq\bar{\varepsilon}.$$ Let us prove that $\{(\phi_{1,n},\phi_{2,n})\}$ satisfies and . Then Lemma \[lem:stability\_compact\] provides a contradicton to , thus concluding the proof.
By Lemma \[lem:stability\_compact\], $G_{\bar \alpha}$ is compact. Therefore, implies the existence of $(u_1,u_2)\in G_{\bar \alpha}$ such that, up to a subsequence, $$\label{eq:strongconvergence_stability}
(\psi_{1,n},\psi_{2,n})\to (u_{1},u_{2}) \qquad \text{ in } H^1_0(\Omega;{{\mathbb{C}}}^2).$$ This, combined with the continuity of Sobolev embeddings, implies that $(\psi_{1,n},\psi_{2,n})$ satisfies . Then the conservation of the mass and of the energy imply that $$\int_\Omega |\phi_{i,n}|^2=\int_\Omega |\psi_{i,n}|^2\to \rho_i \text{ for } i=1,2,\quad \text{ and } \quad {{\mathcal{E}}}(\phi_{1,n},\phi_{2,n})={{\mathcal{E}}}(\psi_{1,n},\psi_{2,n})\to c_{\bar \alpha},$$ as $n\to+\infty$, so that $(\phi_{1,n},\phi_{2,n})$ also satisfies .
To conclude the proof, let us check that, at least for a subsequence, $(\phi_{1,n},\phi_{2,n})$ satisfies , that is, we claim that, for $n$ sufficiently large, $$\label{eq:stability_contr2}
\int_\Omega |\nabla \phi_{1,n}|^2 + |\nabla \phi_{2,n}|^2 \leq(\rho_1+\rho_2) \bar \alpha + \text{o}(1)$$ By contradiction, assume there exists $\bar n\in {{\mathbb{N}}}$ and $\bar {\varepsilon}>0$ such that $$\int_\Omega |\nabla \phi_{1,n}|^2 + |\nabla \phi_{2,n}|^2 \geq (\rho_1+\rho_2)\bar \alpha +\bar {\varepsilon}.$$ Since $$\begin{gathered}
\int_\Omega |\nabla \Psi_{1,n}(0,\cdot)|^2 + |\nabla \Psi_{2,n}(0,\cdot)|^2 =\int_\Omega |\nabla \psi_{1,n}|^2 + |\nabla \psi_{2,n}|^2 \\
\leq \int_\Omega |\nabla u_{1,n}|^2 + |\nabla u_{2,n}|^2 + \text{o}(1) \leq (\rho_1+\rho_2)\bar\alpha + \text{o}(1)\end{gathered}$$ for $n$ large, then there exists $\bar t_n\in (0,t_n)$ such that $(\Psi_{1,n}(\bar t_n,\cdot), \Psi_{2,n}(\bar t_n,\cdot))$ satisfies and $$\int_\Omega |\nabla \Psi_{1,n}(\bar t_n,\cdot)|^2 + |\nabla \Psi_{2,n}(\bar t_n,\cdot)|^2 = (\rho_1+\rho_2)\bar \alpha + \text{o}(1)$$ and in particular . By Lemma \[lem:stability\_compact\] there exists $(\bar u_1,\bar u_2)\in G_{\bar \alpha}$ such that $$\int_\Omega |\nabla \bar u_{1}|^2 + |\nabla \bar u_2|^2=(\rho_1+\rho_2)\bar \alpha,$$ which contradicts the assumption $c_{\bar \alpha}<\tilde c_{\bar \alpha}$.
We proved Proposition \[prop:stability\] assuming that $c_{\bar \alpha}<\tilde c_{\bar \alpha}$. We now check that, since $c_{\bar \alpha}<\hat c_{\bar \alpha}$, this assumption is satisfied.
\[lem:c\_tilde\] Let $\bar \alpha$ be as above. Then $c_{\bar \alpha}<\tilde c_{\bar \alpha}$.
If by contradiction $\tilde c=c$, then there exists ${\varepsilon}_n\to 0$ and $(v_{1,n},v_{2,n})\in H^1_0(\Omega;{{\mathbb{C}}}^2)$ such that $$\|(v_{1,n},v_{2,n})\|^2_{H^1_0(\Omega;{{\mathbb{C}}}^2)}=\bar\alpha(\rho_1+\rho_2),
\qquad
\int_\Omega |v_{i,n}|^2=\rho_i,
\qquad
c_{\bar \alpha}\leq {{\mathcal{E}}}(v_{1,n},v_{2,n})\leq c_{\bar \alpha}+{\varepsilon}_n,$$ for every $n$, $i=1,2$. Letting $u_{i,n}:=|v_{i,n}|$, $i=1,2$, the diamagnetic inequality implies $$\label{eq:c_tilde1}
\|(u_{1,n},u_{2,n})\|^2_{H^1_0(\Omega;{{\mathbb{R}}}^2)} \leq
\|(v_{1,n},v_{2,n})\|^2_{H^1_0(\Omega;{{\mathbb{C}}}^2)}=(\rho_1+\rho_2)\bar\alpha,$$ so that $(u_{1,n},u_{2,n})$ is an admissible couple for the minimization problem $c$ and then $$\label{eq:c_tilde2}
c_{\bar \alpha}\leq {{\mathcal{E}}}(u_{1,n},u_{2,n})\leq {{\mathcal{E}}}(v_{1,n},v_{2,n})\leq c_{\bar \alpha}+{\varepsilon}_n.$$ In particular, $$\label{eq:c_tilde3}
\frac{1}{2}\left(\|(v_{1,n},v_{2,n})\|^2_{H^1_0(\Omega;{{\mathbb{C}}}^2)}-\|(u_{1,n},u_{2,n})\|^2_{H^1_0(\Omega;{{\mathbb{R}}}^2)}\right)= {{\mathcal{E}}}(v_{1,n},v_{2,n})- {{\mathcal{E}}}(u_{1,n},u_{2,n})\leq {\varepsilon}_n.$$ Then Lemma \[lem:stability\_compact\] applies to both sequences, yielding both $(v_{1,n},v_{2,n})\to (v_{1,\infty},v_{2,\infty})$ and $(u_{1,n},u_{2,n})\to (u_{1,\infty},u_{2,\infty})$, strongly in $H^1_0$. Passing to the limit in and , we infer $${{\mathcal{E}}}(u_{1,\infty},u_{2,\infty})= c_{\bar \alpha}
\qquad\text{and}\qquad
\|(u_{1,\infty},u_{2,\infty})\|^2_{H^1_0(\Omega;{{\mathbb{R}}}^2)}=
\|(v_{1,\infty},v_{2,\infty})\|^2_{H^1_0(\Omega;{{\mathbb{C}}}^2)}=\bar \alpha(\rho_1+\rho_2).$$ Then $(u_{1,\infty},u_{2,\infty})\in {{\mathcal{U}}}_{\bar\alpha}$, contradicting the fact that $c_{\bar \alpha}<\hat c_{\bar \alpha}$.
Recalling the first paragraph of this section, we have to prove that the set $G_{\bar \alpha}$ is (conditionally) orbitally stable. This is a direct consequence of Proposition \[prop:stability\] together with Lemma \[lem:c\_tilde\].
Asymptotic study as β→−∞ {#sec:segregation}
========================
In this section we prove Theorem \[thm\_beta-infty\]. Let $\mu_1,\mu_2>0$, and take $\rho_1,\rho_2>0$ satisfying $$\label{eq:uniform_beta_proof}
\begin{cases}
\rho_1,\rho_2>0 & \quad \text{ if }1<p<1+4/N, \\ \smallbreak
0<\mu_1\rho_1^\frac{2}{N},\mu_2\rho_2^\frac{2}{N}<\frac{N+2}{NC_{N}} &\quad \text{ if } p=1+\frac{4}{N}\\ \smallbreak
\max\{ \mu_1 \rho_1^{2r},\mu_2 \rho_2^{2r}\} \cdot (\rho_1+\rho_2)^{a-1} \le \frac{(a-1)^{a-1}}{a^a}
\lambda_2(\Omega)^{-(a-1)} &\quad \text{ if }1+\frac{4}{N}<p\leq 2^*-1
\end{cases}$$ Observe that all these conditions are independent from $\beta$. Combining Theorems \[prop:subcritical\], \[prop:supercritical\] and \[thm:existence\_bad\_cond\] (see also Remarks \[rem:uniform\_beta1\] and \[rem:uniform\_beta2\]) with the definition of $R$ in , we deduce that, given $\beta<0$, there exist positive functions $u_{1,\beta}, u_{2,\beta}$ and $\omega_{1,\beta},\omega_{2,\beta}\in {{\mathbb{R}}}$ such that $$\label{eq:system_elliptic_beta}
\begin{cases}
-\Delta u_{1,\beta}+ \omega_{1,\beta} u_{1,\beta}=\mu_1 u_{1,\beta}^{p}+\beta u_{1,\beta}^{(p-1)/2} u_{2,\beta}^{(p+1)/2}\\
-\Delta u_{2,\beta}+ \omega_{2,\beta} u_{2,\beta}=\mu_2 u_{2,\beta}^{p}+\beta u_{2,\beta}^{(p-1)/2} u_{1,\beta}^{(p+1)/2}\\
\int_\Omega u_i^2=\rho_i, \quad i=1,2,\\
(u_1,u_2) \in H^1_0(\Omega;{{\mathbb{R}}}^2)
\end{cases}$$ while $$\begin{aligned}
&{{\mathcal{E}}}(u_{1,\beta},u_{2,\beta})=\inf_{{\mathcal{M}}}{{\mathcal{E}}}& \text{ if } 1<p\leq 1+\frac{4}{N}\label{eq_uniformalphabar}\\
&{{\mathcal{E}}}(u_{1,\beta},u_{2,\beta})=\inf_{{{\mathcal{B}}}_{\bar \alpha}} {{\mathcal{E}}},\quad (u_{1,\beta},u_{2,\beta})\in {{\mathcal{B}}}_{\bar \alpha}\setminus {{\mathcal{U}}}_{\bar \alpha} & \text{ if } 1+\frac{4}{N}< p\leq 2^*-1 \label{eq_uniformalphabar2}\end{aligned}$$ where $\bar \alpha:=\frac{a}{a-1}\lambda_2(\Omega)$ in .
\[lemma\_betainfty\_aux\] Under the previous assumptions, there exists a constant $C>0$, independent of $\beta$, such that $$\|u_{i,\beta}\|_{H^1_0(\Omega)} + \|u_{i,\beta}\|_{L^\infty(\Omega)} + |\omega_{i,\beta}| \leq C \qquad \text{ for every $\beta<0$, $i=1,2$.}$$
Take $(\xi_1,\xi_2)\in {{\mathcal{M}}}$ (if $p\leq 1+\frac{1}{N}$) or $(\xi_1,\xi_2)\in {{\mathcal{B}}}_{\bar \alpha}$ (if $p>1+\frac{4}{N}$), with $\xi_1\cdot \xi_2\equiv 0$ in either case. From – we have $$\begin{aligned}
\label{eq:energybound}
{{\mathcal{E}}}(u_{1,\beta},u_{2,\beta})\leq {{\mathcal{E}}}(\xi_1,\xi_2)=\frac{1}{2}\int_\Omega (|\nabla \xi_{1}|^2+|\nabla \xi_{2}|^2)-\frac{1}{p+1}\int_\Omega (\mu_1|\xi_1|^{p+1}+\mu_2 |\xi_2|^{p+1})=:C_1\end{aligned}$$ where $C_1$ is independent of $\beta<0$.
From the first statement in we deduce that $\{(u_{1,\beta},u_{2,\beta})\}_{\beta<0}$ is uniformly bounded in $H^1_0(\Omega)$ for $p> 1+\frac{4}{N}$. In case $1<p<1+\frac{4}{N}$, the $H^1_0$-boundedness follows combining with the estimate $${{\mathcal{E}}}(u_{1,\beta},u_{2,\beta}) \geq
\|\nabla u_{1,\beta}\|_2^{2a} \left( \frac{1}{2} \|\nabla u_{1,\beta}\|_{2}^{2-2a} -\frac{C_{N,p}}{p+1}{\mu_1\rho_1^{2r}}\right) +
\|\nabla u_{2,\beta}\|_2^{2a} \left( \frac{1}{2} \|\nabla u_{2,\beta}\|_{2}^{2-2a} -\frac{C_{N,p}}{p+1}{\mu_2\rho_2^{2r}}\right)$$ ($a:=N(p-1)/4<1$), which corresponds to for $\beta<0$, while for $p=1+\frac{4}{N}$ it follows from and $${{\mathcal{E}}}(u_{1,\beta},u_{2,\beta}) \geq \frac{1}{2}\left(1-\frac{NC_N \mu_1\rho_1^{2/N}}{N+2}\right)\|\nabla u_{1,\beta}\|_2^2 + \frac{1}{2}\left(1-\frac{NC_N \mu_2\rho_2^{2/N}}{N+2}\right)\|\nabla u_{2,\beta}\|_2^2.$$ (see with $\beta<0$).
By the Sobolev embedding $H^1_0(\Omega)\hookrightarrow L^{p+1}(\Omega)$, we have that $\{(u_{1,\beta},u_{2,\beta})\}_{\beta<0}$ is uniformly bounded in the $L^{p+1}$-norm. In particular, $$\begin{aligned}
0\leq \frac{2(-\beta)}{p+1}\int_\Omega (u_{1,\beta} u_{2,\beta})^{(p+1)/2} \leq {{\mathcal{E}}}(u_{1,\beta},u_{2,\beta}) + \frac{1}{p+1} \int_\Omega \mu_1 u_{1,\beta}^{p+1}+\mu_2 u_{2,\beta}^{p+1}\leq C_2.\end{aligned}$$ By testing the first equation in by $u_{1,\beta}$ and the second one by $u_{2,\beta}$, and usign the previous estimates, we have, for $i,j\in \{1,2\}$, $i\neq j$, $$\begin{gathered}
\rho_i|\omega_{i,\beta}|=\left|\int_\Omega \mu_i u_{i,\beta}^{p+1}+\beta (u_{1,\beta}u_{2,\beta})^{(p+1)/2}- |\nabla u_{i,\beta}|^2\right| \\
\leq \int_\Omega \mu_i u_{i,\beta}^{p+1}+|\beta| (u_{1,\beta}u_{2,\beta})^{(p+1)/2}+ |\nabla u_{i,\beta}|^2 \le C_3.\end{gathered}$$ Now we can use a Brezis-Kato-Moser type argument exactly as in [@MR2928850 pp. 1264–1265], obtaining uniform $L^\infty$–bounds for $\{(u_{1,\beta},u_{2,\beta})\}_{\beta<0}$.
By Lemma \[lemma\_betainfty\_aux\], $\{(u_{1,\beta},v_{2,\beta})\}_{\beta<0}$ satisfies the assumptions of [@STTZ Theorems 1.3 and 1.5]. Therefore this sequence is uniformly bounded in $C^{0,\alpha}(\overline \Omega)$ for every $0<\alpha<1$, and there exist $(u_1,u_2)\in C^{0,1}(\overline \Omega)$ with $u_1,u_2\geq 0$ in $\Omega$, and $(\omega_1,\omega_2)\in {{\mathbb{R}}}^2$ such that, up to subsequences, as $\beta\to -\infty$ we have $$u_{i,\beta}\to u_i \text{ in } C^{0,\alpha}(\overline \Omega)\cap H^1_0(\Omega), \qquad \omega_i \to \omega_i.$$ (see also [@MR2599456; @SZ15]). By [@DWZ Theorem 1.2] (which is stated for $p=3$, but holds also for a general $p$ without any extra efford), we have $$-\Delta (u_1-u_2) + \omega_1 u_1-\omega_2 u_2 \geq \mu_1 u_1^{p} - \mu_2 u_2^p\quad \text{ and }\quad -\Delta (u_2-u_1) + \omega_2 u_2-\omega u_1 \geq \mu_2 u_2^p-\mu_1 u_1^p \quad \text{ in } \Omega.$$ We can now conclude by taking $w:=u_1-u_2$.
Acknowledgments {#acknowledgments .unnumbered}
===============
All authors are partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”. H. Tavares is partially supported by FCT (Portugal) grant UID/MAT/04561/2013. G. Verzini is partially supported by the PRIN-2015KB9WPT Grant: “Variational methods, with applications to problems in mathematical physics and geometry”. B. Noris and G. Verzini are partially supported by the INDAM-GNAMPA group.
[10]{}
G. P. Agrawal. . Springer, 2000.
A. Ambrosetti and E. Colorado. Standing waves of some coupled nonlinear [S]{}chr[ö]{}dinger equations. , 75(1):67–82, 2007.
A. Ambrosetti and G. Prodi. , volume 34 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 1993.
T. Bartsch and L. Jeanjean. Normalized solutions for nonlinear [S]{}chrödinger systems. , 148(2):225–242, 2018.
T. Bartsch, L. Jeanjean, and N. Soave. Normalized solutions for a system of coupled cubic [S]{}chrödinger equations on [$\mathbb{R}^3$]{}. , 106(4):583–614, 2016.
T. [Bartsch]{} and N. [Soave]{}. . , (arXiv:1703.02832), 2017.
T. Bartsch and N. Soave. A natural constraint approach to normalized solutions of nonlinear [S]{}chrödinger equations and systems. , 272(12):4998–5037, 2017.
T. Bartsch and Z.-Q. Wang. Note on ground states of nonlinear [S]{}chr[ö]{}dinger systems. , 19(3):200–207, 2006.
J. Bellazzini, N. Boussaïd, L. Jeanjean, and N. Visciglia. Existence and stability of standing waves for supercritical [NLS]{} with a partial confinement. , 353(1):229–251, 2017.
D. Bonheure, L. Jeanjean, and B. Noris. Orbital stability of the ground states for the nonlinear [S]{}chrödinger equation with harmonic potential. , 2018.
H. Br[é]{}zis and E. H. Lieb. A relation between pointwise convergence of functions and convergence of functionals. , 88(3):486–490, 1983.
H. Brézis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical [S]{}obolev exponents. , 36(4):437–477, 1983.
T. Cazenave. , volume 10 of [*Courant Lecture Notes in Mathematics*]{}. New York University Courant Institute of Mathematical Sciences, New York, 2003.
S.-M. Chang, C.-S. Lin, T.-C. Lin, and W.-W. Lin. Segregated nodal domains of two-dimensional multispecies [B]{}ose-[E]{}instein condensates. , 196(3-4):341–361, 2004.
M. Cirant and G. Verzini. Bifurcation and segregation in quadratic two-populations mean field games systems. , 23(3):1145–1177, 2017.
M. Conti, S. Terracini, and G. Verzini. Nehari’s problem and competing species systems. , 19(6):871–888, 2002.
E. N. Dancer, K. Wang, and Z. Zhang. The limit equation for the [G]{}ross-[P]{}itaevskii equations and [S]{}. [T]{}erracini’s conjecture. , 262(3):1087–1131, 2012.
E. N. Dancer, J. Wei, and T. Weth. A priori bounds versus multiple existence of positive solutions for a nonlinear [S]{}chr[ö]{}dinger system. , 27(3):953–969, 2010.
G. Fibich and F. Merle. Self-focusing on bounded domains. , 155(1-2):132–158, 2001.
R. Fukuizumi, F. H. Selem, and H. Kikuchi. Stationary problem related to the nonlinear schr[ö]{}dinger equation on the unit ball. , 25(8):2271, 2012.
T. Gou and L. Jeanjean. Existence and orbital stability of standing waves for nonlinear [S]{}chrödinger systems. , 144:10–22, 2016.
T. Gou and L. Jeanjean. Multiple positive normalized solutions for nonlinear schr[ö]{}dinger systems. , 31(5):2319, 2018.
M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry, i. , 74(1):160–197, 1987.
L. Jeanjean. Existence of solutions with prescribed norm for semilinear elliptic equations. , 28(10):1633–1659, 1997.
M. K. Kwong. Uniqueness of positive solutions of [$\Delta u-u+u^p=0$]{} in [${\bf
R}^n$]{}. , 105(3):243–266, 1989.
E. H. Lieb and M. Loss. , volume 14 of [*Graduate Studies in Mathematics*]{}. American Mathematical Society, Providence, RI, second edition, 2001.
T.-C. Lin and J. Wei. Ground state of [$N$]{} coupled nonlinear [S]{}chrödinger equations in [$\mathbf{R}^n$]{}, [$n\leq 3$]{}. , 255(3):629–653, 2005.
T.-C. Lin and J. Wei. Erratum: “[G]{}round state of [$N$]{} coupled nonlinear [S]{}chrödinger equations in [${\bf R}^n$]{}, [$n\leq3$]{}” \[[C]{}omm. [M]{}ath. [P]{}hys. [**255**]{} (2005), no. 3, 629–653; mr2135447\]. , 277(2):573–576, 2008.
L. A. Maia, E. Montefusco, and B. Pellacci. Positive solutions for a weakly coupled nonlinear [S]{}chr[ö]{}dinger system. , 229(2):743–767, 2006.
B. Noris, H. Tavares, S. Terracini, and G. Verzini. Uniform [H]{}[ö]{}lder bounds for nonlinear [S]{}chr[ö]{}dinger systems with strong competition. , 63(3):267–302, 2010.
B. Noris, H. Tavares, S. Terracini, and G. Verzini. Convergence of minimax structures and continuation of critical points for singularly perturbed systems. , 14(4):1245–1273, 2012.
B. Noris, H. Tavares, and G. Verzini. Existence and orbital stability of the ground states with prescribed mass for the [$L^2$]{}-critical and supercritical [NLS]{} on bounded domains. , 7(8):1807–1838, 2014.
B. Noris, H. Tavares, and G. Verzini. Stable solitary waves with prescribed [$L^2$]{}-mass for the cubic [S]{}chrödinger system with trapping potentials. , 35(12):6085–6112, 2015.
D. Pierotti and G. Verzini. Normalized bound states for the nonlinear [S]{}chrödinger equation in bounded domains. , 56(5):Art. 133, 27, 2017.
H. A. Rose and M. I. Weinstein. On the bound states of the nonlinear [S]{}chrödinger equation with a linear potential. , 30(1-2):207–218, 1988.
B. Sirakov. Least energy solitary waves for a system of nonlinear [S]{}chrödinger equations in [$\mathbb{R}^n$]{}. , 271(1):199–221, 2007.
N. Soave, H. Tavares, S. Terracini, and A. Zilio. H[ö]{}lder bounds and regularity of emerging free boundaries for strongly competing [S]{}chr[ö]{}dinger equations with nontrivial grouping. , 138:388–427, 2016.
N. Soave and A. Zilio. Uniform bounds for strongly competing systems: the optimal [L]{}ipschitz case. , 218(2):647–697, 2015.
M. Struwe. , volume 34 of [*Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics \[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics\]*]{}. Springer-Verlag, Berlin, fourth edition, 2008. Applications to nonlinear partial differential equations and Hamiltonian systems.
E. Timmermans. Phase separation of bose-einstein condensates. , 81:5718–5721, Dec 1998.
M. I. Weinstein. Nonlinear [S]{}chrödinger equations and sharp interpolation estimates. , 87(4):567–576, 1982/83.
`benedetta.noris@u-picardie.fr`\
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée\
Université de Picardie Jules Verne\
33 Rue Saint-Leu, 80039 Amiens (France)
`hrtavares@fc.ul.pt`\
CMAFCIO & Departamento de Matemática\
Faculdade de Ciências da Universidade de Lisboa\
Edifício C6, Piso 1, Campo Grande 1749–016 Lisboa (Portugal)
`gianmaria.verzini@polimi.it`\
Dipartimento di Matematica, Politecnico di Milano\
Piazza Leonardo da Vinci 32, 20133 Milano (Italy)
|
---
abstract: 'We report thermodynamic values of four-point renormalized coupling constant calculated by Monte Carlo simulations in the continuum limits of the lattice versions of the two-dimensional O(2) and O(3) non-linear sigma models. In each case the critical index of the coupling constant vanishes, which leads to hyperscaling ( non-triviality).'
author:
- 'Jae-Kwon Kim'
---
[Scaling behavior of the four-point renormalized coupling constant in the two dimensional O(2) and O(3) non-linear sigma models]{}
[Department of Physics, University of California, Los Angeles, CA 90024, and\
Department of Physics, University of Arizona, Tucson, AZ 85721]{}
A general problem of (Euclidean) quantum field theory (QFT) is whether the n-point ($n>2$) correlation functions, which are constructed so as to satisfy Osterwalder-Schrader axioms[@OST], are those of the Gaussian model in the continuum limit. If so this QFT is non-interacting, and is referred to be a [*trivial*]{} theory[@SOK]. It is rigorously proven that one cannot construct an interacting continuum limit of the $\lambda \phi^{4}$ theory in the symmetric phase in dimensions larger than 4 ($D > 4$)[@AIZ], and is widely conjectured that this is the case even in four dimensions[@HARA]. A trivial theory, which might be interacting only when a finite cutoff is imposed in the theory, cannot be a candidate of a genuine QFT, so it is very important to find out whether a theory is trivial or not. Without any proof, it is generally believed that a QFT which is not asymptotic free in 4D would be trivial. An important example of such a theory is the 4D non-compact QED, and the issue of triviality in this theory remains unresolved with variant conclusions of studies.
As far as the issue of the triviality is concerned, studying the scaling behavior of the four-point renormalized coupling constant ($g^{(4)}$) is primarily important. A rigorous theorem[@NEW] states that vanishing of $g^{(4)}$ is sufficient for triviality in the Ising like ferromagnetic systems, and it is conjectured that the same holds for other type of ferromagnetic lattice models[@SOK]. While much (numerical) studies on the scaling behavior of $g_{R}^{(4)}$ have been done for the $\lambda \phi^{4}$ theories in various dimensions[@WEIN][@KIMD] and for the Yukawa type interaction[@HAS], little has been studied for the 2D O(N) models for $N \ge 2$[@LUS], which share many similarities with 4D lattice gauge theories. In particular, the 2D O(2) non-linear sigma model is not asymptotic free in the context of perturbation theory, while O($N\ge 3$) non-linear sigma models are asymptotic free[@POL].
In this article, we investigate the scaling behaviors of $g^{(4)}$ for the 2D O(N) (N=2,3) non-linear sigma models on a square lattice of linear size L with periodic boundary condition, based on Monte Carlo simulations. The model is defined by the action $$A=-\beta \sum_{<i,j>} \sigma_{i} \cdot \sigma_{j},$$ where $\beta$ is the inverse temperature ($\beta=1/T$), and $\sigma_{i}$ is an N-dimensional unit vector at lattice site $i$. We report compelling numerical evidences that the continuum limits of both models are not trivial, irrespective of the property of perturbative asymptotic freedom.
The four- point renormalized coupling constant ($g^{(4)}$) which is basically the connected four-point function defined at zero-momentum, and which is constructed so as to be independent of rescaling of field strength, can be written in a system with translation symmetry as $$g^{(4)} = \lim_{L \to \infty} g_{L}^{(4)}\equiv
\lim_{L \to \infty} U_{L}~~ (\xi_{L}/L)^{D}, \label{eq:def}$$ Here, $g_{L}^{(4)}$ and $\xi_{L}$ are respectively the renormalized coupling constant and correlation length[@DEF] defined on a finite lattice of linear size L, and $U_{L}$ is a modified (N component) Binder’s cumulant ratio defined as $U_{L}\equiv [(1+2/N)<S^{2}>^{2}-)<S^{4}>]/<S^{2}>^{2}$, with $S^{2}\equiv |\sum_{i} \sigma_{i}|^{2}$[@BER].
For a system displaying power-law type scaling behavior, $g^{(4)}$ scales as $g^{(4)}(t) \sim t^{-2\Delta+\gamma+D\nu}$, where the notations are standard ($t$, for example, is the deviation of the dimensionless temperature from the coupling defined as $t\equiv (\beta_{c}-\beta)/\beta_{c}$). The inequality, $2\Delta \le \gamma+D\nu$ holds[@BAK] in general for ferromagnetic systems with power-law singularities. For the $\lambda \phi^{4}$ models, the inequality has been rigorously proven for $D >4$ so that $\lim_{t \to 0} g^{(4)} \to 0$, whereas the equality holds for $ D \le 4$ (hyperscaling)[@AIZ]. In 4D, it is rigorously proven under some mild assumptions[@HARA] that the scaling behavior of $g^{(4)}(t)$ has such a multiplicative logarithmic correction that $\lim_{t \to 0} g^{(4)}(t) \to 0$.
For the 2D O(N) (N=2,3) models, we have calculated $g^{(4)}$ by employing Wolff’s single cluster algorithm [@WOL] on square lattice with periodic boundary condition. For each model, our continuum limit is achieved by adjusting the value of $\beta$ (or, $T$) so that the correlation length in unit of lattice spacing starts to be diverging; at the same time, the lattice spacing scales with $\beta$ so that the correlation length multiplied by the lattice spacing remains a constant. In general the auto-correlation time for $g^{(4)}$ is much larger than that of $\chi$ or $\xi$, so much more computational efforts are required for the precise determination of $g^{(4)}$. For a given $\beta$ and L, 20-80 different [*bins*]{} were obtained for our calculations, with each bin being composed of 10 000 measurements each of which was separated by 8-15 consecutive one cluster updating. The statistical errors were estimated by jack-knife method.
In order to monitor the effect of finite size in the measurements of $g^{(4)}$, for each model at an arbitrary temperature in the scaling region, we measured $g_{L}^{(4)}$ by varying L by 10 from L= 10 until $g_{L}^{(4)}$ did not vary with further increasing of L. For each model, we conclude that $g_{L}^{(4)}$ (as well as $\chi$ and $\xi$) converges to its thermodynamic value (within the statistical error of very precise Monte Carlo data) on the condition that $L /\xi_{\infty} \ge 7$ (Figure(1)). According to the theory of finite size scaling[@KIML], the following relation holds for the renormalized four-point coupling: $$g_{L}^{(4)}(\beta)= g^{(4)}(\beta) f_{g}(s),~~s\equiv L/\xi_{\infty}(\beta)
\label{eq:fun},$$ with $f_{g}$ representing a scaling function characterized by $g^{(4)}$. $f_{g}(s)$ has no explicit temperature dependence so that the [*thermodynamic*]{} condition holds for any temperature.
We thus chose L such that $ L /\xi_{\infty} \simeq 7$ for the direct measurement of the thermodynamic $g^{(4)}$ (T=1.19, 1.10, 1.04, and 1.02 for the O(2) model; $\beta=1.5, 1.6$, and 1.7 for the O(3) model). For the temperatures where the corresponding $\xi$ becomes very large, we fix $L/\xi_{\infty}$ to be a value much smaller than 7 so that, according to Eq.(\[eq:fun\]), $g_{L}^{(4)}$ thus obtained is exactly proportional to its corresponding thermodynamic value. When the corresponding $\xi_{\infty}(\beta)$ is known at a certain $\beta$, this procedure enables one to obtain accurate thermodynamic values of other physical quantities without using sufficiently large L required in direct measurements[@KIML]. Let us illustrate this for the 2D O(2) model, where very accurate values of $\xi_{\infty}$ are already available up to $T=0.98$[@GUP]. At $T=1.02$ ($\xi_{\infty}=26.20(20)$), we obtained $g^{(4)}=8.86(13)$ using L= 184 and $g_{55}^{(4)}=4.79(2)$, so $f_{g}(s)$ is calculated to be 0.541(10) for $s= L/\xi_{\infty}=2.10(2)$. At $T=0.98$ ($\xi \simeq 70.5(7)$), choosing L=148 makes the value of $s$ the same as at $T=1.02$ within the statistical errors, so with our Monte Carlo data, $g_{148}^{(4)}=4.81(2)$, we extract $g^{(4)}=8.89(20)$ for $T=0.98$. For the O(3) model, we fixed the value of $s=1.165(21)$ by choosing L=40, 75, and 142 for $\beta=1.7, 1.8,$ and 1.9 respectively. Our final results are summerized in Table (1).
For such broad ranges of correlation length ($5.01(2) \le \xi \le 70.5(9)$ for the O(2); $11.1(1) \le \xi \le 121.9(7)$ for the O(3)) we observe that $g^{(4)}(t) \sim t^{0}$. These behaviors are qualitatively the same as those in the 2D and 3D Ising models[@KIMD]. We thus conclude that the continuum limits are non-trivial for both models. In particular, the perturbative property of the non-asymptotic freedom in the 2D O(2) model and that of the asymptotic freedom in the 2D O(3) model do not affect the non-trivialities in these models. Especially, because of the asymptotic freedom in the 2D O(3) model, the value of the renormalized coupling constant defined at any other value of the momentum must be smaller than our value, $g^{(4)}=6.6(1)$. Another remarkable conclusion of the current study is the existence of hyperscaling in these models. Hyperscaling relations cannot be defined in our models due to the exponential singularities in the critical behaviors of $\xi$ and $\chi$; nevertheless, the fundamental claim of the hyperscaling in the sense that the thermodynamic correlation length is the only relevant scale in the scaling region, still holds in two dimensions irrespective of the type of critical behavior.
The author would like to thank Chuck Buchanan for hospitality; Ghi-Ryang Shin and Jaeshin Lee for their support in computing. The Monte Carlo simulations were carried out on Convex C240 at the University of Arizona.
------ ----------- ---------- ---------- ---------- ---------- ----------
O(2) T 1.19 1.10 1.04 1.02 0.98
$g^{(4)}$ 8.70(16) 8.80(12) 8.71(10) 8.86(13) 8.89(20)
O(3) $\beta$ 1.5 1.6 1.7 1.8 1.9
$g^{(4)}$ 6.58(13) 6.70(15) 6.50(6) 6.50(11) 6.57(13)
------ ----------- ---------- ---------- ---------- ---------- ----------
: $g^{(4)}$ for the 2D O(2) and O(3) non-linear sigma models. For the O(3) model, we fixed $s=L/\xi_{\infty}=1.165$ by choosing L=40, 75, and 142 for $\beta=1.7, 1.8,$ and 1.9 respectively. The corresponding values of $g_{L}^{(4)}$ are 2.56(1), 2.56(1), and 2.59(2) respectively, showing the constancy of $g^{(4)}$ in this range of $\beta$ as well. $f_{g}(s)$ is calculated to be 0.394(5). $\xi_{\infty}(\beta)$ =64.6(5) and 121.7(8) for $\beta$=1.8 and 1.9 [@FOX].
[99]{} K. Osterwalder and R. Schrader, Comm. Math. Phys. [**31**]{} (1973) 83; [*ibid*]{}, [**42**]{} (1975) 281 For a review, see, R. Fernández, J. Fröhlich, and A.D. Sokal, [*Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory*]{}, (Springer-Verlag, Berlin, 1992) M. Aizenman, Phys. Rev. Lett.[**47**]{} (1981) 1; J. Frölich, Nucl. Phys. [**200**]{} \[FS4\] (1982) 281 T. Hara, J. Stat. Phys. [**47**]{} (1987) 57; T. Hara and H. Tasaki, J. Stat. Phys. [**47**]{} (1987) 99 C.M. Newman, Comm. Math. Phys. [**41**]{} (1975) 75 B. Freedmann, P. Smolensky and D. Weingarten, Phys. Lett. [**113 B**]{} (1982) 481 J.-K. Kim and A. Patrascioiu, Phys. Rev. D, [**47**]{} (1993) 2588 See, for example, R. E. Shrock, in [*Quantum Fields on the Computer*]{} edited by M. Creutz (World Scientific, Singapore, 1992) For a numerical demonstration of the non-triviality of the S-matrix in the 2D O(3) model, however, see M. Lüscher and U. Wolff, Nucl. Phys. B [**339**]{} (1990) 222, A.M. Polyakov, Phys. Lett. [**59B**]{} (1975) 79 For the definition of $\xi_{L}$, see for example Ref.[@KIMD] W. Bernreuther and M. Göckler, Phys. Lett. [**214B**]{} (1988) 109 G.A. Baker Jr, Phys. Rev. Lett [**20**]{} (1968) 990; R. Schrader, Phys. Rev. B [**14**]{} (1976) 172 U. Wolff, Phys. Rev. Lett. [**62**]{} (1989) 361 J.-K. Kim, Phys. Rev. Lett. [**70**]{} (1993) 1735 R. Gupta and C.F. Baillie, Phys. Rev. B [**45**]{} (1992) 2883 J. Apostolakis, C.F. Baillie, and G. Fox, Phys. Rev. D [**43**]{} (1991) 2687; The correlation lengths are measured up to 15000 in the 2D O(3) model. J.-K. Kim, Phys. Rev. D [**50**]{} (1994) 4663
\
Figure(1): $g_{L}^{(4)}/g^{(4)}$ as a function of $L/\xi_{\infty}$ at $T=1.10$ ($\xi_{\infty}=9.32(2)$) for the 2D O(2) model, and at $\beta=1.5$ ($\xi_{\infty}=11.05(2)$) for the 2D O(3) model. The range of L is \[10,80\] for the O(2), and is \[10,100\] for the O(3), varied by 10.
|
Beside Linear Momentum (LM) and Spin Angular Momentum (SAM), photons possess transverse degrees of freedom: Orbital Angular Momentum (OAM) and radial profile distribution [@arnold:08]. The light OAM received a lot of attentions in both classical and quantum regimes during last two decades [@gibson:04; @mair:01; @nagali:09b; @karimi:10a]. The light OAM is characterized by a vortex optical phase, whose topological charge is given by the azimuthal index $m=0,\pm1,\pm2,\dots$. On the contrary, the Radial Index (RI) $p=0,1,2,\dots$, which is associated to the intensity distribution of the light in the transverse plane, has been out of interest almost completely. Nevertheless, these two transverse degrees of freedom, i.e. OAM and radial intensity distribution, are strictly correlated, and different OAM generators produce specific (and different) distributions of radial modes. For instance, cylindrical mode converters generate pure Laguerre-Gauss (LG) modes starting from the Hermite-Gauss (HG) modes, spiral phase plates and pitch-fork holograms generate Hypergeometric-Gaussian (HyGG) modes, and $q$-plates generate other kind of radial HyGG modes as well [@padgett:04; @berry:04; @bekshaev:08; @karimi:07; @karimi:09a]. Furthermore, some type of OAM projectors such as the single mode optical fibers widely used in the light quantum regime, affect the radial profile independently of the OAM value. In fact, optical fibers are not good OAM detectors since they are sensible only to specific radial indices [@karimi:09b; @molinaterriza:07a].\
Unlike LM and OAM photon eigenstates, the radial modes exhibits a more complicate structure, which render their manipulation less obvious. In this respect, to overcome the difficulties inherent to the radial profile of paraxial beans, a deeper understanding of the underlying symmetry is advisable. Such hidden symmetry, in fact, can be used to unveil novel and interesting features of the radial modes. In particular, the radial profile operator and *its conjugate-variable* can be found from the symmetry of the state. Among the possible radial states, of particular interest is the coherent state(CS). The CS, in fact, is the quantum state closest to the classical one and provides a minimum uncertainty between two conjugate observables [@perelomov:86]. Erwin Schrödinger was the first to derive a CS for the case of the harmonic oscillator in 1926, when he was looking for a quantum wave-packet with minimum momentum-position uncertainty [@schrodinger:26]. Later on, generalized coherent states (GCS) were developed for different dynamical systems and symmetries. In order to define the GCS, several strategies have been used: minimizing the uncertainty relations of two conjugate observables, thus creating *ideal* states, finding the group lowering operator eigenvalue, thus creating *Intelligent* States (IS), or introducing a suitable displacement operator acting on some reference state (ground state, usually), thus creating proper *Coherent* States (CS) [@perelomov:86; @barut:71].\
In this paper, we studied the dynamical (or hidden) symmetry of the scalar paraxial wave equation, by considering the group algebra of the LG modes. Both IS and CS radial modes were then constructed starting from the algebra and some features of these modes were described in the position representation of the state. The most common optical beams generated by laser sources obeys the scalar Paraxial Wave Equation (PWE) $$\begin{aligned}
\label{eq:pwe}
\left(\mathbf{\nabla}^2_{\bot}+4i\partial_\zeta\right)\psi(\rho,\phi,\zeta)=0,\end{aligned}$$ where $(\rho,\phi,\zeta)$ and $\mathbf{\nabla}^2_{\bot}$ are dimensionless cylindrical coordinate and the transverse Laplacian, respectively [@note]. The field at any given $\zeta$-plane can be calculated by applying a unitary propagator ${\widehat{U}}$ to the pupil wave-function, ${|\psi\rangle}_\zeta={\widehat{U}}(\zeta,\zeta'){|\psi\rangle}_{\zeta'}$, where ${\widehat{U}}(\zeta,\zeta')=\exp{\left(\frac{i}{2}\,(\zeta-\zeta')\mathbf{\nabla}^2_{\bot}\right)}$. In the position representation the operator ${\widehat{U}}$ is the Fresnel propagation kernel $K(\rho,\phi,\zeta;\rho',\phi',\zeta')={\langle \rho,\phi|}{\widehat{U}}(\zeta,\zeta'){|\rho',\phi'\rangle}$. Hereafter, without loss of generality, we consider the pupil solution of the PWE, and keep whole calculations in $\zeta=0$ plane. LG modes, $\mbox{LG}_{p,m}(\rho,\phi,0):={\langle\rho,\phi|\mbox{p,m}\rangle}$, are a solutions of the scalar PWE with pupil function given by
![\[fig:lg\_basis\] (Color online) Simulated spectra of different RCS in the basis of LG modes. (a) $m=1$, $|\alpha|=0.2$ (b) $m=3$, $|\alpha|=0.2$ (c) $m=3$, $|\alpha|=0.5$, and (d) $m=3$, $|\alpha|=0.9$. ](f1.eps)
$$\begin{aligned}
\label{eq:paraxial_lg}
\mbox{LG}_{p,m}(\rho,\phi,0)=\sqrt{\frac{ 2^{|m|+1}p!}{\pi(p+|m|)!}}\,\rho^{|m|} e^{-\rho^2} L_{p}^{|m|}\left(2\rho^2\right),\nonumber\end{aligned}$$
where $m$ is an integer number defining the OAM eigenvalue, $p\geq0$ is a not negative integer number defining the radial nodes in the beam transverse plane, and $L_{p}^{|m|}(.)$ is the generalized Laguerre polynomial. The LG modes are orthogonal set of PWE’s solution, i.e. ${\langle\mbox{p}',\mbox{m}'|\mbox{p,m}\rangle}=\delta_{p,p'}\delta_{m,m'}$, and carry a finite power [@siegman:86]. Here, we focus our attention to the radial node Hilbert space keeping the OAM number $m$ fixed. In order to find the dynamical (or hidden) symmetry lying behind the radial nodes of LG modes, we need the algebra of the node space. A straightforward calculation shows that the radial nodes ladder operators for the LG modes are given by $$\begin{aligned}
\label{eq:ladder_def}
{\widehat{\cal{P}}}_+{|\mbox{p,m}\rangle}&=&{\cal{P}}_+{|\mbox{p+1,m}\rangle}\cr
{\widehat{\cal{P}}}_-{|\mbox{p,m}\rangle}&=&{\cal{P}}_-{|\mbox{p-1,m}\rangle}.\end{aligned}$$ where ${\widehat{\cal{P}}}_{\pm}={\widehat{\cal{Q}}}\mp\left(\frac{|m|}{2}-\rho^2\right)$, ${\widehat{\cal{Q}}}=\frac{1}{2}\rho\,\partial_\rho+2\rho^2-({\widehat{p}}+|m|+1)$ are the raising and lowering operators with eigenvalues ${\cal P}_{+}=\sqrt{(p+1)(p+|m|+1)}$ and ${\cal P}_{-}=\sqrt{p(p+|m|)}$, respectively. It can be easily checked that the ladder operators obey the su(1,1) Lie algebra $$\begin{aligned}
\label{eq:algebra}
[{\widehat{\cal{P}}}_+,{\widehat{\cal{P}}}_-]=-2{\widehat{\cal{P}}}_0\quad [{\widehat{\cal{P}}}_0,{\widehat{\cal{P}}}_\pm]=\pm{\widehat{\cal{P}}}_\pm,\end{aligned}$$ where ${\widehat{\cal{P}}}_0=\left(2{\widehat{p}}+|m|+1\right)/2$. In the equations above, ${\widehat{p}}$ is the second-order differential operator $$\begin{aligned}
\label{eq:algebra}
{\widehat{p}}=-\frac{1}{8\rho}\partial_\rho(\rho\,\partial_\rho)+\left(\frac{|m|^2}{8\rho^2}+\frac{\rho^2}{2}-\frac{|m|+1}{2}\right).\end{aligned}$$ The states ${|\mbox{p,m}\rangle}$ are eigenstates of ${\widehat{p}}$ with eigenvalue $p$, i.e. we have ${\widehat{p}}\,{|\mbox{p,m}\rangle}=p{|\mbox{p,m}\rangle}$. The operators acting in the $\zeta$-plane can be obtained by the corresponding operators in the pupil plane $\zeta=0$ from ${\widehat{\cal{N}}}_\zeta={\widehat{U}}(\zeta,0)^{\dag}{\widehat{\cal{N}}}_0{\widehat{U}}(\zeta,0)$, where ${\widehat{\cal{N}}}$ is any operator in the node Hilbert space in the pupil plane. The Casimir operator is given by ${\widehat{\cal{C}}}={\widehat{\cal{P}}}_0^2-1/2\left({\widehat{\cal{P}}}_+{\widehat{\cal{P}}}_-+{\widehat{\cal{P}}}_-{\widehat{\cal{P}}}_+\right)$, where ${\widehat{\cal{C}}}{|\mbox{p,m}\rangle}=(m^2-1)/4{|\mbox{p,m}\rangle}$ [@perelomov:86]. By knowing the underlying symmetry, one can try to exploit further properties of the radial node space. We here introduce the CS of the Lie group as the state resulting formally from the displacement of the ground state [@perelomov:86]. Later, we will introduce also the IS as the eigenvector of the lowering operator [@barut:71]. Unlike in the case of the harmonic oscillator, in the case of su(1,1) dynamics these two states are not coincident.\
![\[fig:propagation\] (Color online) Simulated propagation of two different RCS: (a) $m=1$, and (b) $m=2$ with $\alpha=0.6$.](f2.eps)
\(i) *Radial coherent state*: the ground state of radial nodes is ${|\mbox{0,m}\rangle}$, so, the CS is $$\begin{aligned}
\label{eq:gcs_def}
{|\alpha\rangle}_m=e^{\left(\alpha{\widehat{\cal{P}}}_+-\alpha^{\star}{\widehat{\cal{P}}}_-\right)}{|\mbox{0,m}\rangle},\end{aligned}$$ where $\alpha=\tanh{(\xi/2)}\,e^{i\theta}$ is a complex number and is given in terms of the two real quantities $(\xi,\theta)$. The coherent parameter for su(1,1) is bounded by the unit Poincaré disk, i.e. $|\alpha|<1$. We may expand the CS in terms of the LG modes: $$\begin{aligned}
\label{eq:gcs_lg_exp}
{|\alpha\rangle}_m=(1-|\alpha|^2)^{\left(\frac{|m|+1}{2}\right)}\sum_{p=0}^{\infty}\sqrt{\frac{\left(p+|m|\right)!}{|m|!p!}}\,\alpha^p{|p,m\rangle},\end{aligned}$$ where $|m|!$ is a normalization factor. The radial CS (RCS) depends on the OAM value as well. Therefore, for each values of the OAM, there will be a specific set of RCS. The RCS is not an orthogonal set of modes: ${}_{m'}\!{\langle\alpha'|\alpha\rangle}_{m}=\left((1-|\alpha|^2)(1-|\alpha'|^2)/(1-\alpha\alpha'^{\ast})^2\right)^{(|m|+1)/2}\, \delta_{m,m'}$, nevertheless, the set is complete $$\begin{aligned}
\label{eq:gcs_completeness}
\frac{|m|}{\pi}\int \frac{d^2\alpha}{(1-|\alpha|^2)^2}\,{|\alpha\rangle}_m\,{}_{m}\!{\langle \alpha|}={\widehat{\mathbb{I}}},\end{aligned}$$ where $\frac{|m|}{(1-|\alpha|^2)^2}$ is the weight of measure. Therefore, the RCS is an over-complete set solution of the radial modes.\
The RCS is a solution of the PWE and, hence, provides a novel type of the paraxial modes. It can be shown that $\sum_{p=0}^{\infty}|{\langle\mbox{p,m}|\alpha\rangle}_m|^2=1$, so, the RCS carries a finite power. Fig (\[fig:lg\_basis\]) shows the projection of different RCS in the basis of the first hundred p number of the LG modes.\
The position representation of RCS at the pupil, $\mbox{RCS}_{\alpha,m}(\rho,\phi,0)={\langle\rho,\phi,0|\alpha\rangle}_m$, is given by $$\begin{aligned}
\label{eq:gcs_position_rep_pupil}
\mbox{RCS}_{\alpha,m}(\rho,\phi,0)=\sqrt{\frac{2^{|m|+1}}{\pi\,|m|!}}\left(\frac{1-\alpha^{\ast}}{1-\alpha}\right)^{\frac{|m|+1}{2}}e^{i\,m\phi-\frac{(1+\alpha)}{1-\alpha}\,\rho^2}\rho^{|m|}.\nonumber\end{aligned}$$ Due to the presence of $e^{im\phi}$, photons in the RCS mode ${|\alpha\rangle}_m$ carry a well-defined OAM value of $m\hbar$ per photon. Propagation of two different RCSs is shown in Fig. (\[fig:propagation\]). The mode intensity vanishes as $\rho^{-2|m|}€$ at the beam center, so that that this mode can be generated starting from a Gaussian beam by simply combining a phase singularity of charge $m$ and parabolic transmission filter of order $|m|$. When $\alpha=0$ the RCS reduces to a subfamily of (type-I) HyGG modes with $p=0$ (the first class of the so-called modified LG modes) [@karimi:07]. As discussed in Ref. [@karimi:07], the HyGG modes are over-complete set of modes yet non-orthogonal solutions of the PWE as well.
\(ii) *Radial intelligent state*: the eigenvector of lowering operator is named the IS. Here we are seeking for a special radial intelligent state (RIS), which obeys the following eigenvalues problem $$\begin{aligned}
\label{eq:intelligent_def}
{\widehat{\cal{P}}}_-{|\eta\rangle}_m=\eta{|\eta\rangle}_m.\end{aligned}$$
![\[fig:prop\_bg\] (Color online) Simulated propagation of two different RISs: (a) $m=1$ with $\eta=1$ and (b) $m=2$ with $\eta=10$, respectively. The later case has several rings at the pupils, which during propagation collapses to a doughnut shape.](f3.eps)
Equation (\[eq:intelligent\_def\]) can be solved in the ${|\mbox{p,m}\rangle}$ basis yielding $$\begin{aligned}
\label{eq:intelligent_exp}
{|\eta\rangle}_m=\frac{\eta^{\frac{|m|}{2}}}{\sqrt{I_{|m|}\left(2|\eta|\right)}}\sum_{p=0}^{\infty}\frac{\eta^p}{\sqrt{p!\left(p+|m|\right)!}}\,{|\mbox{p,m}\rangle}\end{aligned}$$ where $\eta$ is a free complex parameter, and $I_{n}(.)$ is the modified Bessel function of order $n$. Like the RCS modes, the RIS are a not orthogonal set of modes, since we have ${}_{m'}\!{\langle\eta'|\eta\rangle}_{m}=I_{|m|}\left(2\sqrt{\eta\,\eta'^{\ast}}\right)/\sqrt{\left(I_{|m|}\left(2|\eta|\right)I_{|m|}\left(2|\eta'|\right)\right)} \delta_{m,m'}$. Nevertheless, they are a complete set since we have $$\begin{aligned}
\label{eq:is_completeness}
\frac{2}{\pi}\int {d^2\eta}\,K_m\left(2|\eta|\right) I_m\left(2|\eta|\right)\,{|\eta\rangle}_m\,{}_{m}\!{\langle \eta|}={\widehat{\mathbb{I}}},\end{aligned}$$ where $K_{n}(.)$ is the $n$-order modified Bessel function of the second kind. In order to have an easy and clear picture of the RIS, we may calculate its representation in the position space, i.e. $\mbox{RIS}(\rho,\phi,0)={\langle\rho,\phi,0|\eta\rangle}_m$ $$\begin{aligned}
\label{eq:ris_position_rep_pupil}
\mbox{RIS}_{\eta,m}(\rho,\phi,0)=\sqrt{\frac{2}{\pi\, I_{|m|}(2|\eta|)}}\,e^{im\phi+\eta-\rho^2}\,J_{|m|}\left(2\sqrt{2\eta}\rho\right).\nonumber\end{aligned}$$ We see that the RIS is the same as the so-called Bessel-Gauss beam. The Bessel-Gauss beam is then the intelligent state of radial modes and is the eigenstate of the lowering operator in the LG base representation. Like the RCS, the RIS is an eigenstate of the light OAM as well. Nevertheless, unlike RCS, it is not shape invariant under free-air propagation for general parameter $\eta$. Fig. (\[fig:prop\_bg\]) shows the propagation of two RIS beams with different $\eta$. We see that the intensity of one of the beams is fraught with a dramatic change during propagation.
In conclusion, we have shown that the dynamical symmetry of the radial node index of the scalar PWE is related to the su(1,1) Lie algebra. Based on this algebra, we studied two kinds of CS associated to the radial profile. Such CSs, indeed, may be used for minimizing the uncertainty relation between radial index number and its conjugate variable.
We acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 255914- PHORBITECH.
[99]{}
S. Franke-Arnold, L. Allen, and M. J. Padgett, Laser Photonics Rev. [**2**]{}, 299, (2008).
G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pasko, S. M. Barnett, and S. Franke-Arnold, , 5448 (2004).
A. Mair, A. Vaziri, G. Welhs, and A. Zeilinger, , 313 (2001).
E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, Nat. Photon. [**3**]{}, 712 (2009).
E. Karimi, S. Sluserenko, B. Piccirillo, L. Marrucci, and E. Santamato, , 053813 (2010).
M. Padgett, J. Courtial, and L. Allen, Physics Today [**57**]{}, 35 (2004).
M. V. Berry, J. Optics. A [**6**]{}, 259 (2004).
A. Ya. Bekshaev, A. I. Karamoch, 1366 (2008).
E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, , 3053 (2007).
E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, , 1225 (2009).
E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, , 231124 (2009).
G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, S. Carrasco, J. Europ. Opt. Soc. Rap. Public. [**2**]{}, 07014 (2007).
A. Perelomov, [*Generalized coherent states and their applications*]{}, Springer(1986).
E. Schrödinger , Naturwissenschaften [**14**]{}, 664 (1926).
A. O. Barut, L. Girardello, Commun. Math. Phys. [**21**]{}, 41 (1971).
The following dimensionless cylindrical and cartesian coordinates have been used $(\rho=r/w_0,\phi,\zeta=z/z_R)$ and $(u=x/w_0,v=y/w_0,\zeta=z/z_R)$, respectively, where $w_0$ is the beam waist and $z_R=\pi{w_0}^2/\lambda$ is the beam Rayleigh range.
A. E. Siegman, [*Lasers*]{}, University Science Books (1986).
|
---
abstract: 'The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear ODE of superior order is disscused.'
author:
- |
Mircea Neagu\
[July 2007; Revised June 2009 (minor revisions)]{}
title: Jet Geometrical Objects Produced by Linear ODEs Systems and Superior Order ODEs
---
**Mathematics Subject Classification (2000):** 53C43, 53C07, 83C22.
**Key words and phrases:** 1-jet spaces, jet least squares Lagrangian functions, Riemann-Lagrange geometry, linear ODEs systems, superior order ODEs.
Introduction
============
According to Olver’s opinion expressed in \[7\] and in private discussions, we point out that the 1-jet spaces are main mathematical models necessary for the study of classical or quantum field theories. In a such context, the contravariant differential geometry of the 1-jet spaces was intensively studied by authors like Asanov \[1\] or Saunders \[9\].
Situated in the direction initiated by Asanov \[1\], it has been recently developed the *Riemann-Lagrange geometry of 1-jet spaces* \[2\], \[4\], which is a geometrical theory on 1-jet spaces analogous with the well known *Lagrange geometry of the tangent bundle* developed by Miron and Anastasiei \[3\].
It is important to note that the Riemann-Lagrange geometry of the 1-jet spaces allows the regarding of the solutions of a given ODEs (respectively, PDEs) system as *geodesics* \[10\] (respectively, *generalized harmonic maps* \[6\] or *potential maps* \[11\]) in a convenient Riemann-Lagrange geometrical structure on 1-jet spaces. In this way, it was given a final solution for an open problem suggested by Poincaré \[8\] (*find the geometric structure which transforms the field lines of a given vector field into geodesics*) and generalized by Udrişte \[10\] (*find the geometrical structure which converts the solutions of a given first order PDEs system into harmonic maps*).
In this context, using the Riemann-Lagrange geometrical methods, it was constructed an entire contravariant differential geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting only with a given ODEs \[5\] (respectively, PDEs \[6\]) system of order one and a pair of Riemannian metrics.
Jet Riemann-Lagrange geometry produced by a non-linear ODEs system of order one and a pair of Riemannian metrics
================================================================================================================
In this Section we present the main jet Riemann-Lagrange geometrical ideas used for the geometrical study of a given non-linear first order ODEs system. For more details, the reader is invited to consult the works \[4\], \[5\] and \[11\].
Let $T=[a,b]\subset \mathbb{R}$ be a compact interval of the set of real numbers and let us consider the jet fibre bundle of order one$$J^{1}(T,\mathbb{R}^{n})\rightarrow T\times \mathbb{R}^{n},\text{ }n\geq 2,$$whose local coordinates $(t,x^{i},x_{1}^{i}),$ $i=\overline{1,n},$ transform after the rules$$\widetilde{t}=\widetilde{t}(t),\text{ }\widetilde{x}^{i}=\widetilde{x}^{i}(x^{j}),\text{ }\widetilde{x}_{1}^{i}=\frac{\partial \widetilde{x}^{i}}{\partial x^{j}}\frac{dt}{d\widetilde{t}}\cdot x_{1}^{j}.$$
From a physical point of view, in the 1-jet space of **physical events** the coordinate $t$ has the physical meaning of **relativistic time**, the coordinates $(x^{i})_{i=\overline{1,n}}$ represent **spatial coordinates** and the coordinates $(x_{1}^{i})_{i=\overline{1,n}}$ have the physical meaning of **relativistic velocities**.
Let $X=\left( X_{(1)}^{(i)}(t,x^{k})\right) $ be an arbitrary given d-tensor field on the first order jet space $J^{1}(T,\mathbb{R}^{n})$, which produces the jet non-linear ODEs system of order one (*jet dynamical system*)$$x_{1}^{i}=X_{(1)}^{(i)}(t,x^{k}(t)),\text{ }\forall \text{ }i=\overline{1,n},
\label{ODEs}$$where $c(t)=(x^{i}(t))$ is an unknown curve on $\mathbb{R}^{n}$ and we use the notations$$x_{1}^{i}\overset{not}{=}\dot{x}^{i}=\frac{dx^{i}}{dt},\text{ }\forall \text{
}i=\overline{1,n}.$$
Suppose now that we fixed *a priori* two Riemannian structures $(T,h_{11}(t))$ and $(\mathbb{R}^{n},\varphi _{ij}(x))$, where $x=(x^{k})_{k=\overline{1,n}}$, together with their attached Christoffel symbols $H_{11}^{1}(t)$ and $\gamma _{jk}^{i}(x)$. Automatically, the jet non-linear ODEs system of order one (\[ODEs\]), together with the pair of Riemannian metrics$$\mathcal{P}=(h_{11}(t),\varphi _{ij}(x)),$$produce the *jet least squares Lagrangian function* $$JLS_{\mathcal{P}}^{\text{ODEs}}:J^{1}(T,\mathbb{R}^{n})\rightarrow \mathbb{R}_{+},$$expressed by$$JLS_{\mathcal{P}}^{\text{ODEs}}(t,x^{k},x_{1}^{k})=h^{11}(t)\varphi _{ij}(x)\left[ x_{1}^{i}-X_{(1)}^{(i)}(t,x)\right] \left[
x_{1}^{j}-X_{(1)}^{(j)}(t,x)\right] .$$
It is obvious that the *global minimum points* of the *jet least squares energy action*$$\mathbb{E}_{\mathcal{P}}^{\text{ODEs}}(c(t))=\int_{a}^{b}JLS_{\mathcal{P}}^{\text{ODEs}}(t,x^{k}(t),\dot{x}^{k}(t))\sqrt{h_{11}(t)}dt$$are exactly the solutions of class $C^{2}$ of the jet non-linear ODEs system of order one (\[ODEs\]). In other words, we have
The solutions of class $C^{2}$ of the first order ODEs system (\[ODEs\]) verify the second order Euler-Lagrange equations produced by the jet *least squares Lagrangian function* $JLS_{\mathcal{P}}^{\text{ODEs}}$, namely (**jet geometric dinamics**)$$\frac{\partial \left[ JLS_{\mathcal{P}}^{\text{ODEs}}\right] }{\partial x^{i}}-\frac{d}{dt}\left( \frac{\partial \left[ JLS_{\mathcal{P}}^{\text{ODEs}}\right] }{\partial \dot{x}^{i}}\right) =0,\text{ }\forall \text{ }i=\overline{1,n}. \label{E-L-P}$$
Conversely, the above statement does not hold good because there exist solutions for the second order Euler-Lagrange ODEs system (\[E-L-P\]) which are not global minimum points for the jet least squares energy action $\mathbb{E}_{\mathcal{P}}^{\text{ODEs}}$, that is which are not solutions for the jet first order ODEs system (\[ODEs\]).
As a conclusion, we believe that we may regard $JLS_{\mathcal{P}}^{\text{ODEs}}$ as a natural geometrical substitut on $J^{1}(T,\mathbb{R}^{n})$ for the jet first order ODEs system (\[ODEs\]).
But, we point out that a Riemann-Lagrange geometry on $J^{1}(T,\mathbb{R}^{n})$ produced by the jet least squares Lagrangian function $JLS_{\mathcal{P}}^{\text{ODEs}}$, via its second order Euler-Lagrange equations (\[E-L-P\]), geometry in the sense of non-linear connection, generalized Cartan connection, d-torsions and d-curvatures, is now completely done in the papers \[4\], \[5\] and \[6\]. Moreover, a distinguished jet electromagnetic 2-form, characterized by some natural generalized Maxwell equations and a geometric jet Yang-Mills energy \[5\], is constructed from the jet least squares Lagrangian function $JLS_{\mathcal{P}}^{\text{ODEs}}$.
Any geometrical object on $J^{1}(T,\mathbb{R}^{n})$, which is produced by the jet least squares Lagrangian function $JLS_{\mathcal{P}}^{\text{ODEs}}$, via the Euler-Lagrange equations (\[E-L-P\]), is called **geometrical object produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$.
In this context, we give the following jet Riemann-Lagrange geometrical result, which is proved in \[5\] and, for the multi-time general case, in \[6\]. For more details, the reader is invited to consult the book \[4\].
\[MainThODEs\] (i) The **canonical non-linear connection on** $J^{1}(T,\mathbb{R}^{n})$** produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$ is$$\Gamma _{\mathcal{P}}^{\text{ODEs}}=\left(
M_{(1)1}^{(i)},N_{(1)j}^{(i)}\right) ,$$whose local components are given by$$M_{(1)1}^{(i)}=-H_{11}^{1}x_{1}^{i}\text{ and }N_{(1)j}^{(i)}=\gamma
_{jk}^{i}x_{1}^{k}-\frac{1}{2}\left[ X_{(1)||j}^{(i)}-\varphi
^{ir}X_{(1)||r}^{(s)}\varphi _{sj}\right] ,$$where$$X_{(1)||j}^{(i)}=\frac{\partial X_{(1)}^{(i)}}{\partial x^{j}}+X_{(1)}^{(m)}\gamma _{mj}^{i}.$$
\(ii) The **canonical generalized Cartan connection** $C\Gamma _{\mathcal{P}}^{\text{ODEs}}$** produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$ has the adapted components$$C\Gamma _{\mathcal{P}}^{\text{ODEs}}=(H_{11}^{1},0,\gamma _{jk}^{i},0).$$
\(iii) The effective adapted components of the **torsion** d-tensor **T**$_{\mathcal{P}}^{\text{ODEs}}$ of the canonical generalized Cartan connection $C\Gamma _{\mathcal{P}}^{\text{ODEs}}$ **produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$ are$$R_{(1)1j}^{(i)}=\frac{1}{2}\left[ X_{(1)||j//1}^{(i)}-\varphi
^{ir}X_{(1)||r//1}^{(s)}\varphi _{sj}\right]$$and$$R_{(1)jk}^{(i)}=r_{jkm}^{i}x_{1}^{m}-\frac{1}{2}\left[ X_{(1)||j||k}^{(i)}-\varphi ^{ir}X_{(1)||r||k}^{(s)}\varphi _{sj}\right] ,$$where $r_{ijk}^{l}(x)$ are the components of the curvature tensor of the Riemannian metric $\varphi _{ij}(x)$ and$$\begin{array}{l}
X_{(1)||j//1}^{(i)}=\dfrac{\partial X_{(1)||j}^{(i)}}{\partial t}-X_{(1)||j}^{(i)}H_{11}^{1},\medskip \\
X_{(1)||j||k}^{(i)}=\dfrac{\partial X_{(1)||j}^{(i)}}{\partial x^{k}}+X_{(1)||j}^{(m)}\gamma _{mk}^{i}-X_{(1)||m}^{(i)}\gamma _{jk}^{m}.\end{array}$$
\(iv) The effective adapted components of the **curvature** d-tensor **R**$_{\mathcal{P}}^{\text{ODEs}}$ of the canonical generalized Cartan connection $C\Gamma _{\mathcal{P}}^{\text{ODEs}}$ **produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$ are only $R_{ijk}^{l}=r_{ijk}^{l}.$
\(v) The **geometric electromagnetic distinguished 2-form produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$ has the expression$$F_{\mathcal{P}}^{\text{ODEs}}=F_{(i)j}^{(1)}\delta x_{1}^{i}\wedge dx^{j},$$where$$\delta x_{1}^{i}=dx_{1}^{i}+M_{(1)1}^{(i)}dt+N_{(1)k}^{(i)}dx^{k}$$and, if $h^{11}=1/h_{11}$, then$$F_{(i)j}^{(1)}=\frac{h^{11}}{2}\left[ \varphi _{im}X_{(1)||j}^{(m)}-\varphi
_{jm}X_{(1)||i}^{(m)}\right] .$$
\(vi) The adapted components of the electromagnetic d-form $F_{\mathcal{P}}^{\text{ODEs}}$ produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics $\mathcal{P}$ verify the **generalized Maxwell equations**$$\left\{
\begin{array}{l}
F_{(i)j//1}^{(1)}=\dfrac{1}{4}\mathcal{A}_{\{i,j\}}\left\{ h^{11}\varphi
_{im}\left[ X_{(1)||j//1}^{(m)}-\varphi ^{mr}X_{(1)||r//1}^{(s)}\varphi _{sj}\right] \right\} \medskip \\
\sum_{\{i,j,k\}}F_{(i)j||k}^{(1)}=0,\end{array}\right.$$where $\mathcal{A}_{\{i,j\}}$ represents an alternate sum, $\sum_{\{i,j,k\}}$ means a cyclic sum and$$F_{(i)j//1}^{(1)}=\dfrac{\partial F_{(i)j}^{(1)}}{\partial t}+F_{(i)j}^{(1)}H_{11}^{1}\text{ and }F_{(i)j||k}^{(1)}=\frac{\partial
F_{(i)j}^{(1)}}{\partial x^{k}}-F_{(m)j}^{(1)}\gamma
_{ik}^{m}-F_{(i)m}^{(1)}\gamma _{jk}^{m}$$have the geometrical meaning of the horizontal local covariant derivatives $"_{//1}"$ and $"_{||k}"$ produced by the Berwald linear connection $B\Gamma
_{0}$ on $J^{1}(T,\mathbb{R}^{n}).$ For more details, please consult \[4\].
\(vii) The **geometric jet Yang-Mills energy produced by the jet first order ODEs system (\[ODEs\]) and the pair of Riemannian metrics** $\mathcal{P}$ is defined by the formula$$EYM_{\mathcal{P}}^{\text{ODEs}}(t,x)=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\left[
F_{(i)j}^{(1)}\right] ^{2}.$$
Now, let us consider on $T\times \mathbb{R}^{n}$ the particular pair of Euclidian metrics$$\Delta =(h_{11}(t)=1,\varphi _{ij}(x)=\delta _{ij}),$$where $\delta _{ij}$ are the Kronecker symbols. Then we obtain the particular jet least squares Lagrangian function $$JLS_{\Delta }^{\text{ODEs}}:J^{1}(T,\mathbb{R}^{n})\rightarrow \mathbb{R}_{+},$$defined by$$\begin{aligned}
JLS_{\Delta }^{\text{ODEs}}(t,x^{k},x_{1}^{k}) &=&\delta _{ij}\left[
x_{1}^{i}-X_{(1)}^{(i)}(t,x)\right] \left[ x_{1}^{j}-X_{(1)}^{(j)}(t,x)\right] = \\
&=&\sum_{i=1}^{n}\left[ x_{1}^{i}-X_{(1)}^{(i)}(t,x)\right] ^{2}.\end{aligned}$$
In this new context, we introduce the following concept:
Any geometrical object on $J^{1}(T,\mathbb{R}^{n})$, which is produced by the jet least squares Lagrangian function $JLS_{\Delta }^{\text{ODEs}}$, via its attached second order Euler-Lagrange equations, is called **geometrical object produced by the jet first order ODEs system (\[ODEs\])**.
As a consequence, particularizing the Theorem \[MainThODEs\] for the pair of Euclidian metrics $\mathcal{P}=\Delta $ and taking into account that we have $H_{11}^{1}(t)=0$ and $\gamma _{ij}^{k}(x)=0$, we immediately get the following jet geometrical result:
\[MainCor\] (i) The **canonical non-linear connection on** $J^{1}(T,\mathbb{R}^{n})$** produced by the jet first order ODEs system ([ODEs]{})** has the local components$$\Gamma _{\Delta }^{\text{ODEs}}=\left( \bar{M}_{(1)1}^{(i)},\bar{N}_{(1)j}^{(i)}\right) ,$$where$$\bar{M}_{(1)1}^{(i)}=0\text{ and }\bar{N}_{(1)j}^{(i)}=-\frac{1}{2}\left[
\frac{\partial X_{(1)}^{(i)}}{\partial x^{j}}-\frac{\partial X_{(1)}^{(j)}}{\partial x^{i}}\right] ,\text{ }\forall \text{ }i,j=\overline{1,n}.$$
\(ii) All adapted components of the **canonical generalized Cartan connection** $C\Gamma _{\Delta }^{\text{ODEs}}$** produced by the jet first order ODEs system (\[ODEs\])** vanish.
\(iii) The effective adapted components of the **torsion** d-tensor **T**$_{\Delta }^{\text{ODEs}}$ of the canonical generalized Cartan connection $C\Gamma _{\Delta }^{\text{ODEs}}$ **produced by the jet first order ODEs system (\[ODEs\])** are$$\bar{R}_{(1)1j}^{(i)}=\frac{1}{2}\left[ \frac{\partial ^{2}X_{(1)}^{(i)}}{\partial t\partial x^{j}}-\frac{\partial ^{2}X_{(1)}^{(j)}}{\partial
t\partial x^{i}}\right] ,\text{ }\forall \text{ }i,j=\overline{1,n},$$and$$\bar{R}_{(1)jk}^{(i)}=-\frac{1}{2}\left[ \frac{\partial ^{2}X_{(1)}^{(i)}}{\partial x^{k}\partial x^{j}}-\frac{\partial ^{2}X_{(1)}^{(j)}}{\partial
x^{k}\partial x^{i}}\right] ,\text{ }\forall \text{ }i,j,k=\overline{1,n}.$$
\(iv) All adapted components of the **curvature** d-tensor **R**$_{\Delta }^{\text{ODEs}}$ of the canonical generalized Cartan connection $C\Gamma _{\Delta }^{\text{ODEs}}$ **produced by the jet first order DEs system (\[ODEs\])** vanish.
\(v) The **geometric electromagnetic distinguished 2-form produced by the jet first order ODEs system (\[ODEs\])** has the form$$F_{\Delta }^{\text{ODEs}}=\bar{F}_{(i)j}^{(1)}\delta x_{1}^{i}\wedge dx^{j},$$where$$\delta x_{1}^{i}=dx_{1}^{i}+\bar{N}_{(1)k}^{(i)}dx^{k},\text{ }\forall \text{
}i=\overline{1,n},$$and$$\bar{F}_{(i)j}^{(1)}=\frac{1}{2}\left[ \frac{\partial X_{(1)}^{(i)}}{\partial x^{j}}-\frac{\partial X_{(1)}^{(j)}}{\partial x^{i}}\right] ,\text{
}\forall \text{ }i,j=\overline{1,n}.$$
\(vi) The adapted components $\bar{F}_{(i)j}^{(1)}$ of the electromagnetic d-form $F_{\Delta }^{\text{ODEs}}$ produced by the jet first order ODEs system (\[ODEs\])** **verify the **generalized Maxwell equations**$$\left\{
\begin{array}{l}
\bar{F}_{(i)j//1}^{(1)}=\dfrac{1}{4}\mathcal{A}_{\{i,j\}}\left[ \dfrac{\partial ^{2}X_{(1)}^{(i)}}{\partial t\partial x^{j}}-\dfrac{\partial
^{2}X_{(1)}^{(j)}}{\partial t\partial x^{i}}\right] =\dfrac{1}{2}\left[
\dfrac{\partial ^{2}X_{(1)}^{(i)}}{\partial t\partial x^{j}}-\dfrac{\partial
^{2}X_{(1)}^{(j)}}{\partial t\partial x^{i}}\right] \medskip \\
\sum_{\{i,j,k\}}\bar{F}_{(i)j||k}^{(1)}=0,\end{array}\right.$$where $\mathcal{A}_{\{i,j\}}$ represents an alternate sum, $\sum_{\{i,j,k\}}$ means a cyclic sum and$$\bar{F}_{(i)j//1}^{(1)}=\dfrac{\partial \bar{F}_{(i)j}^{(1)}}{\partial t}\text{ and }\bar{F}_{(i)j||k}^{(1)}=\frac{\partial \bar{F}_{(i)j}^{(1)}}{\partial x^{k}},\text{ }\forall \text{ }i,j,k=\overline{1,n}.$$
\(vii) The **geometric jet Yang-Mills energy produced by the jet first order ODEs system (\[ODEs\])** has the expression$$EYM_{\Delta }^{\text{ODEs}}(t,x)=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\left[ \bar{F}_{(i)j}^{(1)}\right] ^{2}=\frac{1}{4}\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\left[
\frac{\partial X_{(1)}^{(i)}}{\partial x^{j}}-\frac{\partial X_{(1)}^{(j)}}{\partial x^{i}}\right] ^{2}.$$
\[Formulas\] If we use the matriceal notations
- $J\left( X_{(1)}\right) =\left( \dfrac{\partial X_{(1)}^{(i)}}{\partial x^{j}}\right) _{i,j=\overline{1,n}}$ - the **Jacobian matrix**,
- $\bar{N}_{(1)}=\left( \bar{N}_{(1)j}^{(i)}\right) _{i,j=\overline{1,n}} $ - the **non-linear connection matrix**,
- $\bar{R}_{(1)1}=\left( \bar{R}_{(1)1j}^{(i)}\right) _{i,j=\overline{1,n}},$ - the **temporal torsion matrix**,
- $\bar{R}_{(1)k}=\left( \bar{R}_{(1)jk}^{(i)}\right) _{i,j=\overline{1,n}},$ $\forall $ $k=\overline{1,n},$ - the **spatial torsion matrices**,
- $\bar{F}^{(1)}=\left( \bar{F}_{(i)j}^{(1)}\right) _{i,j=\overline{1,n}} $ - the **electromagnetic matrix**,
then the following matriceal geometrical relations attached to the jet first order ODEs system (\[ODEs\]) hold good:
1. $\bar{N}_{(1)}=-\dfrac{1}{2}\left[ J\left( X_{(1)}\right) -\text{ }^{T}J\left( X_{(1)}\right) \right] ;$
2. $\bar{R}_{(1)1}=-\dfrac{\partial }{\partial t}\left[ \bar{N}_{(1)}\right] ;$
3. $\bar{R}_{(1)k}=\dfrac{\partial }{\partial x^{k}}\left[ \bar{N}_{(1)}\right] ,$ $\forall $ $k=\overline{1,n};$
4. $\bar{F}^{(1)}=-\bar{N}_{(1)};$
5. $EYM_{\Delta }^{\text{ODEs}}(t,x)=\dfrac{1}{2}\cdot Trace\left[ \bar{F}^{(1)}\cdot \text{ }^{T}\bar{F}^{(1)}\right] ,$
that is the jet electromagnetic Yang-Mills energy coincides with the square of the norm of the skew-symmetric electromagnetic matrix $\bar{F}^{(1)}$ in the Lie algebra $o(n)=L(O(n)).$
Note that the spatial torsion matrix $\bar{R}_{(1)k}$ does not coincide for $k=1$ with the temporal torsion matrix $\bar{R}_{(1)1}$. We have only an overlap of notations.
Jet Riemann-Lagrange geometry produced by a non-homogenous linear ODEs system of order one
==========================================================================================
In this Section we apply the preceding jet Riemann-Lagrange geometrical results for a non-homogenous linear ODEs system of order one. In this way, let us consider the following non-homogenous linear first order ODEs system locally described, in a convenient chart on $J^{1}(T,\mathbb{R}^{n})$, by the differential equations$$\frac{dx^{i}}{dt}=\sum_{k=1}^{n}a_{(1)k}^{(i)}(t)x^{k}+f_{(1)}^{(i)}(t),\text{ }\forall \text{ }i=\overline{1,n}, \label{LODEs}$$where the local components $a_{(1)k}^{(i)}$ and $f_{(1)}^{(i)}$ transform after the tensorial rules$$a_{(1)k}^{(i)}=\dfrac{\partial x^{i}}{\partial \widetilde{x}^{j}}\dfrac{d\widetilde{t}}{dt}\cdot \widetilde{a}_{(1)k}^{(j)},\text{ }\forall \text{ }k=\overline{1,n},$$and$$f_{(1)}^{(i)}=\frac{\partial x^{i}}{\partial \widetilde{x}^{j}}\frac{d\widetilde{t}}{dt}\cdot \widetilde{f}_{(1)}^{(j)}.$$
We suppose that the product manifold $T\times \mathbb{R}^{n}\subset J^{1}(T,\mathbb{R}^{n})$ is endowed **a priori** with the pair of Euclidian metrics $\Delta =(1,\delta _{ij}),$ with respect to the coordinates $(t,x^{i})$.
It is obvious that the non-homogenous linear ODEs system (\[LODEs\]) is a particular case of the jet first order non-linear ODEs system (\[ODEs\]) for$$X_{(1)}^{(i)}(t,x)=\sum_{k=1}^{n}a_{(1)k}^{(i)}(t)x^{k}+f_{(1)}^{(i)}(t),\text{ }\forall \text{ }i=\overline{1,n}. \label{XLODEs}$$
In order to expose the main jet Riemann-Lagrange geometrical objects that characterize the non-homogenous linear ODEs system (\[LODEs\]), we use the matriceal notation$$A_{(1)}=\left( a_{(1)j}^{(i)}(t)\right) _{i,j=\overline{1,n}}.$$
In this context, applying our preceding jet geometrical Riemann-Lagrange theory to the non-homogenous linear ODEs system (\[LODEs\]) and the pair of Euclidian metrics $\Delta =(1,\delta _{ij})$, we get:
\(i) The **canonical non-linear connection on** $J^{1}(T,\mathbb{R}^{n})$** produced by the non-homogenous linear ODEs system (\[LODEs\])** has the local components$$\hat{\Gamma}=\left( 0,\hat{N}_{(1)j}^{(i)}\right) ,$$where $\hat{N}_{(1)j}^{(i)}$ are the entries of the matrix $$\hat{N}_{(1)}=\left( \hat{N}_{(1)j}^{(i)}\right) _{i,j=\overline{1,n}}=-\frac{1}{2}\left[ A_{(1)}-\text{ }^{T}A_{(1)}\right] .$$
\(ii) All adapted components of the **canonical generalized Cartan connection** $C\hat{\Gamma}$** produced by the non-homogenous linear ODEs system (\[LODEs\])** vanish.
\(iii) The effective adapted components $\hat{R}_{(1)1j}^{(i)}$ of the **torsion** d-tensor **T** of the canonical generalized Cartan connection $C\hat{\Gamma}$ **produced by the non-homogenous linear ODEs system (\[LODEs\])** are the entries of the matrices$$\hat{R}_{(1)1}=\left( \hat{R}_{(1)1j}^{(i)}\right) _{i,j=\overline{1,n}}=\frac{1}{2}\left[ \dot{A}_{(1)}-\text{ }^{T}\dot{A}_{(1)}\right] ,$$where$$\dot{A}_{(1)}=\frac{d}{dt}\left[ A_{(1)}\right] .$$
\(iv) All adapted components of the **curvature** d-tensor **R** of the canonical generalized Cartan connection $C\hat{\Gamma}$ **produced by the non-homogenous linear ODEs system (\[LODEs\])** vanish.
\(v) The **geometric electromagnetic distinguished 2-form produced by the non-homogenous linear ODEs system (\[LODEs\])** is given by$$\hat{F}=\hat{F}_{(i)j}^{(1)}\delta x_{1}^{i}\wedge dx^{j},$$where$$\delta x_{1}^{i}=dx_{1}^{i}-\frac{1}{2}\left[ a_{(1)k}^{(i)}-a_{(1)i}^{(k)}\right] dx^{k},\text{ }\forall \text{ }i=\overline{1,n},$$and the adapted components $\hat{F}_{(i)j}^{(1)}$ are the entries of the matrix$$\hat{F}^{(1)}=\left( \hat{F}_{(i)j}^{(1)}\right) _{i,j=\overline{1,n}}=-\hat{N}_{(1)}=\frac{1}{2}\left[ A_{(1)}-\text{ }^{T}A_{(1)}\right] ,$$that is $$\hat{F}_{(i)j}^{(1)}=\frac{1}{2}\left[ a_{(1)j}^{(i)}-a_{(1)i}^{(j)}\right] .$$
\(vi) The ** jet Yang-Mills energy produced by the non-homogenous linear ODEs system (\[LODEs\])** is given by the formula$$EYM^{\text{NHLODEs}}(t)=\frac{1}{4}\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\left[
a_{(1)j}^{(i)}-a_{(1)i}^{(j)}\right] ^{2}.$$
Using the relations (\[XLODEs\]), we easily deduce that we have the Jacobian matrix$$J\left( X_{(1)}\right) =A_{(1)}.$$
Consequently, applying the Corollary \[MainCor\] to the non-homogenous linear ODEs system (\[LODEs\]), together with the Remark \[Formulas\], we obtain the required results.
The entire jet Riemann-Lagrange geometry produced by the non-homogenous linear ODEs system (\[LODEs\]) does not depend on the non-homogeneity terms $f_{(1)}^{(i)}(t)$.
The ** **jet Yang-Mills energy produced by the non-homogenous linear ODEs system (\[LODEs\]) vanishes if and only if the matrix $A_{(1)}$ is a symmetric one. In this case, the entire jet Riemann-Lagrange geometry produced by the non-homogenous linear ODEs system (\[LODEs\]) vanish, so it does not offer geometrical informations about the system (\[LODEs\]). However, it is important to note that in this particular situation we have the symetry of the matrix $A_{(1)},$ which implies that the matrix $A_{(1)}$ is diagonalizable.
All torsion adapted components of a non-homogenous linear ODEs system with constant coefficients $a_{(1)j}^{(i)}$ are zero.
Jet Riemann-Lagrange geometry produced by a superior order ODE
==============================================================
Let us consider the superior order ODE expressed by$$y^{(n)}(t)=f(t,y(t),y^{\prime }(t),...,y^{(n-1)}(t)),\text{ }n\geq 2,
\label{SODE}$$where $y(t)$ is an unknown function, $y^{(k)}(t)$ is the derivative of order $k$ of the unknown function $y(t)$ for each $k\in \{0,1,...,n\}$ and $f$ is a given differentiable function depending on the distinct variables $t,$ $y(t),$ $y^{\prime }(t),...,$ $y^{(n-1)}(t).$
It is well known the fact that, using the notations$$x^{1}=y,\text{ }x^{2}=y^{\prime },\text{ }...,\text{ }x^{n}=y^{(n-1)},$$the superior order ODE (\[SODE\]) is equivalent with the non-linear ODEs system of order one$$\left\{
\begin{array}{l}
\dfrac{dx^{1}}{dt}=x^{2}\medskip \\
\dfrac{dx^{2}}{dt}=x^{3}\medskip \\
\cdot \\
\cdot \\
\cdot \\
\dfrac{dx^{n-1}}{dt}=x^{n}\medskip \\
\dfrac{dx^{n}}{dt}=f(t,x^{1},x^{2},...,x^{n}).\end{array}\right. \label{NLSODE}$$
But, the first order non-linear ODEs system (\[NLSODE\]) can be regarded, in a convenient local chart, as a particular case of the jet non-linear ODEs system of order one (\[ODEs\]), taking$$\begin{array}{lll}
X_{(1)}^{(1)}(t,x)=x^{2}, & X_{(1)}^{(2)}(t,x)=x^{3}, & \cdot \cdot \cdot
\medskip \\
\cdot \cdot \cdot & X_{(1)}^{(n-1)}(t,x)=x^{n}, &
X_{(1)}^{(n)}(t,x)=f(t,x^{1},x^{2},...,x^{n}),\end{array}
\label{XSODE}$$where we suppose that the geometrical object $X=\left(
X_{(1)}^{(i)}(t,x)\right) $ behaves as a d-tensor on $J^{1}(T,\mathbb{R}^{n}) $.
We assume that the product manifold $T\times \mathbb{R}^{n}\subset J^{1}(T,\mathbb{R}^{n})$ is endowed **a priori** with the pair of Euclidian metrics $\Delta =(1,\delta _{ij}),$ with respect to the coordinates $(t,x^{i})$.
Any geometrical object on $J^{1}(T,\mathbb{R}^{n})$, which is produced by the ** first order non-linear ODEs system (\[NLSODE\])** is called **geometrical object produced by the superior order ODE (\[SODE\])**.
In this context, the Riemann-Lagrange geometrical behavior on the 1-jet space $J^{1}(T,\mathbb{R}^{n})$ of the superior order ODE (\[SODE\]) is described in the following result:
\[MainThSODE\] (i) The **canonical non-linear connection on** $J^{1}(T,\mathbb{R}^{n})$** produced by the superior order ODE ([SODE]{})** has the local components$$\check{\Gamma}=\left( 0,\check{N}_{(1)j}^{(i)}\right) ,$$where $\check{N}_{(1)j}^{(i)}$ are the entries of the matrix $\check{N}_{(1)}=\left( \check{N}_{(1)j}^{(i)}\right) _{i,j=\overline{1,n}}=$$$=-\frac{1}{2}\left(
\begin{array}{cccccccc}
0 & 1 & 0 & \cdot & \cdot & 0 & 0 & -\dfrac{\partial f}{\partial x^{1}}\medskip \\
-1 & 0 & 1 & \cdot & \cdot & 0 & 0 & -\dfrac{\partial f}{\partial x^{2}}\medskip \\
0 & -1 & 0 & \cdot & \cdot & 0 & 0 & -\dfrac{\partial f}{\partial x^{3}}\medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
0 & 0 & 0 & \cdot & \cdot & 0 & 1 & -\dfrac{\partial f}{\partial x^{n-2}}\medskip \\
0 & 0 & 0 & \cdot & \cdot & -1 & 0 & 1-\dfrac{\partial f}{\partial x^{n-1}}\medskip \\
\dfrac{\partial f}{\partial x^{1}} & \dfrac{\partial f}{\partial x^{2}} &
\dfrac{\partial f}{\partial x^{3}} & \cdot & \cdot & \dfrac{\partial f}{\partial x^{n-2}} & -1+\dfrac{\partial f}{\partial x^{n-1}} & 0\end{array}\right) .$$
\(ii) All adapted components of the **canonical generalized Cartan connection** $C\check{\Gamma}$** produced by the superior order ODE (\[SODE\])** vanish.
\(iii) The effective adapted components of the **torsion** d-tensor **Ť** of the canonical generalized Cartan connection $C\check{\Gamma}$ **produced by the superior order ODE (\[SODE\])** are the entries of the matrices$$\check{R}_{(1)1}=\frac{1}{2}\left(
\begin{array}{cccccc}
0 & 0 & \cdot & \cdot & 0 & -\dfrac{\partial ^{2}f}{\partial t\partial x^{1}}\medskip \\
0 & 0 & \cdot & \cdot & 0 & -\dfrac{\partial ^{2}f}{\partial t\partial x^{2}}\medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
0 & 0 & \cdot & \cdot & 0 & -\dfrac{\partial ^{2}f}{\partial t\partial
x^{n-1}}\medskip \\
\dfrac{\partial ^{2}f}{\partial t\partial x^{1}} & \dfrac{\partial ^{2}f}{\partial t\partial x^{2}} & \cdot & \cdot & \dfrac{\partial ^{2}f}{\partial
t\partial x^{n-1}} & 0\end{array}\right)$$and$$\check{R}_{(1)k}=-\frac{1}{2}\left(
\begin{array}{cccccc}
0 & 0 & \cdot & \cdot & 0 & -\dfrac{\partial ^{2}f}{\partial x^{k}\partial
x^{1}}\medskip \\
0 & 0 & \cdot & \cdot & 0 & -\dfrac{\partial ^{2}f}{\partial x^{k}\partial
x^{2}}\medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
0 & 0 & \cdot & \cdot & 0 & -\dfrac{\partial ^{2}f}{\partial x^{k}\partial
x^{n-1}}\medskip \\
\dfrac{\partial ^{2}f}{\partial x^{k}\partial x^{1}} & \dfrac{\partial ^{2}f}{\partial x^{k}\partial x^{2}} & \cdot & \cdot & \dfrac{\partial ^{2}f}{\partial x^{k}\partial x^{n-1}} & 0\end{array}\right) ,$$where $k\in \{1,2,...,n\}$.
\(iv) All adapted components of the **curvature** d-tensor **Ř** of the canonical generalized Cartan connection $C\check{\Gamma}$ **produced by the superior order ODE (\[SODE\])** vanish.
\(v) The **geometric electromagnetic distinguished 2-form produced by the superior order ODE (\[SODE\])** has the form$$\check{F}=\check{F}_{(i)j}^{(1)}\delta x_{1}^{i}\wedge dx^{j},$$where$$\delta x_{1}^{i}=dx_{1}^{i}+\check{N}_{(1)k}^{(i)}dx^{k},\text{ }\forall
\text{ }i=\overline{1,n},$$and the adapted components $\check{F}_{(i)j}^{(1)}$ are the entries of the matrix$$\check{F}^{(1)}=\left( \check{F}_{(i)j}^{(1)}\right) _{i,j=\overline{1,n}}=-\check{N}_{(1)}.$$
\(vi) The **jet geometric Yang-Mills energy produced by the superior order ODE (\[SODE\])** is given by the formula$$EYM^{\text{SODE}}(t,x)=\frac{1}{4}\left[ n-1-2\frac{\partial f}{\partial
x^{n-1}}+\sum_{j=1}^{n-1}\left( \frac{\partial f}{\partial x^{j}}\right) ^{2}\right] .$$
By partial derivatives, the relations (\[XSODE\]) lead to the Jacobian matrix$$J\left( X_{(1)}\right) =\left(
\begin{array}{ccccccc}
0 & 1 & 0 & \cdot & \cdot & 0 & 0\medskip \\
0 & 0 & 1 & \cdot & \cdot & 0 & 0\medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \medskip \\
0 & 0 & 0 & \cdot & \cdot & 0 & 1\medskip \\
\dfrac{\partial f}{\partial x^{1}} & \dfrac{\partial f}{\partial x^{2}} &
\dfrac{\partial f}{\partial x^{3}} & \cdot & \cdot & \dfrac{\partial f}{\partial x^{n-1}} & \dfrac{\partial f}{\partial x^{n}}\end{array}\right) .$$
In conclusion, the Corollary \[MainCor\], together with the Remark [Formulas]{}, applied to first order non-linear ODEs system (\[NLSODE\]),** **give what we were looking for.
Riemann-Lagrange geometry produced by a non-homogenous linear ODE of superior order
===================================================================================
If we consider the non-homogenous linear ODE of order $n\in
\mathbb{N}$, $n\geq 2$, expressed by$$a_{0}(t)y^{(n)}+a_{1}(t)y^{(n-1)}+...+a_{n-1}(t)y^{\prime }+a_{n}(t)y=b(t),
\label{NHLSODE}$$where $b(t)$ and $a_{i}(t)$, $\forall $ $i=\overline{0,n}$, are given differentiable real functions and $a_{0}(t)\neq 0$, $\forall $ $t\in \lbrack
a,b]$, then we recover the superior order ODE (\[SODE\]) for the particular function$$f(t,x)=\frac{b(t)}{a_{0}(t)}-\frac{a_{n}(t)}{a_{0}(t)}\cdot x^{1}-\frac{a_{n-1}(t)}{a_{0}(t)}\cdot x^{2}-...-\frac{a_{1}(t)}{a_{0}(t)}\cdot x^{n},
\label{fNHLSODE}$$where we recall that we have$$y=x^{1},y^{\prime }=x^{2},...,y^{(n-1)}=x^{n}.$$
Consequently, we can derive the jet Riemann-Lagrange geometry attached to the non-homogenous linear superior order ODE (\[NHLSODE\]).
\(i) The **canonical non-linear connection on** $J^{1}(T,\mathbb{R}^{n})$** produced by the non-homogenous linear superior order ODE ([NHLSODE]{})** has the local components$$\tilde{\Gamma}=\left( 0,\tilde{N}_{(1)j}^{(i)}\right) ,$$where $\tilde{N}_{(1)j}^{(i)}$ are the entries of the matrix $$\begin{aligned}
\tilde{N}_{(1)} &=&\left( \tilde{N}_{(1)j}^{(i)}\right) _{i,j=\overline{1,n}}= \\
&=&-\frac{1}{2}\left(
\begin{array}{cccccccc}
0 & 1 & 0 & \cdot & \cdot & 0 & 0 & \dfrac{a_{n}}{a_{0}}\medskip \\
-1 & 0 & 1 & \cdot & \cdot & 0 & 0 & \dfrac{a_{n-1}}{a_{0}}\medskip \\
0 & -1 & 0 & \cdot & \cdot & 0 & 0 & \dfrac{a_{n-2}}{a_{0}}\medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot
\medskip \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot
\medskip \\
0 & 0 & 0 & \cdot & \cdot & 0 & 1 & \dfrac{a_{3}}{a_{0}}\medskip \\
0 & 0 & 0 & \cdot & \cdot & -1 & 0 & 1+\dfrac{a_{2}}{a_{0}}\medskip \\
-\dfrac{a_{n}}{a_{0}} & -\dfrac{a_{n-1}}{a_{0}} & -\dfrac{a_{n-2}}{a_{0}} &
\cdot & \cdot & -\dfrac{a_{3}}{a_{0}} & -1-\dfrac{a_{2}}{a_{0}} & 0\end{array}\right) .\end{aligned}$$
\(ii) All adapted components of the **canonical generalized Cartan connection** $C\tilde{\Gamma}$** produced by the non-homogenous linear superior order ODE (\[NHLSODE\])** vanish.
\(iii) All adapted components of the **torsion** d-tensor **T** of the canonical generalized Cartan connection $C\tilde{\Gamma}$ **produced by the non-homogenous linear superior order ODE (\[NHLSODE\])** are zero, except the temporal components$$\tilde{R}_{(1)1n}^{(i)}=-\tilde{R}_{(1)1i}^{(n)}=\frac{a_{n-i+1}^{\prime
}a_{0}-a_{n-i+1}a_{0}^{\prime }}{2a_{0}^{2}},\text{ }\forall \text{ }i=\overline{1,n-1},$$where we denoted by $"$ $^{\prime }$ $"$ the derivatives of the functions $a_{k}(t)$.
\(iv) All adapted components of the **curvature** d-tensor **R** of the canonical generalized Cartan connection $C\tilde{\Gamma}$ **produced by the non-homogenous linear superior order ODE (\[NHLSODE\])** vanish.
\(v) The **geometric electromagnetic distinguished 2-form produced by the non-homogenous linear superior order ODE (\[NHLSODE\])** has the expression$$\tilde{F}=\tilde{F}_{(i)j}^{(1)}\delta x_{1}^{i}\wedge dx^{j},$$where$$\delta x_{1}^{i}=dx_{1}^{i}+\tilde{N}_{(1)k}^{(i)}dx^{k},\text{ }\forall
\text{ }i=\overline{1,n},$$and the adapted components $\tilde{F}_{(i)j}^{(1)}$ are the entries of the matrix$$\tilde{F}^{(1)}=\left( \tilde{F}_{(i)j}^{(1)}\right) _{i,j=\overline{1,n}}=-\tilde{N}_{(1)}.$$
\(vi) The **jet geometric Yang-Mills electromagnetic energy produced by the non-homogenous linear superior order ODE (\[NHLSODE\])** has the form$$EYM^{\text{NHLSODE}}(t)=\frac{1}{4}\left[ n-1+2\frac{a_{2}}{a_{0}}+\sum_{j=2}^{n}\frac{a_{j}^{2}}{a_{0}^{2}}\right] .$$
We apply the Theorem \[MainThSODE\] for the particular function ([fNHLSODE]{}) and we use the relations$$\frac{\partial f}{\partial x^{j}}=-\frac{a_{n-j+1}}{a_{0}},\text{ }\forall
\text{ }j=\overline{1,n}.$$
The entire jet Riemann-Lagrange geometry produced by the non-homogenous linear superior order ODE (\[NHLSODE\]) is independent by the term of non-homogeneity $b(t)$. In author’s opinion, this fact emphasizes that the most important role in the study of the ODE (\[NHLSODE\]) is played by its attached homogenous linear superior order ODE.
The law of motion without friction (**harmonic oscillator**) of a material point of mass $m>0$, which is placed on a spring having the constant of elasticity $k>0$, is given by the homogenous linear ODE of order two$$\frac{d^{2}y}{dt^{2}}+\omega ^{2}y=0, \label{Oscilator}$$where the coordinate $y$ measures the distance from the mass’s equlibrium point and $\omega ^{2}=k/m.$ It follows that we have$$n=2,\text{ }a_{0}(t)=1,\text{ }a_{1}(t)=0\text{ and }a_{2}(t)=\omega ^{2},$$that is the **harmonic oscillator** second order ODE (\[Oscilator\]) provides the **jet geometric Yang-Mills electromagnetic energy**$$EYM^{\text{Harmonic Oscillator}}=\frac{1}{4}\left( 1+\omega ^{2}\right) ^{2}.$$
**Open problem.** There exists a real physical interpretation for the previous jet geometric Yang-Mills electromagnetic energy attached to the harmonic oscillator?
**Acknowledgements.** A version of this paper was presented at *Conference of Differential Geometry dedicated to the Memory of Professor Kostake Teleman (1933-2007)*, University of Bucharest, May 15-17, 2009.
The present work was supported by Contract with Sinoptix No. 8441/2009.
[99]{} G. S. Asanov, *Jet Extension of Finslerian Gauge Approach*, Fortschritte der Physik **38**, No. **8** (1990), 571-610.
V. Balan, *Generalized Maxwell and Lorentz Equations on First Order Geometrized Jet Spaces*, Proceeding of the International Conference in Geometry and Topology, Cluj-Napoca, Romania, 1-5 October 2002; University of Cluj-Napoca Editors (2004), 11-24.
R. Miron, M. Anastasiei, *The Geometry of Lagrange Spaces: Theory and Applications*, Kluwer Academic Publishers, 1994.
M. Neagu, *Riemann-Lagrange Geometry on 1-Jet Spaces*, Matrix Rom, Bucharest, 2005.
M. Neagu, I. R. Nicola, *Geometric Dynamics of Calcium Oscillations ODEs Systems*, Balkan Journal of Geometry and Its Applications, Vol. **9**, No. **2** (2004), 36-67.
M. Neagu, C. Udrişte, *From PDEs Systems and Metrics to Geometric Multi-Time Field Theories*, Seminarul de Mecanică, Sisteme Dinamice Diferenţiale, No. **79** (2001), Timişoara, Romania.
P. J. Olver, *Applications of Lie Groups to Differential Equations*, Springer-Verlag, 1986.
H. Poincaré, *Sur les Courbes Definies par les Equations Différentielle*, C.R. Acad. Sci., Paris **90** (1880), 673-675.
D. Saunders, *The Geometry of Jet Bundles*, Cambridge University Press, New York, London, 1989.
C. Udrişte, *Geometric Dynamics*, Kluwer Academic Publishers, 2000.
C. Udrişte, M. Postolache, *Atlas of Magnetic Geometric Dynamics*, Geometry Balkan Press, Bucharest, 2001.
**Author’s address:** Mircea N[EAGU]{}
University Transilvania of Braşov
Faculty of Mathematics and Informatics
Department of Algebra, Geometry and Differential Equations
B-dul Eroilor, Nr. 29, 500036 Braşov, Romania.
E-mail: mircea.neagu@unitbv.ro
Website: http://www.2collab.com/user:mirceaneagu
|
---
abstract: 'Let $M$ to be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Given a positive integer $k$, $M$ is $k$-locked if the number of its locked subsets is $O(|E|^k)$. $\mathcal L_k$ is the class of $k$-locked matroids (for a fixed k). In this paper, we give a new axiom system for matroids based on locked subsets. We deduce that the matroid isomorphism problem (MIP) for $\mathcal L_k$ is polynomially time reducible to the graph isomorphism problem (GIP). $\mathcal L_k$ is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. MIP is coNP-hard even for linear matroids.'
author:
- |
[**Brahim Chaourar**]{}\
[ Department of Mathematics and Statistics,\
Al Imam Mohammad Ibn Saud Islamic University (IMSIU)\
P.O. Box 90950, Riyadh 11623, Saudi Arabia ]{}\
[Correspondence address: P. O. Box 287574, Riyadh 11323, Saudi Arabia]{}
title: '**On the Matroid Isomorphism Problem**'
---
[**2010 Mathematics Subject Classification:**]{} Primary 05B35, Secondary 90C27, 52B40. locked subsets, matroid isomorphism, graph isomorphism, self duality, polynomially locked matroid.
Introduction
============
Sets and their characterisitic vectors will not be distinguished. We refer to Oxley [@Oxley; @1992] and Schrijver [@Schrijver; @1986] about matroids and polyhedra terminolgy and facts, respectively.
Given two matroids $M_1$ and $M_2$, the Matroid Isomorphism Problem (MIP) is to find a bijection $\varphi:E(M_1) \rightarrow E(M_2)$ such that the class of bases of both matroids $\mathcal B(M_1)$ and $\mathcal B(M_2)$ are isomorphic, i.e., for any base $B_1\in \mathcal B(M_1)$ there exists a unique base $B_2\in \mathcal B(M_2)$ such that $\varphi (B_1)=B_2$. This problem is in $\Sigma_2^p$ [@Rao; @and; @Sarma; @2009]. Even for linear matroids, MIP is in $\Sigma_2^p$-complete and coNP-hard [@Rao; @and; @Sarma; @2009]. For matroids and linear matroids with matroid rank bounded by a constant, MIP is polynomially time equivalent to the Graph Isomorphism Problem (GIP)[@Rao; @and; @Sarma; @2009]. Nonsuccinct approaches of MIP has been studied in [@Mayhew; @2008]: MIP is GI-complete if we use the list of all independent sets (for example) in the input instead of a matroid oracle to access the matroid. Unfortunately, this approach requires an exponential input size (see [@Robinson; @and; @Welsh; @1980]) and then an exponential time on the size of $|E|$. In this paper, we prove that MIP is reducible to Graph Isomorphism Problem for a large class of matroids denoted $\mathcal L_k$. This class is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids, i.e., $M(K_4)$, $W^3$, $Q_6$ and $P_6$ [@Chaourar; @2011]. It follows that this class contains strictly 2-sums of uniform matroids. Let’s give some definitions. Given a matroid $M$ defined on a finite set $E$. Suppose that $M$ (and $M^*$) is 2-connected. A subset $L\subset E$ is called a locked subset of $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and their corresponding ranks are at least 2, i.e., $min\{r(L), r^*(E\backslash L)\} \geq 2$. It is not difficult to see that if $L$ is locked then both $L$ and $E\backslash L$ are closed, respectively, in $M$ and $M^*$ (That is why we call them locked). We denote by $\mathcal{L}(M)$ and $\ell(M)$, respectively, the class of locked subsets of $M$ and its cardinality, which is called the locked number of $M$. Given a positive integer $k$ ($k$ does not depend on $M$ or $|E|$), we say that $M$ is $k$-locked if $\ell(M) \in O(|E|^k)$. $\mathcal L_k$ is the class of $k$-locked matroids. $M$ is 0-locked if $\mathcal L(M) = \varnothing$, i.e., $\ell(M)$ = 0 and the class of such matroids is $\mathcal L_0$. For a given nonegative integer $k$, $\mathcal L_k$ is called also a polynomially locked class of matroids. It is not difficult to see that the class of lockeds subsets of a matroid $M$ is the union of lockeds subsets of the 2-connected components of $M$. The locked structure of $M$ is the quadruple ($\mathcal P(M)$, $\mathcal S(M)$, $\mathcal L(M)$, $\rho$ ), where $\mathcal P(M)$ and $\mathcal S(M)$ are, respectively, the class of parallel and coparallel closures, and $\rho$ is the rank function restricted to $\mathcal P(M)\cup \mathcal S(M)\cup \mathcal L(M)\cup \{\varnothing , E\}$. A matroid can be completly characterized by its locked structure through its bases polytope [@Chaourar; @2017]. We will give in this paper a new axiom system for defining matroids based on this quadruple. The remainder of this paper is organized as follows. In section 2, we give a new axiom system for defining matroids, then, in section 3, we prove that MIP is polynomially time reducible to GI for polynomially locked classes of matroids, and give some large classes of such classes of matroids. Finally, we will conclude in section 4.
The Locked Axioms for a Matroid
===============================
Given a finite set $E$, $M=(E, \mathcal P, \mathcal S, \mathcal L, r)$ is a locked system defined on $E$ if: (L1) $E\neq \varnothing$, (L2) $\mathcal P$ and $\mathcal S$ are partitions of $E$, (L3) For any $(P, S)\in \mathcal P\times \mathcal S$, if $P\cap S\neq \varnothing$ then $|P|=1$ or $|S|=1$, (L4) $\mathcal L$ is a class of nonempty and proper subsets of $E$ such that $\mathcal L\cap \mathcal P=\mathcal L\cap \mathcal S=\varnothing$, (L5) For any $(X, L)\in (\mathcal P\cup \mathcal S)\times \mathcal L$, $X\cap L= \varnothing$ or $X\subset L$, (L6) $r$ is a nonegative function defined on $2^E$, (L7) $r(\varnothing)=0$ and $r(E)\geq r(X)$ for any $X\subseteq E$, (L8) $r(P)=min\{1, r(E)\}$ for any $P\in \mathcal P$, (L9) $r(E\backslash P)=min\{|E\backslash P|, r(E)\}$ for any $P\in \mathcal P$, (L10) $r(S)=min\{|S|, r(E)\}$ for any $S\in \mathcal S$, (L11) $r(E\backslash S)=min\{|E\backslash S|, r(E)+1-|S|\}$ for any $S\in \mathcal S$, (L12) $r(L)\geq max\{2, r(E)+2-|E\backslash L|$ for any $L\in \mathcal L$, (L13) $r$ is increasing on $\mathcal P\cup \mathcal L\cup \{\varnothing, E\}$, (L14) $r$ is submodular on $\mathcal P\cup \mathcal S\cup \mathcal L\cup \{\varnothing, E\}$, (L15) For any $L\in \mathcal L$, if $L=X\cup Y$ and $X\cap Y=\varnothing$ then $r(L)<r(X)+r(Y)$, (L16) For any $L\in \mathcal L$, if $L=X\cap Y$ and $X\cup Y=E$ then $r(L)<r(X)+r(Y)-r(E)$, (L17) For any $X\not \in \mathcal P\cup \mathcal S\cup \mathcal L\cup \{\varnothing, E\}$, one of the following holds:
[=0.5in ]{}
(P1) There exists $L\in \mathcal L$ such that $L\subset X$ , $r(X)=r(L)+r(X\backslash L)$, and $X\backslash L$ verifies (P1) or (P2),
(P2) There exists $P\in \mathcal P$ such that $P\cap X\neq \varnothing$ , $r(X)=r(P)+r(X\backslash P)$, and $X\backslash P$ verifies (P1) or (P2),
(P3) There exists $L\in \mathcal L$ such that $X\subset L$ , $r(X)=r(L)+r(X\cup (E\backslash L))-r(E)$, and $X\cup (E\backslash L$ verifies (P3) or (P4),
(P4) There exists $S\in \mathcal S$ such that $(E\backslash S)\cup X\neq E$ , $r(X)=r(E\backslash S)+r(X\cup S)+|S\cap X|-r(E)$, and $X\cup S$ verifies (P3) or (P4),
(L18) For any $(L_1, L_2)\in \mathcal L^2$, if $L_1\cap L_2\neq \varnothing$ and $L_1\cap L_2\not \in \mathcal L$ then $L_1\cap L_2$ verifies (P1) or (P2) of (L17), (L19) For any $(L_1, L_2)\in \mathcal L^2$, if $L_1\cup L_2\neq E$ and $L_1\cup L_2\not \in \mathcal L$then $L_1\cup L_2$ verifies (P3) or (P4) of (L17), Note that the axiom (L17) gives a way on how to compute the values of the function $r$ outside $\mathcal P\cup \mathcal S\cup \mathcal L\cup \{\varnothing, E\}$. So we do not need to verify it for a locked system realization. Note also that some axioms will not be used partially or completly later but were introduced in order to define a dual system if needed. Without loss of generality, we can replace axioms (L8)-(L11) by the following axioms respectively: (LL8) $r(P)=1$ for any $P\in \mathcal P$, (LL9) $r(E\backslash P)=r(E)$ for any $P\in \mathcal P$, (LL10) $r(S)=|S|$ for any $S\in \mathcal S$, (LL11) $r(E\backslash S)=r(E)+1-|S|$ for any $S\in \mathcal S$. Let’s give the following polyhedra associated to the locked system $M$: $P(M)$ is the set of all $x\in R^E$ satisfying the following inequalities: $$x(E)=r(E) \eqno (1)$$ $$x(P) \leq 1 \> \> \mathrm{for\>any}\> P\in \mathcal P \eqno (2)$$ $$x(S) \geq |S|-–1 \> \mathrm{for\>any}\> S\in \mathcal S \eqno (3)$$ $$x(L) \leq r(L) \> \mathrm{for\> any}\> L\in \mathcal L \eqno (4)$$ Now, we can start our process to prove the main theorem.
If $x\in P(M)$ then $$0\leq x(e) \leq 1 \> \> \mathrm{for\>any}\>e\in E \eqno (5)$$
Let $e\in E$. Since $\mathcal P$ and $\mathcal S$ are partitions of E (L2) then there exist a pair $(P, S)\in \mathcal P\times\mathcal S$ such that $\{e\}=P\cap S$ (L3). **Case 1:** if $|P|=|S|=1$ then inequalities (2) and (3) imply inequalties (5). **Case 2:** if $|S|\geq 2$ then $\{f\}\in \mathcal P$ for any $f\in S$ (L3). Inequalities (2) imply $x(f)\leq 1$ for any $f\in S$. In particular $x(e)\leq 1$. It follows that $x(S\backslash \{e\})\leq |S|-1$, then $x(e)=x(S)-x(S\backslash \{e\})\geq (|S|-1)-(|S|-1)=0$ **Case 3:** if $|P|\geq 2$ then $\{f\}\in \mathcal S$ for any $f\in P$ (L3). Inequalities (3) imply $x(f)\geq 0$ for any $f\in P$. In particular $x(e)\geq 0$. It follows that $x(P\backslash \{e\})\geq 0$, then $x(e)=x(P)-x(P\backslash \{e\})\leq x(P)\leq 1$.
If $x\in P(M)$ then $$x(A) \leq r(A) \> \> \mathrm{for\>any}\>A\subseteq E \eqno (6)$$
We have the following cases: **Case 1:** if $A=\varnothing$ then $x(A)=0\leq 0=r(A)$ (L7). **Case 2:** if $A=E$ then $x(A)=r(A)\leq r(A)$ (inequality 1). **Case 3:** if $A\in \mathcal P$ then $x(A) \leq 1=r(A)$ (inequality 2 and LL8). **Case 4:** if $A\in \mathcal S$ then $x(A)\leq |A|=r(A)$ (Lemma 2.1 and LL10). **Case 5:** if $E\backslash A\in \mathcal S$ then $x(A)=x(E)-x(E\backslash A)\leq r(E)-|E\backslash A|+1=r(A)$ (inequality 3 and LL11). **Case 6:** if $A\not \in \mathcal P\cup \mathcal S\cup \mathcal L\cup \{\varnothing, E\}$ and $E\backslash A\not \in \mathcal S$ then the axiom (L17) implies one of the following subcases: **Subcase 6.1:** There exists $L\in \mathcal L$ such that $L\subset A$ , $r(A)=r(L)+r(A\backslash L)$, and $A\backslash L$ verifies (P1) or (P2). So by induction on $|A|$, $x(A)=x(L)+x(A\backslash L)\leq r(L)+r(A\backslash L)=r(A)$ because $|A\backslash L|<|A|$ and inequality 4. **Subcase 6.2:** There exists $P\in \mathcal P$ such that $P\cap A\neq \varnothing$ , $r(A)=r(P)+r(A\backslash P)$, and $A\backslash P$ verifies (P1) or (P2). So by induction on $|A|$, $x(A)\leq x(P)+x(A\backslash P)\leq r(P)+r(A\backslash P)=r(A)$ because $|A\backslash P|<|A|$, Lemma 2.1 and Case 3, **Subcase 6.3:** There exists $L\in \mathcal L$ such that $A\subset L$ , $r(A)=r(L)+r(A\cup (E\backslash L))-r(E)$, and $A\cup (E\backslash L)$ verifies (P3) or (P4). So by induction on $|E\backslash A|$, $x(A)=x(L)+x(A\cup (E\backslash L))-x(E)\leq r(L)+r(A\cup (E\backslash L))-r(E)=r(A)$ because $|E\backslash(A\cup (E\backslash L))|=|(E\backslash A)\cap L|<|E\backslash A|$ and inequality 4, **Subcase 6.4:** There exists $S\in \mathcal S$ such that $(E\backslash S)\cup A\neq E$ , $r(A)=r(E\backslash S)+r(A\cup S)+|S\cap A|-r(E)$, and $A\cup S$ verifies (P3) or (P4). So by induction on $|E\backslash A|$, $$x(A)=x((E\backslash S)+x(A\cup S)+x(S\cap A)-x(E)\leq r(E\backslash S)+r(A\cup S)+|S\cap A|-r(E)=r(A)$$ because $|E\backslash(A\cup S)|=|(E\backslash A)\cap (E\backslash S)|<|E\backslash A|$, Lemma 2.1, Case 5, and inequality 4.
Let $Q(M)$ be the set of $x\in R^E$ such that $x$ verifies the inequalities (1), (5) and (6).
$P(M)=Q(M)$.
Lemma 2.1 and 2.2 imply that $P(M)\subseteq Q(M)$. We need to prove the inverse inclusion. Let $x\in Q(M)$. It is clear that $x$ verifies the inequalities (2) and (4) by using inequality (6) and axiom (LL8). Let $S\in \mathcal S$ then, by using inequalities (1), (6) and axiom (LL11), $x(S)=x(E)-x(E\backslash S)\geq r(E)-r(E\backslash S)=r(E)-r(E)-1+|S|=|S|-1$, which is inequality (3).
Let $x\in P(M)$ such that $x(L_i)=r(L_i)$, for some $L_i\in \mathcal L, i=1, 2$. If $L_1\cap L_2\neq \varnothing$ then there exists $L\in\mathcal P\cup \mathcal L$ such that $L\subseteq L_1\cap L_2$ and $x(L)=r(L)$.
By using Lemma 2.2 and axiom (L14), we have: $$r(L_1)+r(L_2)=x(L_1)+x(L_2)=x(L_1\cap L_2)+x(L_1\cup L_2)\leq r(L_1\cap L_2)+r(L_1\cup L_2)\leq r(L_1)+r(L_2).$$ It follows that $x(L_1\cap L_2)=r(L_1\cap L_2)$ and $x(L_1\cup L_2)=r(L_1\cup L_2)$. If $L_1\cap L_2\in \mathcal L$ then $L=L_1\cap L_2$. Otherwise, by using axiom (L18), we have two cases: **Case 1:** There exists $L\in \mathcal L$ such that $L\subset L_1\cap L_2$ , $r(L_1\cap L_2)=r(L)+r((L_1\cap L_2)\backslash L)$, and $(L_1\cap L_2)\backslash L$ verifies (P1) or (P2) of axiom (L17). It is not difficult to see, by a similar argument as hereabove, that $x(L)=r(L)$ and $x((L_1\cap L_2)\backslash L)=r((L_1\cap L_2)\backslash L)$. **Case 2:** There exists $P\in \mathcal P$ such that $P\cap (L_1\cap L_2)\neq \varnothing$ , $r(L_1\cap L_2)=r(P)+r((L_1\cap L_2)\backslash P)$, and $(L_1\cap L_2)\backslash P$ verifies (P1) or (P2) of axiom (L17). Axiom (L5) implies that $P\subseteq L_1\cap L_2$. It is not difficult to see, by a similar argument as hereabove, that $x(P)=r(P)$ and $x((L_1\cap L_2)\backslash P)=r((L_1\cap L_2)\backslash P)$.
$P(M)$ is integral.
Let $x\in P(M)$ be a fractional extreme point and $F=\{g\in E$ such that $0<x(g)<1\}$. Since $x$ is fractional and $x(E)=r(E)$ is integral then $|F|\geq 2$. Let $\mathcal P_x=\{P\in \mathcal P$ such that $x(P)=1\}$, $\mathcal S_x=\{S\in \mathcal S$ such that $x(S)=|S|-1\}$, and $\mathcal L_x=\{L\in \mathcal L$ such that $x(L)=r(L)\}$, i.e., the corresponding tight inequalities of $x$. **Case 1:** There exists $X\in \mathcal P\cup \mathcal S$ such that $|X\cap F|\geq 2$. Let $\{e, f\}\subseteq X\cap F$. It follows that there exists $\varepsilon >0$ such that $0<x(e)-\varepsilon <1$ and $0<x(f)+\varepsilon <1$. Let $x_\varepsilon \in R^E$ such that: $$x_\varepsilon (g)= \left\{ \begin{array}{ll}
x(g) & \mbox{if $g\not \in \{e, f\}$};\\
x(e)-\varepsilon & \mbox{if $g=e$ };\\
x(f)+\varepsilon & \mbox{if $g=f$}.
\end{array} \right.$$ It is clear that $x_\epsilon(E)=r(E)$. Axioms (L2), (L3) and (L5) imply that $\mathcal P_x=\mathcal P_{x_\varepsilon}$, $\mathcal S_x=\mathcal S_{x_\varepsilon}$, and $\mathcal L_x=\mathcal L_{x_\varepsilon}$, i.e., $x_\varepsilon$ verifies the same tight constraints as $x$, a contradiction. **Case 2:** For any $X\in \mathcal P\cup \mathcal S$, we have $|X\cap F|\leq 1$. It follows that for any $X\in \mathcal P_x\cup \mathcal S_x$, we have $X\cap F=\varnothing$. **Subcase 2.1:** There exists $L\in \mathcal L_x$ such that $|L\cap F|\geq 2$, and $\{ e, f\}\subseteq L\cap F$ such that if $L'\in \mathcal L_x$ then $\{ e, f\}\subseteq L'$ or $\{ e, f\}\cap L'=\varnothing$. So we proceed as in Case 1 and we conclude. **Subcase 2.2:** For any $L\in \mathcal L_x$ such that $|L\cap F|\geq 2$, and any $\{ e, f\}\subseteq L\cap F$, there exists $L'\in \mathcal L_x$ such that $|\{ e, f\}\cap L'|=1$. Suppose that $f\in L'$. So we have: $$r(L)+r(L')=x(L)+x(L')=x(L\cap L')+x(L\cup L')\leq r(L\cap L')+r(L\cup L')\leq r(L)+r(L').$$ It follows that $x(L\cap L')=r(L\cap L')$ and $x(L\cup L')=r(L\cup L')$. It follows that $|L\cap L'\cap F|\geq 2$. By using Lemma 2.4, and since $L\cap L'\neq \varnothing$ then there exists $X\in \mathcal P\cup \mathcal L$ such that $X\subseteq L\cap L'$ and $x(X)=r(X)$. By induction on $|L\cap L'|$, we have $X\cap F\neq \varnothing$ (otherwise we do the same for $(L_1\cap L_2)\backslash X$), i.e., $|X\cap F|\geq 2$. Induction on $|L|$ and axiom (L13) imply that $r(X)=1$, i.e., $X\in \mathcal P$, a contradiction.
Now we can state our main theorem as follows.
The extreme points of $P(M)$ are the bases of a matroid defined on $E$, and $\mathcal P, \mathcal S, \mathcal L, r$ are, respectively, the class of parallel and coparallel closures, locked subsets and rank function of this matroid.
Lemma 2.1 and Theorem 2.5 imply that the extreme points of $P(M)$ are in $\{0, 1\}^E$. We remind here that we will not distinguish between sets and $\{0, 1\}$-vectors. Inequality (1) implies that extreme points of $P(M)$ have the same cardinality $r(E)$. We only need to prove the basis exchange axiom. We will do it by contradiction. Let $x$ and $x'$ to be two extreme points of $P(M)$ and $e\in x\backslash x'$ such that for any $f\in x'\backslash x$, $x-e+f$ is not an extreme point, i.e., $x-e+f\not \in P(M)$. It is clear that $|x'\backslash x|\geq 2$. Let $x_f=x-e+f$. **Case 1:** $x_f$ violates an inequality of type (2), i.e., there exists $P_f\in \mathcal P$ such that $x_f(P_f)\geq 2$. It follows that $e\not \in P_f$, $f\in P_f$, $x(P_f)=1$, and $x_f(P_f)=2$. Thus there exists $f'\in P_f\cap x_f\cap x$ such that $f'\neq f$. **Claim:** If $f_1\neq f_2$ then $f'_1\neq f'_2$. Suppose, by contradiction, that $f'=f'_1=f'_2$. Since $f'\in P_{f_i}\cap x_{f_i}, i=1, 2$, then $f'\in P_{f_1}\cap P_{f_2}$. Axiom (L2) implies that $P_{f_1}=P_{f_2}=P$ and $\{f_1, f_2\}\subseteq P\cap x'$. It follows that $x'(P)\geq 2$, a contradiction. Since $|x\backslash x'|=|x'\backslash x|$ then $x\backslash x'=\bigcup \limits_{i=1}^{|x\backslash x'|} \{f'_i\}\subseteq \bigcup \limits_{i=1}^{|x\backslash x'|} P_{f_i}$ but $e\not \in P_{f_i}, i=1, 2, ..., |x\backslash x'|$, a contradiction. **Case 2:** $x_f$ violates an inequality of type (3), i.e., there exists $S_f\in \mathcal S$ such that $x_f(S_f)\leq |S|-2$. It follows that $e\in S_f$, $f\not \in S_f$, $x(S_f)=|S_f|-1$, and $x_f(S_f)=|S_f|-2$. Since $e\in S_f$ for any $f\in x'\backslash x$, and by using axiom (L2), we have $S_f=S$, i.e., for distinct $f_1$ and $f_2$, $S_{f_1}=S_{f_2}=S$. It follows that $(x'\backslash x)\cap S=\varnothing$. But $x'(S)\geq |S|-1$ because $x'\in P(M)$, then $(x'\cap x)(S)\geq |S|-1$. It follows that $(x\backslash x')\cap S=\varnothing$, a contradiction with $e\in S$. **Case 3:** $x_f$ violates an inequality of type (4), i.e., there exists $L_f\in \mathcal L$ such that $x_f(L_f)\geq r(L_f)+1$. It follows that $e\not \in L_f$, $f\in L_f$, $x(L_f)=r(L_f)$, and $x_f(L_f)=r(L_f)+1$. We choose $L_f$ maximal for this property. **Subcase 3.1:** There are $f_1\neq f_2$ such that $x_{f_1}(L_{f_2})=r(L_{f_2})$, i.e., $f_1\notin L_{f_2}$. As shown in the proof of Lemma 2.4, $x(L_{f_1}\cup (L_{f_2})=r(L_{f_1}\cup (L_{f_2})$. Since $L_{f_2}$ is maximal then $L_{f_1}\cup L_{f_2}\notin \mathcal L$. Since $e\not \in L_{f_1}\cup L_{f_2}$, and by using axiom (L19), there exists $S\in \mathcal S$ such that $(E\backslash S)\cup (L_{f_1}\cup L_{f_2})\neq E$ , $r(L_{f_1}\cup L_{f_2})=r(E\backslash S)+r(L_{f_1}\cup L_{f_2}\cup S)+|S\cap (L_{f_1}\cup L_{f_2})|-r(E)$, and $(L_{f_1}\cup L_{f_2})\cup S$ verifies (P4) (property (P3) cannot be verified because of maximality of $L_{f_2}$). By a similar argument as in the proof of Lemma 2.4, we have: **(1)** $x(E\backslash S)=r(E\backslash S)$ which imply that $x(S)=|S|-1$, i.e., $S=x\backslash \{e'\}\cup \{ f'\}$ for some $e'\in x$ and $f'\notin x$, and **(2)** $x(L_{f_1}\cup L_{f_2}\cup S)=r(L_{f_1}\cup L_{f_2}\cup S)$. If $e\in S$ (i.e. $e\neq e'$) then at least one the $x_{f_i}(S)=|S|-2$ (i.e. $f_i\neq f'$) and we are in Case 2). Else $e\notin S$, i.e. $e\notin L_{f_1}\cup L_{f_2}\cup S$ and by induction on $|E\backslash X|$ where $X=L_{f_1}\cup L_{f_2}$, we get a contradiction. **Subcase 3.2:** For any $f_1\neq f_2$, $x_{f_1}(L_{f_2})=r(L_{f_2})+1$, i.e., $f_1\in L_{f_2}$. It follows that there exists $L\in \mathcal L$ such that $x'\backslash x\subseteq L$, $e\notin L$, and $x(L)=r(L)$. We have then: $r(L)\geq x'(L)=(x'\backslash x)(L)+(x'\cap x)(L)=|x'\backslash x|+(x'\cap x)(L)=|x\backslash x'|+(x'\cap x)(L)\geq (x\backslash x')(L)+(x'\cap x)(L)=x(L)=r(L)$. It follows that $(x\backslash x')(L)=|x\backslash x'|$, i.e., $x\backslash x'\subseteq L$, a contradiction with $e\in x\backslash x'$.
Actually this gives a new proof for the bases polytope of a matroid and its facets based on the locked structure only.
MIP is reducible to GI for polynomially locked matroids
=======================================================
For any matroid, we can constrcut an [*augmented*]{} lattice of locked subsets veretx-labeled where we add a vertex (representing) for every coparallel closure adjacent to the root (the empty set) and a vertex (representing) for every parallel closure adjacent to a veretx representing a coparallel closure intersectig this parallel closure. Every vertex is labeled by two numbers: its cardinality and its rank. This lattice is called the locked lattice of this matroid. For example we give herebelow the locked lattice of the graphical matroid $M(K_4)$. The labels are the same for every vertex of a same level. The root ($\varnothing$, level 0) is labeled with (0, 0); the coparallel (level 1) and parallel closures (level 2) are labeled with (1, 1); the locked subsets (level 3) are labeled with (3, 2); and finally the ground set ($E$, sink or level 4) is labeled with (6, 3).
\[scale=.8,auto=center\] (2,4) circle (4pt);
(2,2) circle (4pt);
(0,0) circle (4pt);
(4,0) circle (4pt);
(2,4) – (2,2); (2,4) – (0,0); (2,4) – (4,0); (2,2) – (0,0); (2,2) – (4,0); (0,0) – (4,0);
at (0.5,2.2) ; at (3.5,2.2) ; at (1.5,0.8) ; at (2.5,0.8) ; at (2,-0.5) ; at (1.7,2.7) ;
\[scale=.8,auto=center,every node/.style=[rectangle,fill=green!50]{}\] (n1) at (5,0) [$\varnothing$]{}; (n2) at (0,2) [{ a}]{}; (n3) at (2,2) [{ b}]{}; (n4) at (4,2) [{ c}]{}; (n5) at (6,2) [{ d}]{}; (n6) at (8,2) [{ e}]{}; (n7) at (10,2) [{ f}]{};
/in [n1/n2,n1/n3,n1/n4,n1/n5,n1/n6,n1/n7]{} () – ();
(n8) at (0,4) [{ a}]{}; (n9) at (2,4) [{ b}]{}; (n10) at (4,4) [{ c}]{}; (n11) at (6,4) [{ d}]{}; (n12) at (8,4) [{ e}]{}; (n13) at (10,4) [{ f}]{};
/in [n2/n8,n3/n9,n4/n10,n5/n11,n6/n12,n7/n13]{} () – ();
(n14) at (0,6) [{a, b, d}]{}; (n15) at (3,6) [{a, c, f}]{}; (n16) at (7,6) [{b, c, e}]{}; (n17) at (10,6) [{d, e, f}]{};
(n8)–(n14); (n8)–(n15); (n9)–(n14); (n9)–(n16); (n10)–(n15); (n10)–(n16); (n11)–(n17); (n11)–(n14); (n12)–(n16); (n12)–(n17); (n13)–(n15); (n13)–(n17);
(n18) at (5,8) [$E$]{};
/in [n14/n18,n15/n18,n16/n18,n17/n18]{} () – ();
at (0,-1) ; at (8,-1) ;
It is not difficult to generalize the notion of isorphism between graphs to directed vertex-labeled graphs. Since the locked structure is completly described by the locked lattice, then we can state the following proposition.
Two matroids are isomorphic if and only if their corresponding locked lattices are isomorphic.
We can now reduce the locked lattice to the following. We do not need to label coparallel and parallel closures by their ranks because they are equal, respectively, to the cardinality and 1. We do not need to label any locked subset by its cardinality because it can be computed as the maximum flow from the root to the vertex representing this locked subset in the locked lattice with capacities equal to 1 for arcs between coparallel and parallel closures, and infinity otherwise. This reduced lattice is a directed acyclic vertex-labeled graph and the labels are nonegative numbers bounded by $|E|$. We call it also the locked lattice of the given matroid. And the previous proposition holds for this locked lattice. For reducing MIP to GI we need the following theorem [@Zemlyachenko; @et; @al.; @1985].
The Isomorphism Problem in acyclic directed graphs (ADGI) is GI-complete.
We can now state the main result of this section.
MIP is polynomial time reducible to GI for polynoimally locked classes of matroids.
We will reduce ismorphism of locked lattices to ADGI. Consider a matroid M and its locked lattice $L(M)$. If we replace each vertex in $L(M)$ by a number of series arcs equal to its label, then $L(M)$ become an acyclic directed graph without any vertex-labeling. The number of the vertices and the edges in the new graph is bounded, respectively, by $(\ell (M)+2|E|+2)(|E|+1)$ and by $(\ell (M)+2|E|+2)^2(|E|+1)^2$ which is polynomial on the size of the ground set $E$ if $M$ is polynomially locked.
The special case in $\mathcal L_0$ is polynomial.
MIP is polynomial for matroids in $\mathcal L_0$.
Locked lattices in $\mathcal L_0$ are completly described by coparallel and parallel closures. Each of the later forms a partition of the ground set $E$. Axiom (L3) reduces locked lattices in this case to one level only: coparallel closures (or parallel closures). So locked lattices can be represented by a sequence of at most $|E|$ positive nondecreasing numbers bounded by $|E|$. It is clear that checking numbers of two such sequences will give an answer for MIP in this case. This checking requires a linear runing time complexity.
Another interesting problem related to MIP is testing self-duality (TSD). Jensen and Korte [@Jensen; @and; @Korte; @1982] proved that there exists no polynomial algorithm in which the matroid is represented by an independence test oracle (or an oracle polynomially related to an independence test oracle) for TSD. Chaourar [@Chaourar; @2017] introduced the following matroid oracle which reduces (partially) this question to GI (for polynomially locked classes of matroids).
The $k$-locked oracle
[\*6l]{} Input: & a nonegative integer $k$ and a matroid $M$ defined on $E$.\
Output:& (1) No if $\ell(M)\notin O(|E|^k)$, and\
& (2) ($\mathcal P(M)$, $\mathcal S(M)$, $\mathcal L(M)$, $\rho$ ) if $\ell(M)\in O(|E|^k)$.\
“Note that this oracle has time complexity $O(|E|^{k+1})$ because we need to count at most $|E|^{k+1}$ members of $\mathcal L(M)$ in order to know that $M$ is not $k$-locked, even if the memory complexity can be $O(|E|+\ell (M))$. Actually this matroid oracle permits to recognize if a given matroid is $k$-locked or not for a given nonegative integer $k$ (which does not depend on $M$ or $|E|$)” [@Chaourar; @2017], and this matroid oracle is stronger then the rank and the independence oracles for polynomially locked matroids. We can now state our second main result for this section.
TSD is polynomial time reducible to GI for polynomially locked matroids represented by a $k$-locked oracle.
If a matroid $M$ is $k$-locked for a given nonegative $k$ (which does not depend on $M$ or $|E|$), then the $k$-locked oracle gives its locked structure. Since $\mathcal P(M^*)=\mathcal S(M)$, $\mathcal S(M^*)=\mathcal P(M)$, $\mathcal L(M^*)=\{ E\backslash L$ such that $L\in \mathcal L(M)\}$, $\rho ^*(X)=\rho (E\backslash X)+|X|-r(E)$, then $L(M^*)$ can be (computed) constructed in polynomial time from $L(M)$. Since $\ell (M)=\ell (M^*)$ then both matroids are polynomially locked and TSD is reducible to MIP which is reduicible to GI.
Finally we give some large classes of polynomially locked matroids [@Chaourar; @2008; @Chaourar; @2011; @Chaourar; @2017].
The following properties hold for matroids. (1) 2-sums preserves $k$-lockdness for a given positive integer $k$; (2) A fixed number of $p$-sums preserves $k$-lockdness for positive integers $p$ and $k$; (3) 2-sums of uniform matroids are 1-locked; (4) All matroids on 6 elements are 1-locked, in particular, all excluded minors of 2-sums of uniform matroids; (5) The Vámos matroid is 1-locked; (6) $\mathcal L_1$ contains strictly the class of 2-sums of uniform matroids.
Conclusion
==========
We have given a new system of axioms for defining a matroid based essentially on locked subsets. We have proved that MIP and TSD are reducible to GI for polynomially locked classes of matroids. We have given some large classes of polynomially locked matroids. Future investigations can be characterizing partially or completly polynomially locked matroids.
[1]{}
B. Chaourar (2008), [*On the Kth Best Basis of a Matroid*]{}, Operations Research Letters 36 (2), 239-242.
B. Chaourar (2011), [*A Characterization of Uniform Matroids*]{}, ISRN Algebra, Vol. 2011, Article ID 208478, 4 pages, doi:10.5402/2011/208478.
B. Chaourar (2017), [*The Facets of the Bases Polytope of a Matroid and Two Consequences*]{}, arXiv 1702.07128.
P. M. Jensen and B. Korte, [*Complexity of Matroid Property Algorithms*]{}, SIAM J. COMPUT. 11 (1): 184-190.
D. Mayhew (2008), [*Matroid complexity and nonsuccinct descriptions*]{}, SIAM Journal on Discrete Mathematics, 22 (2): 455–466.
J. G. Oxley (1992), [*Matroid Theory*]{}, Oxford University Press, Oxford.
B.V. R. Rao and M. N. J. Sarma (2009), [*On the Complexity of the Matroid Isomorphism Problems*]{}, In: A. Frid, A. Morozov, A. Rybalchenko, K. W. Wagner (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, Vol. 5675, Springer, Berlin, Heidelberg.
G. C. Robinson and D. J. A. Welsh (1980), [*The computational complexity of matroid properties*]{}, Math. Proc. Cambridge Phil. Society 87, 29-45.
A. Schrijver (1986), [*Theory of Linear and Integer Programming*]{}, John Wiley and Sons, Chichester.
V. N. Zemlyachenko, N. M. Korneenko, and R. I. Tyshkevich (1985), [*Graph isomorphism problem*]{}, Journal of Mathematical Sciences 29 (4): 1426-1481.
|
---
abstract: 'The goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient.'
address: 'Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfengarten 1, 30167 Hannover, Germany'
author:
- Stefan Tappe
title: Invariance of closed convex cones for stochastic partial differential equations
---
[^1]
Introduction {#sec-intro}
============
Consider a semilinear stochastic partial differential equation (SPDE) of the form $$\begin{aligned}
\label{SPDE}
\left\{
\begin{array}{rcl}
dr_t & = & ( A r_t + \alpha(r_t) ) dt + \sigma(r_t) dW_t + \int_E \gamma(r_{t-},x) (\mu(dt,dx) - F(dx)dt) \medskip
\\ r_0 & = & h_0
\end{array}
\right.\end{aligned}$$ driven by a trace class Wiener process $W$ and a Poisson random measure $\mu$ on some mark space $E$ with compensator $dt \otimes F(dx)$. The state space of the SPDE (\[SPDE\]) is a separable Hilbert space $H$, and the operator $A$ is the generator of a strongly continuous semigroup $(S_t)_{t \geq 0}$ on $H$. We refer to Section \[sec-ass\] for more details concerning the mathematical framework.
In applications, one is often interested in the question when a certain subset of the state space is invariant for the SPDE (\[SPDE\]), and frequently it turns out that this subset is a closed convex cone. For example, when modeling the evolution of interest rate curves, a desirable feature is that the model produces nonnegative interest curves; or when modeling multiple yield curves, it is desirable to have spreads which are ordered with respect to different tenors.
In order to translate these ideas into mathematical terms, let $K \subset H$ be a closed convex cone of the state space $H$. We say that the cone $K$ is invariant for the SPDE (\[SPDE\]) if for each starting point $h_0 \in K$ the solution process $r$ to (\[SPDE\]) stays in $K$. The goal of this paper is to clarify when the cone $K$ is invariant for the SPDE (\[SPDE\]), and to provide conditions on the parameters $(A,\alpha,\sigma,\gamma)$ – or, equivalently, on $((S_t)_{t \geq 0},\alpha,\sigma,\gamma)$ – of the SPDE (\[SPDE\]), which are necessary and sufficient.
Stochastic invariance of a given subset $K \subset H$ for jump-diffusion SPDEs (\[SPDE\]) has already been studied in the literature, mostly for diffusion SPDEs $$\begin{aligned}
\label{SPDE-Wiener}
\left\{
\begin{array}{rcl}
dr_t & = & ( A r_t + \alpha(r_t) ) dt + \sigma(r_t) dW_t \medskip
\\ r_0 & = & h_0
\end{array}
\right.\end{aligned}$$ without jumps. The classes of subsets $K \subset H$, for which stochastic invariance has been investigated, can roughly be divided as follows:
- For a finite dimensional submanifold $K \subset H$ the stochastic invariance has been studied in [@Filipovic-inv] and [@Nakayama] for diffusion SPDEs (\[SPDE-Wiener\]), and in [@FTT-manifolds] for jump-diffusion SPDEs (\[SPDE\]). Here a related problem is the existence of a finite dimensional realization (FDR), which means that for each starting point $h_0 \in H$ a finite dimensional invariant manifold $K \subset H$ with $h_0 \in K$ exists. This problem has mostly been studied for the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation from mathematical finance, and we refer, for example, to [@Bj_Sv; @Bj_La; @Filipovic; @Filipovic-Teichmann-royal; @Tappe-Wiener; @Tappe-affine] for the existence of FDRs for diffusion SPDEs (\[SPDE-Wiener\]), and, for example, to [@Tappe-Levy; @Platen-Tappe; @Tappe-affin-real] for the existence of FDRs for SPDEs driven by Lévy processes, which are particular cases of jump-diffusion SPDEs (\[SPDE\]).
- For an arbitrary closed subset $K \subset H$ the stochastic invariance has been studied for PDEs in [@Jachimiak-note], and for diffusion SPDEs (\[SPDE-Wiener\]) in [@Jachimiak] and – based on the support theorem presented in [@Nakayama-Support] – in [@Nakayama]. Both authors obtain the so-called stochastic semigroup Nagumo’s condition (SSNC) as a criterion for stochastic invariance, which is necessary and sufficient. An indispensable assumption for the formulation of the SSNC is that the volatility $\sigma$ is sufficiently smooth; it must be two times continuously differentiable.
- For a closed convex cone $K \subset H$ – as in our paper – the stochastic invariance has been studied in two particular situations on function spaces. In [@Milian] the state space $H$ is an $L^2$-space, $K$ is the closed convex cone of nonnegative functions, and its stochastic invariance is investigated for diffusion SPDEs (\[SPDE-Wiener\]). In [@Positivity] the state space $H$ is a Hilbert space consisting of continuous functions, $K$ is also the closed convex cone of nonnegative functions, and its stochastic invariance is investigated for jump-diffusion SPDEs (\[SPDE\]); a particular application in [@Positivity] is the positivity preserving property of interest rate curves from the aforementioned HJMM equation, which appears in mathematical finance.
In this paper, we provide a general investigation of the stochastic invariance problem for an arbitrary closed convex cone $K \subset H$, contained in an arbitrary separable Hilbert space $H$, for jump-diffusion SPDEs (\[SPDE\]). Taking advantage of the structural properties of closed convex cones, we do not need smoothness of the volatility $\sigma$, as it is required in [@Jachimiak] and [@Nakayama], and also in [@Positivity].
In order to present our main result of this paper, let $K \subset H$ be a closed convex cone, and let $K^* \subset H$ be its dual cone $$\begin{aligned}
\label{dual-cone}
K^* = \bigcap_{h \in K} \{ h^* \in H : \langle h^*,h \rangle \geq 0 \}.\end{aligned}$$ Then the cone $K$ has the representation $$\begin{aligned}
\label{cone-repr}
K = \bigcap_{h^* \in K^*} \{ h \in H : \langle h^*,h \rangle \geq 0 \}.\end{aligned}$$ We fix a generating system $G^*$ of the cone $K$; that is, a subset $G^* \subset K^*$ such that the cone admits the representation $$\begin{aligned}
\label{cone-G}
K = \bigcap_{h^* \in G^*} \{ h \in H : \langle h^*,h \rangle \geq 0 \}.\end{aligned}$$ In particular, we could simply take $G^* = K^*$. However, for applications we will choose a generating system $G^*$ which is as convenient as possible. Throughout this paper, we make the following assumptions:
- The semigroup $(S_t)_{t \geq 0}$ is pseudo-contractive; see Assumption \[ass-pseudo-contractive\].
- The coefficients $(\alpha,\sigma,\gamma)$ are locally Lipschitz and satisfy the linear growth condition, which ensures existence and uniqueness of mild solutions to the SPDE (\[SPDE\]); see Assumption \[ass-loc-Lip-LG\].
- The cone $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$; see Assumption \[ass-cone-semigroup\].
- The cone $K$ is generated by an unconditional Schauder basis; see Assumption \[ass-Schauder-basis\].
We refer to Section \[sec-ass\] for the precise mathematical framework. We define the set $D \subset G^* \times K$ as $$\begin{aligned}
\label{def-D}
D := \bigg\{ (h^*,h) \in G^* \times K : \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} < \infty \bigg\}.\end{aligned}$$ Since the cone $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$, for all $(h^*,h) \in G^* \times K$ the limes inferior in (\[def-D\]) exists with value in $\overline{{\mathbb{R}}}_+ = [0,\infty]$. Now, our main result reads as follows.
\[thm-main\] Suppose that Assumptions \[ass-pseudo-contractive\], \[ass-loc-Lip-LG\], \[ass-cone-semigroup\] and \[ass-Schauder-basis\] are fulfilled. Then the following statements are equivalent:
1. The closed convex cone $K$ is invariant for the SPDE (\[SPDE\]).
2. We have $$\begin{aligned}
\label{main-1}
h + \gamma(h,x) \in K \quad \text{for $F$-almost all $x \in E$,} \quad \text{for all $h \in K$,}\end{aligned}$$ and for all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{main-3}
&\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha(h) \rangle - \int_E \langle h^*,\gamma(h,x) \rangle F(dx) \geq 0,
\\ \label{main-4} &\langle h^*,\sigma^j(h) \rangle = 0, \quad j \in {\mathbb{N}}.\end{aligned}$$
Conditions (\[main-1\])–(\[main-4\]) are geometric conditions on the coefficients of the SPDE (\[SPDE\]); condition (\[main-1\]) concerns the behaviour of the solution process in the cone, and conditions (\[main-3\]) and (\[main-4\]) concern the behaviour of the solution process at boundary points of the cone:
- Condition (\[main-1\]) is a condition on the jumps; it means that the cone $K$ is invariant for the functions $h \mapsto h + \gamma(h,x)$ for $F$-almost all $x \in E$.
- Condition (\[main-3\]) means that the drift is inward pointing at boundary points of the cone.
- Condition (\[main-4\]) means that the volatilities are parallel at boundary points of the cone.
Figure \[fig-geometric\] illustrates conditions (\[main-1\])–(\[main-4\]). Let us provide further explanations regarding the drift condition (\[main-3\]). For this purpose, we fix an arbitrary pair $(h^*,h) \in D$. By the definition (\[def-D\]) of the set $D$, we have $\langle h^*,h \rangle = 0$, indicating that we are at the boundary of the cone.
- The drift condition (\[main-3\]) implies $$\begin{aligned}
\label{main-2}
\int_E \langle h^*,\gamma(h,x) \rangle F(dx) < \infty.\end{aligned}$$ This means that the jumps of the solution process at boundary points of the cone are of finite variation, unless they are parallel to the boundary.
- If $h \in {\mathcal{D}}(A)$, then the drift condition (\[main-3\]) is fulfilled if and only if $$\begin{aligned}
\label{FV-cond-3}
\langle h^*, Ah + \alpha(h) \rangle - \int_E \langle h^*,\gamma(h,x) \rangle F(dx) \geq 0.\end{aligned}$$ In view of condition (\[FV-cond-3\]), we point out that $K \cap {\mathcal{D}}(A)$ is dense in $K$.
- If $h^* \in {\mathcal{D}}(A^*)$, then the drift condition (\[main-3\]) is fulfilled if and only if $$\begin{aligned}
\label{FV-cond-4}
\langle A^* h^*,h \rangle + \langle h^*, \alpha(h) \rangle - \int_E \langle h^*,\gamma(h,x) \rangle F(dx) \geq 0.\end{aligned}$$ In particular, if $A^*$ is a local operator, then the drift condition (\[main-3\]) is equivalent to $$\begin{aligned}
\label{FV-cond-5}
\langle h^*, \alpha(h) \rangle - \int_E \langle h^*,\gamma(h,x) \rangle F(dx) \geq 0.\end{aligned}$$ In any case, condition (\[FV-cond-5\]) implies the drift condition (\[main-3\]).
![Illustration of the invariance conditions.[]{data-label="fig-geometric"}](Invariance-cond-english-crop.pdf){width="50.00000%"}
We refer to Section \[sec-ass\] for the proofs of these and of further statements. We emphasize that for $(h^*,h) \in G^* \times K$ with $\langle h^*,h \rangle = 0$ it may happen that $(h^*,h) \notin D$. In this case, conditions (\[main-3\]) – and hence (\[main-2\]) – and (\[main-4\]), the two boundary conditions illustrated in Figure \[fig-geometric\], do not need to be fulfilled. Intuitively, at such a boundary point $h$ of the cone, there is an infinite drift pulling the process in the interior of the half space $\{ h \in H : \langle h^*,h \rangle \geq 0 \}$, whence we can skip conditions (\[main-3\]) and (\[main-4\]) in this situation. This phenomenon is typical for SPDEs, as for norm continuous semigroups $(S_t)_{t \geq 0}$ (in particular, if $A = 0$) the limes inferior appearing in (\[def-D\]) is always finite.
Now, let us outline the essential ideas for the proof of Theorem \[thm-main\]:
- In Theorem \[thm-nec\] we will prove that conditions (\[main-1\])–(\[main-4\]) are necessary for invariance of the cone $K$, where the main idea is to perform a short-time analysis of the sample paths of the solution processes. We emphasize that for this implication we do not need the assumption that $K$ is generated by an unconditional Schauder basis; that is, we can skip Assumption \[ass-Schauder-basis\] here.
- In order to show that conditions (\[main-1\])–(\[main-4\]) are sufficient for invariance of the cone $K$, we perform several steps:
1. First, we show that the cone $K$ is invariant for diffusion SPDEs (\[SPDE-Wiener\]) with smooths volatilities $\sigma^j \in C_b^2(H)$, $j \in {\mathbb{N}}$; see Theorem \[thm-diffusion-C2\]. The essential idea is to verify the aforementioned SSNC.
2. Then, we show that the cone $K$ is invariant for diffusion SPDEs (\[SPDE-Wiener\]) with Lipschitz coefficients without imposing smoothness on the volatilities; see Theorem \[thm-diffusion\]. The main idea is to approximate the volatility $\sigma$ by a sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ of smooth volatilities, and to apply a stability result (see Proposition \[prop-K-stability\]) for SPDEs.
3. Then, we show that the cone $K$ is invariant for general jump-diffusion SPDEs (\[SPDE\]) with Lipschitz coefficients; see Theorem \[thm-suff\]. This is done by using the so-called method to switch on the jumps – also used in [@Positivity] – and the aforementioned stability result for SPDEs.
4. Finally, we show that the cone $K$ is invariant for the SPDE (\[SPDE\]) in the general situation, where the coefficients are locally Lipschitz and satisfy the linear growth condition; see Theorem \[thm-suff-general\]. This is done by approximating the parameters $(\alpha,\sigma,\gamma)$ of the SPDE (\[SPDE\]) by a sequence $(\alpha_n,\sigma_n,\gamma_n)_{n \in {\mathbb{N}}}$ of globally Lipschitz coefficients, and to argue by stability. In order to ensure that the modified coefficients $(\alpha_n,\sigma_n,\gamma_n)$ also satisfy the required invariance conditions (\[main-1\])–(\[main-4\]), the structural properties of closed convex cones are essential.
The most challenging is the second step, where we approximate the volatility $\sigma$ by a sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ of smooth volatilities. In particular, for an application of our stability result (Proposition \[prop-K-stability\]) we must ensure that all $\sigma_n$ are Lipschitz continuous with a joint Lipschitz constant. We can roughly divide the approximation procedure into the following steps:
1. First, we approximate $\sigma$ by a sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ of bounded volatilities with finite dimensional range; see Propositions \[prop-sigma-FDR\] and \[prop-sigma-bounded\]. We construct similar approximations $(\alpha_n)_{n \in {\mathbb{N}}}$ for the drift $\alpha$; see Propositions \[prop-alpha-FDR\] and \[prop-alpha-bounded\].
2. Then, we approximate a bounded volatility $\sigma$ with finite dimensional range by a sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ from $C_b^{1,1}$. This is done by the so-called sup-inf convolution technique from [@Lasry-Lions]; see Proposition \[prop-C-b-1-1\]. Although we do not use it in this paper, we mention the related article [@Johanis], which shows how a Lipschitz function can be approximated by uniformly Gâteaux differentiable functions.
3. Finally, we approximate a volatility $\sigma$ from $C_b^{1,1}$ by a sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ from $C_b^2$; see Proposition \[prop-C-b-2\]. This is done by a generalization of the mollifying technique in infinite dimension. For this procedure, we follow the construction provided in [@Fry], which constitutes a generalization of a result from Moulis (see [@Moulis]), whence we also refer to this method as Moulis’ method. Concerning smooth approximations in infinite dimensional spaces, we also mention the related papers [@Approx; @Fine-Approx; @Hajek-Johanis-G; @Hajek-Johanis].
We emphasize that we cannot directly apply Moulis’ method in step (b), because for a Lipschitz continuous function $\sigma$ this would only provide a sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ from $C^2$ – in fact, even $C^{\infty}$ – but the second order derivatives might be unbounded. Applying the sup-inf convolution technique before ensures that we obtain a sequence from $C_b^2$. We mention that a combination of the sup-inf convolution technique and Moulis’ method has also been used in [@Approx] in order to prove that every Lipschitz continuous function defined on a (possibly infinite dimensional) separable Riemannian manifold can be uniformly approximated by smooth Lipschitz functions.
Besides the aforementioned required joint Lipschitz constant, we have to take care that the respective approximations $(\sigma_n)_{n \in {\mathbb{N}}}$ of the volatility $\sigma$ remain parallel at boundary points of the cone; that is, condition (\[main-4\]) must be preserved, which is expressed by Definition \[def-parallel\]. The situation is similar for the approximations $(\alpha_n)_{n \in {\mathbb{N}}}$ of the drift $\alpha$. They must remain inward pointing at boundary points of the cone; that is, condition (\[main-3\]) must be preserved, which is expressed by Definition \[def-inward-pointing\].
It arises the problem that we can generally not ensure in steps (b) and (c) that the approximating volatilities remain parallel. In order to illustrate the situation in step (c), where we apply Moulis’ method, let us assume for the sake of simplicity that the state space is $H = {\mathbb{R}}^d$. Then the construction of the approximating sequence $(\sigma_n)_{n \in {\mathbb{N}}}$ becomes simpler than in the infinite dimensional situation in [@Fry], and it is given by the well-known construction $$\begin{aligned}
\sigma_n : {\mathbb{R}}^d \to {\mathbb{R}}^d, \quad \sigma_n(h) := \int_{{\mathbb{R}}^d} \sigma(h-g) \varphi_n(g) dg,\end{aligned}$$ where $(\varphi_n)_{n \in {\mathbb{N}}} \subset C^{\infty}({\mathbb{R}}^d,{\mathbb{R}}_+)$ is an appropriate sequence of mollifiers. Then, for $(h^*,h) \in D$, which implies $\langle h^*,h \rangle = 0$, we generally have $$\begin{aligned}
\langle h^*,\sigma_n(h) \rangle = \int_{{\mathbb{R}}^d} \langle h^*, \sigma(h-g) \rangle \varphi_n(g) dg \neq 0,\end{aligned}$$ because we only have $\langle h^*, \sigma(h) \rangle = 0$, but generally not $\langle h^*, \sigma(h-g) \rangle = 0$ for all $g \in {\mathbb{R}}^d$ from a neighborhood of $0$. This problem leads to the notion of locally parallel functions (see Definition \[def-locally-par\]), which have the desired property that $\langle h^*, \sigma(h-g) \rangle = 0$ for all $g \in {\mathbb{R}}^d$ from an appropriate neighborhood of $0$. In order to implement this concept, we have to show that a parallel function can be approximated by a sequence of locally parallel functions. The idea is to approximate a function $\sigma : {\mathbb{R}}^d \to {\mathbb{R}}^d$ for $\epsilon > 0$ by taking $\sigma \circ \Phi_{\epsilon}$, where $$\begin{aligned}
\Phi_{\epsilon} : {\mathbb{R}}^d \to {\mathbb{R}}^d, \quad \Phi_{\epsilon}(h) := (\phi_{\epsilon}(h_1),\ldots,\phi_{\epsilon}(h_d)),\end{aligned}$$ and where the function $\phi_{\epsilon} : {\mathbb{R}}\to {\mathbb{R}}$ is defined as $$\begin{aligned}
\label{def-psi-intro}
\phi_{\epsilon}(x) := (x + \epsilon) \mathbbm{1}_{(-\infty,-\epsilon]}(x) + (x - \epsilon) \mathbbm{1}_{[\epsilon,\infty)}(x),\end{aligned}$$ see Figure \[fig-approx\]. We can also establish this procedure in infinite dimension; see Proposition \[prop-locally-par\].
![Approximation with locally parallel functions.[]{data-label="fig-approx"}](Approx-crop.pdf){width="50.00000%"}
The remainder of this paper is organized as follows. In Section \[sec-ass\] we present the mathematical framework and preliminary results. In Section \[sec-nec\] we prove that our invariance conditions are necessary for invariance of the cone. In Section \[sec-Schauder\] we provide the required background about closed convex cones generated by unconditional Schauder basis. Afterwards, we start with the proof that our invariance conditions are sufficient for invariance of in the cone. In Section \[sec-proof-1\] we prove this for diffusion SPDEs with smooth volatilities, in Section \[sec-proof-2\] for diffusion SPDEs with Lipschitz coefficients without imposing smoothness on the volatility, in Section \[sec-proof-3\] for general jump-diffusion SPDEs with Lipschitz coefficients, and in Section \[sec-proof-4\] for the general situation of jump-diffusion SPDEs with coefficients being locally Lipschitz and satisfying the linear growth condition. In Section \[sec-example\] we provide an example illustrating our main result. In Appendix \[app-function-spaces\] we collect the function spaces which we use throughout this paper, and in Appendix \[app-stability\] we present the required stability result for SPDEs. In Appendix \[app-drift\] we provide the required results about inward pointing functions, and in Appendix \[app-volatility\] about parallel functions.
Mathematical framework and preliminary results {#sec-ass}
==============================================
In this section, we present the mathematical framework and preliminary results. Let $(\Omega,{\mathcal{F}},({\mathcal{F}}_t)_{t \in {\mathbb{R}}_+},{\mathbb{P}})$ be a filtered probability space satisfying the usual conditions. Let $H$ be a separable Hilbert space and let $A : {\mathcal{D}}(A) \subset H \to H$ be the infinitesimal generator of a $C_0$-semigroup $(S_t)_{t \geq 0}$ on $H$.
\[ass-pseudo-contractive\] We assume that the semigroup $(S_t)_{t \geq 0}$ is pseudo-contractive; that is, there exists a constant $\beta \geq 0$ such that $$\begin{aligned}
\label{growth-semigroup}
\| S_t \| \leq e^{\beta t} \quad \text{for all $t \geq 0$.}\end{aligned}$$
In view of condition (\[growth-semigroup\]), we emphasize that for $h \in H$ we denote by $\| h \|$ the Hilbert space norm, and that for a bounded linear operator $T \in L(H)$ we denote by $\| T \|$ the operator norm $$\begin{aligned}
\| T \| = \inf \{ M \geq 0 : \| Th \| \leq M \| h \| \text{ for all } h \in H \}.\end{aligned}$$ Let $U$ be a separable Hilbert space, and let $W$ be an $U$-valued $Q$-Wiener process for some nuclear, self-adjoint, positive definite linear operator $Q \in L(U)$; see [@Da_Prato pages 86, 87]. There exist an orthonormal basis $\{ e_j \}_{j \in {\mathbb{N}}}$ of $U$ and a sequence $(\lambda_j)_{j \in {\mathbb{N}}} \subset (0,\infty)$ with $\sum_{j \in {\mathbb{N}}} \lambda_j < \infty$ such that $$\begin{aligned}
Q e_j = \lambda_j e_j \quad \text{for all $j \in {\mathbb{N}}$.}\end{aligned}$$ Let $(E,{\mathcal{E}})$ be a Blackwell space, and let $\mu$ be a homogeneous Poisson random measure with compensator $dt \otimes F(dx)$ for some $\sigma$-finite measure $F$ on $(E,{\mathcal{E}})$; see [@Jacod-Shiryaev Def. II.1.20]. The space $U_0 := Q^{1/2}(U)$, equipped with the inner product $$\begin{aligned}
\label{inner-prod-U0}
\langle u,v \rangle_{U_0} := \langle Q^{-1/2}u, Q^{-1/2}v \rangle_U,\end{aligned}$$ is another separable Hilbert space. We denote by $L_2^0(H) := L_2(U_0,H)$ the space of all Hilbert-Schmidt operators from $U_0$ into $H$. We fix the orthonormal basis $\{ g_j \}_{j \in {\mathbb{N}}}$ of $U_0$ given by $g_j := \sqrt{\lambda_j} e_j$ for each $j \in {\mathbb{N}}$, and for each $\sigma \in L_2^0(H)$ we set $\sigma^j := \sigma g_j$ for $j \in {\mathbb{N}}$. Furthermore, we denote by $L^2(F) := L^2(E,{\mathcal{E}},F;H)$ the space of all square-integrable functions from $E$ into $H$. Let $\alpha : H \to H$, $\sigma : H \to L_2^0(H)$ and $\gamma : H \to L^2(F)$ be measurable functions. Concerning the upcoming notation, we remind the reader that in Appendix \[app-function-spaces\] we have collected the function spaces used in this paper.
\[ass-loc-Lip-LG\] We suppose that $$\begin{aligned}
\alpha &\in {{\rm Lip}}^{{{\rm loc}}}(H) \cap {{\rm LG}}(H),
\\ \sigma &\in {{\rm Lip}}^{{{\rm loc}}}(H,L_2^0(H)) \cap {{\rm LG}}(H,L_2^0(H)),
\\ \gamma &\in {{\rm Lip}}^{{{\rm loc}}}(H,L^2(F)) \cap {{\rm LG}}(H,L^2(F)).\end{aligned}$$
Assumption \[ass-loc-Lip-LG\] ensures that for each $h_0 \in H$ the SPDE (\[SPDE\]) has a unique mild solution; that is, an $H$-valued càdlàg adapted process $r$, unique up to indistinguishability, such that $$\label{mild-solution}
\begin{aligned}
r_t &= S_t h_0 + \int_0^t S_{t-s} \alpha(r_s) ds + \int_0^t S_{t-s} \sigma(r_s) dW_s
\\ &\quad + \int_0^t S_{t-s} \gamma(r_{s-},x) (\mu(ds,dx) - F(dx)ds), \quad t \in {\mathbb{R}}_+.
\end{aligned}$$ The sequence $(\beta^j)_{j \in {\mathbb{N}}}$ defined as $$\begin{aligned}
\label{beta-j}
\beta^j := \frac{1}{\sqrt{\lambda_j}} \langle W,e_j \rangle, \quad j \in {\mathbb{N}}\end{aligned}$$ is a sequence of real-valued standard Wiener processes, and we can write (\[mild-solution\]) equivalently as $$\label{mild-solution-beta}
\begin{aligned}
r_t &= S_t h_0 + \int_0^t S_{t-s} \alpha(r_s) ds + \sum_{j \in {\mathbb{N}}} \int_0^t S_{t-s} \sigma^j(r_s) d\beta_s^j
\\ &\quad + \int_0^t S_{t-s} \gamma(r_{s-},x) (\mu(ds,dx) - F(dx)ds), \quad t \in {\mathbb{R}}_+.
\end{aligned}$$ Note that Assumption \[ass-loc-Lip-LG\] is implied by the slightly stronger conditions $$\begin{aligned}
\alpha \in {{\rm Lip}}(H), \quad \sigma &\in {{\rm Lip}}(H,L_2^0(H)) \quad \text{and} \quad \gamma \in {{\rm Lip}}(H,L^2(F)).\end{aligned}$$ Under such global Lipschitz conditions, we refer the reader to [@Da_Prato; @Prevot-Roeckner; @Atma-book; @Liu-Roeckner] for diffusion SPDEs, to [@P-Z-book] for Lévy driven SPDEs, and to [@MPR; @SPDE] for general jump-diffusion SPDEs. Under the local Lipschitz and linear growth conditions from Assumption \[ass-loc-Lip-LG\], we refer to [@Tappe-refine].
A subset $K \subset H$ is called *invariant* for the SPDE (\[SPDE\]) if for each $h_0 \in K$ we have $r \in K$ up to an evanescent set, where $r$ denotes the mild solution to (\[SPDE\]) with $r_0 = h_0$.
\[def-cone\] A subset $K \subset H$ is called a *cone* if we have $\lambda h \in K$ for all $\lambda \geq 0$ and all $h \in K$.
\[def-conv-cone\] A cone $K \subset H$ is called a *convex cone* if we have $h+g \in K$ for all $h,g \in H$.
Note that a convex cone $K \subset H$ is indeed a convex subset of $H$.
A convex cone $K \subset H$ is called a *closed convex cone* if it is closed as a subset of $H$.
For what follows, we fix a closed convex cone $K \subset H$. Denoting by $K^* \subset H$ its dual cone (\[dual-cone\]), the cone $K$ has the representation (\[cone-repr\]).
A subset $G^* \subset K^*$ is called a *generating system* of the cone $K$ if we have the representation (\[cone-G\]).
Of course $G^* = K^*$ is a generating system of the cone $K$. However, for applications we will choose the generating system $G^*$ as convenient as possible. In this respect, we mention that, by Lindelöf’s lemma, the cone $K$ admits a generating system $G^*$ which is at most countable. For what follows, we fix a generating system $G^* \subset K^*$.
For a function $f : H \to H$ we say that $K$ is *$f$-invariant* if $f(K) \subset K$.
The closed convex cone $K$ is called *invariant* for the semigroup $(S_t)_{t \geq 0}$ if $K$ is $S_t$-invariant for all $t \geq 0$.
According to [@Pazy Cor. 1.10.6] the adjoint semigroup $(S_t^*)_{t \geq 0}$ is a $C_0$-semigroup on $H$ with infinitesimal generator $A^*$.
\[lemma-cone-adj\] The following statements are equivalent:
1. $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$.
2. $K^*$ is invariant for the adjoint semigroup $(S_t^*)_{t \geq 0}$.
For all $(h^*,h) \in K^* \times K$ and all $t \geq 0$ we have $$\begin{aligned}
\langle h^*,S_t h \rangle = \langle S_t^* h^*,h \rangle,\end{aligned}$$ and hence, the representations (\[cone-repr\]) and (\[dual-cone\]) of $K$ and $K^*$ prove the claimed equivalence.
For $\lambda > \beta$, where the constant $\beta \geq 0$ stems from the growth estimate (\[growth-semigroup\]), we define the resolvent $R_{\lambda} := (\lambda - A)^{-1}$. We consider the abstract Cauchy problem $$\begin{aligned}
\label{PDE-Cauchy}
\left\{
\begin{array}{rcl}
dr_t & = & A r_t dt \medskip
\\ r_0 & = & h_0.
\end{array}
\right.\end{aligned}$$
\[lemma-cone-semi-inv\] The following statements are equivalent:
1. $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$.
2. $K$ is invariant for the abstract Cauchy problem (\[PDE-Cauchy\]).
3. $K$ is $R_{\lambda}$-invariant for all $\lambda > \beta$.
\(i) $\Leftrightarrow$ (ii): This equivalence follows, because for each $h_0 \in K$ the mild solution to the abstract Cauchy problem (\[PDE-Cauchy\]) is given by $r_t = S_t h_0$ for $t \geq 0$.\
(i) $\Rightarrow$ (iii): For each $\lambda > \beta$ and each $h \in K$ we have $$\begin{aligned}
R_{\lambda} h = \int_0^{\infty} e^{-\lambda t} S_t h \, dt \in K.\end{aligned}$$ (iii) $\Rightarrow$ (i): Let $t > 0$ and $h \in K$ be arbitrary. By the exponential formula (see [@Pazy Thm. 1.8.3]) we have $$\begin{aligned}
S_t h = \lim_{n \to \infty} \bigg( \frac{n}{t} R_{n/t} \bigg)^n h \in K,\end{aligned}$$ completing the proof.
From now on, we make the following assumption.
\[ass-cone-semigroup\] We assume that the cone $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$; that is, any of the equivalent conditions from Lemma \[lemma-cone-semi-inv\] is fulfilled.
For all $(h^*,h) \in G^* \times K$ we have $$\begin{aligned}
\liminf_{t \downarrow 0} \frac{\langle h^*, S_t h \rangle}{t} \in \overline{{\mathbb{R}}}_+.\end{aligned}$$
Since $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$, we have $\langle h^*,S_t h \rangle \geq 0$ for all $t \geq 0$, which establishes the proof.
For $g,h \in H$ we write $g \leq_K h$ if $h-g \in K$.
Recall the set $D \subset G^* \times K$ defined in (\[def-D\]). We define the function $$\begin{aligned}
a : D \to {\mathbb{R}}_+, \quad a(h^*,h) := \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t}.\end{aligned}$$
\[lemma-fct-a\] For each $(h^*,h) \in D$ the following statements are true:
1. We have $\langle h^*,h \rangle = 0$.
2. For all $\lambda \geq 0$ we have $(h^*,\lambda h) \in D$ and $$\begin{aligned}
\label{id-a}
a(h^*,\lambda h) = \lambda a(h^*,h).\end{aligned}$$
3. For all $g \in K$ with $g \leq_K h$ we have $(h^*,g) \in D$ and $$\begin{aligned}
\label{id-a-g}
a(h^*,g) \leq a(h^*,h).\end{aligned}$$
For each $(h^*,h) \in G^* \times K$ with $\langle h^*,h \rangle > 0$ we have $$\begin{aligned}
\lim_{t \downarrow 0} \langle h^*,S_t h \rangle = \langle h^*,h \rangle > 0,\end{aligned}$$ and hence $$\begin{aligned}
\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} = \infty,\end{aligned}$$ showing that $(h^*,h) \notin D$. This proves the first statement, and we proceed with the second statement. Since $K$ is a cone, we have $\lambda h \in K$. Furthermore, we have $$\begin{aligned}
\liminf_{t \downarrow 0} \frac{\langle h^*,S_t(\lambda h) \rangle}{t} = \lambda \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} < \infty,\end{aligned}$$ showing $(h^*,\lambda h) \in D$ and the identity (\[id-a\]). For the proof of the third statement, let $t \geq 0$ be arbitrary. By Lemma \[lemma-cone-adj\] we have $S_t^* h^* \in K^*$. Since $g \leq_K h$, we obtain $\langle S_t^* h^*,h-g \rangle \geq 0$, and hence $$\begin{aligned}
\langle h^*,S_t g \rangle = \langle S_t^* h^*,g \rangle \leq \langle S_t^* h^*,h \rangle = \langle h^*,S_t h \rangle.\end{aligned}$$ Consequently, we have $$\begin{aligned}
\label{compare-semi}
\langle h^*,S_t g \rangle \leq \langle h^*,S_t h \rangle \quad \text{for all $t \geq 0$.}\end{aligned}$$ There exists a sequence $(t_n)_{n \in {\mathbb{N}}} \subset (0,\infty)$ with $t_n \downarrow 0$ such that the sequence $(b_n)_{n \in {\mathbb{N}}} \subset {\mathbb{R}}_+$ defined as $$\begin{aligned}
b_n := \frac{\langle h^*,S_{t_n} h \rangle}{t_n}, \quad n \in {\mathbb{N}}\end{aligned}$$ converges to $a(h^*,h) \in {\mathbb{R}}_+$. Defining the sequence $(a_n)_{n \in {\mathbb{N}}} \subset {\mathbb{R}}_+$ as $$\begin{aligned}
a_n := \frac{\langle h^*,S_{t_n} g \rangle}{t_n}, \quad n \in {\mathbb{N}},\end{aligned}$$ by (\[compare-semi\]) we have $0 \leq a_n \leq b_n$ for each $n \in {\mathbb{N}}$. Hence, the sequence $(a_n)_{n \in {\mathbb{N}}}$ is bounded, and by the Bolzano-Weierstrass theorem there exists a subsequence $(n_k)_{k \in {\mathbb{N}}}$ such that $(a_{n_k})_{k \in {\mathbb{N}}}$ converges to some $a \in {\mathbb{R}}_+$ with $a \leq a(h^*,h)$, which proves $(h^*,g) \in D$ and (\[id-a-g\]).
\[lemma-liminf-domain\] Let $(h^*,h) \in G^* \times K$ with $\langle h^*,h \rangle = 0$ be arbitrary. Then the following statements are true:
1. If $h \in {\mathcal{D}}(A)$, then we have $(h^*,h) \in D$ and $$\begin{aligned}
\label{liminf-domain}
\liminf_{t \downarrow 0} \frac{\langle h^*, S_t h \rangle}{t} = \langle h^*,A h \rangle.\end{aligned}$$
2. If $h^* \in {\mathcal{D}}(A^*)$, then we have $(h^*,h) \in D$ and $$\begin{aligned}
\label{liminf-domain-2}
\liminf_{t \downarrow 0} \frac{\langle h^*, S_t h \rangle}{t} = \langle A^* h^*,h \rangle.\end{aligned}$$
3. If the semigroup $(S_t)_{t \geq 0}$ is norm continuous, then we have $(h^*,h) \in D$ as well as (\[liminf-domain\]) and (\[liminf-domain-2\]).
If $h \in {\mathcal{D}}(A)$, then we have $$\begin{aligned}
\frac{\langle h^*, S_t h \rangle}{t} = \frac{\langle h^*, S_t h \rangle - \langle h^*,h \rangle}{t} = \frac{\langle h^*, S_t h - h \rangle}{t} = \bigg\langle h^*, \frac{S_t h - h}{t} \bigg\rangle \to \langle h^*,A h \rangle\end{aligned}$$ as $t \downarrow 0$, showing the first statement. Furthermore, if $h^* \in {\mathcal{D}}(A^*)$, then we obtain $$\begin{aligned}
\frac{\langle h^*,S_t h \rangle}{t} &= \frac{\langle S_t^* h^*,h \rangle}{t} = \frac{\langle S_t^* h^*,h \rangle - \langle h^*,h \rangle}{t}
\\ &= \frac{\langle S_t^* h^* - h^*,h \rangle}{t} = \bigg\langle \frac{S_t^* h^* - h^*}{t}, h \bigg\rangle \to \langle A^* h^*,h \rangle\end{aligned}$$ as $t \downarrow 0$, showing the second statement. The third statement is an immediate consequence of the first and the second statement.
The following definition is inspired by [@Milian Lemma 5].
\[def-local-op\] We call $A^*$ a *local operator* if $G^* \subset {\mathcal{D}}(A^*)$, and for all $(h^*,h) \in D$ we have $\langle A^* h^*,h \rangle = 0$.
\[prop-conditions\] Suppose that condition (\[main-1\]) is fulfilled. Then for all $(h^*,h) \in D$ the following statements are true:
1. We have $$\begin{aligned}
\langle h^*,\gamma(h,x) \rangle \geq 0 \quad \text{for $F$-almost all $x \in E$.}\end{aligned}$$
2. We have $$\begin{aligned}
\int_E \langle h^*,\gamma(h,x) \rangle F(dx) \in \overline{{\mathbb{R}}}_+.\end{aligned}$$
3. If condition (\[main-3\]) is satisfied, then we have (\[main-2\]).
4. If $h \in {\mathcal{D}}(A)$, then conditions (\[main-3\]) and (\[FV-cond-3\]) are equivalent.
5. If $h^* \in {\mathcal{D}}(A^*)$, then conditions (\[main-3\]) and (\[FV-cond-4\]) are equivalent.
6. If $A^*$ is a local operator, then conditions (\[main-3\]) and (\[FV-cond-5\]) are equivalent.
7. Condition (\[FV-cond-5\]) implies (\[main-3\]).
By (\[main-1\]), for $F$-almost all $x \in E$ we have $$\begin{aligned}
\langle h^*,\gamma(h,x) \rangle = \langle h^*,h \rangle + \langle h^*,\gamma(h,x) \rangle = \langle h^*,h+\gamma(h,x) \rangle \geq 0,\end{aligned}$$ which establishes the first statement. The second statement is an immediate consequence, and the third statement is obvious. The fourth and the fifth statement follow from Lemma \[lemma-liminf-domain\]. Taking into account Definition \[def-local-op\], the sixth statement is an immediate consequence of the fifth statement. Finally, the last statement follows from the first statement.
In view of condition (\[FV-cond-3\]), we emphasize that $K \cap {\mathcal{D}}(A)$ is dense is $K$, which follows from the next result.
\[lemma-cone-dense\] We have $K = \overline{K \cap {\mathcal{D}}(A)}$.
Since $K$ is closed, we have $\overline{K \cap {\mathcal{D}}(A)} \subset K$. In order to prove the converse inclusion, let $h \in K$ be arbitrary. For $t > 0$ we set $h_t := \frac{1}{t} \int_0^t S_s h ds$. Then we have $h_t \in {\mathcal{D}}(A)$ for each $t > 0$, and we have $h_t \to h$ for $t \downarrow 0$. It remains to show that $h_t \in K$ for each $t > 0$. For this purpose, let $t > 0$ and $h^* \in G^*$ be arbitrary. Since $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$, we obtain $$\begin{aligned}
\langle h^*,h_t \rangle = \bigg\langle h^*,\frac{1}{t} \int_0^t S_s h ds \bigg\rangle = \frac{1}{t} \int_0^t \langle h^*,S_s h \rangle ds \geq 0,\end{aligned}$$ showing that $h_t \in K$.
Necessity of the invariance conditions {#sec-nec}
======================================
In this section, we prove the necessity of our invariance conditions.
\[thm-nec\] Suppose that Assumptions \[ass-pseudo-contractive\], \[ass-loc-Lip-LG\] and \[ass-cone-semigroup\] are fulfilled. If the closed convex cone $K$ is invariant for the SPDE (\[SPDE\]), then we have (\[main-1\]), and for all $(h^*,h) \in D$ we have (\[main-3\]) and (\[main-4\]).
Condition (\[main-1\]) follows from [@FTT-appendix Lemma 2.11]. Let $(h^*,h) \in D$ be arbitrary, and denote by $r$ the mild solution to (\[SPDE\]) with $r_0 = h$. Since the measure space $(E,{\mathcal{E}},F)$ is $\sigma$-finite, there exists an increasing sequence $(B_n)_{n \in {\mathbb{N}}} \subset {\mathcal{E}}$ with $F(B_n) < \infty$ for each $n \in {\mathbb{N}}$ such that $E = \bigcup_{n \in {\mathbb{N}}} B_n$. Let $n \in {\mathbb{N}}$ be arbitrary. According to [@FTT-appendix Lemma 2.20] the mapping $T_n : \Omega \to \overline{{\mathbb{R}}}_+$ given by $$\begin{aligned}
T_n := \inf \{ t \in {\mathbb{R}}_+ : \mu([0,t] \times B_n) = 1 \}\end{aligned}$$ is a strictly positive stopping time. We denote by $r^n$ the mild solution to the SPDE $$\begin{aligned}
\left\{
\begin{array}{rcl}
dr_t^n & = & (A r_t^n + \alpha(r_t^n) - \int_{B_n} \gamma(r_t^n,x) F(dx)) dt + \sigma(r_t^n) dW_t
\\ && + \int_{B_n^c} \gamma(r_{t-}^n,x) (\mu(dt,dx) - F(dx) dt) \medskip
\\ r_0^n & = & h.
\end{array}
\right.\end{aligned}$$ Since $K$ is a closed subset of $H$, by [@FTT-appendix Prop. 2.21] we obtain $(r^n)^{T_n} \in K$ up to an evanescent set. We define the strictly positive, bounded stopping time $$\begin{aligned}
T &:= \inf \{ t \in {\mathbb{R}}_+ : \| r_t^n \| > 1 + \| h \| \} \wedge T_n \wedge 1.\end{aligned}$$ Furthermore, for every stopping time $R \leq T$ we define the processes $A^n(R)$ and $M^n(R)$ as $$\begin{aligned}
A^n(R)_t &:= \int_0^t \bigg\langle h^*,S_{R-s} \bigg( \alpha(r_s^n) - \int_{B_n} \gamma(r_s^n,x) F(dx) \bigg) \bigg\rangle {\mathbbm{1}}_{\{ R \geq s \}} ds, \quad t \in {\mathbb{R}}_+,
\\ M^n(R)_t &:= \int_0^t \langle h^*,S_{R-s} \sigma(r_s^n) \rangle {\mathbbm{1}}_{\{ R \geq s \}} dW_s
\\ &\quad + \int_0^t \int_{B_n} \langle h^*,S_{R-s} \gamma(r_{s-}^n,x) \rangle {\mathbbm{1}}_{\{ R \geq s \}} (\mu(ds,dx) - F(dx)ds), \quad t \in {\mathbb{R}}_+.\end{aligned}$$ Then, by the Cauchy-Schwarz inequality and Assumptions \[ass-pseudo-contractive\], \[ass-loc-Lip-LG\] we have $A^n(R) \in {\mathcal{A}}$ and $M^n(R) \in {\mathcal{H}}^2$ for each stopping time $R \leq T$, where ${\mathcal{A}}$ denotes the space of all finite variation processes with integrable variation (see [@Jacod-Shiryaev I.3.7]) and ${\mathcal{H}}^2$ denotes the space of all square-integrable martingales (see [@Jacod-Shiryaev Def. I.1.41]). Moreover, we have ${\mathbb{P}}$-almost surely $$\begin{aligned}
0 \leq \langle h^*,r_{T \wedge t}^n \rangle = \langle h^*,S_{T \wedge t} h \rangle + A^n(T \wedge t)_{T \wedge t} + M^n(T \wedge t)_{T \wedge t} \quad \text{for all $t \in {\mathbb{R}}_+$.}\end{aligned}$$ Let $(t_k)_{k \in {\mathbb{N}}} \subset (0,\infty)$ be a sequence with $t_k \downarrow 0$ such that $$\begin{aligned}
\label{liminf-nec}
\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} =
\lim_{k \to \infty} \frac{\langle h^*,S_{t_k} h \rangle}{t_k}.\end{aligned}$$ By Lebesgue’s dominated convergence theorem we obtain $$\begin{aligned}
0 &\leq \lim_{k \to \infty} \frac{1}{t_k} {\mathbb{E}}[ \langle h^*,r_{T \wedge t_k}^n \rangle ] = \lim_{k \to \infty} \frac{1}{t_k} {\mathbb{E}}[\langle h^*,S_{T \wedge t_k} h \rangle] + \lim_{k \to \infty} \frac{1}{t_k} {\mathbb{E}}[A^n(T \wedge t_k)_{T \wedge t_k}]
\\ &= \lim_{k \to \infty} \frac{\langle h^*,S_{t_k} h \rangle}{t_k} + \langle h^*,\alpha(h) \rangle - \int_{B_n} \langle h^*,\gamma(h,x) \rangle F(dx),\end{aligned}$$ showing that $$\begin{aligned}
\label{drift-nec-1}
\liminf_{t \downarrow 0} \frac{1}{t} \langle h^*,S_t h \rangle + \langle h^*,\alpha(h) \rangle - \int_{B_n} \langle h^*,\gamma(h,x) \rangle F(dx) \geq 0.\end{aligned}$$ Furthermore, by the monotone convergence theorem and Proposition \[prop-conditions\] we have $$\begin{aligned}
\label{drift-nec-2}
\int_E \langle h^*,\gamma(h,x) \rangle F(dx) = \lim_{n \to \infty} \int_{B_n} \langle h^*,\gamma(h,x) \rangle F(dx).\end{aligned}$$ Combining (\[drift-nec-1\]) and (\[drift-nec-2\]), we arrive at (\[main-3\]).
Now, suppose that condition (\[main-4\]) is not fulfilled. Then there exist $j \in {\mathbb{N}}$ and $(h^*,h) \in D$ such that $\langle h^*,\sigma^j(h) \rangle \neq 0$. We define $\eta,\Phi \in {\mathbb{R}}$ by $$\begin{aligned}
\label{def-eta}
\eta := \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha(h) \rangle \quad \text{and} \quad \Phi := -\frac{\eta + 1}{\langle h^*,\sigma^j(h) \rangle}.\end{aligned}$$ Note that, by (\[main-3\]) and Proposition \[prop-conditions\] we have $\eta \in {\mathbb{R}}_+$. The stochastic exponential $$\begin{aligned}
Z := {\mathcal{E}}(\Phi \beta^j),\end{aligned}$$ where the Wiener process $\beta^j$ is given by (\[beta-j\]), is a strictly positive, continuous local martingale. We define the strictly positive, bounded stopping time $$\begin{aligned}
T &:= \inf \{ t \in {\mathbb{R}}_+ : \| r_t \| > 1 + \| h \| \} \wedge \inf \{ t \in {\mathbb{R}}_+ : | Z_t | > 2 \}
\\ &\quad \wedge \inf \{ t \in {\mathbb{R}}_+ : \langle Z,Z \rangle_t > 1 \} \wedge 1.\end{aligned}$$ For every stopping time $R \leq T$ we define the processes $A(R)$, $M(R)$ and $N(R)$ as $$\begin{aligned}
A(R)_t &:= \int_0^t \langle h^*,S_{R-s} \alpha(r_s) \rangle {\mathbbm{1}}_{\{ R \geq s \}} ds, \quad t \in {\mathbb{R}}_+,
\\ M(R)_t &:= \int_0^t \langle h^*,S_{R-s} \sigma(r_s) \rangle {\mathbbm{1}}_{\{ R \geq s \}} dW_s
\\ &\quad + \int_0^t \int_E \langle h^*,S_{R-s} \gamma(r_{s-},x) \rangle {\mathbbm{1}}_{\{ R \geq s \}} (\mu(ds,dx) - F(dx)ds), \quad t \in {\mathbb{R}}_+,
\\ N(R)_t &:= \int_0^t ( A(R)_{s-} + M(R)_{s-} ) {\mathbbm{1}}_{\{ R \geq s \}} dZ_s + \int_0^t Z_s {\mathbbm{1}}_{\{ R \geq s \}} dM(R)_s, \quad t \in {\mathbb{R}}_+.\end{aligned}$$ Then, by Assumptions \[ass-pseudo-contractive\], \[ass-loc-Lip-LG\] we have $A(R) \in {\mathcal{A}}$ and $M(R),N(R) \in {\mathcal{H}}^2$ for each stopping time $R \leq T$. Moreover, we have ${\mathbb{P}}$-almost surely $$\begin{aligned}
0 \leq \langle h^*,r_{T \wedge t} \rangle = \langle h^*,S_{T \wedge t} h \rangle + A(T \wedge t)_{T \wedge t} + M(T \wedge t)_{T \wedge t} \quad \text{for all $t \in {\mathbb{R}}_+$.}\end{aligned}$$ Let $R \leq T$ be an arbitrary stopping time. By [@Jacod-Shiryaev Prop. I.4.49] we have $[A(R),Z^R] = 0$, and by [@Jacod-Shiryaev Thm. I.4.52] we have $[M(R),Z^R] = \langle M(R)^c,Z^R \rangle$. Therefore, and since $$\begin{aligned}
Z_t^R = 1 + \Phi \int_0^t Z_s {\mathbbm{1}}_{\{ R \geq s \}} d\beta_s^j, \quad t \in {\mathbb{R}}_+,\end{aligned}$$ by [@Jacod-Shiryaev Def. I.4.45] we obtain $$\label{int-by-parts}
\begin{aligned}
&( A(R)_t + M(R)_t ) Z_t^R = N(R)_t + \int_0^t Z_s {\mathbbm{1}}_{\{ R \geq s \}} dA(R)_s + \langle M(R)^c,Z^R \rangle
\\ &= N(R)_t + \int_0^t \langle h^*, S_{R-s} ( \alpha(r_s) + \Phi \sigma^j(r_s) ) \rangle Z_s {\mathbbm{1}}_{\{ R \geq s \}} ds, \quad t \in {\mathbb{R}}_+.
\end{aligned}$$ Let $(t_k)_{k \in {\mathbb{N}}} \subset (0,\infty)$ be a sequence with $t_k \downarrow 0$ such that we have (\[liminf-nec\]). By (\[int-by-parts\]), Lebesgue’s dominated convergence theorem and (\[def-eta\]) we obtain $$\begin{aligned}
0 &\leq \lim_{k \to \infty} \frac{1}{t_k} {\mathbb{E}}[ \langle h^*,r_{T \wedge t_k}^n \rangle Z_{T \wedge t_k} ] = \lim_{k \to \infty} \frac{1}{t_k} {\mathbb{E}}[\langle h^*,S_{T \wedge t_k} h \rangle Z_{T \wedge t_k}]
\\ &\quad + \lim_{k \to \infty} \frac{1}{t_k} {\mathbb{E}}[(A(T \wedge t_k)_{T \wedge t_k} + M(T \wedge t_k)_{T \wedge t_k} )Z_{T \wedge t_k}^{T \wedge t_k}]
\\ &= \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha(h) + \Phi \sigma^j(h) \rangle
\\ &= \eta + \Phi \langle h^*,\sigma^j(h) \rangle = \eta - (\eta + 1) = -1,\end{aligned}$$ a contradiction.
Cones generated by unconditional Schauder bases {#sec-Schauder}
===============================================
In this section, we provide the required background about closed convex cones generated by unconditional Schauder bases. Let $\{ e_k \}_{k \in {\mathbb{N}}}$ be an unconditional Schauder basis of the Hilbert space $H$; that is, for each $h \in H$ there is a unique sequence $(h_k)_{k \in {\mathbb{N}}} \subset {\mathbb{R}}$ such that $$\begin{aligned}
\label{series-h}
h = \sum_{k \in {\mathbb{N}}} h_k e_k,\end{aligned}$$ and the series (\[series-h\]) converges unconditionally. Without loss of generality, we assume that $\| e_k \| = 1$ for all $k \in {\mathbb{N}}$.
Every orthonormal basis of the Hilbert space $H$ is an unconditional Schauder basis. Of course, the converse statement is not true, but for every unconditional Schauder basis of the Hilbert space $H$ there is an equivalent inner product on $H$ under which the unconditional Schauder basis is an orthonormal basis; see [@Bari].
There are unique elements $\{ e_k^* \}_{k \in {\mathbb{N}}} \subset H$ such that $$\begin{aligned}
\langle e_k^*,h \rangle = h_k \quad \text{for each $h \in H$,}\end{aligned}$$ where we refer to the series representation (\[series-h\]); see [@Fabian page 164]. Given these coordinate functionals $\{ e_k^* \}_{k \in {\mathbb{N}}}$, we also call $\{ e_k^*,e_k \}_{k \in {\mathbb{N}}}$ an unconditional Schauder basis of $H$. Recall that, throughout this paper, we consider a closed convex cone $K \subset H$ with representation (\[cone-G\]) for some generating system $G^* \subset K^*$. Now, we make an additional assumption on the generating system $G^*$ of the cone.
\[ass-Schauder-basis\] We assume there is an unconditional Schauder basis $\{ e_k^*,e_k \}_{k \in {\mathbb{N}}}$ of $H$ such that $$\begin{aligned}
G^* \subset \{ \theta e_k^* : \theta \in \{ -1,1 \} \text{ and } k \in {\mathbb{N}}\}.\end{aligned}$$
Equivalently, we could demand $G^* \subset \bigcup_{k \in {\mathbb{N}}} \langle e_k^* \rangle$. Assumption \[ass-Schauder-basis\] ensures that the generating system $G^*$ becomes minimal.
We define the sequence $(E_n)_{n \in {\mathbb{N}}_0}$ of finite dimensional subspaces $E_n \subset H$ as $E_n := \langle e_1,\ldots,e_n \rangle$. Furthermore, we define the sequence $(\Pi_n)_{n \in {\mathbb{N}}_0}$ of projections $\Pi_n \in L(H,E_n)$ as $$\begin{aligned}
\label{Pi-def}
\Pi_n h = \sum_{k=1}^n \langle e_k^*,h \rangle e_k = \sum_{k=1}^n h_k e_k, \quad h \in H,\end{aligned}$$ where we refer to the series representation (\[series-h\]) of $h$. We denote by ${{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}:= \sup_{n \in {\mathbb{N}}} \| \Pi_n \|$ the basis constant of the Schauder basis $\{ e_k \}_{k \in {\mathbb{N}}}$. Since the Schauder basis is unconditional, by [@Fabian Prop. 6.31] there is a constant $C \in {\mathbb{R}}_+$ such for all $m \in {\mathbb{N}}$, all $\lambda_1,\ldots,\lambda_m \in {\mathbb{R}}$ and all $\epsilon_1,\ldots,\epsilon_m \in \{ -1,1 \}$ we have $$\begin{aligned}
\label{ubc-est}
\bigg\| \sum_{k=1}^m \epsilon_k \lambda_k e_k \bigg\| \leq C \bigg\| \sum_{k=1}^m \lambda_k e_k \bigg\|.\end{aligned}$$ The smallest possible constant $C \in {\mathbb{R}}_+$ such that the inequality (\[ubc-est\]) is fulfilled, is called the unconditional basis constant, and is denoted by ${{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}$.
\[lemma-norm-g-star\] The following statements are true:
1. We have $1 \leq {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}\leq {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}$.
2. For each $k \in {\mathbb{N}}$ we have $\| \langle e_k^*,\cdot \rangle \| \leq 2 {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}$.
3. For all $h \in H$ with representation (\[series-h\]) and every bounded sequence $(\lambda_k)_{k \in {\mathbb{N}}}$ we have $$\begin{aligned}
g := \sum_{k \in {\mathbb{N}}} \lambda_k h_k e_k \in H\end{aligned}$$ with norm estimate $$\begin{aligned}
\| g \| \leq {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}\Big( \sup_{k \in {\mathbb{N}}} |\lambda_k| \Big) \| h \|.\end{aligned}$$
The first statement follows the proof of [@Fabian Prop. 6.31]. Noting that $\| e_k \| = 1$, by the Cauchy-Schwarz inequality, Assumption \[ass-Schauder-basis\] and the identity $$\begin{aligned}
\| e_k^* \| \, \| e_k \| \leq 2 {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}\end{aligned}$$ from [@Fabian page 164], for each $h \in H$ we obtain $$\begin{aligned}
|\langle e_k^*,h \rangle| \leq \| e_k^* \| \, \| h \| \leq 2 {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}\| h \|.\end{aligned}$$ The third statement follows from [@Fabian Lemma 6.33].
\[lemma-proj-par\] The following statements are true:
1. We have $\Pi_n \to {\rm Id}_H$ as $n \to \infty$.
2. For all $k,n \in {\mathbb{N}}$, all $h^* \in \langle e_k^* \rangle$ and all $h \in H$ we have $$\begin{aligned}
\langle h^*,\Pi_n h \rangle = \langle h^*,h \rangle \mathbbm{1}_{\{ k \leq n \}}.\end{aligned}$$
The first statement follows from [@Fabian Lemma 6.2.iii], and the second statement follows from the definition (\[Pi-def\]) of the projection $\Pi_n$.
Sufficiency of the invariance conditions for diffusion SPDEs with smooth volatilities {#sec-proof-1}
=====================================================================================
In this section, we prove the sufficiency of our invariance conditions for diffusion SPDEs (\[SPDE-Wiener\]) with smooth volatilities. Recall that the distance function $d_K : H \to {\mathbb{R}}_+$ of the cone $K$ is given by $$\begin{aligned}
d_K(h) := \inf_{g \in K} \| h - g \|.\end{aligned}$$
\[lemma-distance\] The following statements are true:
1. For all $\lambda \geq 0$ and $h \in H$ we have $$\begin{aligned}
\label{distance-1}
d_K(\lambda h) = \lambda d_K( h ).\end{aligned}$$
2. For all $h \in H$ and $g \in K$ we have $$\begin{aligned}
d_K(h+g) \leq d_K(h).\end{aligned}$$
Let $h \in H$ be arbitrary. For $\lambda = 0$ both sides in (\[distance-1\]) are zero, and for $\lambda > 0$, by Definition \[def-cone\] we obtain $$\begin{aligned}
d_K(\lambda h) = \inf_{g \in K} \| \lambda h - g \| = \inf_{f \in K} \| \lambda h - \lambda f \| = \lambda \inf_{f \in K} \| h - f \| = \lambda d_K( h ),\end{aligned}$$ proving the first statement. For the proof of the second statement, let $h \in H$ and $g \in K$ be arbitrary. Note that $K \subset K - \{ g \}$. Indeed, for each $f \in K$ by Definition \[def-conv-cone\] we have $f + g \in K$, and hence $f = (f+g)-g \in K - \{ g \}$. This gives us $$\begin{aligned}
d_K(h+g) &= \inf_{f \in K} \| (h+g) - f \| = \inf_{f \in K} \| h - (f-g) \|
\\ &= \inf_{e \in K - \{ g \}} \| h - e \| \leq \inf_{e \in K} \| h - e \| = d_K(h),\end{aligned}$$ establishing the second statement.
The following result ensures that the stochastic semigroup Nagumo’s condition (SSNC) is fulfilled in our situation.
\[prop-SSNC\] Let $\Sigma \in {{\rm F}}(H)$ be such that for all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{drift-Nagumo}
\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\Sigma(h) \rangle \geq 0.\end{aligned}$$ Then, for each $h \in K$ we have $$\begin{aligned}
\label{SSNC}
\liminf_{t \downarrow 0} \frac{1}{t} d_K(S_t h + t \Sigma(h)) = 0.\end{aligned}$$
Since $\Sigma \in {{\rm F}}(H)$, there is an index $n \in {\mathbb{N}}$ such that $\Sigma(H) \subset E_n$. Let $h \in K$ be arbitrary. We set ${\mathbb{N}}_n := \{ 1,\ldots,n \}$ and $$\begin{aligned}
{\mathbb{N}}_n^1 &:= \{ k \in {\mathbb{N}}_n : (e_k^*,h) \in D \text{ or } (-e_k^*,h) \in D \},
\\ {\mathbb{N}}_n^2 &:= \{ k \in {\mathbb{N}}_n : e_k^* \in G^* \text{ or } -e_k^* \in G^* \}
\\ &\qquad \cap \{ k \in {\mathbb{N}}_n : (e_k^*,h) \notin D \text{ and } (-e_k^*,h) \notin D \},
\\ {\mathbb{N}}_n^3 &:= \{ k \in {\mathbb{N}}_n : e_k^* \notin G^* \text{ and } -e_k^* \notin G^* \}.\end{aligned}$$ Then we have the decomposition ${\mathbb{N}}_n = {\mathbb{N}}_n^1 \cup {\mathbb{N}}_n^2 \cup {\mathbb{N}}_n^3$, for each $k \in {\mathbb{N}}_n^1$ there exists $\theta_k \in \{ -1,1 \}$ such that $(\theta_k e_k^*,h) \in D$, and for each $k \in {\mathbb{N}}_n^2$ there exists $\theta_k \in \{ -1,1 \}$ such that $\theta_k e_k^* \in G^*$ and $(\theta_k e_k^*,h) \notin D$. Furthermore, we set $\theta_k := 1$ for each $k \in {\mathbb{N}}_n^3$. There is a sequence $(t_m)_{m \in {\mathbb{N}}} \subset (0,\infty)$ with $t_m \downarrow 0$ such that $$\begin{aligned}
\label{SSNC-proof-1}
c_m(k) \geq 0 \quad \text{for all $m \in {\mathbb{N}}$ and all $k \in {\mathbb{N}}_n^2.$}\end{aligned}$$ where we agree on the notation $$\begin{aligned}
c_m(k) := \frac{\langle \theta_k e_k^*,S_{t_m} h + t_m \Sigma(h) \rangle}{t_m} \quad \text{for all $m \in {\mathbb{N}}$ and all $k \in {\mathbb{N}}_n$.}\end{aligned}$$ Inductively, we define the subsequences $(m(k)_p)_{p \in {\mathbb{N}}}$ for $k \in \{ 0 \} \cup {\mathbb{N}}_n^1$ as follows:
1. For $k = 0$ we set $m(0)_p := p$ for each $p \in {\mathbb{N}}$.
2. Let $k \in {\mathbb{N}}_n^1$ be arbitrary, and suppose that we have defined $(m(l)_p)_{p \in {\mathbb{N}}}$, where $l$ denotes the largest integer from $\{ 0 \} \cup {\mathbb{N}}_n^1$ with $l < k$. We distinguish two cases:
- If $\liminf_{p \to \infty} c_{m(l)_p}(k) = \infty$, then we choose a subsequence $(m(k)_p)_{p \in {\mathbb{N}}}$ of $(m(l)_p)_{p \in {\mathbb{N}}}$ such that $c_{m(k)_p}(k) \geq 0$ for all $p \in {\mathbb{N}}$.
- Otherwise, we choose a subsequence $(m(k)_p)_{p \in {\mathbb{N}}}$ of $(m(l)_p)_{p \in {\mathbb{N}}}$ such that $c_{m(k)_p}(k)$ converges to a finite limit for $p \to \infty$.
Now, we define the subsequence $(m_p)_{p \in {\mathbb{N}}}$ as $m_p := m(k)_p$ for each $p \in {\mathbb{N}}$, where $k$ denotes the largest integer from $\{ 0 \} \cup {\mathbb{N}}_n^1$. Furthermore, we define the sets $$\begin{aligned}
{\mathbb{N}}_n^{1a} &:= \Big\{ k \in {\mathbb{N}}_n^1 : \liminf_{p \to \infty} c_{m_p}(k) < \infty \Big\},
\\ {\mathbb{N}}_n^{1b} &:= \Big\{ k \in {\mathbb{N}}_n^1 : \liminf_{p \to \infty} c_{m_p}(k) = \infty \Big\}.\end{aligned}$$ Then we have the decomposition ${\mathbb{N}}_n^1 = {\mathbb{N}}_n^{1a} \cup {\mathbb{N}}_n^{1b}$, and by (\[drift-Nagumo\]) we have $$\begin{aligned}
\label{SSNC-proof-2}
\lim_{p \to \infty} c_{m_p}(k) &\in {\mathbb{R}}_+ \quad \text{for all $k \in {\mathbb{N}}_n^{1a},$}
\\ \label{SSNC-proof-3} c_{m_p}(k) &\geq 0 \quad \text{for all $p \in {\mathbb{N}}$ and all $k \in {\mathbb{N}}_n^{1b}.$}\end{aligned}$$ Since $\Sigma(H) \subset E_n$, and $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$ and $({\rm Id} - \Pi_n)$-invariant, by Lemma \[lemma-distance\] and (\[SSNC-proof-1\]), (\[SSNC-proof-3\]), for each $p \in {\mathbb{N}}$ we obtain $$\begin{aligned}
&\frac{1}{t_{m_p}} d_K ( S_{t_{m_p}} h + t_{m_p} \Sigma(h) ) = \frac{1}{t_{m_p}} d_K \big( \underbrace{({\rm Id} - \Pi_n) S_{t_{m_p}} h}_{\in K} + \Pi_n(S_{t_{m_p}} h + t_{m_p} \Sigma(h)) \big)
\\ &\leq \frac{1}{t_{m_p}} d_K \big( \Pi_n(S_{t_{m_p}} h + t_{m_p} \Sigma(h)) \big) = d_K \bigg( \Pi_n \frac{S_{t_{m_p}} h + t_{m_p} \Sigma(h)}{t_{m_p}} \bigg)
\\ &= d_K \bigg( \sum_{k \in {\mathbb{N}}_n^{1a}} c_{m_p}(k) \theta_k e_k + \underbrace{\sum_{k \in {\mathbb{N}}_n^{1b} \cup {\mathbb{N}}_n^2 \cup {\mathbb{N}}_n^3} c_{m_p}(k) \theta_k e_k}_{\in K} \bigg) \leq d_K \bigg( \sum_{k \in {\mathbb{N}}_n^{1a}} c_{m_p}(k) \theta_k e_k \bigg),\end{aligned}$$ and by the continuity of the distance function $d_K$ and (\[SSNC-proof-2\]) we have $$\begin{aligned}
\lim_{p \to \infty} d_K \bigg( \sum_{k \in {\mathbb{N}}_n^{1a}} c_{m_p}(k) \theta_k e_k \bigg) = d_K \bigg( \underbrace{\sum_{k \in {\mathbb{N}}_n^{1a}} \lim_{p \to \infty} c_{m_p}(k) \theta_k e_k}_{\in K} \bigg) = 0,\end{aligned}$$ completing the proof.
\[thm-diffusion-C2\] Suppose that Assumptions \[ass-pseudo-contractive\], \[ass-cone-semigroup\] and \[ass-Schauder-basis\] are fulfilled, and that $$\begin{aligned}
\alpha &\in {\rm Lip}(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H),
\\ \sigma &\in {{\rm G}}(H,L_2^0(H)) \cap {{\rm F}}(H,L_2^0(H)) \cap C_b^2(H,L_2^0(H)).\end{aligned}$$ If we have $$\begin{aligned}
\label{drift-cond-C2}
\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha(h) \rangle \geq 0 \quad \text{for all $(h^*,h) \in D$,}\end{aligned}$$ and for all $(h^*,h) \in D$ and each $j \in {\mathbb{N}}$ there exists $\epsilon = \epsilon(h^*,h,j) > 0$ such that $$\begin{aligned}
\label{loc-par-prop-result}
\langle h^*, \sigma^j(h-g) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon$,}\end{aligned}$$ then the closed convex cone $K$ is invariant for the SPDE (\[SPDE-Wiener\]).
Condition (\[loc-par-prop-result\]) just means that for each $j \in {\mathbb{N}}$ the function $\sigma^j : H \to H$ is weakly locally parallel in the sense of Definition \[def-locally-par-weak\], which allows us to apply Lemma \[lemma-rho-parallel\] in the sequel. Let $\rho : H \to H$ be the function defined in (\[def-rho\]). According to our hypotheses and Lemma \[lemma-rho\], all assumptions from [@Nakayama] are satisfied. Let $u \in U_0$ be arbitrary, and define the function $\Sigma : H \to H$ as $$\begin{aligned}
\Sigma(h) := \alpha(h) - \rho(h) + \sigma(h)u, \quad h \in H.\end{aligned}$$ Since $\alpha \in {{\rm F}}(H)$ and $\sigma \in {{\rm F}}(H,L_2^0(H))$, we have $\Sigma \in {{\rm F}}(H)$. Let $(h^*,h) \in D$ be arbitrary. Then, by (\[drift-cond-C2\]) and Lemmas \[lemma-rho-parallel\], \[lemma-sigma-u\] we deduce that condition (\[drift-Nagumo\]) is fulfilled. Therefore, by Proposition \[prop-SSNC\] the SSNC (\[SSNC\]) is fulfilled. Consequently, applying [@Nakayama Prop. 1.1] yields that the closed convex cone $K$ is invariant for the SPDE (\[SPDE-Wiener\]).
Sufficiency of the invariance conditions for diffusion SPDEs with Lipschitz coefficients {#sec-proof-2}
========================================================================================
In this section, we prove that our invariance conditions are sufficient for diffusion SPDEs (\[SPDE-Wiener\]) with Lipschitz coefficients, without imposing smoothness on the volatility.
\[thm-diffusion\] Suppose that Assumptions \[ass-pseudo-contractive\], \[ass-cone-semigroup\] and \[ass-Schauder-basis\] are fulfilled, and that $\alpha \in {\rm Lip}(H)$ and $\sigma \in {\rm Lip}(H,L_2^0(H))$. If for all $(h^*,h) \in D$ we have (\[drift-cond-C2\]) and (\[main-4\]), then the closed convex cone $K$ is invariant for the SPDE (\[SPDE-Wiener\]).
For the proof of this result, we will apply the results from Appendices \[app-drift\] and \[app-volatility\]. Note that Assumption \[ass-D\] is fulfilled by virtue of Lemma \[lemma-fct-a\]. Concerning the drift $\alpha$, we use the approximation results from Appendix \[app-drift\] as follows:
1. Condition (\[drift-cond-C2\]) just means that $(a,\alpha)$ is inward pointing in the sense of Definition \[def-inward-pointing\].
2. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-alpha-FDR\] we may assume that $$\begin{aligned}
\alpha \in {\rm Lip}(H) \cap {{\rm F}}(H).\end{aligned}$$
3. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-alpha-bounded\] we may assume that $$\begin{aligned}
\alpha \in {\rm Lip}(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H).\end{aligned}$$
Furthermore, concerning the volatility $\sigma$, we use the approximation results from Appendix \[app-volatility\] as follows:
1. Condition (\[main-4\]) just means that for each $j \in {\mathbb{N}}$ the volatility $\sigma^j : H \to H$ is parallel in the sense of Definition \[def-parallel\].
2. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-sigma-finite\] we may assume that $$\begin{aligned}
\sigma \in {\rm Lip}(H,L_2^0(H)) \cap {{\rm G}}(H,L_2^0(H)).\end{aligned}$$ This allows us to apply the remaining results from Appendix \[app-volatility\] (Propositions \[prop-sigma-FDR\]–\[prop-C-b-2\]), which are all stated for volatilities of the form $\sigma : H \to H$.
3. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-sigma-FDR\] we may assume that $$\begin{aligned}
\sigma \in {\rm Lip}(H,L_2^0(H)) \cap {{\rm F}}(H,L_2^0(H)).\end{aligned}$$
4. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-sigma-bounded\] we may assume that $$\begin{aligned}
\sigma \in {\rm Lip}(H,L_2^0(H)) \cap {{\rm F}}(H,L_2^0(H)) \cap {{\rm B}}(H,L_2^0(H)).\end{aligned}$$
5. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-locally-par\] we may assume that for each $j \in {\mathbb{N}}$ the volatility $\sigma^j : H \to H$ is locally parallel in the sense of Definition \[def-locally-par\].
6. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-C-b-1-1\] we may assume that $$\begin{aligned}
\sigma \in {{\rm F}}(H,L_2^0(H)) \cap C_b^{1,1}(H,L_2^0(H)),\end{aligned}$$ and that $\sigma^j : H \to H$ is locally parallel for each $j \in {\mathbb{N}}$.
7. By our stability result for SPDEs (Proposition \[prop-K-stability\]) and Proposition \[prop-C-b-2\] we may assume that $$\begin{aligned}
\sigma \in {{\rm F}}(H,L_2^0(H)) \cap C_b^2(H,L_2^0(H)),\end{aligned}$$ and that for each $j \in {\mathbb{N}}$ the volatility $\sigma^j : H \to H$ is weakly locally parallel in the sense of Definition \[def-locally-par-weak\].
Consequently, applying Theorem \[thm-diffusion-C2\] completes the proof.
Sufficiency of the invariance conditions for SPDEs with Lipschitz coefficients {#sec-proof-3}
==============================================================================
In this section, we prove that our invariance conditions are sufficient for general jump-diffusion SPDEs (\[SPDE\]) with Lipschitz coefficients.
\[thm-suff\] Suppose that Assumptions \[ass-pseudo-contractive\], \[ass-cone-semigroup\] and \[ass-Schauder-basis\] are fulfilled, and that $\alpha \in {{\rm Lip}}(H)$, $\sigma \in {{\rm Lip}}(H,L_2^0(H))$ and $\gamma \in {{\rm Lip}}(H,L^2(F))$. If we have (\[main-1\]), and for all $(h^*,h) \in D$ we have (\[main-3\]) and (\[main-4\]), then the closed convex cone $K$ is invariant for the SPDE (\[SPDE\]).
Since the measure $F$ is $\sigma$-finite, by our stability result (Proposition \[prop-K-stability\]) it suffices to prove that for each $B \in {\mathcal{E}}$ with $F(B) < \infty$ the cone $K$ is invariant for the SPDE $$\begin{aligned}
\left\{
\begin{array}{rcl}
dr_t & = & ( A r_t + \alpha(r_t) - \int_B \gamma(r_t,x) F(dx) ) dt + \sigma(r_t) dW_t \medskip
\\ && + \int_B \gamma(r_{t-},x) \mu(dt,dx) \medskip
\\ r_0 & = & h_0.
\end{array}
\right.\end{aligned}$$ Moreover, by the jump condition (\[main-1\]) and [@FTT-appendix Lemmas 2.12 and 2.20], it suffices to prove that the cone $K$ is invariant for the SPDE $$\begin{aligned}
\label{SPDE-B-2}
\left\{
\begin{array}{rcl}
dr_t & = & ( A r_t + \alpha_B(r_t) ) dt + \sigma(r_t) dW_t \medskip
\\ r_0 & = & h_0.
\end{array}
\right.\end{aligned}$$ where $\alpha_B : H \to H$ is given by $$\begin{aligned}
\alpha_B(h) := \alpha(h) - \int_B \gamma(h,x) F(dx), \quad h \in H.\end{aligned}$$ Note that by the Cauchy-Schwarz inequality we have $\alpha_B \in {{\rm Lip}}(H)$. Let $(h^*,h) \in D$ be arbitrary. By (\[main-3\]) and Proposition \[prop-conditions\] we obtain $$\begin{aligned}
&\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*, \alpha_B(h) \rangle = \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha(h) \rangle
\\ &\quad - \int_E \langle h^*,\gamma(h,x) \rangle F(dx) + \int_{E \setminus B} \langle h^*,\gamma(h,x) \rangle F(dx) \geq 0.\end{aligned}$$ Therefore, applying Theorem \[thm-diffusion\] yields that the cone $K$ is invariant for the SPDE (\[SPDE-B-2\]), completing the proof.
Sufficiency of the invariance conditions and proof of the main result {#sec-proof-4}
=====================================================================
In this section, we prove that our invariance conditions are sufficient for jump-diffusion SPDEs (\[SPDE-Wiener\]) with coefficients being locally Lipschitz and satisfying the linear growth condition.
\[thm-suff-general\] Suppose that Assumptions \[ass-pseudo-contractive\], \[ass-loc-Lip-LG\], \[ass-cone-semigroup\] and \[ass-Schauder-basis\] are fulfilled. If we have (\[main-1\]), and for all $(h^*,h) \in D$ we have (\[main-3\]) and (\[main-4\]), then the closed convex cone $K$ is invariant for the SPDE (\[SPDE\]).
Let $h_0 \in K$ be arbitrary. Let $(R_n)_{n \in {\mathbb{N}}}$ be the sequence of retractions $R_n : H \to H$ defined according to Definition \[def-retract-X\]. We define the sequences of functions $(\alpha_n)_{n \in {\mathbb{N}}}$, $(\sigma_n)_{n \in {\mathbb{N}}}$ and $(\gamma_n)_{n \in {\mathbb{N}}}$ as $$\begin{aligned}
\alpha_n := \alpha \circ R_n, \quad \sigma_n := \sigma \circ R_n \quad \text{and} \quad \gamma_n := \gamma \circ R_n.\end{aligned}$$ Let $n \in {\mathbb{N}}$ be arbitrary. Then, by Lemma \[lemma-retract-X\] we have $$\begin{aligned}
\alpha_n \in {{\rm Lip}}(H), \quad \sigma_n \in {{\rm Lip}}(H,L_2^0(H)) \quad \text{and} \quad \gamma \in {{\rm Lip}}(H,L^2(F)),\end{aligned}$$ and hence, there exists a unique mild solution $r^n$ to the SPDE (\[SPDE-n\]) with $r_0^n = h_0$. Now, we check that conditions (\[main-1\])–(\[main-4\]) are fulfilled with $(\alpha,\sigma,\gamma)$ replaced by $(\alpha_n,\sigma_n,\gamma_n)$. Following the notation from Definition \[def-retract-X\], there is a function $\lambda_n : H \to (0,1]$ such that $$\begin{aligned}
R_n(h) = \lambda_n(h) h \quad \text{for all $h \in H$.}\end{aligned}$$ Let $h \in K$ be arbitrary. By the properties of the closed convex cone $K$ we have $\lambda_n(h) h \in K$ and $(1-\lambda_n(h)) h \in K$, and hence, since condition (\[main-1\]) is satisfied for $\gamma$, we obtain $$\begin{aligned}
h + \gamma_n(h,x) = h + \gamma(\lambda_n(h) h,x) = \underbrace{(1 - \lambda_n(h)) h}_{\in K} + \underbrace{\lambda_n(h) h + \gamma(\lambda_n(h) h,x)}_{\in K} \in K\end{aligned}$$ for $F$-almost all $x \in E$, showing (\[main-1\]) with $\gamma$ replaced by $\gamma_n$. Now, let $h^* \in G^*$ be such that $(h^*,h) \in D$. Then, by Lemma \[lemma-fct-a\] we also have $(h^*,\lambda_n(h) h) \in D$, and since condition (\[main-4\]) is satisfied for $\sigma$, we obtain $$\begin{aligned}
\langle h^*,\sigma_n^j(h) \rangle = \langle h^*,\sigma^j(\lambda_n(h)h) \rangle = 0, \quad j \in {\mathbb{N}},\end{aligned}$$ showing (\[main-4\]) with $\sigma$ replaced by $\sigma_n$. Furthermore, since condition (\[main-3\]) is satisfied for $(\alpha,\gamma)$, we obtain $$\begin{aligned}
&\liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha_n(h) \rangle - \int_E \langle h^*,\gamma_n(h,x) \rangle F(dx)
\\ &= \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \langle h^*,\alpha(\lambda_n(h)h) \rangle - \int_E \langle h^*,\gamma(\lambda_n(h)h,x) \rangle F(dx)
\\ &\geq (1-\lambda_n(h)) \liminf_{t \downarrow 0} \frac{\langle h^*,S_t h \rangle}{t} + \liminf_{t \downarrow 0} \frac{\langle h^*,S_t (\lambda_n(h)h) \rangle}{t}
\\ &\quad + \langle h^*,\alpha(\lambda_n(h)h) \rangle - \int_E \langle h^*,\gamma(\lambda_n(h)h,x) \rangle F(dx) \geq 0,\end{aligned}$$ showing (\[main-3\]) with $(\alpha,\gamma)$ replaced by $(\alpha_n,\gamma_n)$. Consequently, by Theorem \[thm-suff\] we have $r^n \in K$ up to an evanescent set. Now, we define the increasing sequence $(T_n)_{n \in {\mathbb{N}}_0}$ of stopping times by $T_0 := 0$ and $$\begin{aligned}
T_n := \inf \{ t \in {\mathbb{R}}_+ : \| r_t^n \| > n \} \quad \text{for all $n \in {\mathbb{N}}$.}\end{aligned}$$ Then we have ${\mathbb{P}}(T_n \to \infty) = 1$, and the mild solution $r$ to (\[SPDE\]) with $r_0 = h_0$ is given by $$\begin{aligned}
\label{r-lin-growth}
r = h_0 \mathbbm{1}_{[\![ T_0 ]\!]} + \sum_{n \in \mathbb{N}} r^n \mathbbm{1}_{]\!] T_{n-1}, T_n ]\!]},\end{aligned}$$ showing that $r \in K$ up to an evanescent set.
Now, we are ready to provide the proof of our main result, which concludes the paper.
\(i) $\Rightarrow$ (ii): This implication follows from Theorem \[thm-nec\].
\(ii) $\Rightarrow$ (i): This implication follows from Theorem \[thm-suff-general\].
An example {#sec-example}
==========
In this section, we provide an example illustrating our main result. Let $H = \ell^2({\mathbb{N}})$ be the Hilbert space consisting of all sequences $h = (h_k)_{k \in {\mathbb{N}}} \subset {\mathbb{R}}$ such that $\sum_{k \in {\mathbb{N}}} |h_k|^2 < \infty$. As in [@Pazy Example 2.5.4], let $(S_t)_{t \geq 0}$ be the semigroup given by $$\begin{aligned}
\label{def-semigroup}
S_t h := (e^{-kt} h_k)_{k \in {\mathbb{N}}} \quad \text{for $t \geq 0$ and $h = (h_k)_{k \in {\mathbb{N}}} \in H$.}\end{aligned}$$ Then $(S_t)_{t \geq 0}$ is a $C_0$-semigroup with infinitesimal generator $A : {\mathcal{D}}(A) \subset H \to H$ defined on the domain $$\begin{aligned}
{\mathcal{D}}(A) = \{ (h_k)_{k \in {\mathbb{N}}} \in H : (k h_k)_{k \in {\mathbb{N}}} \in H \},\end{aligned}$$ and given by $$\begin{aligned}
Ah = (-k h_k)_{k \in {\mathbb{N}}} \quad \text{for $h = (h_k)_{k \in {\mathbb{N}}} \in {\mathcal{D}}(A)$.}\end{aligned}$$ We consider the closed convex cone $$\begin{aligned}
K := \{ h = (h_k)_{k \in {\mathbb{N}}} \in H : h_k \geq 0 \text{ for all } k \in {\mathbb{N}}\}\end{aligned}$$ consisting of all nonnegative sequences.
Suppose that Assumption \[ass-loc-Lip-LG\] is fulfilled. Then the following statements are equivalent:
1. The closed convex cone $K$ is invariant for the SPDE (\[SPDE\]).
2. We have $$\begin{aligned}
h + \gamma(h,x) \in K \quad \text{for $F$-almost all $x \in E$,} \quad \text{for all $h \in K$,}\end{aligned}$$ and for all $(k,h) \in {\mathbb{N}}\times K$ with $h_k = 0$ we have $$\begin{aligned}
& \alpha_k(h) - \int_E \gamma_k(h,x) F(dx) \geq 0,
\\ &\sigma_k^j(h) = 0, \quad j \in {\mathbb{N}}.\end{aligned}$$
By definition (\[def-semigroup\]) the semigroup $(S_t)_{t \geq 0}$ is a semigroup of contractions, and the cone $K$ is invariant for the semigroup $(S_t)_{t \geq 0}$, showing that Assumptions \[ass-pseudo-contractive\] and \[ass-cone-semigroup\] are fulfilled. Moreover, the cone $K$ is self-dual; that is $K^* = K$, and we have the representation $$\begin{aligned}
K = \bigcap_{h^* \in G^*} \{ h \in H : \langle h^*,h \rangle \geq 0 \},\end{aligned}$$ where $G^* \subset K^*$ is given by $G^* = \{ e_k : k \in {\mathbb{N}}\}$, showing that Assumption \[ass-Schauder-basis\] is satisfied. Furthermore, for all $(k,h) \in {\mathbb{N}}\times K$ we have $$\begin{aligned}
\liminf_{t \downarrow 0} \frac{e^{-kt} h_k}{t} < \infty \quad \text{if and only if} \quad h_k = 0,\end{aligned}$$ and in this case the limes inferior vanishes. Consequently, applying Theorem \[thm-main\] completes the proof.
Function spaces {#app-function-spaces}
===============
In this appendix, we collect the function spaces used in this paper. Let $X$ and $Y$ be two normed spaces.
We introduce the following notions:
1. For a constant $L \in {\mathbb{R}}_+$ a function $f : X \to Y$ is called *$L$-Lipschitz* if $$\begin{aligned}
\label{Lip-cond-A}
\| f(x) - f(y) \| \leq L \| x-y \| \quad \text{for all $x,y \in X$.}\end{aligned}$$
2. For a constant $L \in {\mathbb{R}}_+$ we define the space $$\begin{aligned}
{\rm Lip}_L(X,Y) := \{ f : X \to Y : \text{$f$ is $L$-Lipschitz} \}.\end{aligned}$$
3. A function $f \in {\rm Lip}_L(X,Y)$ is called *Lipschitz continuous*.
4. We define the space ${\rm Lip}(X,Y) := \bigcup_{L \in {\mathbb{R}}_+} {\rm Lip}_L(X,Y)$.
5. For a constant $L \in {\mathbb{R}}_+$ we define the space ${\rm Lip}_L(X) := {\rm Lip}_L(X,X)$.
6. We define the space ${\rm Lip}(X) := {\rm Lip}(X,X)$.
We introduce the following notions:
1. A function $f : X \to Y$ is called *locally Lipschitz* if for each $C \in {\mathbb{R}}_+$ there is a constant $L(C) \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\| f(x) - f(y) \| \leq L(C) \| x-y \| \quad \text{for all $x,y \in X$ with $\| x \|, \| y \| \leq C$.}\end{aligned}$$
2. We denote by ${{\rm Lip}}^{{{\rm loc}}}(X,Y)$ the space of all locally Lipschitz functions $f : X \to Y$.
3. We define the space ${{\rm Lip}}^{\rm loc}(X) := {{\rm Lip}}^{{{\rm loc}}}(X,X)$.
We introduce the following notions:
1. We say that a function $f : X \to Y$ satisfies the *linear growth condition* if there is a finite constant $C \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\label{LG-A}
\| f(x) \| \leq C (1 + \| x \|) \quad \text{for all $x \in X$.}\end{aligned}$$
2. We denote by ${{\rm LG}}(X,Y)$ the space of all functions $f : X \to Y$ satisfying the linear growth condition.
3. We define the space ${{\rm LG}}(X) := {{\rm LG}}(X,Y)$.
Note that ${{\rm Lip}}(X,Y) \subset {{\rm Lip}}^{\rm loc}(X,Y) \cap {{\rm LG}}(X,Y)$. Indeed, if (\[Lip-cond-A\]) is fulfilled, setting $C := \max \{ L, \| f(0) \| \}$, for all $x \in X$ we obtain $$\begin{aligned}
\| f(x) \| \leq \| f(x) - f(0) \| + \| f(0) \| \leq L \| x \| + \| f(0) \| \leq C \| x \| + C = C (1 + \| x \|),\end{aligned}$$ showing (\[LG-A\]).
We introduce the following notions:
1. A function $f : X \to Y$ is called *bounded* if there is a constant $M \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\| f(x) \| \leq M \quad \text{for all $x \in X$.}\end{aligned}$$
2. We denote by ${{\rm B}}(X,Y)$ the space of all bounded functions $f : X \to Y$.
3. We define the space ${{\rm B}}(X) := {{\rm B}}(X,X)$.
We introduce the following notions:
1. A function $f : X \to Y$ is called *locally bounded* if for each $C \in {\mathbb{R}}_+$ there is a constant $M(C) \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\label{loc-bounded-A}
\| f(x) \| \leq M(C) \quad \text{for all $x \in X$ with $\| x \| \leq C$.}\end{aligned}$$
2. We denote by ${{\rm B}}^{\rm loc}(X,Y)$ the space of all locally bounded functions $f : X \to Y$.
3. We define the space ${{\rm B}}^{\rm loc}(X) := {{\rm B}}^{\rm loc}(X,X)$.
Note that ${{\rm LG}}(X,Y) \subset {{\rm B}}^{{{\rm loc}}}(X,Y)$. Indeed, if (\[LG-A\]) is satisfied, for each $C \in {\mathbb{R}}_+$ we set $M(C) := C(1+C)$, and then for all $x \in X$ with $\| x \| \leq C$ we obtain $$\begin{aligned}
\| f(x) \| \leq C(1 + \| x \|) \leq C(1 + C) = M(C),\end{aligned}$$ showing (\[loc-bounded-A\]).
We introduce the following notions:
1. We denote by $C(X,Y)$ the space of all continuous functions $f : X \to Y$.
2. We define the space $C_b(X,Y) := C(X,Y) \cap {{\rm B}}(X,Y)$.
3. We define the spaces $C(X) := C(X,X)$ and $C_b(X) := C_b(X,X)$.
Note that ${{\rm Lip}}^{{{\rm loc}}}(X,Y) \subset C(X,Y)$. For the next definition, we agree about the convention $\overline{{\mathbb{N}}} := {\mathbb{N}}\cup \{ \infty \}$, where ${\mathbb{N}}= \{ 1,2,3,\ldots \}$ denotes the natural numbers.
Let $p \in \overline{{\mathbb{N}}}$ be arbitrary.
1. We denote by $C^p(X,Y)$ the space of all $p$-times continuously differentiable functions $f : X \to Y$.
2. We denote by $C_b^p(X,Y)$ the space of all $f \in C^p(X,Y)$ such that $f$ is bounded and the derivatives $D^k f$, $k=1,\ldots,p$ are bounded.
3. We define the spaces $C^p(X) := C^p(X,X)$ and $C_b^p(X) := C_b^p(X,X)$.
Note that $C_b^1(X,Y) \subset {\rm Lip}(X,Y) \cap {{\rm B}}(X,Y)$.
We introduce the following notions:
1. We denote by $C_b^{1,1}(X,Y)$ the space of all $f \in C_b^1(X,Y)$ such that $Df \in {\rm Lip}(X,L(X,Y))$.
2. We define the space $C_b^{1,1}(X) := C_b^{1,1}(X,X)$.
Note that $C_b^2(X,Y) \subset C_b^{1,1}(X,Y) \subset C_b^1(X,Y)$. For the rest of this section, let $H$ be a Hilbert space.
\[def-retract-X\] For each $n \in {\mathbb{N}}$ we define the retraction $$\begin{aligned}
R_n : H \to H, \quad R_n(h) := \lambda_n(h) h,\end{aligned}$$ where the function $\lambda_n : H \to (0,1]$ is given by $$\begin{aligned}
\lambda_n(h) := {\mathbbm{1}}_{\{ \| h \| \leq n \}} + \frac{n}{\| h \|} {\mathbbm{1}}_{\{ \| h \| > n \}}, \quad h \in H.\end{aligned}$$
\[lemma-retract-X\] The following statements are true:
1. We have $R_n \to {\rm Id}_H$ as $n \to \infty$.
2. For each $n \in {\mathbb{N}}$ we have $R_n \in {{\rm Lip}}_1(H) \cap {{\rm B}}(H)$.
The first statement directly follows from Definition \[def-retract-X\]. For the proof of the second statement, let $n \in {\mathbb{N}}$ be arbitrary. Then we have $\| R_n(h) \| \leq n$ for all $h \in H$, and hence $R_n \in {{\rm B}}(H)$. Furthermore, the ball $K_n := \{ h \in H : \| h \| \leq n \}$ is a closed convex set. Let $h \in H$ and $g \in K_n$ be arbitrary. If $h \in K_n$, then we have $R_n(h) = h$, and hence $$\begin{aligned}
\langle h - R_n(h),g - R_n(h) \rangle = 0.\end{aligned}$$ Now, suppose that $h \in H \setminus K_n$. By the Cauchy Schwarz inequality we have $$\begin{aligned}
\langle h,g \rangle \leq |\langle h,g \rangle| \leq \| h \| \, \| g \| \leq n \| h \|.\end{aligned}$$ Moreover, we have $\lambda_n(h) \| h \| = n$, and it follows that $$\begin{aligned}
\langle h - R_n(h),g - R_n(h) \rangle &= \langle h - \lambda_n(h) h, g - \lambda_n(h) h \rangle = \langle (1 - \lambda_n(h)) h, g - \lambda_n(h) h \rangle
\\ &= (1 - \lambda_n(h)) \big( \langle h,g \rangle - \lambda_n(h) \| h \|^2 \big)
\\ &\leq (1 - \lambda_n(h)) \big( n \| h \| - \lambda_n(h) \| h \|^2 \big)
\\ &= (1 - \lambda_n(h)) \| h \| ( n - \lambda_n(h) \| h \| ) = 0.\end{aligned}$$ Consequently, the mapping $R_n$ is the metric projection onto the closed convex set $K_n$, and therefore we have $R_n \in {{\rm Lip}}_1(H)$.
Stability result for SPDEs {#app-stability}
==========================
In this appendix, we present the required stability result for SPDEs. The mathematical framework is that of Section \[sec-ass\]. Apart from the SPDE (\[SPDE\]), we consider the sequence of SPDEs given by $$\begin{aligned}
\label{SPDE-n}
\left\{
\begin{array}{rcl}
dr_t^n & = & (A r_t^n + \alpha_n(r_t^n)) dt + \sigma_n(r_t^n) dW_t + \int_E \gamma_n(r_{t-}^n,x) (\mu(dt,dx) - F(dx) dt) \medskip
\\ r_0^n & = & h_0
\end{array}
\right.\end{aligned}$$ for each $n \in {\mathbb{N}}$.
\[ass-stability\] We suppose that the following conditions are fulfilled:
1. There exists $L \in {\mathbb{R}}_+$ such that $\alpha_n \in {\rm Lip}_L(H)$, $\sigma_n \in {\rm Lip}_L(H,L_2^0(H))$ and $\gamma_n \in {\rm Lip}_L(H,L^2(F))$ for all $n \in {\mathbb{N}}$.
2. We have $\alpha_n \to \alpha$, $\sigma_n \to \sigma$ and $\gamma_n \to \gamma$ for $n \to \infty$.
\[prop-stability\] Suppose that Assumption \[ass-stability\] is fulfilled. Then, for each $h_0 \in H$ we have $$\begin{aligned}
{\mathbb{E}}\bigg[ \sup_{t \in [0,T]} \| r_t - r_t^n \|^2 \bigg] \to 0 \quad \text{for every $T \in {\mathbb{R}}_+$,}\end{aligned}$$ where $r$ denotes the mild solution to (\[SPDE\]) with $r_0 = h_0$, and for each $n \in {\mathbb{N}}$ the process $r^n$ denotes the mild solution to (\[SPDE-n\]) with $r_0^n = h_0$.
This is a consequence of [@SPDE Prop. 9.1.2].
\[prop-K-stability\] Suppose that Assumption \[ass-stability\] is fulfilled, and that for each $n \in {\mathbb{N}}$ the closed convex cone $K$ is invariant for the SPDE (\[SPDE-n\]). Then $K$ is also invariant for the SPDE (\[SPDE\]).
Let $h_0 \in K$ be arbitrary. We denote by $r$ the mild solution to (\[SPDE\]) with $r_0 = h_0$, and for each $n \in {\mathbb{N}}$ we denote by $r^n$ the mild solution to (\[SPDE-n\]) with $r_0^n = h_0$. Then, for each $n \in {\mathbb{N}}$ there is an event $\tilde{\Omega}_n \in {\mathcal{F}}$ with ${\mathbb{P}}(\tilde{\Omega}_n) = 1$ such that $r_t^n(\omega) \in K$ for all $(\omega,t) \in \tilde{\Omega}_n \times {\mathbb{R}}_+$. Setting $\tilde{\Omega} := \bigcap_{n \in {\mathbb{N}}} \tilde{\Omega}_n \in {\mathcal{F}}$ we have ${\mathbb{P}}(\tilde{\Omega}) = 1$ and $r_t^n(\omega) \in K$ for all $(\omega,t) \in \tilde{\Omega} \times {\mathbb{R}}_+$ and all $n \in {\mathbb{N}}$. Now, let $N \in {\mathbb{N}}$ be arbitrary. By Proposition \[prop-stability\] we have $$\begin{aligned}
{\mathbb{E}}\bigg[ \sup_{t \in [0,N]} \| r_t - r_t^n \|^2 \bigg] \to 0,\end{aligned}$$ and hence, there is a subsequence $(n_k)_{k \in {\mathbb{N}}}$ such that ${\mathbb{P}}$-almost surely $$\begin{aligned}
\sup_{t \in [0,N]} \| r_t - r_t^{n_k} \| \to 0.\end{aligned}$$ Since $K$ is closed, there is an event $\bar{\Omega}_N \in {\mathcal{F}}$ with ${\mathbb{P}}(\bar{\Omega}_N) = 1$ such that $r_t(\omega) \in K$ for all $(\omega,t) \in \bar{\Omega}_N \times [0,N]$. Therefore, setting $\bar{\Omega} := \bigcap_{N \in {\mathbb{N}}} \bar{\Omega}_N \in {\mathcal{F}}$ we obtain ${\mathbb{P}}(\bar{\Omega}) = 1$ and $r_t(\omega) \in K$ for all $(\omega,t) \in \bar{\Omega} \times {\mathbb{R}}_+$, showing that $K$ is invariant for (\[SPDE\]).
Inward pointing functions {#app-drift}
=========================
In this appendix, we provide the required results about inward pointing functions, which we need for the proof of Theorem \[thm-diffusion\]. As in Section \[sec-ass\], let $H$ be a separable Hilbert space, let $K \subset H$ be a closed convex cone, and let $G^* \subset K^*$ be a generating system of the cone such that Assumption \[ass-Schauder-basis\] is fulfilled. Let $D \subset G^* \times K$ be a subset, and let $a : D \to {\mathbb{R}}_+$ be a function.
\[ass-D\] We suppose that for each $(h^*,h) \in D$ the following conditions are fulfilled:
1. We have $\langle h^*,h \rangle = 0$.
2. For all $\lambda \geq 0$ we have $(h^*,\lambda h) \in D$ and $$\begin{aligned}
a(h^*,\lambda h) = \lambda a(h^*,h).\end{aligned}$$
3. For all $g \in K$ with $g \leq_K h$ we have $(h^*,g) \in D$ and $$\begin{aligned}
a(h^*,g) \leq a(h^*,h).\end{aligned}$$
\[def-inward-pointing\] Let $\alpha : H \to H$ be a function. We call the pair $(a,\alpha)$ *inward pointing at the boundary of $K$* (in short *inward pointing*) if for all $(h^*,h) \in D$ we have $$\begin{aligned}
a(h^*,h) + \langle h^*, \alpha(h) \rangle \geq 0.\end{aligned}$$
\[def-parallel\] A function $\sigma : H \to H$ is called *parallel at the boundary of $K$* (in short *parallel*) if for all $(h^*,h) \in D$ we have $$\begin{aligned}
\langle h^*, \sigma(h) \rangle = 0.\end{aligned}$$
\[def-inv-set\] Let $\sigma : H \to H$ be a function. Then the set $D$ is called *$({\rm Id}_H,\sigma)$-invariant* if $$\begin{aligned}
(h^*,\sigma(h)) \in D \quad \text{for all $(h^*,h) \in D$.}\end{aligned}$$
Let $\sigma : H \to H$ be a function. If $D$ is $({\rm Id}_H,\sigma)$-invariant, then $\sigma$ is parallel.
\[lemma-Pi-alpha\] Let $\alpha : H \to H$ be a function such that $(a,\alpha)$ is inward pointing. Then, for each $n \in {\mathbb{N}}$ the pair $(a,\Pi_n \circ \alpha)$ is inward pointing, too.
Let $(h^*,h) \in D$ be arbitrary. By Assumption \[ass-Schauder-basis\] we have $h^* \in \langle e_k^* \rangle$ for some $k \in {\mathbb{N}}$. Thus, by Lemma \[lemma-proj-par\], and since $a$ is nonnegative, we obtain $$\begin{aligned}
a(h^*,h) + \langle h^*,\Pi_n(\alpha(h)) \rangle = a(h^*,h) + \langle h^*,\alpha(h) \rangle \mathbbm{1}_{\{ k \leq n \}} \geq 0,\end{aligned}$$ finishing the proof.
\[def-F-spaces\] We introduce the following spaces:
1. For each $n \in {\mathbb{N}}$ we denote by ${{\rm F}}_n(H)$ the space of all functions $\alpha : H \to E_n$.
2. We set ${{\rm F}}(H) := \bigcup_{n \in {\mathbb{N}}} {{\rm F}}_n(H)$.
\[prop-alpha-FDR\] Let $\alpha \in {{\rm Lip}}(H)$ be a function such that $(a,\alpha)$ is inward pointing. Then, there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
\label{alpha-n-FDR}
(\alpha_n)_{n \in {\mathbb{N}}} \subset {{\rm Lip}}_L(H) \cap {{\rm F}}(H) \end{aligned}$$ such that $(a,\alpha_n)$ is inward pointing for each $n \in {\mathbb{N}}$, and we have $\alpha_n \to \alpha$.
We set $\alpha_n := \Pi_n \circ \alpha$ for each $n \in {\mathbb{N}}$. Then, by construction for each $n \in {\mathbb{N}}$ we have $\alpha_n \in {{\rm F}}(H)$. By hypothesis there exists a constant $M \in {\mathbb{R}}_+$ such that $\alpha \in {\rm Lip}_M(H)$. Setting $L := M {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}$, we have $\alpha_n \in {\rm Lip}_L(H)$ for each $n \in {\mathbb{N}}$, showing (\[alpha-n-FDR\]). Furthermore, by Lemma \[lemma-Pi-alpha\], for each $n \in {\mathbb{N}}$ the pair $(a,\alpha_n)$ is inward pointing, and by Lemma \[lemma-proj-par\] we have $\alpha_n \to \alpha$.
\[lemma-inward-comp\] Let $\alpha,\beta : H \to H$ be two functions such that the following conditions are fulfilled:
1. $(a,\alpha)$ is inward pointing.
2. $D$ is $({\rm Id}_H,\beta)$-invariant, and for all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{inward-comp-drift}
a(h^*,\beta(h)) \leq a(h^*,h).\end{aligned}$$
Then the pair $(a,\alpha \circ \beta)$ is inward pointing.
Let $(h^*,h) \in D$ be arbitrary. Since the set $D$ is $({\rm Id}_H,\beta)$-invariant, we have $(h^*,\beta(h)) \in D$. Therefore, by (\[inward-comp-drift\]), and since $(a,\alpha)$ is inward pointing, we obtain $$\begin{aligned}
a(h^*,h) + \langle h^*,\alpha(\beta(h)) \rangle \geq a(h^*,\beta(h)) + \langle h^*,\alpha(\beta(h)) \rangle \geq 0,\end{aligned}$$ finishing the proof.
We denote $(R_n)_{n \in {\mathbb{N}}}$ the retractions $R_n : H \to H$ defined according to Definition \[def-retract-X\]. We will need the following auxiliary result.
\[lemma-retract-par\] Let $n \in {\mathbb{N}}$ be arbitrary. Then $D$ is $({\rm Id}_H,R_n)$-invariant, and for all $(h^*,h) \in D$ we have $$\begin{aligned}
a(h^*,R_n(h)) \leq a(h^*,h).\end{aligned}$$
Let $n \in {\mathbb{N}}$ be arbitrary. Recalling the notation from Definition \[def-retract-X\], there is a function $\lambda_n : H \to (0,1]$ such that $$\begin{aligned}
R_n(h) = \lambda_n(h) h \quad \text{for each $h \in H$.}\end{aligned}$$ By Assumption \[ass-D\], for all $(h^*,h) \in D$ we obtain $(h^*,R_n(h)) = (h^*,\lambda_n(h) h) \in D$ and $$\begin{aligned}
a(h^*,R_n(h)) = a(h^*,\lambda_n(h) h) = \lambda_n(h) a(h^*,h) \leq a(h^*,h),\end{aligned}$$ completing the proof.
\[prop-alpha-bounded\] Let $\alpha \in {\rm Lip}(H) \cap {{\rm F}}(H)$ be a function such that $(a,\alpha)$ is inward pointing. Then there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
\label{alpha-bounded}
(\alpha_n)_{n \in {\mathbb{N}}} \subset {\rm Lip}_L(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)\end{aligned}$$ such that $(a,\alpha_n)$ is inward pointing for each $n \in {\mathbb{N}}$, and we have $\alpha_n \to \alpha$.
We set $\alpha_n := \alpha \circ R_n$ for each $n \in {\mathbb{N}}$. Let $n \in {\mathbb{N}}$ be arbitrary. Then we have $\alpha_n \in {{\rm F}}(H)$, because $\alpha \in {{\rm F}}(H)$. By hypothesis there exists a constant $L \in {\mathbb{R}}_+$ such that $\alpha \in {\rm Lip}_L(H)$, and by Lemma \[lemma-retract-X\] and the inclusion ${{\rm Lip}}_L(H) \subset {{\rm B}}^{{{\rm loc}}}(H)$ it follows that $\alpha_n \in {\rm Lip}_L(H) \cap {{\rm B}}(H)$, showing (\[alpha-bounded\]). Combining Lemmas \[lemma-inward-comp\] and \[lemma-retract-par\], we obtain that $(a,\alpha_n)$ is inward pointing. Furthermore, by Lemma \[lemma-retract-X\] we have $\alpha_n \to \alpha$.
Parallel functions {#app-volatility}
==================
In this appendix, we provide the required results about parallel function, which we need for the proofs of Theorems \[thm-diffusion-C2\] and \[thm-diffusion\]. The general mathematical framework is that of Appendix \[app-drift\]. First, we will extend the Definition \[def-parallel\] of a parallel function.
\[def-locally-par\] A function $\sigma : H \to H$ is called *locally parallel to the boundary of $K$* (in short *locally parallel*) if there exists $\epsilon > 0$ such that for all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{loc-par-prop}
\langle h^*, \sigma(h-g) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon$.}\end{aligned}$$
\[def-locally-par-weak\] A function $\sigma : H \to H$ is called *weakly locally parallel to the boundary of $K$* (in short *weakly locally parallel*) if for all $(h^*,h) \in D$ there exists $\epsilon = \epsilon(h^*,h) > 0$ such that we have (\[loc-par-prop\]).
Let $\sigma : H \to H$ be a function. Then the set $D$ is called *locally $({\rm Id}_H,\sigma)$-invariant* if there exists $\epsilon > 0$ such that for all $(h^*,h) \in D$ we have $$\begin{aligned}
(h^*,\sigma(h-g)) \in D \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon$.}\end{aligned}$$
Let $\sigma : H \to H$ be a function.
1. If $\sigma$ is locally parallel, then it weakly locally parallel, too.
2. If $D$ is locally $({\rm Id}_H,\sigma)$-invariant, then $\sigma$ is locally parallel.
As in Section \[sec-ass\], let $U$ be a separable Hilbert space, and let $Q \in L(U)$ be a nuclear, self-adjoint, positive definite linear operator. Recall that $U_0 := Q^{1/2}(U)$ equipped with the inner product (\[inner-prod-U0\]) is another separable Hilbert space, and that $L_2^0(H) := L_2(U_0,H)$ denotes the space of Hilbert-Schmidt operators from $U_0$ into $H$. Furthermore, recall that we have fixed an orthonormal basis $\{ g_j \}_{j \in {\mathbb{N}}}$ of $U_0$, and that for each $\sigma \in L_2^0(H)$ we set $\sigma^j := \sigma g_j$ for $j \in {\mathbb{N}}$. With this notation, the Hilbert-Schmidt norm is given by $$\begin{aligned}
\label{norm-HS}
\| \sigma \| = \sqrt{\sum_{j \in {\mathbb{N}}} \| \sigma^j \|^2} \quad \text{for each $\sigma \in L_2^0(H)$.}\end{aligned}$$
We denote by ${{\rm F}}(H,L_2^0(H))$ the space of all functions $\sigma : H \to L_2^0(H)$ such that for some $n \in {\mathbb{N}}$ we have $\sigma^j(H) \subset E_n$ for all $j \in {\mathbb{N}}$.
We denote by ${{\rm G}}(H,L_2^0(H))$ the space of all functions $\sigma : H \to L_2^0(H)$ such that for some index $N \in {\mathbb{N}}$ we have $\sigma^j = 0$ for all $j \in {\mathbb{N}}$ with $j > N$.
In view of the following Lemma \[lemma-rho\] and later results such as Lemma \[lemma-norm-Tn\], we emphasize that for a bounded linear operator $T$ we denote by $\| T \|$ the usual operator norm. As an exception, we agree that in the particular situation $\sigma \in L_2^0(H)$ we denote by $\| \sigma \|$ the Hilbert-Schmidt norm defined in (\[norm-HS\]), unless stated otherwise.
\[lemma-rho\] Let $\sigma \in C_b^2(H,L_2^0(H)) \cap {{\rm G}}(H,L_2^0(H))$ be arbitrary. Then the following statements are true:
1. For each $h \in H$ we have $\sum_{j \in {\mathbb{N}}} \| D \sigma^j(h) \sigma^j(h) \| < \infty$.
2. The function $\rho : H \to H$ defined as $$\begin{aligned}
\label{def-rho}
\rho(h) := \frac{1}{2} \sum_{j \in {\mathbb{N}}} D \sigma^j(h) \sigma^j(h), \quad h \in H,\end{aligned}$$ belongs to ${\rm Lip}(H) \cap {{\rm B}}(H)$.
By assumption, there exists a constant $C \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\max \{ \| \sigma(h) \|, \| D \sigma(h) \|, \| D^2 \sigma(h) \| \} \leq C \quad \text{for all $h \in H$.}\end{aligned}$$ Furthermore, there exists an index $N \in {\mathbb{N}}$ such that $\sigma^j(h) = 0$ for all $h \in H$ and all $j \in {\mathbb{N}}$ with $j > N$. Noting that for each $j \in {\mathbb{N}}$ the norm of the linear operator $L_2^0(H) \to H$, $\sigma \mapsto \sigma^j$ is bounded by $1$, by the chain rule and the Cauchy Schwarz inequality, for each $h \in H$ we obtain $$\begin{aligned}
&\sum_{j \in {\mathbb{N}}} \| D \sigma^j(h) \sigma^j(h) \| \leq \sum_{j=1}^N \| D \sigma^j(h) \| \, \| \sigma^j(h) \|
\\ &\leq \bigg( \sum_{j=1}^N \| D \sigma(h) \|^2 \bigg)^{1/2} \bigg( \sum_{j=1}^N \| \sigma^j(h) \|^2 \bigg)^{1/2} \leq \sqrt{N} \| D \sigma(h) \| \, \| \sigma(h) \| \leq \sqrt{N} C^2.\end{aligned}$$ proving the first statement and $\rho \in {{\rm B}}(H)$. For the proof of the second statement, let $h_1,h_2 \in H$ be arbitrary. By the chain rule and Cauchy Schwarz inequality we obtain $$\begin{aligned}
&\| \rho(h_1) - \rho(h_2) \| \leq \frac{1}{2} \sum_{j=1}^N \| D \sigma^j(h_1) \sigma^j(h_1) - D \sigma^j(h_2) \sigma^j(h_2) \|
\\ &\leq \frac{1}{2} \sum_{j=1}^N \| D \sigma^j(h_1) \| \, \| \sigma^j(h_1) - \sigma^j(h_2) \| + \frac{1}{2} \sum_{j=1}^N \| \sigma^j(h_2) \| \, \| D \sigma^j(h_1) - D \sigma^j(h_2) \|
\\ &\leq \frac{1}{2} \bigg( \sum_{j=1}^N \| D \sigma(h_1) \|^2 \bigg)^{1/2} \bigg( \sum_{j=1}^N \| \sigma^j(h_1) - \sigma^j(h_2) \|^2 \bigg)^{1/2}
\\ &\quad + \frac{1}{2} \bigg( \sum_{j=1}^N \| \sigma^j(h_2) \|^2 \bigg)^{1/2} \bigg( \sum_{j=1}^N \| D \sigma(h_1) - D \sigma(h_2) \|^2 \bigg)^{1/2},\end{aligned}$$ and hence $$\begin{aligned}
\| \rho(h_1) - \rho(h_2) \| &\leq \frac{\sqrt{N}}{2} \| D \sigma(h_1) \| \, \| \sigma(h_1) - \sigma(h_2) \|
\\ &\quad + \frac{\sqrt{N}}{2} \| \sigma(h_2) \| \, \| D \sigma(h_1) - D \sigma(h_2) \|
\\ &\leq \sqrt{N} C^2 \| h_1 - h_2 \|, \end{aligned}$$ showing that $\rho \in {\rm Lip}(H)$.
\[lemma-rho-parallel\] Let $\sigma \in C_b^2(H,L_2^0(H)) \cap {{\rm G}}(H,L_2^0(H))$ be such that for each $j \in {\mathbb{N}}$ the function $\sigma^j : H \to H$ is weakly locally parallel. Then the function $\rho : H \to H$ defined in (\[def-rho\]) is parallel.
Let $(h^*,h) \in D$ be arbitrary. Furthermore, let $j \in {\mathbb{N}}$ be arbitrary. Since $\sigma^j$ is locally parallel, there exists $\epsilon > 0$ such that $$\begin{aligned}
\langle h^*, \sigma^j(h-g) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon$.}\end{aligned}$$ We define $\delta > 0$ as $$\begin{aligned}
\delta :=
\begin{cases}
\epsilon / \| \sigma^j(h) \|, & \text{if $\sigma^j(h) \neq 0$,}
\\ 1, & \text{if $\sigma^j(h) = 0$.}
\end{cases}\end{aligned}$$ Then we have $$\begin{aligned}
\langle h^*, \sigma^j(h + t \sigma^j(h)) \rangle = 0 \quad \text{for all $t \in [-\delta,\delta]$.}\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
\langle h^*, D \sigma^j(h) \sigma^j(h) \rangle &= \Big\langle h^*, \lim_{t \to 0} \frac{\sigma^j(h + t \sigma^j(h)) - \sigma^j(h)}{t} \Big\rangle
\\ &= \lim_{t \to 0} \frac{\langle h^*,\sigma^j(h + t \sigma^j(h)) \rangle - \langle h^*, \sigma^j(h) \rangle}{t} = 0.\end{aligned}$$ This implies $$\begin{aligned}
\langle h^*,\rho(h) \rangle = \bigg\langle h^*, \frac{1}{2} \sum_{j \in {\mathbb{N}}} D \sigma^j(h) \sigma^j(h) \bigg\rangle = \frac{1}{2} \sum_{j \in {\mathbb{N}}} \langle h^*,D \sigma^j(h) \sigma^j(h) \rangle = 0,\end{aligned}$$ showing that $\rho$ is parallel.
\[lemma-sigma-u\] Let $\sigma : H \to L_2^0(H)$ be such that for each $j \in {\mathbb{N}}$ the function $\sigma^j : H \to H$ is parallel. Then, for each $u \in U_0$ the function $\sigma(\cdot)u : H \to H$ is parallel.
Recall that we have fixed an orthonormal basis $\{ g_j \}_{j \in {\mathbb{N}}}$ of $U_0$. Let $u \in U_0$ be arbitrary, and let $(h^*,h) \in D$ be arbitrary. Since for each $j \in {\mathbb{N}}$ the function $\sigma^j : H \to H$ is parallel, we obtain $$\begin{aligned}
\langle h^*, \sigma(h)u \rangle = \Big\langle h^*, \sigma(h) \sum_{j \in {\mathbb{N}}} \langle u,g_j \rangle_{U_0} g_j \Big\rangle = \sum_{j \in {\mathbb{N}}} \langle u,g_j \rangle_{U_0} \langle h^*, \sigma^j(h) \rangle = 0,\end{aligned}$$ showing that $\sigma(\cdot)u$ is parallel.
For each $n \in {\mathbb{N}}$ let $G_n \subset U_0$ be the finite dimensional subspace $G_n := \langle g_1,\ldots,g_n \rangle$, denote by $\pi_n : U_0 \to G_n$ the corresponding projection $$\begin{aligned}
\pi_n u = \sum_{j=1}^n \langle u,g_j \rangle_{U_0} g_j, \quad u \in U_0,\end{aligned}$$ and let $T_n : L_2^0(H) \to L_2^0(H)$ be the linear operator given by $T_n \sigma := \sigma \circ \pi_n$ for each $\sigma \in L_2^0(H)$. Note that for each $n \in {\mathbb{N}}$ and each $\sigma \in L_2^0(H)$ we have $$\begin{aligned}
\label{finite}
(T_n \sigma)^j = \sigma(\pi_n(g_j)) = \sigma^j {\mathbbm{1}}_{\{ j \leq n \}}, \quad j \in {\mathbb{N}}.\end{aligned}$$
\[lemma-norm-Tn\] The following statements are true:
1. For each $n \in {\mathbb{N}}$ we have $\| T_n \| \leq 1$.
2. For each $\sigma \in L_2^0(H)$ we have $T_n \sigma \to \sigma$ as $n \to \infty$.
Let $\sigma \in L_2^0(H)$ be arbitrary. Noting (\[norm-HS\]) and (\[finite\]), for each $n \in {\mathbb{N}}$ we have $$\begin{aligned}
\| T_n \sigma \| = \sqrt{\sum_{j=1}^n \| \sigma^j \|^2} \leq \sqrt{\sum_{j \in {\mathbb{N}}} \| \sigma^j \|^2} = \| \sigma \|,\end{aligned}$$ showing that $\| T_n \| \leq 1$. Furthermore, by (\[norm-HS\]) and (\[finite\]) we obtain $$\begin{aligned}
\| T_n \sigma - \sigma \| = \sqrt{\sum_{j > n} \| \sigma^j \|^2} \to 0 \quad \text{as $n \to \infty$,}\end{aligned}$$ showing that $T_n \sigma \to \sigma$.
\[prop-sigma-finite\] Let $\sigma \in {\rm Lip}(H,L_2^0(H))$ be such that for each $j \in {\mathbb{N}}$ the function $\sigma^j : H \to H$ is parallel. Then there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
\label{sigma-finite}
(\sigma_n)_{n \in {\mathbb{N}}} \subset {\rm Lip}_L(H,L_2^0(H)) \cap {{\rm G}}(H,L_2^0(H))\end{aligned}$$ such that for all $n,j \in {\mathbb{N}}$ the function $\sigma_n^j : H \to H$ is parallel, and we have $\sigma_n \to \sigma$.
We set $\sigma_n := T_n \circ \sigma$ for each $n \in {\mathbb{N}}$. By noting (\[finite\]), we have $(\sigma_n)_{n \in {\mathbb{N}}} \subset {{\rm G}}(H,L_2^0(H))$, and for all $n,j \in {\mathbb{N}}$ the function $\sigma_n^j : H \to H$ is parallel. By hypothesis, there is a constant $L \in {\mathbb{R}}_+$ such that $\sigma \in {\rm Lip}_L(H,L_2^0(H))$, and by Lemma \[lemma-norm-Tn\], it follows that $\sigma_n \in {\rm Lip}_L(H,L_2^0(H))$ for each $n \in {\mathbb{N}}$, showing (\[sigma-finite\]), and that $\sigma_n \to \sigma$.
\[lemma-Pi-sigma\] Let $\sigma : H \to H$ be a parallel function. Then, for each $n \in {\mathbb{N}}$ the function $\Pi_n \circ \sigma$ is parallel, too.
Let $(h^*,h) \in D$ be arbitrary. By Assumption \[ass-Schauder-basis\] we have $h^* \in \langle e_k^* \rangle$ for some $k \in {\mathbb{N}}$. Therefore, by Lemma \[lemma-proj-par\] we obtain $$\begin{aligned}
\langle h^*,\Pi_n(\sigma(h)) \rangle = \langle h^*,\sigma(h) \rangle \mathbbm{1}_{\{ k \leq n \}} = 0,\end{aligned}$$ finishing the proof.
In view of the following results, recall the Definition \[def-F-spaces\] of ${{\rm F}}(H)$.
\[prop-sigma-FDR\] Let $\sigma \in {\rm Lip}(H)$ be a parallel function. Then there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
\label{sigma-FDR}
(\sigma_n)_{n \in {\mathbb{N}}} \subset {{\rm Lip}}_L(H) \cap {{\rm F}}(H)\end{aligned}$$ such that $\sigma_n$ is parallel for each $n \in {\mathbb{N}}$, and we have $\sigma_n \to \sigma$.
We set $\sigma_n := \Pi_n \circ \sigma$ for each $n \in {\mathbb{N}}$. Then, by construction for each $n \in {\mathbb{N}}$ we have $\sigma_n \in {{\rm F}}(H)$. By hypothesis there exists a constant $M \in {\mathbb{R}}_+$ such that $\sigma \in {\rm Lip}_M(H)$. Setting $L := M {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}$, we have $\sigma_n \in {\rm Lip}_L(H)$ for each $n \in {\mathbb{N}}$, showing (\[sigma-FDR\]). Furthermore, by Lemma \[lemma-Pi-sigma\], for each $n \in {\mathbb{N}}$ the function $\sigma_n$ is parallel, and by Lemma \[lemma-proj-par\] we have $\sigma_n \to \sigma$.
\[lemma-inward-comp-2\] Let $\sigma,\tau : H \to H$ be two functions such that the following conditions are fulfilled:
1. $\sigma$ is parallel.
2. $D$ is $({\rm Id}_H,\tau)$-invariant.
Then $\sigma \circ \tau$ is parallel.
Let $(h^*,h) \in D$ be arbitrary. Then we have $(h^*,\tau(h)) \in D$, because $D$ is $({\rm Id}_H,\tau)$-invariant. Therefore, and since $\sigma$ is parallel, we obtain $$\begin{aligned}
\langle h^*,\sigma(\tau(h)) \rangle = 0,\end{aligned}$$ finishing the proof.
\[prop-sigma-bounded\] Let $\sigma \in {\rm Lip}(H) \cap {{\rm F}}(H)$ be a parallel function. Then there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
\label{sigma-bounded}
(\sigma_n)_{n \in {\mathbb{N}}} \subset {{\rm Lip}}_L(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)\end{aligned}$$ such that $\sigma_n$ is parallel for each $n \in {\mathbb{N}}$, and we have $\sigma_n \to \sigma$.
We set $\sigma_n := \sigma \circ R_n$ for each $n \in {\mathbb{N}}$. Let $n \in {\mathbb{N}}$ be arbitrary. Then we have $\sigma_n \in {{\rm F}}(H)$, because $\sigma \in {{\rm F}}(H)$. By hypothesis there exists a constant $L \in {\mathbb{R}}_+$ such that $\sigma \in {\rm Lip}_L(H)$, and by Lemma \[lemma-retract-X\] and the inclusion ${{\rm Lip}}_L(H) \subset {{\rm B}}^{{{\rm loc}}}(H)$ it follows that $\sigma_n \in {\rm Lip}_L(H) \cap {{\rm B}}(H)$, showing (\[sigma-bounded\]). Combining Lemmas \[lemma-inward-comp-2\] and \[lemma-retract-par\], we obtain that $\sigma_n$ is parallel. Furthermore, by Lemma \[lemma-retract-X\] we have $\sigma_n \to \sigma$.
\[lemma-tau-sigma-local\] Let $\sigma,\tau : H \to H$ be two functions such that the following conditions are fulfilled:
1. $\sigma$ is parallel.
2. $D$ is locally $({\rm Id}_H,\tau)$-invariant.
Then $\sigma \circ \tau$ is locally parallel.
By assumption, there exists $\epsilon > 0$ such that for all $(h^*,h) \in D$ we have $$\begin{aligned}
(h^*,\tau(h-g)) \in D \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon$.}\end{aligned}$$ Let $(h^*,h) \in D$ be arbitrary. Since $\sigma$ is parallel, we obtain $$\begin{aligned}
\langle h^*, \sigma(\tau(h-g)) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon$,}\end{aligned}$$ completing the proof.
For $\epsilon > 0$ let $\phi_{\epsilon} : {\mathbb{R}}\to {\mathbb{R}}$ be the function given by (\[def-psi-intro\]); see Figure \[fig-approx\]. Then we have $\phi_{\epsilon} \in {\rm Lip}_1({\mathbb{R}})$ and $$\begin{aligned}
\phi_{\epsilon}(x) &= 0 \quad \text{for all $x \in [-\epsilon,\epsilon]$,}
\\ \phi_{\epsilon}(x) &\geq 0 \quad \text{for all $x \in [-\epsilon,\infty)$,}
\\ \label{psi-2} | \phi_{\epsilon}(x) - x | &\leq \epsilon \quad \text{for all $x \in {\mathbb{R}}$,}
\\ \label{psi-3} \bigg| \frac{\phi_{\epsilon}(x) - \phi_{\epsilon}(y)}{x-y} \bigg| &\leq 1 \quad \text{for all $x,y \in {\mathbb{R}}$ with $x \neq y$.}\end{aligned}$$ Furthermore, for each $\theta \in \{ -1,1 \}$ we have $$\begin{aligned}
\label{psi-theta-1}
\theta \phi_{\epsilon}(\theta y) &\geq 0 \quad \text{for all $y \in [-\epsilon, \infty)$,}
\\ \label{psi-theta-2} x - \theta \phi_{\epsilon}(\theta y) &\geq 0 \quad \text{for all $x \in {\mathbb{R}}_+$ and $y \in {\mathbb{R}}$ with $|x-y| \leq \epsilon$.}\end{aligned}$$
\[lemma-Psi-n\] There exist a constant $L \in {\mathbb{R}}_+$ and a sequence $(\Phi_n)_{n \in {\mathbb{N}}} \subset {\rm Lip}_L(H)$ such that for each $n \in {\mathbb{N}}$ the set $D$ is locally $({\rm Id}_H,\Phi_n)$-invariant, and we have $\Phi_n \to {\rm Id}_H$.
We set $L := 2 {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}$. Let $n \in {\mathbb{N}}$ be arbitrary. We define the function $$\begin{aligned}
\label{def-Psi-n}
\Phi_n : H \to H, \quad \Phi_n(h) := \sum_{k=1}^n \phi_{2^{-n}}(h_k) e_k,\end{aligned}$$ where we refer to the series representation (\[series-h\]) of $h$. Let $h,g \in H$ be arbitrary. We define the sequence $(\lambda_k)_{k \in {\mathbb{N}}} \subset {\mathbb{R}}$ as $$\begin{aligned}
\lambda_k := \frac{\phi_{2^{-n}}(h_k)-\phi_{2^{-n}}(g_k)}{h_k - g_k} \mathbbm{1}_{\{ h_k \neq g_k \}} {\mathbbm{1}}_{\{ k \leq n \}}, \quad k \in {\mathbb{N}}.\end{aligned}$$ By (\[psi-3\]) we have $|\lambda_k| \leq 1$ for all $k \in {\mathbb{N}}$, and by Lemma \[lemma-norm-g-star\] we obtain $$\begin{aligned}
\| \Phi_n(h) - \Phi_n(g) \| &= \bigg\| \sum_{k=1}^n (\phi_{2^{-n}}(h_k) - \phi_{2^{-n}}(g_k)) e_k \bigg\|
\\ &= \bigg\| \sum_{k \in {\mathbb{N}}} \lambda_k (h_k - g_k) e_k \bigg\| \leq L \bigg\| \sum_{k \in {\mathbb{N}}} (h_k - g_k) e_k \bigg\| = L \| h-g \|,\end{aligned}$$ showing that $\Phi_n \in {\rm Lip}_L(H)$. Let $h \in H$ be arbitrary. Then, by (\[psi-2\]) we obtain $$\begin{aligned}
&\| \Phi_n(h) - h \| = \bigg\| \sum_{k = 1}^n \phi_{2^{-n}}(h_k) e_k - \sum_{k \in {\mathbb{N}}} h_k e_k \bigg\|
\\ &= \bigg\| \sum_{k = 1}^n (\phi_{2^{-n}}(h_k) - h_k) e_k - \sum_{k = n+1}^{\infty} h_k e_k \bigg\|
\leq \sum_{k=1}^n | \phi_{2^{-n}}(h_k) - h_k | + \bigg\| \sum_{k = n+1}^{\infty} h_k e_k \bigg\|
\\ &\leq n \cdot 2^{-n} + \bigg\| \sum_{k = n+1}^{\infty} h_k e_k \bigg\| \to 0 \quad \text{a $n \to \infty$,}\end{aligned}$$ showing that $\Phi_n \to {\rm Id}_H$. Let $n \in {\mathbb{N}}$ be arbitrary. In order to show that $D$ is locally $({\rm Id}_H,\Phi_n)$-invariant, we set $\epsilon := 2^{-n} / L$. Let $(h^*,h) \in D$ be arbitrary, and let $g \in H$ with $\| g \| \leq \epsilon$ be arbitrary. We will show that $(h^*,\Phi_n(h-g)) \in D$. For this purpose, let $g^* \in G^*$ be arbitrary. Since $\| g \| \leq \epsilon$, by Lemma \[lemma-norm-g-star\] we have $$\begin{aligned}
\label{h-g-rel}
|\langle g^*,g \rangle| \leq L \| g \| \leq L \epsilon = 2^{-n}.\end{aligned}$$ Since $h \in K$, we have $\langle g^*,h \rangle \geq 0$, and hence, we obtain $$\begin{aligned}
\label{g-g-rel}
\langle g^*,h-g \rangle = \langle g^*,h \rangle - \langle g^*,g \rangle \geq -L\epsilon = -2^{-n}.\end{aligned}$$ By Assumption \[ass-Schauder-basis\] we have $g^* = \theta e_k^*$ for some $\theta \in \{ -1,1 \}$ and some $k \in {\mathbb{N}}$. Thus, by the definition (\[def-Psi-n\]) of $\Phi_n$ and relations (\[g-g-rel\]) and (\[psi-theta-1\]) we deduce $$\begin{aligned}
\langle g^*,\Phi_n(h-g) \rangle = \theta \phi_{2^{-n}}(\theta \langle g^*,h-g \rangle) {\mathbbm{1}}_{\{ k \leq n \}} \geq 0,\end{aligned}$$ showing that $\Phi_n(h-g) \in K$. Furthermore, noting that $h \in K$, by the definition (\[def-Psi-n\]) of $\Phi_n$ and relations (\[h-g-rel\]) and (\[psi-theta-2\]) we obtain $$\begin{aligned}
\langle g^*,h-\Phi_n(h-g) \rangle = \langle g^*,h \rangle - \theta \phi_{2^{-n}}(\theta \langle g^*,h-g \rangle) {\mathbbm{1}}_{\{ k \leq n \}} \geq 0,\end{aligned}$$ showing that $h-\Phi_n(h-g) \in K$, and hence $\Phi_n(h-g) \leq_K h$. By Assumption \[ass-D\] we deduce that $(h^*,\Phi_n(h-g)) \in D$, showing that $D$ is locally $({\rm Id}_H,\Phi_n)$-invariant.
\[prop-locally-par\] Let $\sigma \in {\rm Lip}(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ be a parallel function. Then there are a constant $L \in {\mathbb{R}}$ and a sequence $$\begin{aligned}
\label{locally-par}
(\sigma_n)_{n \in {\mathbb{N}}} \subset {\rm Lip}_L(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)\end{aligned}$$ such that $\sigma_n$ is locally parallel for each $n \in {\mathbb{N}}$, and we have $\sigma_n \to \sigma$.
According to Lemma \[lemma-Psi-n\], there exist a constant $M \in {\mathbb{R}}$ and a sequence $(\Phi_n)_{n \in {\mathbb{N}}} \subset {\rm Lip}_M(H)$ such that for each $n \in {\mathbb{N}}$ the set $D$ is locally $({\rm Id}_H,\Phi_n)$-invariant, and we have $\Phi_n \to {\rm Id}_H$. Therefore, setting $\sigma_n := \sigma \circ \Phi_n$ for each $n \in {\mathbb{N}}$, we have (\[locally-par\]) for some $L \in {\mathbb{R}}$, and applying Lemma \[lemma-tau-sigma-local\] shows that $\sigma_n$ is locally parallel for each $n \in {\mathbb{N}}$.
For our next step, we apply the sup-inf convolution technique from [@Lasry-Lions].
Let $\sigma : H \to {\mathbb{R}}$ be arbitrary.
1. For each $\lambda > 0$ we define $$\begin{aligned}
\sigma_{\lambda} : H \to {\mathbb{R}}, \quad \sigma_{\lambda}(h) := \inf_{g \in H} \bigg( \sigma(g) + \frac{1}{2 \lambda} \| h-g \|^2 \bigg).\end{aligned}$$
2. For each $\mu > 0$ we define $$\begin{aligned}
\sigma^{\mu} : H \to {\mathbb{R}}, \quad \sigma^{\mu}(h) := \sup_{g \in H} \bigg( \sigma(g) - \frac{1}{2 \mu} \| h-g \|^2 \bigg).\end{aligned}$$
Let $\sigma : H \to {\mathbb{R}}$ and $\lambda,\mu > 0$ be arbitrary. A straightforward calculation shows that $$\begin{aligned}
(\sigma_{\lambda})^{\mu}(h) = \sup_{f \in H} \inf_{g \in H} \bigg( \sigma(g) + \frac{1}{2 \lambda} \| f-g \|^2 - \frac{1}{2 \mu} \| f-h \|^2 \bigg) \quad \text{for all $h \in H$.}\end{aligned}$$ Therefore, the function $(\sigma_{\lambda})^{\mu}$ is also called *sup-inf convolution*.
Let $\sigma \in {{\rm F}}(H)$ be arbitrary.
1. For each $\lambda > 0$ we define $\sigma_{\lambda} : H \to H$ as $$\begin{aligned}
\sigma_{\lambda} := \sum_{k \in {\mathbb{N}}} (\sigma_k)_{\lambda} e_k.\end{aligned}$$
2. For each $\mu > 0$ we define $\sigma^{\mu} : H \to H$ as $$\begin{aligned}
\sigma^{\mu} := \sum_{k \in {\mathbb{N}}} (\sigma_k)^{\mu} e_k.\end{aligned}$$
3. For all $\lambda,\mu > 0$ we define $(\sigma_{\lambda})^{\mu} : H \to H$ as $$\begin{aligned}
(\sigma_{\lambda})^{\mu} := \sum_{k \in {\mathbb{N}}} ((\sigma_k)_{\lambda})^{\mu} e_k.\end{aligned}$$
\[lemma-sup-inf-2\] Let $\sigma \in {\rm Lip}_L(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ be arbitrary. Then, for each $\epsilon > 0$ there are $\lambda_0,\mu_0 > 0$ such that for all $\lambda \in (0,\lambda_0]$ and $\mu \in (0,\mu_0]$ with $\mu < \lambda$ we have $$\begin{aligned}
\sup_{h \in H} \| (\sigma_{\lambda})^{\mu}(h) - \sigma(h) \| \leq \epsilon.\end{aligned}$$
This follows from the theorem on pages 260, 261 in [@Lasry-Lions]; in particular relation (12) therein.
\[lemma-Lip-components-pre\] There is a constant $C \in {\mathbb{R}}_+$ such that for all $L \in {\mathbb{R}}_+$ and all $\sigma \in {{\rm Lip}}_L(H)$ we have $\sigma_k \in {{\rm Lip}}_{CL}(H,{\mathbb{R}})$ for each $k \in {\mathbb{N}}$.
Setting $C := 2 {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}$, this is an immediate consequence of Lemma \[lemma-norm-g-star\].
\[lemma-Lip-components\] Let $L \in {\mathbb{R}}_+$ and $\sigma \in {{\rm F}}(H)$ be such that $\sigma_k \in {{\rm Lip}}_L(H,{\mathbb{R}}_+)$ for all $k = 1,\ldots,N$, where $N := \dim \langle \sigma(H) \rangle$. Then we have $\sigma \in {{\rm Lip}}_{NL}(H)$.
For all $h,g \in H$ we have $$\begin{aligned}
\| \sigma(h) - \sigma(g) \| = \bigg\| \sum_{k=1}^N (\sigma_k(h) - \sigma_k(g)) e_k \bigg\| \leq \sum_{k=1}^N |\sigma_k(h) - \sigma_k(g)| \leq NL \| h-g \|,\end{aligned}$$ completing the proof.
\[lemma-sup-inf-1\] There exists a constant $C \in {\mathbb{R}}_+$ such that for all $L \in {\mathbb{R}}_+$, all $\sigma \in {\rm Lip}_L(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ and all $\lambda,\mu > 0$ with $\mu < \lambda$ we have $$\begin{aligned}
(\sigma_{\lambda})^{\mu} \in {\rm Lip}_{CNL}(H) \cap {{\rm F}}(H) \cap C_b^{1,1}(H),\end{aligned}$$ where $N := \dim \langle \sigma(H) \rangle$.
Let $\lambda,\mu > 0$ with $\mu < \lambda$ be arbitrary. For all $k \in {\mathbb{N}}$ with $\sigma_k = 0$ we have $((\sigma_k)_{\lambda})^{\mu} = 0$, showing that $(\sigma_{\lambda})^{\mu} \in {{\rm F}}(H)$. The remaining assertions follow from Lemmas \[lemma-Lip-components-pre\], \[lemma-Lip-components\] and the theorem on pages 260, 261 in [@Lasry-Lions]; in particular relations (11), (13) and (15) therein.
\[lemma-sup-inf-parallel\] Let $\sigma \in {\rm Lip}(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ be a locally parallel function. Then the following statements are true:
1. There exists $\lambda_0 > 0$ such that $\sigma_{\lambda}$ is locally parallel for each $\lambda \in (0,\lambda_0]$.
2. There exists $\mu_0 > 0$ such that $\sigma^{\mu}$ is locally parallel for each $\mu \in (0,\mu_0]$.
3. There exist $\lambda_0,\mu_0 > 0$ such that $(\sigma_{\lambda})^{\mu}$ is locally parallel for all $\lambda \in (0,\lambda_0]$ and $\mu \in (0,\mu_0]$ with $\mu < \lambda$.
Since $\sigma$ is locally parallel, there exists $\epsilon > 0$ such that for all $(h^*,h) \in D$ we have (\[loc-par-prop\]). Furthermore, since $\sigma \in {{\rm B}}(H)$, there exists a finite constant $C > 0$ such that $$\begin{aligned}
\label{sup-inf-sigma-bounded}
\| \sigma(h) \| \leq C \quad \text{for all $h \in H$.}\end{aligned}$$ We define the constants $M,\lambda_0 > 0$ as $$\begin{aligned}
M := 2 C {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}\quad \text{and} \quad \lambda_0 := \frac{\epsilon^2}{8M}.\end{aligned}$$ Let $\lambda \in (0,\lambda_0]$ be arbitrary. We will show that $\sigma_{\lambda}$ is locally parallel. For this purpose, let $(h^*,h) \in D$ be arbitrary. By Assumption \[ass-Schauder-basis\] there exist $\theta \in \{ -1,1 \}$ and $k \in {\mathbb{N}}$ such that $h^* = \theta e_k^*$. Let $g \in H$ with $\| g \| \leq \epsilon / 2$ be arbitrary. We define the function $$\begin{aligned}
\Sigma : H \to {\mathbb{R}}, \quad \Sigma(f) := \sigma_k(f) + \frac{1}{2 \lambda} \| (h-g)-f \|^2.\end{aligned}$$ Then we have $$\begin{aligned}
\label{sup-inf-proof-3}
\Sigma \geq 0 \quad \text{and} \quad \Sigma(h-g) = 0.\end{aligned}$$ Indeed, by (\[loc-par-prop\]) we have $\sigma_k(h-g) = 0$, and hence $\Sigma(h-g) = 0$. In order to show that $\Sigma \geq 0$, let $f \in H$ be arbitrary. We distinguish two cases:
- Suppose that $\| h - f \| \leq \epsilon$. Since $f = h - (h-f)$, by (\[loc-par-prop\]) we have $\sigma_k(f) = 0$, showing $\Sigma(f) \geq 0$.
- Suppose that $\| h - f \| > \epsilon$. Since $\| g \| \leq \epsilon / 2$, by the inverse triangle inequality we obtain $$\begin{aligned}
\| (h-g) - f \| = \| (h-f) - g \| \geq | \, \| h-f \| - \| g \| \, | \geq \epsilon / 2.\end{aligned}$$ Furthermore, by (\[sup-inf-sigma-bounded\]) and Lemma \[lemma-norm-g-star\] we have $$\begin{aligned}
| \sigma_k | = |\langle e_k^*,\sigma \rangle| \leq 2 {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}\| \sigma \| \leq M,\end{aligned}$$ and hence $$\begin{aligned}
\Sigma(f) &= \sigma_k(f) + \frac{1}{2 \lambda} \| (h-g)-f \|^2 \geq -M + \frac{1}{2 \lambda_0} \frac{\epsilon^2}{4} = 0.\end{aligned}$$
Consequently, we have (\[sup-inf-proof-3\]), and thus, we obtain $$\begin{aligned}
\langle h^*, \sigma_{\lambda}(h-g) \rangle = \theta \inf_{f \in H} \Sigma(f) = 0,\end{aligned}$$ showing that $\sigma_{\lambda}$ is locally parallel. This provides the proof of the first statement. The proof of the second statement is analogous, and the third statement follows from the first and the second statement.
\[prop-C-b-1-1\] Let $\sigma \in {\rm Lip}(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ be a locally parallel function. Then there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
(\sigma_n)_{n \in {\mathbb{N}}} \subset {{\rm Lip}}_L(H) \cap {{\rm F}}(H) \cap C_b^{1,1}(H)\end{aligned}$$ such that $\sigma_n$ is locally parallel for each $n \in {\mathbb{N}}$, and we have $\sigma_n \to \sigma$.
This is an immediate consequence of Lemmas \[lemma-sup-inf-2\], \[lemma-sup-inf-1\] and \[lemma-sup-inf-parallel\].
For our last step, we use Moulis’ method, as presented in [@Fry]. For this purpose, we introduce some notation. Let $\varphi \in C^{\infty}({\mathbb{R}},[0,1])$ be a smooth function such that the following conditions are fulfilled:
- We have $\varphi(t) = 1$ for all $t \in (-\frac{1}{2},\frac{1}{2})$.
- We have $\varphi(t) = 0$ for all $t \in {\mathbb{R}}$ with $|t| \geq 1$.
- We have $\varphi'(t) \in [-3,0]$ for all $t \in {\mathbb{R}}_+$.
- We have $\varphi(-t) = \varphi(t)$ for all $t \in {\mathbb{R}}_+$.
Let $\sigma \in {{\rm F}}(H) \cap C_b^{1,1}(H)$ be arbitrary. We fix a sequence $a = (a_n)_{n \in {\mathbb{N}}} \subset (0,\infty)$ and a constant $r > 0$. We define the sequence $(\Sigma_n)_{n \in {\mathbb{N}}}$ of functions $\Sigma_n : H \to H$ as $$\begin{aligned}
\label{def-cap-sigma}
\Sigma_n(h) := \frac{(a_n)^n}{c_n} \int_{E_n} \sigma(h-g) \varphi(a_n \| g \|) dg, \quad h \in H,\end{aligned}$$ where the sequence $(c_n)_{n \in {\mathbb{N}}} \subset (0,\infty)$ is given by $$\begin{aligned}
\label{def-c-n-const}
c_n := \int_{E_n} \varphi(\| g \|) dg.\end{aligned}$$
\[lemma-cap-sigma\] The following statements are true:
1. We have $\Sigma_n \in C^{\infty}(H)$ for each $n \in {\mathbb{N}}$.
2. There is a constant $C \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\label{cap-sigma-n-est}
\max \{ \| \Sigma_n(h) \|, \| D \Sigma_n(h) \|, \| D^2 \Sigma_n(h) \| \} \leq C\end{aligned}$$ for all $n \in {\mathbb{N}}$ and all $h \in H$.
The first statement follows from the definition (\[def-cap-sigma\]). Since $\sigma \in C_b^{1,1}(H)$, there is a constant $C \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\max \{ \| \sigma(h) \| + \| D \sigma(h) \| \} &\leq C \quad \text{for all $h \in H$,}
\\ \| D \sigma(h) - D \sigma(g) \| &\leq C \| h-g \| \quad \text{for all $h,g \in H$.}\end{aligned}$$ Thus, arguing as in [@Fry page 602], we see that (\[cap-sigma-n-est\]) is fulfilled.
Now, we define the sequence $(\hat{\sigma}_n)_{n \in {\mathbb{N}}}$ of functions $\hat{\sigma}_n : H \to H$ as $$\begin{aligned}
\label{def-hat-sigma}
\hat{\sigma}_n(h) := \frac{(b_n)^n}{c_n} \int_{E_n} \Sigma_n(h-g) \varphi(b_n \| g \|) dg, \quad h \in H,\end{aligned}$$ where the sequence $b = (b_n)_{n \in {\mathbb{N}}} \subset (0,\infty)$ is chosen large enough such that $$\begin{aligned}
\label{diff-hat-cap}
\max \{ \| \hat{\sigma}_n(h) - \Sigma_n(h) \|, \| D \hat{\sigma}_n(h) - D \Sigma_n(h) \|, \| D^2 \hat{\sigma}_n(h) - D^2 \Sigma_n(h) \| \} \leq 2^{-n}\end{aligned}$$ for all $n \in {\mathbb{N}}$ and all $h \in H$. Inductively, we define the sequence $(\bar{\sigma}_n)_{n \in {\mathbb{N}}_0}$ of functions $\bar{\sigma}_n : H \to H$ by $$\begin{aligned}
\label{def-sigma-0-bar}
\bar{\sigma}_0 &:= \sigma(0) \quad \text{and}
\\ \label{def-sigma-n-bar} \bar{\sigma}_n &:= \hat{\sigma}_n + \bar{\sigma}_{n-1} \circ \Pi_{n-1} - \hat{\sigma}_n \circ \Pi_{n-1} \quad \text{for all $n \in {\mathbb{N}}$.}\end{aligned}$$
\[lemma-bar-sigma\] The following statements are true:
1. We have $\bar{\sigma}_n|_{E_n} = \bar{\sigma}_{n-1}|_{E_n}$ and $\bar{\sigma}_n|_{E_n} \in C^{\infty}(E_n,H)$ for all $n \in {\mathbb{N}}$.
2. There is a constant $C \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\label{est-bar-sigma}
\max \{ \| \bar{\sigma}_n(h) \|, \| D \bar{\sigma}_n(h) \|, \| D^2 \bar{\sigma}_n(h) \| \} \leq C\end{aligned}$$ for all $n \in {\mathbb{N}}$ and all $h \in E_n$.
The first statement follows from [@Fry page 602]. Using (\[diff-hat-cap\]), we prove inductively as in [@Fry] that $$\begin{aligned}
\max \{ \| \bar{\sigma}_n(h) - \Sigma_n(h) \|, \| D \bar{\sigma}_n(h) - D \Sigma_n(h) \|, \| D^2 \bar{\sigma}_n(h) - D^2 \Sigma_n(h) \| \} \leq 2 (1 - 2^{-n})\end{aligned}$$ for all $n \in {\mathbb{N}}$ and all $h \in H$. Together with Lemma \[lemma-cap-sigma\], this proves the second statement.
Now, we define $\bar{\sigma} : E^{\infty} \to H$ as $$\begin{aligned}
\label{def-sigma-bar}
\bar{\sigma} := \lim_{n \to \infty} \bar{\sigma}_n,\end{aligned}$$ where $E^{\infty} := \bigcup_{n \in {\mathbb{N}}} E_n$. In view of Lemma \[lemma-bar-sigma\], we have $$\begin{aligned}
\label{restr-bar-sigma}
\bar{\sigma}|_{E_n} = \bar{\sigma}_n|_{E_n} \quad \text{for all $n \in {\mathbb{N}}$.}\end{aligned}$$ Now, we define the function $$\begin{aligned}
\label{def-Psi-final}
\Psi : H \to E^{\infty}, \quad \Psi(h) := \sum_{k \in {\mathbb{N}}} \chi_k(h) h_k e_k,\end{aligned}$$ where we refer to the series representation (\[series-h\]) of $h$, and where for each $k \in {\mathbb{N}}$ the function $\chi_k : H \to [0,1]$ is given by $$\begin{aligned}
\label{def-chi}
\chi_k(h) := 1 - \varphi(\| T_k h \|),\end{aligned}$$ where $T_k \in L(H)$ denotes the linear operator $$\begin{aligned}
\label{def-T-k}
T_k := \frac{{{\rm Id}}_H - \Pi_{k-1}}{r}\end{aligned}$$ with $r > 0$ denoting the constant from above.
\[lemma-Psi-local\] The following statements are true:
1. We have $\Psi \in {{\rm Lip}}(H,E^{\infty}) \cap C^{\infty}(H,E^{\infty})$.
2. For each $h \in H$ there exist $n \in {\mathbb{N}}$ and $\delta > 0$ such that $$\begin{aligned}
\label{Psi-local}
\Psi(h-g) \in E_n \quad \text{for all $g \in H$ with $\| g \| \leq \delta$.}\end{aligned}$$
This follows from [@Fine-Approx page 17].
Now, we define the function $$\begin{aligned}
\label{def-sigma-final}
\sigma^{(a,b,r)} : H \to H, \quad \sigma^{(a,b,r)} := \bar{\sigma} \circ \Psi.\end{aligned}$$ Note that we emphasize the dependence on the sequences $a$ and $b$, and on the constant $r$. For two sequences $a = (a_n)_{n \in {\mathbb{N}}} \subset {\mathbb{R}}$ and $b = (b_n)_{n \in {\mathbb{N}}} \subset {\mathbb{R}}$ we agree to write $a \leq_{{\mathbb{N}}} b$ if $a_n \leq b_n$ for all $n \in {\mathbb{N}}$.
\[lemma-approx-C-b-2-a\] Let $\sigma \in {{\rm Lip}}(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ be arbitrary. Then, for each $\epsilon > 0$ there are sequences $a^0, b^0 \in (0,\infty)^{{\mathbb{N}}}$, where $b^0$ is chosen such that (\[diff-hat-cap\]) is fulfilled with $b$ replaced by $b^0$, and a constant $r^0 > 0$ such that for all sequences $a,b \in (0,\infty)^{{\mathbb{N}}}$ with $a^0 \leq_{{\mathbb{N}}} a$ and $b^0 \leq_{{\mathbb{N}}} b$ and all $r > 0$ with $r \leq r^0$ we have $$\begin{aligned}
\sup_{h \in H} \| \sigma^{(a,b,r)}(h) - \sigma(h) \| \leq \epsilon.\end{aligned}$$
This follows from [@Fry Thm. 1] and its proof.
\[lemma-C-b-2-pre\] There exists a constant $C \in {\mathbb{R}}_+$ such that for all $L \in {\mathbb{R}}_+$, all $\sigma \in {\rm Lip}_L(H) \cap {{\rm F}}(H) \cap {{\rm B}}(H)$ and all sequences $a,b \in (0,\infty)^{{\mathbb{N}}}$, where $b$ is chosen such that (\[diff-hat-cap\]) is fulfilled, and every constant $r > 0$ we have $$\begin{aligned}
\sigma^{(a,b,r)} \in {\rm Lip}_{CNL}(H) \cap {{\rm F}}(H) \cap C^{\infty}(H),\end{aligned}$$ where $N := \dim \langle \sigma(H) \rangle$.
Let $a,b$ be arbitrary sequences, where $b$ is chosen such that (\[diff-hat-cap\]) is fulfilled, and let $r > 0$ be arbitrary. By the construction (\[def-cap-sigma\])–(\[def-sigma-final\]), for all $k \in {\mathbb{N}}$ with $\sigma_k = 0$ we have $\sigma_k^{(a,b,r)} = 0$, showing that $\sigma^{(a,b,r)} \in {{\rm F}}(H)$. The remaining assertions follow from Lemmas \[lemma-Lip-components-pre\], \[lemma-Lip-components\] and [@Fry Thm. 1].
Lemma \[lemma-C-b-2-pre\] does not ensure that $\sigma^{(a,b,r)} \in C_b^2(H)$; that is, it remains to show that the second order derivative is bounded. For this purpose, we prepare some auxiliary results. For the next two results, we fix a constant $r > 0$. Note that the functions $\chi_k$, $k \in {\mathbb{N}}$ defined in (\[def-chi\]) and $\Psi$ defined in (\[def-Psi-final\]) depend on the choice of $r$.
\[lemma-chi-est\] The following statements are true:
1. We have $\chi_k \in C^{\infty}(H,{\mathbb{R}})$ for each $k \in {\mathbb{N}}$.
2. There is a constant $C \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\label{est-chi-k}
\max \{ \| \chi_k(h) \|, r \| D \chi_k(h) \|, r^2 \| D^2 \chi_k(h) \| \} \leq C\end{aligned}$$ for all $k \in {\mathbb{N}}$ and all $h \in H$.
Let $U \subset H$ be the open set $U := \{ \| \cdot \| > \frac{1}{4} \}$. For the norm function $\eta : U \to {\mathbb{R}}_+$ given by $\eta(h) := \| h \|$ we have $\eta \in C^{\infty}(U,{\mathbb{R}})$ with derivatives $$\begin{aligned}
D\eta(h)g &= \frac{\langle h,g \rangle}{\| h \|}, \quad \text{$h \in U$ and $g \in H$,}
\\ D^2 \eta(h)(g,f) &= \frac{\langle g,f \rangle}{\| h \|} - \frac{\langle h,g \rangle \langle h,f \rangle}{\| h \|^3}, \quad \text{$h \in U$ and $g,f \in H$.} \end{aligned}$$ Therefore, for all $h \in U$ we obtain $$\begin{aligned}
\label{norm-est}
\| D \eta(h) \| \leq 1 \quad \text{and} \quad
\| D^2 \eta(h) \| &\leq \frac{2}{\| h \|} \leq 8.\end{aligned}$$ We define the constant $L \in {\mathbb{R}}_+$ as $$\begin{aligned}
L := 1 + {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}.\end{aligned}$$ Then, by the definition (\[def-T-k\]) of $T_k$ we have $$\begin{aligned}
\label{T-k-est}
\| T_k \| \leq L/r \quad \text{for all $k \in {\mathbb{N}}$.} \end{aligned}$$ There is a constant $M \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\max \{ \varphi(t), \varphi'(t), \varphi''(t) \} \leq M \quad \text{for all $t \in {\mathbb{R}}$.} \end{aligned}$$ Now, we define the constant $C \in {\mathbb{R}}_+$ as $$\begin{aligned}
C := \max \{ 1, ML, ML^2 + 8 M^2 L^2 \}.\end{aligned}$$ Let $k \in {\mathbb{N}}$ be arbitrary. By the definition (\[def-chi\]) of $\chi_k$ we have $$\begin{aligned}
\chi_k = 1 - \varphi \circ \eta \circ T_k,\end{aligned}$$ and hence $$\begin{aligned}
\| \chi_k(h) \| \leq 1 \leq C \quad \text{for all $h \in H$.} \end{aligned}$$ We define the open sets $U_k,V_k \subset H$ as $$\begin{aligned}
U_k := \{ \| T_k \| > 1/4 \} \quad \text{and} \quad V_k := \{ \| T_k \| < 1/2 \}. \end{aligned}$$ Then we have $H = U_k \cup V_k$ and $\chi_k(h) = 0$ for all $h \in V_k$. This shows $\chi_k \in C^{\infty}(H,{\mathbb{R}})$, proving the first statement, and regarding the second statement, it suffices to show (\[est-chi-k\]) for all $k \in {\mathbb{N}}$ and all $h \in U_k$. Let $k \in {\mathbb{N}}$ and all $h \in U_k$ be arbitrary. By (\[norm-est\]) and (\[T-k-est\]) we obtain $$\begin{aligned}
\| D(\eta \circ T_k)(h) \| &\leq \| D\eta(T_k h) \| \, \| DT_k h \| \leq \| T_k \| \leq L/r,
\\ \| D^2(\eta \circ T_k)(h) \| &\leq \| D^2 \eta(T_k h) \| \, \| D T_k h \|^2 + \| D\eta (T_k h) \|^2 \| D^2 T_k h \| \leq 8 L^2 / r^2,\end{aligned}$$ and hence $$\begin{aligned}
\| D \chi_k(h) \| &= \| D(\varphi \circ \eta \circ T_k)(h) \| \leq \| D \varphi (\eta(T_k h)) \| \, \| D (\eta \circ T_k)(h) \| \leq ML/r \leq C/r,
\\ \| D^2 \chi_k(h) \| &= \| D^2(\varphi \circ \eta \circ T_k)(h) \| \leq \| D^2 \varphi(\| T_k h \|) \| \, \| D(\eta \circ T_k)(h) \|^2
\\ &\quad + \| D \varphi(\| T_k h \|) \|^2 \| D^2 (\eta \circ T_k)(k) \| \leq M L^2 / r^2 + 8 M^2 L^2 / r^2 \leq C / r^2,\end{aligned}$$ completing the proof.
The following auxiliary result extends Fact 7 in [@Fine-Approx].
\[lemma-C-b-2-part-2\] There exists a constant $M \in {\mathbb{R}}_+$ such that $$\begin{aligned}
\label{est-Psi-final}
\max \{ \| D \Psi(h) \|, r \| D^2 \Psi(h) \| \} \leq M \quad \text{for all $h \in H$.}\end{aligned}$$
Let $C \in {\mathbb{R}}_+$ be the constant from Lemma \[lemma-chi-est\]. We define the constant $M \in {\mathbb{R}}_+$ as $$\begin{aligned}
M := 3 {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C.\end{aligned}$$ Let $h \in H$ be arbitrary. Noting that $T_k h \to 0$ for $k \to \infty$, let $n \in {\mathbb{N}}$ be the smallest index such that $$\begin{aligned}
\label{norm-T-1}
\| T_n h \| \leq 1. \end{aligned}$$ Then we have $\| T_k h \| > 1$ for all $k = 1,\ldots,n-1$. By the continuity of the linear operators $T_1,\ldots,T_{n-1}$, there exists $\delta > 0$ such that $$\begin{aligned}
\| T_k (h-g) \| > 1 \quad \text{for all $k = 1,\ldots,n-1$ and all $g \in H$ with $\| g \| \leq \delta$.}\end{aligned}$$ By the definition (\[def-chi\]) of $\chi_k$ we obtain $$\begin{aligned}
\chi_k(h-g) = 1 \quad \text{for all $k = 1,\ldots,n-1$ and all $g \in H$ with $\| g \| \leq \delta$,}\end{aligned}$$ and it follows that $$\begin{aligned}
\label{chi-diff-zero}
D \chi_k(h) = 0 \quad \text{and} \quad D^2 \chi_k(h) = 0 \quad \text{for all $k = 1,\ldots,n-1$.}\end{aligned}$$ Furthermore, by the definition (\[def-Psi-final\]) of $\Psi$ we have $$\begin{aligned}
D \Psi(h) &= \sum_{k \in {\mathbb{N}}} D \chi_k(h) \langle e_k^*,h \rangle e_k + \sum_{k \in {\mathbb{N}}} \chi_k(h) \langle e_k^*, \cdot \rangle e_k,
\\ D^2 \Psi(h) &= \sum_{k \in {\mathbb{N}}} D^2 \chi_k(h) \langle e_k^*,h \rangle e_k + 2 \sum_{k \in {\mathbb{N}}} D \chi_k(h) \langle e_k^*,\cdot \rangle e_k,\end{aligned}$$ and hence, by (\[chi-diff-zero\]), Lemmas \[lemma-norm-g-star\], \[lemma-chi-est\] and (\[norm-T-1\]) we obtain $$\begin{aligned}
\| D \Psi(h) \| &\leq \bigg\| \sum_{k \geq n} D \chi_k(h) \langle e_k^*,h \rangle e_k \bigg\| + \bigg\| \sum_{k \in {\mathbb{N}}} \chi_k(h) \langle e_k^*,\cdot \rangle e_k \bigg\|
\\ &\leq {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C/r \bigg\| \sum_{k \geq n} \langle e_k^*,h \rangle e_k \bigg\| + {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C \bigg\| \sum_{k \in {\mathbb{N}}} \langle e_k^*,\cdot \rangle e_k \bigg\|
\\ &\leq {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C \| T_n h \| + {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C \leq M,\end{aligned}$$ and similarly $$\begin{aligned}
\| D^2 \Psi(h) \| &\leq \bigg\| \sum_{k \geq n} D^2 \chi_k(h) \langle e_k^*,h \rangle e_k \bigg\| + 2 \bigg\| \sum_{h \in {\mathbb{N}}} D \chi_k(h) \langle e_k^*,\cdot \rangle e_k \bigg\|
\\ &\leq {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C / r^2 \bigg\| \sum_{k \geq n} \langle e_k^*,h \rangle e_k \bigg\| + 2 {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C / r \bigg\| \sum_{k \in {\mathbb{N}}} \langle e_k^*,\cdot \rangle e_k \bigg\|
\\ &\leq {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C/r \| T_n h \| + 2 {{\rm ubc}(\{ e_l \}_{l \in {\mathbb{N}}})}C/r \leq M/r,\end{aligned}$$ completing the proof.
\[lemma-approx-C-b-2-b\] For all $\sigma \in {{\rm F}}(H) \cap C_b^{1,1}(H)$ and all sequences $a,b \in (0,\infty)^{{\mathbb{N}}}$, where $b$ is chosen such that (\[diff-hat-cap\]) is fulfilled, and every constant $r > 0$ we have $\sigma^{(a,b,r)} \in C_b^2(H)$.
By Lemmas \[lemma-bar-sigma\] and \[lemma-C-b-2-part-2\] there exist constants $C,M \in {\mathbb{R}}_+$ such that we have (\[est-bar-sigma\]) and (\[est-Psi-final\]). Let $h \in H$ be arbitrary. By Lemma \[lemma-Psi-local\] there exist $n \in {\mathbb{N}}$ and $\delta > 0$ such that we have (\[Psi-local\]). Furthermore, by the definition (\[def-sigma-final\]) of $\sigma^{(a,b,r)}$ and relation (\[restr-bar-sigma\]) we have $$\begin{aligned}
\sigma^{(a,b,r)}(h-g) = \bar{\sigma}_n(\Psi(h-g)) \quad \text{for all $g \in H$ with $\| g \| \leq \delta$.}\end{aligned}$$ Therefore, and by estimates (\[est-bar-sigma\]) and (\[est-Psi-final\]), we obtain $$\begin{aligned}
\| \sigma^{(a,b,r)}(h) \| &= \| \bar{\sigma}_n(\Psi(h)) \| \leq C,
\\ \| D \sigma^{(a,b,r)}(h) \| &= \| D (\bar{\sigma}_n \circ \Psi)(h) \| \leq \| D \bar{\sigma}_n(\Psi(h)) \| \, \| D \Psi(h) \| \leq CM,
\\ \| D^2 \sigma^{(a,b,r)}(h) \| &= \| D^2 (\bar{\sigma}_n \circ \Psi)(h) \| \leq \| D^2 \bar{\sigma}_n (\Psi(h)) \| \, \| D \Psi(h) \|^2
\\ &\quad + \| D \bar{\sigma}_n (\Psi(h)) \|^2 \, \| D^2 \Psi(h) \| \leq CM^2 + C^2 M/r,\end{aligned}$$ finishing the proof.
\[lemma-approx-C-b-2-c\] Let $\sigma \in {{\rm F}}(H) \cap C_b^{1,1}(H)$ be a locally parallel function. Then, there exist a sequences $a^0,b^0 \in (0,\infty)^{{\mathbb{N}}}$, where $b^0$ is chosen such that (\[diff-hat-cap\]) is fulfilled with $b$ replaced by $b^0$, such that for all sequences $a,b \in (0,\infty)^{{\mathbb{N}}}$ with $a^0 \leq_{{\mathbb{N}}} a$ and $b^0 \leq_{{\mathbb{N}}} b$ and every constant $r > 0$ the function $\sigma^{(a,b,r)} : H \to H$ is weakly locally parallel.
Since $\sigma$ is locally parallel, there exists $\epsilon > 0$ such that for all $(h^*,h) \in D$ we have (\[loc-par-prop\]). Let $a^0 \in (0,\infty)^{{\mathbb{N}}}$ be the sequence given by $a_n^0 := 2 / \epsilon$ for each $n \in {\mathbb{N}}$. Furthermore, we choose $b^0 \in (0,\infty)^{{\mathbb{N}}}$ such that $b_n^0 \geq 4 / \epsilon$ for each $n \in {\mathbb{N}}$, and condition (\[diff-hat-cap\]) is fulfilled with $b$ replaced by $b^0$. Let $a,b \in (0,\infty)^{{\mathbb{N}}}$ be arbitrary sequences with $a^0 \leq_{{\mathbb{N}}} a$ and $b^0 \leq_{{\mathbb{N}}} b$, and let $r > 0$ be an arbitrary constant. First, we will show that for all $n \in {\mathbb{N}}$ and all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{cap-sigma-par}
\langle h^*,\Sigma_n(h-g) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon / 2$.}\end{aligned}$$ For this purpose, let $g \in H$ with $\| g \| \leq \epsilon / 2$ be arbitrary. By the definition (\[def-cap-sigma\]) of $\Sigma_n$, relation (\[loc-par-prop\]), and since ${{\rm supp}}(\varphi) \subset [-1,1]$ and $a_n \geq 2 / \epsilon$, we obtain $$\begin{aligned}
\langle h^*,\Sigma_n(h-g) \rangle &= \frac{(a_n)^n}{c_n} \int_{E_n} \langle h^*, \sigma(h-g-f) \rangle \varphi(a_n \| f \|) df
\\ &= \frac{(a_n)^n}{c_n} \int_{E_n} \underbrace{\langle h^*, \sigma(h-(g+f)) \rangle}_{= 0} \varphi(a_n \| f \|) {\mathbbm{1}}_{\{ \| f \| \leq \epsilon / 2 \}} df
\\ &\quad + \frac{(a_n)^n}{c_n} \int_{E_n} \langle g^*, \sigma(h-(g+f)) \rangle \underbrace{\varphi(a_n \| f \|)}_{= 0} {\mathbbm{1}}_{\{ \| f \| > \epsilon / 2 \}} df = 0,\end{aligned}$$ showing (\[cap-sigma-par\]). Noting the definition (\[def-hat-sigma\]) of $\hat{\sigma}_n$, relation (\[cap-sigma-par\]) and that $b_n \geq 4 / \epsilon$, analogously we show that for all $n \in {\mathbb{N}}$ and all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{sigma-hat-par}
\langle h^*,\hat{\sigma}_n(h-g) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \epsilon / 4$.}\end{aligned}$$ Next, we set $M := {{\rm bc}(\{ e_l \}_{l \in {\mathbb{N}}})}\geq 1$. By induction, we will show that for all $n \in {\mathbb{N}}_0$ and all $(h^*,h) \in D$ we have $$\begin{aligned}
\label{sigma-bar-par}
\langle h^*,\bar{\sigma}_n(h-g) \rangle = 0 \quad \text{for all $g \in H$ with $\| g \| \leq \frac{\epsilon}{4 M^n}$.}\end{aligned}$$ Relation (\[sigma-bar-par\]) holds true for $n = 0$. Indeed, since $0 \in K$ and $0 \leq_K h$, by Assumption \[ass-D\] we also have $(h^*,0) \in D$. Therefore, by the definition (\[def-sigma-0-bar\]) of $\bar{\sigma}_0$, and since $\sigma$ is parallel, for all $g \in H$ with $\| g \| \leq \epsilon / 4$ we obtain $$\begin{aligned}
\langle h^*,\bar{\sigma}_0(h-g) \rangle = \langle h^*,\sigma(0) \rangle = 0.\end{aligned}$$ For the induction step, suppose that (\[sigma-bar-par\]) is satisfied for $n-1$. Since $\Pi_{n-1} h \in K$ and $\Pi_{n-1} h \leq_K h$, by Assumption \[ass-D\] we also have $(h^*,\Pi_{n-1} h) \in D$. Let $g \in H$ with $\| g \| \leq \frac{\epsilon}{4 M^n}$ be arbitrary. Then, we have $$\begin{aligned}
\| g \| \leq \frac{\epsilon}{4} \quad \text{and} \quad \| \Pi_{n-1} g \| \leq \frac{\epsilon}{4 M^{n-1}} \leq \frac{\epsilon}{4},\end{aligned}$$ and hence, by the definition (\[def-sigma-n-bar\]) of $\bar{\sigma}_n$, relation (\[sigma-hat-par\]) and the induction hypothesis, we obtain $$\begin{aligned}
&\langle h^*,\bar{\sigma}_n(h-g) \rangle
\\ &= \langle h^*,\hat{\sigma}_n(h-g) \rangle + \langle h^*,\bar{\sigma}_{n-1}(\Pi_{n-1} (h-g)) \rangle + \langle h^*,\hat{\sigma}_n(\Pi_{n-1} (h-g)) \rangle
\\ &= \langle h^*,\hat{\sigma}_n(h-g) \rangle + \langle h^*,\bar{\sigma}_{n-1}(\Pi_{n-1} h - \Pi_{n-1} g) \rangle + \langle h^*,\hat{\sigma}_n(\Pi_{n-1} h - \Pi_{n-1} g) \rangle = 0,\end{aligned}$$ proving (\[sigma-bar-par\]). Now, let $(h^*,h) \in D$ be arbitrary. By the definition (\[def-Psi-final\]) of $\Psi$ we have $\Psi(h) \in K$ and $\Psi(h) \leq_K h$, and hence, by Assumption \[ass-D\] we also have $(h^*,\Psi(h)) \in D$. By Lemma \[lemma-Psi-local\] there exist $n \in {\mathbb{N}}$ and $\delta > 0$ such that we have (\[Psi-local\]), and there exists $C > 0$ such that $$\begin{aligned}
\| \Psi(h-g) - \Psi(h) \| \leq C \| g \| \quad \text{for all $g \in H$.}\end{aligned}$$ We define $\eta > 0$ as $$\begin{aligned}
\eta := \min \bigg\{ \delta, \frac{\epsilon}{4 M^n C} \bigg\}.\end{aligned}$$ Let $g \in H$ with $\| g \| \leq \eta$ be arbitrary. Then we have $$\begin{aligned}
\| \Psi(h-g) - \Psi(h) \| \leq \frac{\epsilon}{4 M^n},\end{aligned}$$ and hence, by the definition (\[def-sigma-final\]) of $\sigma^{(a,b,r)}$, relation (\[restr-bar-sigma\]) and (\[sigma-bar-par\]) we obtain $$\begin{aligned}
\langle h^*,\sigma^{(a,b,r)}(h-g) \rangle &= \langle h^*,\bar{\sigma}(\Psi(h-g)) \rangle = \langle h^*,\bar{\sigma}_n(\Psi(h-g)) \rangle
\\ &= \langle h^*,\bar{\sigma}_n(\Psi(h) - (\Psi(h-g) - \Psi(h)) ) \rangle = 0,\end{aligned}$$ showing that $\sigma^{(a,b,r)}$ is weakly locally parallel.
\[prop-C-b-2\] Let $\sigma \in {{\rm F}}(H) \cap C_b^{1,1}(H)$ be a locally parallel function. Then there are a constant $L \in {\mathbb{R}}_+$ and a sequence $$\begin{aligned}
(\sigma_n)_{n \in {\mathbb{N}}} \subset {{\rm Lip}}_L(H) \cap {{\rm F}}(H) \cap C_b^2(H)\end{aligned}$$ such that $\sigma_n$ is weakly locally parallel for each $n \in {\mathbb{N}}$, and we have $\sigma_n \to \sigma$.
This is an immediate consequence of Lemmas \[lemma-approx-C-b-2-a\], \[lemma-C-b-2-pre\], \[lemma-approx-C-b-2-b\] and \[lemma-approx-C-b-2-c\].
[20]{}
Azagra, D., Ferrera, J., López-Mesas, F., Rangel, Y. (2007): Smooth approximation of Lipschitz functions on Riemannian manifolds. *Journal of Mathematical Analysis and Applications* [**124**]{}(1), 47–66.
Azagra, D., Gil, J. G., Jaramillo, J. A., Lovo, M. (2005): $C^1$-fine approximation of functions on Banach spaces with unconditional basis. *The Quarterly Journal of Mathematics* [**56**]{}(1), 13–20.
Bari, N. K. (1951): Biorthogonal systems and bases in Hilbert Space. *Uch. Zap. Mosk. Gos. Univ.* [**148**]{}, 69–107.
Bj[ö]{}rk, T., Land[é]{}n, C. (2002): On the construction of finite dimensional realizations for nonlinear forward rate models. *Finance and Stochastics* [**6**]{}(3), 303–331.
Bj[ö]{}rk, T., Svensson, L. (2001): On the existence of finite dimensional realizations for nonlinear forward rate models. *Mathematical Finance* [**11**]{}(2), 205–243.
Da Prato, G., Zabczyk, J. (1992): *Stochastic equations in infinite dimensions.* Cambridge University Press, New York.
Fabian, M., Habala, P., Hájek, P., Santalucía, V. M., Pelant, J., Zizler, V. (2001): *Functional analysis and infinite dimensional geometry.* Springer, New York.
Filipović, D. (2000): Invariant manifolds for weak solutions to stochastic equations. *Probability Theory and Related Fields* [**118**]{}(3), 323–341.
Filipović, D., Tappe, S., Teichmann, J. (2010): Jump-diffusions in Hilbert spaces: Existence, stability and numerics. *Stochastics* [**82**]{}(5), 475–520.
Filipović, D., Tappe, S., Teichmann, J. (2010): Term structure models driven by Wiener processes and Poisson measures: Existence and positivity. *SIAM Journal on Financial Mathematics* [**1**]{}(1), 523–554.
Filipović, D., Tappe, S., Teichmann, J. (2014): Invariant manifolds with boundary for jump-diffusions. *Electronic Journal of Probability* [**19**]{}(51), 1–28.
Filipović, D., Tappe, S., Teichmann, J.: Stochastic partial differential equations and submanifolds in Hilbert spaces. Appendix of *Invariant manifolds with boundary for jump-diffusions*, (2014). [(http://arxiv.org/abs/1202.1076v2)]{}
Filipović, D., Teichmann, J. (2003): Existence of invariant manifolds for stochastic equations in infinite dimension. *Journal of Functional Analysis* [**197**]{}(2), 398–432.
Filipović, D., Teichmann, J. (2004): On the geometry of the term structure of interest rates. *Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences* [**460**]{}(2041), 129–167.
Fry, R. (2006): Approximation by $C^p$-smooth Lipschitz functions on Banach spaces. *Journal of Mathematical Analysis and Applications* [**315**]{}(2), 599–605.
Gawarecki, L., Mandrekar, V. (2011): *Stochastic differential equations in infinite dimensions with applications to SPDEs.* Springer, Berlin.
Hájek, P., Johanis, M. (2009): Uniformly Gâteaux smooth approximation on $c_0(\Gamma)$. *Journal of Mathematical Analysis and Applications* [**350**]{}(2), 623–629.
Hájek, P., Johanis, M. (2010): Smooth approximations. *Journal of Functional Analysis* [**259**]{}(3), 561–582.
Jachimiak, W. (1997): A note on invariance for semilinear differential equations. *Bulletin of the Polish Academy of Sciences* [**45**]{}(2).
Jachimiak, W. (1998): Stochastic invariance in infinite dimension. *Polish Academy of Sciences*.
Jacod, J., Shiryaev, A. N. (2003): *Limit theorems for stochastic processes*. Springer, Berlin.
Johanis, M. (2003): Approximation of Lipschitz mappings. *Serdica Mathematical Journal* [**29**]{}(2), 141–148.
Lasry, J. M., Lions, P. L. (1986): A remark on regularization in Hilbert spaces. *Israel Journal of Mathematics* [**55**]{}(3), 257–266.
Liu, W., Röckner, M. (2015): *Stochastic partial differential equations: An introduction* Springer, Heidelberg.
Marinelli, C., Prévôt, C., Röckner, M. (2010): Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. *Journal of Functional Analysis* [**258**]{}(2), 616–649.
Milian, A. (2002): Comparison theorems for stochastic evolution equations. *Stochastics and Stochastic Reports* [**72**]{}(1–2), 79–108.
Moulis, N. (1971): Approximation de fonctions différentiables sur certains espaces de Banach. *Ann. Inst. Fourier (Grenoble)* [**21**]{}(4), 293–345.
Nakayama, T. (2004): Support theorem for mild solutions of SDE’s in Hilbert spaces. *J. Math. Sci. Univ. Tokyo* [**11**]{}(3), 245–311.
Nakayama, T. (2004): Viability Theorem for SPDE’s including HJM framework. *J. Math. Sci. Univ. Tokyo* [**11**]{}(3), 313–324.
Pazy, A. (1983): *Semigroups of linear operators and applications to partial differential equations.* Springer, New York.
Peszat, S., Zabczyk, J. (2007): *Stochastic partial differential equations with Lévy noise*. Cambridge University Press, Cambridge.
Platen, E., Tappe, S. (2015): Real-world forward rate dynamics with affine realizations. *Stochastic Analysis and Applications* [**33**]{}(4), 573–608.
Prévôt, C., Röckner, M. (2007): *A concise course on stochastic partial differential equations.* Springer, Berlin.
Tappe, S. (2010): An alternative approach on the existence of affine realizations for HJM term structure models. *Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences* [**466**]{}(2122), 3033–3060.
Tappe, S. (2012): Existence of affine realizations for Lévy term structure models. *Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences* [**468**]{} (2147), 3685–3704.
Tappe, S. (2012): Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures. *International Journal of Stochastic Analysis*, vol. [**2012**]{}, Article ID 236327, 24 pages.
Tappe, S. (2015): Existence of affine realizations for stochastic partial differential equations driven by Lévy processes. *Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences* [**471**]{}(2178).
Tappe, S. (2016): Affine realizations with affine state processes for stochastic partial differential equations. *Stochastic Processes and Their Applications* [**126**]{}(7), 2062–2091.
[^1]: I am grateful to an anonymous referee for the careful study of my paper and the valuable comments and suggestions.
|
---
abstract: 'A calculation of the mass spectrum in the baryon decuplet sector is presented using the method of QCD sum rules. Sum rules are derived for states of spin-parity $3/2\pm$ and $1/2+$ using both the conventional method and a parity-projection method. The predictive ability of the sum rules is explored by a Monte-Carlo based analysis procedure in which the three phenomenological parameters (mass, coupling, threshold) are treated as free parameters and fitted simultaneously. Taken together, the results give an improved determination of the mass spectrum in this sector from the perspective of non-perturbative QCD.'
author:
- 'Frank X. Lee'
title: Excited decuplet baryons from QCD sum rules
---
Introduction {#intro}
============
A goal of hadronic physics is to understand the baryon spectrum from QCD, the underlying theory of the strong interaction. Experimentally, the push is fueled by the physics program at JLab, and other accelerator facilities. To this end, the time-honored QCD sum rule method [@SVZ79] adds a useful theoretical tool. It is a non-perturbative approach to QCD that reveals a direct connection between hadronic observables and the QCD vacuum structure via a few universal parameters called vacuum condensates (vacuum expectation values of QCD local operators), the most important of which are the quark condensate, the mixed condensate and the gluon condensate. The method is analytical, physically transparent, and has minimal model dependence with well-understood limitations inherent in the operator product expansion (OPE). It provides a complementary view of the same non-perturbative physics to the numerical approach of lattice QCD. The method was applied to the decuplet sector not long after it was introduced [@Bely82; @Bely83; @RRY82; @RRY85; @Chung84]. However, only limited attention has been paid to this sector since then. In Ref. [@Derek90], a systematic study was made that incudes some $\Delta$ states. Later, progress came in the analysis of QCD sum rules by Monte-Carlo sampling of errors [@Derek96]. It was applied to the study of spin-3/2 branch of baryon decuplet [@Lee98], magnetic moments [@Lee98b; @Lee98c] and other spin-3/2 excited baryons [@Lee02].
From a theoretical standpoint, the main issue is how to isolate sum rules that couple to a particular spin-parity, since the interpolating fields used to construct the spin-3/2 states contain both spin-3/2 and spin-1/2 components, and couple to both parities. Separation of the spin-3/2 and spin-1/2 components can be achieved by examining the Dirac structures upon which the sum rules are based, as done in Ref. [@Derek90]. However, the parities are still mixed. In the pole-plus-continuum ansatz of the method, isolation of a particular parity relies on strong cancellations in the excited part of the spectrum. Without such cancellations, the sum rules usually suffer contaminations from higher states, a problem that renders them unstable. A solution to separate the parities exactly was proposed in Ref. [@Jido97] which showed improved convergence in the octet baryon sector. In this work, we apply the method to the decuplet sector, together with the conventional method. The goal is to identify sum rules that couple strongly to the four branches of the decuplet with spin-parity $3/2+$, $3/2-$, $1/2+$, and $1/2-$. In addition to the Delta states, we also examine states that contain the strange quark, which have not been studied in detail. In our calculation, we consistently include operators up to dimension eight, first order strange quark mass corrections, flavor symmetry breaking of the strange quark condensates, anomalous dimension corrections, and possible factorization violation of the four-quark condensate. Furthermore, we try to assess quantitatively the errors in the phenomenological parameters, using the Monte-Carlo based analysis procedure. This procedure incorporates all uncertainties in the QCD input parameters simultaneously, and translates them into uncertainties in the phenomenological parameters, with careful regard to OPE convergence and ground state dominance.
Method {#elem}
======
In the conventional method, the starting point is the time-ordered two-point correlation function in the QCD vacuum $$\Pi(p)=i\int d^4x\; e^{ip\cdot x}\,\langle 0\,|\,
T\{\;\eta(x)\, \bar{\eta}(0)\;\}\,|\,0\rangle. \label{cf2pt-old}$$ where $\eta$ is the interpolating field with the quantum numbers of the baryon under consideration. Assuming SU(2) isospin symmetry in the u and d quarks, we consider the most general current of spin 3/2 and isospin 3/2 for the $\Delta$, $$\eta^{\Delta}_{\mu}(x)=\epsilon^{abc}[
u^{aT}(x)C\sigma_{\mu} u^b(x)]u^c(x)$$ Here $C$ is the charge conjugation operator and the superscript $T$ means transpose. The antisymmetric $\epsilon^{abc}$ and sum over color ensures a color-singlet state. For the other members of the decuplet, we consider, omitting the explicit $x$-dependence $$\eta^{\Sigma^*}_{\mu}=\sqrt{1/3}\epsilon^{abc}\{2[
u^{aT}C\sigma_{\mu} s^b]u^c
+ [u^{aT}C\sigma_{\mu} u^b]s^c \},$$ $$\eta^{\Xi^*}_{\mu}=\sqrt{1/3}\epsilon^{abc}\{2[
s^{aT}C\sigma_{\mu} u^b]s^c
+ [s^{aT}C\sigma_{\mu} s^b]u^c \},$$ $$\eta^{\Omega^-}_{\mu}=\epsilon^{abc}[
s^{aT}C\sigma_{\mu} s^b]s^c.$$ A baryon interpolating field couples to both the ground state and the excited states of a baryon, and to both parities. That leads to sum rules that contain mixed parity states. The parities can be separated by considering the ‘forward-propagating’ version of the correlation function [@Jido97], $$\Pi(p)=i\int d^4x\; e^{ip\cdot x}\,\theta(x_0)\langle 0\,|\,
T\{\;\eta(x)\, \bar{\eta}(0)\;\}\,|\,0\rangle. \label{cf2pt}$$ where the only difference between the above equation and the conventional two-point correlation function in Eq. (\[cf2pt-old\]) is the step function $\theta(x_0)$. Under this condition, the phenomenological representation of the the imaginary part in the rest frame ($\vec{p}=0$) can be written as $$\begin{aligned}
\mbox{Im}\,\Pi(p_0) &=&
\sum_n \left[ (\lambda_n^+)^2 \frac{\gamma_0+1}{2}\delta(p_0-m_n^+)
\nonumber \right. \\
&+& \left. (\lambda_n^-)^2 \frac{\gamma_0-1}{2}\,\delta(p_0-m_n^-) \right]
\nonumber \\
&\equiv& \gamma_0 A(p_0) + B(p_0)
\label{Img_Pi}\end{aligned}$$ where $\lambda^\pm$ and $m^\pm$ are the couplings and masses of the states involved, separated by parity. The functions in the last step are defined by $$A(p_0) =\, \frac{1}{2}\sum_n\,[(\lambda_n^+)^2
\delta(p_0-m_n^+)\,+\,(\lambda_n^-)^2\delta(p_0-m_n^-)],
\label{A}$$ and $$B(p_0) =\, \frac{1}{2}\sum_n\,[(\lambda_n^+)^2
\delta(p_0-m_n^+)\,-\,(\lambda_n^-)^2\delta(p_0-m_n^-)].
\label{B}$$ By considering the combinations $A(p_0)+B(p_0)$ and $A(p_0)-B(p_0)$, positive-parity and negative-parity states are separated exactly, respectively. This method is reminiscent of the one used in lattice QCD under Dirichlet boundary conditions in the time direction [@Lee06].
The construction of sum rules in Borel space proceeds by taking the integral with a weighting factor and truncating all excited states starting at a certain threshold. For positive-parity states, $$\int^{w_+}_0 \left[
A(p_0)+B(p_0) \right] e^{-p_0^2/M^2}\,dp_0
= \lambda_+^2e^{-m_+^2/M^2}
\label{G_sum_rule1}$$ and for positive-parity states, $$\int^{w_-}_0 \left[
A(p_0)-B(p_0) \right] e^{-p_0^2/M^2}\,dp_0
= \lambda_-^2e^{-m_-^2/M^2}.
\label{G_sum_rule2}$$ To separate out the spin components, we need the full Dirac structure of the correlation function for spin-3/2 interpolating fields [@Derek90; @Lee98], $$\begin{aligned}
\Pi_{\mu\nu}(p) &=&
\lambda^2_{3/2}\; \left\{ -g_{\mu\nu}\hat{p}
+{1\over 3}\gamma_\mu \gamma_\nu \hat{p}
\right. \nonumber \\ &-& \left.
{1\over 3}(\gamma_\mu p_\nu-\gamma_\nu p_\mu)
+ {2\over 3}{p_\mu p_\nu \over M^2_{3/2}}
\right. \nonumber \\ &\pm& \left.
M_{3/2} \left[ g_{\mu\nu}
- {1\over 3}\gamma_\mu\gamma_\nu
\right. \right. \nonumber \\ &+& \left. \left.
{1\over 3 M^2_{3/2}}(\gamma_\mu p_\nu-\gamma_\nu p_\mu)\hat{p}
- {2\over 3}{p_\mu p_\nu \over M^2_{3/2}} \right]
\right\} \nonumber \\ &+&
\alpha^2_{1/2}\; \left\{ {16\over M^2_{1/2}} p_\mu p_\nu\hat{p}
-\gamma_\mu \gamma_\nu \hat{p}
\right. \nonumber \\ &-& \left.
6{ \over }(\gamma_\mu p_\nu+\gamma_\nu p_\mu)
+4(\gamma_\mu p_\nu-\gamma_\nu p_\mu)
\right. \nonumber \\ &\pm & \left.
M_{1/2} \left[ -\gamma_\mu\gamma_\nu
- {8 p_\mu p_\nu \over M^2_{1/2}}
\right. \right. \nonumber \\ &+& \left. \left.
{4\over M^2_{1/2}}(\gamma_\mu p_\nu-\gamma_\nu p_\mu)\hat{p}
\right]\,\right\}
+ \cdots,
\label{correlator33}\end{aligned}$$ where the ellipses denote excited state contributions, and $\hat{p}$ denotes $p_\mu \gamma^\mu$. The upper/lower sign corresponds to the explicit positive/negative parity when a state contributes. The entire structure is divided into two parts: one for spin-3/2 with coupling $\lambda^2_{3/2}$, and the other spin-1/2 with coupling $\alpha^2_{1/2}$. The structures are chosen to be effectively orthogonal to each other. Only six of the structures are independent. The two sum rules from $g_{\mu\nu}\hat{p}$ and $g_{\mu\nu}$ couple only to spin-3/2 states because they do not appear in the spin-1/2 part. The two sum rules from $\gamma_\mu p_\nu+\gamma_\nu p_\mu$ and $\gamma_\mu \gamma_\nu+\frac{1}{3}g_{\mu\nu}$ couple only to spin-1/2 states because they do not appear in the spin-3/2 part. By the same token, the sum rule at $\gamma_\mu p_\nu+\gamma_\nu p_\mu$ that couples to spin-1/2 could also be obtained from the combination $\gamma_\mu \gamma_\nu\hat{p}+\frac{1}{3}g_{\mu\nu}\hat{p}$. The two sum rules from $p_\mu p_\nu\hat{p}$ and $p_\mu p_\nu$ couple to both spin-3/2 and spin-1/2 states because they appear in both parts. They will not be considered further. Note that the parities are still mixed in each of these sum rules. It is not known [*a priori*]{} which parity is the dominant state in a given sum rule. That information comes from the specific OPE structure on the left-hand-side.
Parity-projected QCD sum rules for spin-3/2 decuplet states {#qcd33}
===========================================================
The combination of the two tensor structures that couple purely to spin-3/2 states, $g_{\mu\nu}\hat{p}$ and $g_{\mu\nu}$, can be cast into the form $g_{\mu\nu}(\gamma_0 A + B)$ in the rest frame. According to Eq. (\[Img\_Pi\]), it suggests that parity projection can be done exactly in this case. The functions $A(p_0)$ and $B(p_0)$ are readily identified from the usual calculation of the OPE which involves contracting out the quark pairs in the correlation function and substituting the fully-interacting quark propagator from OPE. They are functions of the QCD vacuum condensates and other QCD parameters.
Upon Borel integration in Eq. (\[G\_sum\_rule1\]) and Eq. (\[G\_sum\_rule2\]), the parity-projected sum rules can be written in the general form $${\cal A}(w_+,M)\,+\,{\cal B}(w_+,M)\, =\,
\tilde{\lambda}_{+}^2 e^{-m_{+}^2/M^2},
\label{F_sum_rule1}$$ $${\cal A}(w_-,M)\,-\,{\cal B}(w_-,M)\, =\,
\tilde{\lambda}_{-}^2 e^{-m_{-}^2/M^2},
\label{F_sum_rule2}$$ where $\tilde{\lambda}_{\pm}=(2\pi)^2\lambda_{\pm}$ are the rescaled couplings. The rescaling is done so that no factors of $\pi$ appear explicitly in the sum rules. The rescaled quantities are also ‘natural’ in their numerical values.
For our special case of spin-3/2 states, the ${\cal A}$ function ${\cal A}_{3/2}$ is given by $$\begin{aligned}
{\cal A}_{3/2} &=&
c_1\,[1]\,(2(1-e^{-w^2/M^2})M^6 - 2w^2e^{-w^2/M^2}M^4
\nonumber \\
&-& w^4e^{-w^2/M^2}M^2)\,L^{4/27}
\nonumber \\
&+ & (c_2\,[b] +c_3\,[m_sa])\,(1-e^{-w^2/M^2})M^2 L^{4/27}\,
\nonumber \\
&+& c_4 \,[m_s m_0^2 a]\, L^{-10/27} +\,c_5 \, [\kappa_v a^2]\, L^{28/27}.
\label{sum33A}\end{aligned}$$ In this expression, the vacuum condensates are explicitly isolated in square brackets whose definitions and values will be given below. The Wilson coefficients (which are pure numbers) are given for various members of the decuplet by $$\begin{array}{|c|c|c|c|c|c|c|c}
\hline
{\cal A}_{3/2}&c_1 &c_2 &c_3 &c_4 &c_5 \\
\hline
\Delta &{1\over 20} &{-5\over 144} &0 &0 &{-2\over 3} \\
\hline
\Sigma^* &{1\over 20} &{-5\over 144} &{(4-f_s)\over 6} &{-(14-5f_s)\over 36}&{(2+4f_s)\over 9} \\
\hline
\Xi^* &{1\over 20} &{-5\over 144} &{(2+f_s)\over 3} &{-(7+2f_s)\over 18} &{2f_s(2+f_s)\over 9} \\
\hline
\Omega^- &{1\over 20} &{-5\over 144} &{3f_s\over 2} &{-3f_s\over 4} &{-2f_s^2\over 3}.\\
\hline
\end{array}
%$$ Similarly, the ${\cal B}_{3/2}$ function is given by $$\begin{aligned}
{\cal B}_{3/2} &=&
c_1\,[m_s](I(w)-we^{-w^2/M^2})L^{-8/27}M^4\,
\nonumber \\
&+& c_2 [m_s] w^3\,e^{-w^2/M^2}L^{-8/27}M^2
\nonumber \\
&+& c_3\,[a] (I(w) - we^{-w^2/M^2})L^{16/27}M^2
\nonumber \\
&+& c_4\,[m_s b] I(w) L^{-8/27}
\nonumber \\
&+& c_5\,[m_0^2 a] I(w) L^{2/27},
\label{sum33B}\end{aligned}$$ where the coefficients are given by $$\begin{array}{|c|c|c|c|c|c|}
\hline
{\cal B}_{3/2}&c_1 &c_2 &c_3 &c_4 &c_5 \\
\hline
\Delta & 0 & 0 & {2\over 3} & 0 & {-2\over 3} \\
\hline
\Sigma^* & 3/16 & -1/8 & {2(2+f_s)\over 9} & -1/24 & {-2(2+f_s)\over 9} \\
\hline
\Xi^* & 3/8 & -1/4 & {2(1+2f_s)\over 9} & -1/12 & {-2(1+2f_s)\over 9} \\
\hline
\Omega^- & 9/16 & -3/8 & {2f_s\over 3} & -1/8 & {-2f_s\over 3}. \\
\hline
\end{array}
%$$ The integral $I(w)=\int_0^w e^{-x^2/M^2} dx$ is evaluated numerically in the analysis. Note that the sum rule for $\Omega^-$ reduces to that for the $\Delta$ if $m_s=0$ and $f_s=1$, which serves as a check of the calculation.
Now we explain the meaning of the parameters in the sum rules. The rescaled quark condensate is taken as the positive quantity $a=-(2\pi)^2\,\langle\bar{u}u\rangle=0.52\pm 0.05$ GeV$^3$, corresponding to a central value of $\langle\bar{u}u\rangle=-(236)^3$ MeV$^3$. For the gluon condensate, we use rescaled $b=\langle g^2_c\, G^2\rangle=1.2\pm 0.6$ GeV$^4$. The mixed condensate parameter is placed at $m^2_0=\langle\overline{q}g\sigma\cdot Gq\rangle/\langle\overline{q}q\rangle=0.72\pm 0.08$ GeV$^2$. For the four-quark condensate $\langle\bar{u}u\bar{u}u\rangle=\kappa_v\langle\bar{u}u\rangle^2$, we use $\kappa_v=2\pm 1$ to include possible violation of the factorization approximation. We retain only terms linear in the strange quark mass and set $m_u=m_d=0$. The strange quark mass is taken as $m_s=0.15\pm 0.02$ GeV. We use the ratio $f_s=\langle\bar{s}s\rangle/\langle\bar{u}u\rangle
=\langle\bar{s}g_c\sigma\cdot G s\rangle /\langle\bar{u}g_c\sigma\cdot G u\rangle
=0.83\pm 0.05$ to accounts for the flavor symmetry breaking of the strange quark condensates. The anomalous dimension corrections of the various operators are taken into account via the factors $L^\gamma=\left[{\alpha_s(\mu^2)/ \alpha_s(M^2)}\right]^\gamma
=\left[{\ln(M^2/\Lambda_{QCD}^2)/
\ln(\mu^2/\Lambda_{QCD}^2)}\right]^\gamma$, where $\gamma$ is the appropriate anomalous dimension, $\mu=500$ MeV is the renormalization scale, and $\Lambda_{QCD}=0.15\pm 0.04$ GeV is the QCD scale parameter. We find variations of $\Lambda_{QCD}$ have little effects on the results. The uncertainties are assigned fairly conservatively, ranging from 10% to 100%. They will be mapped into those for the fit parameters in the analysis, giving a realistic estimate of the errors on the parameters.
It is worth pointing out that the $A$ function is chiral-even, meaning that it only involves condensates of even energy dimensions: for example, $b$ of 4, $m_sa$ of 4, $\kappa_va^2$ of 6 and so on. On the other hand, the $B$ function is chiral-odd, with the leading contribution from the quark condensate $a$, the order parameter of spontaneous chiral symmetry breaking of QCD. The difference of the two parity-projected sum rules in Eq. (\[F\_sum\_rule1\]) and Eq. (\[F\_sum\_rule2\]) is the sign in front of the chiral-odd term $B$. So from the point of view of QCD sum rules, the origin of splittings between positive-parity and negative-parity states lies in the chiral-odd vacuum condensates. If the quark condensate and the strange quark mass vanish, then chiral symmetry is restored in the vacuum, and there would be exact parity doubling in the baryon spectrum. This is a valuable insight from the parity-projected QCD sum rules.
The analysis of a QCD sum rule boils down to the following mathematical problem. Given an equation of the general structure $$\mbox{LHS}(M,OPE)=\mbox{RHS}(M,w,m,\lambda^2),
\label{sum}$$ and a set of input QCD parameters denoted by $OPE$, find the best output parameters (the baryon mass of interest $m$, the coupling strength $\lambda^2$ of the interpolating field, and the continuum threshold $w$) by matching the two sides over a region in Borel mass $M$. The LHS has errors arising from our imprecise knowledge of the QCD parameters, as discussed above. From statistical point of view, a $\chi^2$ minimization of the type $$\chi^2=\sum_i{|\mbox{LHS}_i-\mbox{RHS}_i|^2 \over \sigma^2_i},
\label{chi2}$$ offers the least-biased way of finding the unknown parameters. In the Monte-Carlo based procedure used here, first the Borel window in $M$ is divided equally into a grid (we use 51 points). At each point, the uncertainty distribution in the OPE is constructed by randomly-selected, Gaussianly-distributed sets generated from the central values and the uncertainties in the QCD input parameters. Then the $\chi^2$ minimization is applied to the sum rule by fitting the phenomenological parameters. This is done for each QCD parameter set and at each point in $M$, resulting in distributions for phenomenological fit parameters, from which their errors are derived. Usually, 100 such configurations are sufficient for getting stable results. We generally select 500 sets which help resolve more subtle correlations among the QCD parameters and the phenomenological fit parameters. So the same sum rule is fitted not just once, but thousands of times in our analysis. The advantage afforded by the Monte-Carlo is that the entire phase-space of the input QCD parameters is explored. The errors obtained this way represent the most realistic and conservative estimates of the predictive power of a QCD sum rule. This is in contrast to traditional approaches where only a small part of the phase-space is explored at a time. We select the Borel window by trial and error. Our criteria are that the OPE is reasonably convergent by looking at the terms of various dimension, and that the fit results should not be sensitive to small changes in the Borel window. Under these constraints, we seek to make the window as wide as possible. Since the selection of Borel window $M$ and the three phenomenological parameters are inter-dependent in the sum rule, the entire fitting process is iterated until the best solution is found. In general, we seek solutions that are from a three-parameter search in which all three parameters are treated as free simultaneously. We regard such solutions as the best predictions of the QCD sum rule approach. In the absence of such solutions, a two-parameter search is performed in which one of the three parameters is fixed. Usually, it is the continuum threshold which is fixed to values that are larger than the mass in question, or to values suggested by the observed spectrum. Such a two-parameter approach is the one usually adopted in most analyses in the past.
Fig. \[fit1\] shows the matching in the sum rule Eq. (\[F\_sum\_rule1\]) for $3/2+$ states. The two sides match very well over a relatively wide region (1 GeV). The individual contributions of term ${\cal A}$ and term ${\cal B}$ are also shown. The two terms are comparable in size, the sum of which gives rise to the LHS. The error band on the LHS is generated by Monte-Carlo reflecting all the uncertainties assigned to the QCD parameters. The entire uncertainty phase space is then mapped to the output parameters in the fitting process.
------------------------- ------------ ------- ---------------------------- --------------- -------
Region $w$ $\tilde{\lambda}_{3/2+}^2$ Mass Exp.
(GeV) (GeV) (GeV$^6$) (GeV) (GeV)
$\Delta({3\over 2}+)$ 1.4 to 2.4 1.60 2.46$\pm$0.42 1.21$\pm$0.06 1.23
2.32$\pm$0.21 1.21$\pm$0.02
$\Sigma^*({3\over 2}+)$ 1.4 to 2.4 1.84 3.52$\pm$0.43 1.38$\pm$0.06 1.385
3.40$\pm$0.29 1.37$\pm$0.02
$\Xi^*({3\over 2}+)$ 1.4 to 2.4 1.95 4.19$\pm$0.46 1.47$\pm$0.05 1.53
4.10$\pm$0.37 1.47$\pm$0.02
$\Omega^-({3\over 2}+)$ 1.4 to 2.4 2.25 6.94$\pm$0.63 1.68$\pm$0.05 1.67
6.89$\pm$0.56 1.68$\pm$0.02
------------------------- ------------ ------- ---------------------------- --------------- -------
: Results for the $3/2+$ states from the parity-projected sum rule Eq.( \[F\_sum\_rule1\]) in a two-parameter search. The errors are derived from 500 Monte-Carlo sets based on the uncertainties assigned to the QCD input parameters. Two sets of solutions are provided for each case: the first one using the default errors in the OPE input parameters, while the second using uniform 10% errors. The experimental values are taken from the PDG [@pdg04].[]{data-label="tab3p"}
A three-parameter search was tried first. Unfortunately, a solution could not be found. What happens is that the search algorithm keeps returning solutions with threshold smaller than the mass, a clearly unphysical situation. This is an indication that the OPE does not have enough information to resolve all three parameters simultaneously. So we switched to a two-parameter search by fixing the continuum threshold to values suggested by the Particle Data Group [@pdg04]. The extracted parameters are given in Table \[tab3p\]. We offer two sets of results on the fit parameters: one using the default errors assigned to the OPE parameters ranging from 10% to 100%, the other with 10% uniform errors. It sort of gives the worst-case, best-case scenarios, as far as errors on the input parameters are concerned. In the worst-case solution, the errors on the masses are on the order of 5%, and 15% on the couplings. In the best-case solution, the errors are reduced to about 2% on the masses, and less than 10% on the couplings. Note that the central value for the couplings are shifted by a small amount when the errors are reduced, while it is stable for the masses. The computed masses compare favorably with experiment. What is interesting is the fact that the mass pattern emerges under the same Borel window across the particles, and that the continuum thresholds are consistent with the excited states suggested by PDG. In the case of $\Xi^*(3/+)$, the computed mass of 1.47 GeV is slightly below the experimental value of 1.53 GeV. In the PDG, there are two 3-star excited states at 1690 MeV and 1950 MeV with unknown spin-parity. Our sum rule favors a threshold of 1950 MeV to 1690 MeV which leads to a mass that is even smaller than 1.47 GeV. It hints that the spin-parity of 1690 MeV is likely not $3/2+$.
Fig. \[fit2\] shows the matching in the sum rule Eq. (\[F\_sum\_rule2\]) for $3/2-$ states. Here it is the difference between term ${\cal A}$ and term ${\cal B}$ that gives rise to the ground-state pole. The quality of the matching is still fairly good, but we found that the stability of the sum rules is not as good as the corresponding ones for $3/2+$ states in Eq. (\[F\_sum\_rule1\]). A three-parameter search does not work, so a two-parameter search is performed. To get a better understanding of the sum rules, we adopt the following fitting strategy. Instead of fixing the continuum threshold to a value that leads to a known state, we perform two-parameter searches by fixing the mass to values suggested by the PDG, and search for the continuum threshold and coupling. In this way, the searches will provide information on whether a specific sum rule can accommodate a known state with reasonable continuum thresholds and couplings. Such a study is useful since to our knowledge there is no information about these $3/2-$ states from the standpoint of QCD sum rules. The result of the study is given in Table \[tab3n\]. We fix the Borel window to be the same wide window (1.4 to 2.4 GeV) as for the $3/2+$ states for all cases. This choice is quite reasonable judging by the matching plots, which are all of comparable quality as Fig. \[fit2\]. Again, the same worst-case, best-case solutions are provided for each case in order to give some idea about the stability of the fits.
In the case of $\Delta({3/2}-)$, the PDG lists a 4-star state at 1.70 GeV. Using this value as input, a continuum threshold of 2.74 GeV and a coupling of 1.56 are obtained, with relative errors about 20% and 55%, respectively. They decrease to about 5% and 40% in the best-case (10% uniform errors on the input parameters). Note that the central values vary as errors are reduced: the threshold shifts up to 2.9 GeV and the coupling shifts down to 1.43. It is a sign of highly non-linear mapping of the errors from input to output. These errors are much bigger than those for $3/2+$ states, especially on the couplings. The next $\Delta(3/2-)$ state in the PDG is a one-star state at 1.94 GeV, compared to the continuum threshold of 2.7 to 2.9 GeV from this sum rule.
In the case of $\Sigma^*({3/2}-)$, the PDG lists a 2-star state at 1.58 GeV with unknown spin-parity, a 4-star state at 1.67 GeV, and a three-tar state at 1.94 GeV. The corresponding continuum thresholds required to predict these states are 2.87, 2.90, 3.00 GeV.
In the case of $\Xi^*({3/2}-)$, out of the three 3-star states listed by the PDG, 1.69, 1.82, 1.95 GeV, only the 1.82 GeV state is assigned the $3/2-$ spin-parity. The corresponding continuum thresholds required to predict these states are 2.91, 2.95, 3.00 GeV. Interestingly, these values are close to the ones for the $\Sigma^*({3/2}-)$ states.
In the case of $\Omega^-({3/2}-)$, there are three candidates from the PDG, a three-star state at 2.25 GeV, a two-star state at 2.38 GeV, and a two-star state at 2.47 GeV, all with unknown spin-parity. Using the sum rules, they could be assigned $3/2-$ if the the corresponding continuum thresholds are 3.11, 3.18, 3.24 GeV. For the 2.25 GeV state, we could find a solution only when the errors are small.
------------------------- ------------ --------------- ---------------------------- ------- ---------
Region $w$ $\tilde{\lambda}_{3/2-}^2$ Mass Exp.
(GeV) (GeV) (GeV$^6$) (GeV) (GeV)
$\Delta({3\over 2}-)$ 1.4 to 2.4 2.74$\pm$0.47 1.56$\pm$0.84 1.70 1.70
2.90$\pm$0.10 1.43$\pm$0.56
$\Sigma^*({3\over 2}-)$ 1.4 to 2.4 2.68$\pm$0.51 1.15$\pm$0.64 1.58 1.58(?)
2.87$\pm$0.13 1.02$\pm$0.39
2.74$\pm$0.46 1.31$\pm$0.72 1.67 1.67
2.90$\pm$0.11 1.19$\pm$0.46
2.94$\pm$0.30 2.14$\pm$1.20 1.94 1.94
3.00$\pm$0.06 1.94$\pm$0.76
$\Xi^*({3\over 2}-)$ 1.4 to 2.4 2.72$\pm$0.43 1.13$\pm$0.65 1.69 1.69(?)
2.91$\pm$0.12 1.02$\pm$0.42
2.86$\pm$0.36 1.40$\pm$0.81 1.82 1.82
2.95$\pm$0.10 1.28$\pm$0.53
2.94$\pm$0.30 1.80$\pm$1.05 1.95 1.95(?)
3.00$\pm$0.10 1.63$\pm$0.68
$\Omega^-({3\over 2}-)$ 1.4 to 2.4 - - 2.25 2.25(?)
3.11$\pm$0.13 2.31$\pm$1.11
3.18$\pm$0.24 3.47$\pm$2.27 2.38 2.38(?)
3.18$\pm$0.13 3.13$\pm$1.50
3.24$\pm$0.24 4.32$\pm$2.77 2.47 2.47(?)
3.24$\pm$0.14 3.91$\pm$1.82
------------------------- ------------ --------------- ---------------------------- ------- ---------
: Results for the $3/2-$ states from the parity-projected sum rule Eq.( \[F\_sum\_rule2\]) in a two-parameter search. The errors are derived from 500 Monte-Carlo sets based on the uncertainties assigned to the QCD input parameters. Two sets of solutions are provided for each case: the first one using the default errors in the OPE input parameters, while the second using uniform 10% errors. The experimental values are taken from the PDG [@pdg04].[]{data-label="tab3n"}
Conventional QCD sum rules for spin-1/2 decuplet states {#qcd13}
=======================================================
The two sum rules that couple purely to spin-1/2 states are from the Dirac structures $\gamma_\mu p_\nu+\gamma_\nu p_\mu$ and $\gamma_\mu \gamma_\nu+\frac{1}{3}g_{\mu\nu}$. Since they cannot be cast into the $(\gamma_0 A + B)$ form in the rest frame, the parity projection technique cannot be applied to this case. We have to rely on the conventional sum rule method in which the parities are mixed. The spin-$1/2$ decuplets are in fact excited states. They are rarely studied in the QCD sum rule method. Here we have an opportunity to isolate them as the ground-state poles. For this reason, even a conventional analysis is beneficial.
The sum rule from the $\gamma_\mu p_\nu+\gamma_\nu p_\mu$ structure is $$\begin{aligned}
& & c_1\;[1]\; L^{4/27}\; E_2\; M^6
+ c_2\; [b]\; L^{4/27}\; E_0\; M^2
\nonumber \\ & &
+ c_3\; [m_s\,a]\; L^{4/27}\; E_0\; M^2
+ c_4\; [m_s\,m^2_0 a]\; L^{-10/27}
\nonumber \\ & &
+ c_5\; [\kappa_v a^2]\; L^{28/27}
+ c_6\; [m^2_0 a^2]\; L^{14/27}\; {1\over M^2}
\nonumber \\ & &
= \tilde{\alpha}_{1/2-}^2\; e^{-m^2_{1/2-}/M^2}
+ \tilde{\alpha}_{1/2+}^2\; e^{-m^2_{1/2+}/M^2},
\label{sum33c}\end{aligned}$$ where the coefficients are given by $$\begin{array}{|c|c|c|c|c|c|c|}
\hline
&c_1 &c_2 &c_3 & c_4 &c_5 &c_6 \\
\hline
\Delta &{1\over 240}&{5\over 1728}&0 &0 &{-1\over 18} &{7\over 216} \\
\hline
\Sigma^* &{1\over 240}&{5\over 1728}&{(f_s-4)\over 72}&{(7-4f_s)\over 216}&{-(1+2f_s)\over 54} &{7(1+2f_s)\over 648} \\
\hline
\Xi^* &{1\over 240}&{5\over 1728}&{-(2+f_s)\over 36}&{(7-f_s)\over 36}&-{f_s(2+f_s)\over 54}&{7f_s(1+2f_s)\over 648} \\
\hline
\Omega^- &{1\over 240}&{5\over 1728}&{-f_s\over 8} &{f_s\over 24} &{-f_s^2\over 18} &{7f_s^2\over 216} \\
\hline
\end{array}
%$$ This sum rule is chiral-even. Since the parity of the lowest state is not known, we retain one term for each parity in the RHS, and let the OPE reveal which one is lower. The sum rule for $\Delta$ agrees with that given in Ref. [@Derek90], except that the coefficient $c_6$ has the opposite sign. The sum rules for $\Sigma^*$, $\Xi^*$, and $\Omega^-$ are new, to the best of our knowledge. Note that the sum rule for $\Omega^-$ reduces to that for $\Delta$ if the strange quark is turned off ($m_s=0$ and $f_s=1$), as expected. Since they are derived separately, this provides a non-trial check of the calculation. Another check is provided by the fact that $c_5$ and $c_6$ for $\Sigma^*$, $\Xi^*$, and $\Omega^-$ coincide with each other in the limit of $f_s=1$, as expected. The excited state contributions of RHS are modeled using terms on the OPE side surviving $M^2\rightarrow \infty$ under the assumption of duality, and are represented by the factors $E_n(x)=1-e^{-x}\sum_n{x^n/n!}$ with $x=w^2/M^2$ and $w$ an effective continuum threshold.
The sum rule from the $ \gamma_\mu\gamma_\nu + {1\over 3}g_{\mu \nu}$ structure, which is chiral-odd, is given by $$\begin{aligned}
& & c_1\; [ a]\; L^{16/27}\; E_1\; M^4
+c_2\; [m^2_0 a]\; L^{2/27}\; E_0\; M^2
\nonumber \\ & &
+c_3\; [m_s\, b]\; L^{-8/27}\; E_0\; M^2
+c_4\; [a\,b]\; L^{16/27}
\nonumber \\ & &
+c_5\; [m_s\,\kappa_v a^2]\; L^{16/27}
\nonumber \\ & &
= -\tilde{\alpha}_{1/2+}^2 m_{1/2+} e^{-m^2_{1/2+}/M^2}
\nonumber \\ & &
+\tilde{\alpha}_{1/2-}^2 m_{1/2-} e^{-m^2_{1/2-}/M^2},
\label{sum33d}\end{aligned}$$ where the coefficients are given by $$\begin{array}{|c|c|c|c|c|c|}
\hline
& c_1 & c_2 & c_3 & c_4 & c_5 \\
\hline
\Delta &{-1\over 36 }&{1\over 18} & 0 &{-5/864} & 0 \\
\hline
\Sigma^* &{-(2+f_s)\over 108 }&{(2+f_s)\over 54} &{1\over 288}&{-5(2+f_s)\over 2592} &{(f_s-1)\over 18} \\
\hline
\Xi^* &{-(1+2f_s)\over 108}&{(1+2f_s)\over 54}&{1\over 144}&{-5(1+2f_s)\over 2592}&{f_s(f_s-1)\over 18} \\
\hline
\Omega^- &{-f_s\over 36 }&{f_s\over 18} &{1\over 96 }&{-5f_s/864} & 0 \\
\hline
\end{array}
%
\label{coe33d}$$ For $\Delta$, the coefficient $c_4$ is different in both value and sign from that in Ref. [@Derek90]. The sum rules for $\Sigma^*$, $\Xi^*$, and $\Omega^-$ are new, as far as we know. Similarly, the sum rule for $\Omega^-$ reduces to that for $\Delta$ if the strange quark is turned off. Another check is provided by the fact that $c_1$, $c_2$, $c_4$ and $c_5$ for $\Sigma^*$, $\Xi^*$, and $\Omega^-$ coincide with each other in the limit of $f_s=1$.
Now we turn to the analysis of the two sum rules. First, we note that states of opposite parities on the phenomenological side (RHS) are adding up in the chiral-even sum rule, whereas canceling in the chiral-odd sum rule. This is a standard feature of baryon sum rules that leads to the general conclusion that chiral-odd sum rules perform better than chiral-even sum rules in baryon channels. Indeed, we found that the chiral-even sum rule in Eq. (\[sum33c\]) is very poor. The leading term is a perturbative contribution, and the sum rule is almost completely saturated by the continuum. No results could be extracted from this sum rule.
On the other hand, the chiral-odd sum rule in Eq. (\[sum33d\]) has good convergence. The leading term contains the non-perturbative quark condensate. The sign of this term (in $c_1$) indicates that the sum rule is saturated by the positive-parity state: they have the same negative signs, see Eq. (\[sum33d\]) and Eq. (\[coe33d\]). This is an example of how the parity of the ground-state pole is determined in a mixed-parity sum rule. The dominance of the quark-condensate term is further confirmed in our numerical Monte-Carlo analysis. We are able to perform three-parameter searches. It means that the extracted mass, coupling, continuum threshold can be regarded as the true predictions of the sum rule. The results are given in Table \[tab33d\] and the matching of the two sides is shown in Fig. \[fit4\]. In the observed spectrum, the lowest $\Delta(1/2+)$ is a one-star state at 1.75 GeV, followed by a 4-star state at 1.91 GeV. Our prediction favors the latter as the lowest state in this channel. It casts doubts on the existence of the state at 1.75 GeV. For $\Sigma^*(1/2+)$, the PDG lists a 3-star state at 1.66 GeV, followed by a one-star state at 1.77 GeV, then by a 2-star state at 1.88 GeV. Our result seems to favor the state at 1.88 GeV than the state at 1.66 GeV. The spin-parity situation in the $\Xi^*(1/2+)$ channel is not clear in the PDG. It lists two 3-star states with unknown spin-parity, at 1690 MeV and 1950 MeV. Our results is in favor of the 1950 MeV state. In the $\Omega^-$ channel, there are 3 states sitting close to each other in the observed spectrum, at 2250, 2380, and 2470 MeV, whose spin and parity are not clear. Our prediction of 2.37 GeV is in the middle of this range, but the accuracy is not enough to clearly identify with one of them.
-------------------- ------------ --------------- --------------------------- --------------- ---------
Region $w$ $\tilde{\alpha}_{1/2+}^2$ Mass Exp.
(GeV) (GeV) (GeV$^6$) (GeV) (GeV)
$\Delta(1/2+)$ 1.1 to 2.0 2.41$\pm$1.08 0.18$\pm$0.15 2.10$\pm$0.54 1.91
2.52$\pm$0.80 0.18$\pm$0.13 2.17$\pm$0.33
$\Sigma^*(1/2+)$ 1.1 to 2.0 2.15$\pm$1.06 0.13$\pm$0.13 1.97$\pm$0.62 1.88
2.29$\pm$0.83 0.13$\pm$0.11 2.09$\pm$0.37
$\Xi^*(1/2+)$ 1.1 to 2.0 2.42$\pm$1.11 0.16$\pm$0.13 2.11$\pm$0.54 1.95(?)
2.50$\pm$1.50 0.16$\pm$0.12 2.19$\pm$0.35
$\Omega^{-}(1/2+)$ 1.1 to 2.0 3.02$\pm$1.18 0.23$\pm$0.14 2.37$\pm$0.41 2.38(?)
3.11$\pm$1.13 0.23$\pm$0.13 2.39$\pm$0.32
-------------------- ------------ --------------- --------------------------- --------------- ---------
: Results for the $1/2+$ branch of the decuplet states from the chiral-odd sum rule Eq. (\[sum33d\]) in a three-parameter search. The errors are derived from 500 Monte-Carlo sets based on the uncertainties assigned to the QCD input parameters. Two sets of solutions are provided for each case: the first one using the default errors in the OPE input parameters, while the second using uniform 10% errors. The experimental values are taken from the PDG [@pdg04].[]{data-label="tab33d"}
Conclusion {#con}
==========
We have presented a study of the decuplet family using the method of QCD sum rules. New sum rules are derived for the $3/2+$ and $3/2-$ branches using a parity-projection technique. They are more stable and give a better determination of the mass spectrum than the conventional sum rules. The results for the $3/2-$ branch are new as a consequence of the parity separation. The spin-1/2 sector is investigated using the conventional sum rules method. New sum rules are derived for members that contain the strange quark ($\Sigma^*$, $\Xi^*$, and $\Omega^-$), in addition to a careful re-examination of $\Delta$ channel. The chiral-even sum rules of Eq. (\[sum33c\]) are dominated by the continuum. No useful information could be extracted from them. The chiral-odd sum rules of Eq. (\[sum33d\]) have good convergence and allow the only three-parameter searches in this study. The predicted results provide useful information on the $1/2+$ states from a QCD-based standpoint. We could not find sum rules that are saturated by the $1/2-$ states.
This work has been supported by DOE under grant number DE-FG02-95ER40907.
[00]{}
M.A. Shifman, A.I. Vainshtein and Z.I. Zakharov, Nucl. Phys. [**B147**]{}, 385, 448 (1979).
V.M. Belyaev and B.L. Ioffe, Sov. Phys. JETP [**56**]{}, 493 (1982).
V.M. Belyaev and B.L. Ioffe, Sov. Phys. JETP [**57**]{}, 716 (1983).
L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Lett. [**B120**]{}, 209 (1983); [**B122**]{}, 487(E) (1983).
L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. [**127**]{}, 1-97 (1985).
Y. Chung, H.G. Dosch, M. Kremer and D. Schall, Z. Phys. [**C57**]{}, 151 (1984).
D.B. Leinweber, Ann. of Phys. (N.Y.) [**198**]{}, 203 (1990).
D.B. Leinweber, Ann. of Phys. (N.Y.) [**254**]{}, 328 (1997).
F.X. Lee, Phys. Rev. [**C57**]{}, 322 (1998).
F.X. Lee, Phys. Rev. [**D57**]{}, 1801 (1998).
F.X. Lee, Phys. Lett. [**B419**]{}, 14 (1998).
F.X. Lee and X. Liu, Phys. Rev. [**D66**]{}, 014014 (2002).
D. Jido, N. Kodama, and M. Oka, Phys. Rev. D [**54**]{}, 4532 (1996); hep-ph/9611322.
L. Zhou and F.X. Lee, hep-lat/0604023.
S. Eidelman [*et al.*]{}, Phys. Lett. [**B592**]{}, 1 (2004).
|
---
author:
- 'Taichi Kosugi$^{1,2}$'
title: Pauli Equation on a Curved Surface and Rashba Splitting on a Corrugated Surface
---
Much attention has been paid to low-dimensional systems because they show exotic phenomena [@bib:1645_2; @bib:1645_3; @bib:1655; @bib:1657; @bib:1650; @bib:1680; @bib:1683; @bib:1647], essentially distinct from solid state materials. Recent development of nanotechnology facilitates construction of nanostructures with curved geometry [@bib:1685; @bib:1680_3; @bib:1680_4; @bib:1680_9; @bib:1680_13; @bib:1680_15] and those systems can be experimental platforms for study of the interesting phenomena. Curved geometry induces the geometric potential, which affects the dynamics of an electron moving on the curved surface, even when an electrostatic potential is absent.
The Schrödinger equations on curved surfaces have been used as tools for theoretical investigations on such systems. Their formulations have been based on the two methods. The one was proposed by DeWitt [@bib:1681] and the other by da Costa [@bib:1649]. While the former method regards a curved surface to be a fully two-dimensional space and starts from a Lagrangian for a curved space, the latter method regards the curved surface as a two-dimensional system embedded in the flat three-dimensional space. The method established by da Costa, which is called the thin-layer method, has been widely used despite its impossibility of inclusion of arbitrarily oriented magnetic fields. Da Costa also provided the one-dimensional Schrödinger equation for a curved thin tube. Along this line, Takagi [[*et al.*]{} ]{} [@bib:1689] studied the geometry-induced Aharonov-Bohm effect by taking into account the torsion of a tube.
Ferrari [[*et al.*]{} ]{} [@bib:1645] recently adopted the thin-layer approach and rigorously demonstrated by choosing a proper gauge that the separation of the on-surface and transverse dynamics under an electromagnetic field is possible without approximations. Theoretical investigations on nanostructures are thus expected to treat more various situations in future.
It should be pointed out here that there also exists another rich field of studies for formulation of quantum mechanics on a more general constrained system by employing the possibility of inequivalent quantizations (e.g. Refs. 19-22). Ohnuki [[*et al.*]{} ]{} [@bib:1770] analyzed quantum mechanics on $S^D$ embedded in $(D+1)$-dimensional flat space by setting up appropriate fundamental algebra, with gauge potentials induced.
Parallel to the extension of the possibility of geometric arrangement, the role of relativistic effects such as spin-orbit interaction has been increasingly evoking interests in material physics due to the recent rise of spintronics. The relativistic effects and the spin degree of freedom for an electronic system naturally emerge in the Dirac equation [@bib:Sakurai_Advanced]. Since the upper two components of a four-component spinor are much larger than the lower two components in ordinary condensed matter physics, the large part of relativistic electronic structure calculations have been performed using two-component wave functions. The most tractable tool for a quantum mechanical analysis on an electronic system including magnetic properties and relativistic effects is hence the Pauli equation, which is for a charged spin-$1/2$ particle with a nonzero mass and contains the lowest-order relativistic correction term. It will provide deep insights into rich physics on nanosystems.
In this Letter I provide the Pauli equation for a particle confined to a curved surface under an electromagnetic field as an extension of the work done by Ferrari [[*et al.*]{} ]{} [@bib:1645] I reconcile ourselves to two assumptions mentioned below for performing variable separation in the present study. Excepting those assumptions, the derivation of the Pauli equation will proceed on the same strictness as for the Schrödinger case.
The expansion of the Dirac equation for a spin-$1/2$ particle of its mass $m$ and charge $Q$ under an electromagnetic field using the Foldy-Wouthuysen method [@bib:152] leads to the Pauli equation $i \frac{\partial \psi}{\partial t} = H_{\mathrm{P}} \psi$ for the upper two components $\psi$ of the four-component spinor. $H_{\mathrm{P}}$ is the Pauli Hamiltonian, which neglects the mass term $mc^2$ and the terms on the orders higher than $m^{-2}$, given by $$\begin{gathered}
H_{\mathrm{P}} =
\frac{\boldsymbol{\Pi}^2}{2m} + V
- \frac{Q}{mc} \boldsymbol{S} \cdot \boldsymbol{B}
\nonumber \\
- \frac{Q}{4m^2c^2} [
\boldsymbol{\Pi} \cdot \boldsymbol{S} \times \boldsymbol{E}
+ \boldsymbol{S} \times \boldsymbol{E} \cdot \boldsymbol{\Pi}
]
.
\label{Pauli_Cartesian}\end{gathered}$$ $\boldsymbol{\Pi} \equiv -i \nabla - \frac{Q}{c} \boldsymbol{A}$ is the canonical momentum operator and the magnetic field $\boldsymbol{B} = \nabla \times \boldsymbol{A}$ is the rotation of the vector potential. $\boldsymbol{S} = \boldsymbol{\sigma}/2$ is the spin operator and $\boldsymbol{\sigma}$ is the Pauli matrix. The contribution from the divergence of the electric field $\nabla \cdot \boldsymbol{E}$, called the Darwin term, is absorbed into the scalar potential $V$. All spin-orbit interactions which will appear below come from the last term on the right hand side of eq. (\[Pauli\_Cartesian\]). Let us consider a coordinate transformation from the Cartesian coordinates $x_{(a)} (a=x,y,z)$ to the curvilinear coordinates $q_i (i=1,2,3)$. The letter in a parenthesis stands for the Cartesian coordinate and not in a parenthesis for the new coordinate. The dreibein field and its inverse are defined as $$\begin{gathered}
e_i^{(a)} = \frac{\partial x_{(a)}}{\partial q_i} , \
e_{(a)}^i = \frac{\partial q_i}{\partial x_{(a)}}
,\end{gathered}$$ which satisfy the conditions $e_i^{(a)} e_{(a)}^j = \delta_i^j , e_i^{(a)} e_{(b)}^i = \delta_{(b)}^{(a)}$ due to the chain rule of derivative. Summation is implied over the repeated index. The metric tensor in the new coordinate system is given by $G_{ij} = e_i^{(a)} e_j^{(a)}$. We define $v_i \equiv e_i^{(a)} v_{(a)}$ for a vector $\boldsymbol{v} = v_{(a)} \boldsymbol{e}_{(a)}$, where $\boldsymbol{e}_{(a)}$ is the unit vector along $a$ direction in the Cartesian coordinate system, from which it follows that $v_{(a)}=e_{(a)}^i v_i$. Using the Laplacian for the curvilinear coordinate system [@bib:Arfken] and the covariant derivative $D_i \equiv \partial_i - \frac{iQ}{c} A_i$, the Pauli Hamiltonian, eq. (\[Pauli\_Cartesian\]), is rewritten as $$\begin{gathered}
H_{\mathrm{P}} = - \frac{1}{2m} \frac{1}{\sqrt{G}} D_i \sqrt{G} G^{ij} D_j + V
- \frac{Q}{2mc} \sigma_{(a)} \widetilde{B}_{(a)}
\nonumber \\
+ \frac{iQ}{4m^2c^2} h^{ij} E_i D_j
,\end{gathered}$$ where $G \equiv \det G_{ij}$ and $G^{ij}$ is the inverse metric tensor. We have defined $\widetilde{\boldsymbol{B}} \equiv \boldsymbol{B} - \frac{i}{4mc} \frac{\partial \boldsymbol{B}}{\partial t}$ using the Maxwell’s equation $\nabla \times \boldsymbol{E} = - \frac{1}{c} \frac{\partial \boldsymbol{B}}{\partial t}$. We have defined the matrix $h^{ij} \equiv \varepsilon_{(abc)} \sigma_{(a)} e_{(b)}^i e_{(c)}^j = -h^{ji}$, where $\varepsilon_{(abc)}$ is the Levi-Civita symbol. The Pauli equation is obviously invariant under the following gauge transformation with an arbitrary scalar function $\gamma$ [@bib:1645]: $$\begin{gathered}
V \to V' = V - \frac{Q}{c} \frac{\partial \gamma}{\partial t}, \nonumber \\
A_i \to A_i' = A_i + \partial_i \gamma, \nonumber \\
\psi \to \psi' = e^{iQ \gamma/ c} \psi
.\end{gathered}$$ Writing explicitly the spatial derivative with respect to the new coordinates, we write down the Pauli equation as $$\begin{gathered}
i \frac{\partial \psi}{\partial t} = - \frac{1}{2m}
\Bigg[
\frac{1}{\sqrt{G}} \partial_i (\sqrt{G} G^{ij} \partial_j \psi) - \frac{iQ}{ c} \frac{1}{\sqrt{G}} \partial_i ( \sqrt{G} G^{ij} A_j) \psi
\nonumber \\
- \frac{2iQ}{ c} G^{ij} A_i \partial_j \psi - \frac{Q^2}{ c^2} G^{ij} A_i A_j \psi
\Bigg]
\nonumber \\
+ V \psi
- \frac{Q}{2mc} \sigma_{(a)} \widetilde{B}_{(a)} \psi
+ \frac{iQ}{4m^2c^2} h^{ij} E_i D_j \psi
.
\label{Eq_psi}\end{gathered}$$ From here the curved surface $S$ confining the particle is considered. Let us adopt a coordinate transformation such that $S$ is described as $q_3 = 0$ and an arbitrary point $\boldsymbol{r}$ immediately close to the point $\boldsymbol{r}_S$ on $S$ is given by $\boldsymbol{r}(q_1, q_2, q_3) = \boldsymbol{r}_S(q_1, q_2) + q_3 \boldsymbol{n}(q_1, q_2)$, where $\boldsymbol{e}_a \equiv \frac{\partial \boldsymbol{r}_S}{\partial q_a} (a=1,2)$ and $\boldsymbol{n} \equiv \boldsymbol{e}_1 \times \boldsymbol{e}_2 / |\boldsymbol{e}_1 \times \boldsymbol{e}_2|$ is the unit normal vector of $S$ at $\boldsymbol{r}_S$. It obviously follows that $e^{(a)}_i = \boldsymbol{e}_i \cdot \boldsymbol{e}_{(a)} (i=1,2),
e^{(a)}_3 = \boldsymbol{n} \cdot \boldsymbol{e}_{(a)}$ on $S$. The two-dimensional induced metric tensor $g_{ab} = \boldsymbol{e}_a \cdot \boldsymbol{e}_b$ is connected with the three-dimensional one via the following relation: $$\begin{gathered}
G_{ab} = g_{ab} + [ \alpha g + {}^\mathrm{t}(\alpha g)]_{ab} q_3 + (\alpha g {}^\mathrm{t} \alpha)_{ab} q_3^2, \nonumber \\
G_{a3} = G_{3a} = 0, \ G_{33} = 1
.\end{gathered}$$ $\alpha_{ab}$ is the Weingarten matrix [@bib:1649], which satisfies $\frac{\partial \boldsymbol{n}}{\partial q_a} = \alpha_{ab} \boldsymbol{e}_b$. As is done in the case of the Schrödinger equation [@bib:1649; @bib:1645; @bib:1650], we put the wave function in the form $$\begin{gathered}
\psi(q_1, q_2, q_3, t) =
\frac{\chi(q_1, q_2, q_3, t)}{\sqrt{1 + \mathrm{Tr} \, \alpha q_3 + \det \alpha q_3^2 }}
,
\label{def_chi}\end{gathered}$$ which ensures the norm conservation condition $\int |\psi|^2 \sqrt{G} \diff^3q = \int |\chi|^2 \sqrt{g} \diff^3q$. Substituting eq. (\[def\_chi\]) into eq. (\[Eq\_psi\]) and taking the limit $q_3 \to 0$, we obtain the equation for $\chi$: $$\begin{gathered}
i \frac{\partial \chi}{\partial t} = - \frac{1}{2m}
\Bigg[
\frac{1}{\sqrt{g}} \partial_a (\sqrt{g} g^{ab} \partial_b \chi) - \frac{iQ}{ c} \frac{1}{\sqrt{g}} \partial_a ( \sqrt{g} g^{ab} A_b) \chi
\nonumber \\
- \frac{2iQ}{ c} g^{ab} A_a \partial_b \chi - \frac{Q^2}{ c^2} (g^{ab} A_a A_b + A_3^2 ) \chi
\nonumber \\
+ \partial_3^2 \chi - \frac{iQ}{ c} (\partial_3 A_3) \chi - \frac{2iQ}{ c} A_3 \partial_3 \chi
\Bigg]
+ V_S \chi
\nonumber \\
+ V \chi
- \frac{Q}{2mc} \sigma_{(a)} \widetilde{B}_{(a)} \chi
\nonumber \\
+ \frac{iQ}{4m^2c^2} \Bigg( h^{ij} E_i D_j - \frac{1}{2} \mathrm{Tr} \, \alpha h^{i3} E_i \Bigg) \chi
,
\label{Eq_chi1}\end{gathered}$$ where $V_S(q_1, q_2) = -\frac{1}{2m} [ (\mathrm{Tr} \, \alpha)^2/4 - \det \alpha ]$ is the well-known geometric potential. The electromagnetic field is evaluated at $q_3 = 0$. We here apply a gauge transformation with a scalar function $\gamma(q_1, q_2, q_3) \equiv - \int_0^{q_3} A_3(q_1, q_2, z) \diff z$, as done in the Schrödinger case [@bib:1645], so that $A_3'$ and $\partial_3 A_3'$ vanish and $A_1$ and $A_2$ are unchanged on $S$. We divide the scalar potential into the on-surface electric part and the confinement part as $V = V_{\mathrm{el}}(q_1, q_2) + V_{\lambda}(q_3)$. The gauge-transformed equation reads, from eq. (\[Eq\_chi1\]), $$\begin{gathered}
i \frac{\partial \chi}{\partial t} = - \frac{1}{2m}
\Bigg[
\frac{1}{\sqrt{g}} \partial_a (\sqrt{g} g^{ab} \partial_b \chi) - \frac{iQ}{ c} \frac{1}{\sqrt{g}} \partial_a ( \sqrt{g} g^{ab} A_b) \chi
\nonumber \\
- \frac{2iQ}{ c} g^{ab} A_a \partial_b \chi - \frac{Q^2}{ c^2} g^{ab} A_a A_b \chi
\Bigg]
+ V_S \chi
\nonumber \\
+ V_{\mathrm{el}} \chi
- \frac{Q}{2mc} \sigma_{(a)} \widetilde{B}_{(a)} \chi
+ \frac{iQ}{4m^2c^2} h^{ia} E_i D_a \chi
+ H_3 \chi
,
\label{diff_eq_chi}\end{gathered}$$ where $$\begin{gathered}
H_3 \equiv
- \frac{1}{2m} \partial_3^2 + \frac{iQ}{4m^2c^2} h^{i3} E_i ( \partial_3 - \mathrm{Tr} \, \alpha/2 )
+ V_\lambda
\label{Hamiltonian_q3} \end{gathered}$$ is the Hamiltonian describing the dynamics along $q_3$ direction. $H_3$ contains $q_1, q_2$ and $t$ as parameters. The term involving the first-order derivative on the right hand side couples the on-surface and transverse dynamics as a relativistic effect. In the case of the Schrödinger equation, this term were absent and hence the variable separation would have been done. Let the solution of an eigenvalue problem $H_3 \phi = \varepsilon \phi$ of the form $$\begin{gathered}
\phi(q_1,q_2,q_3, t) = f(q_1, q_2, q_3, t)
\begin{pmatrix}
g_\uparrow(q_1, q_2, t) \\
g_\downarrow(q_1, q_2, t) \\
\end{pmatrix}
.\end{gathered}$$ The condition for a nontrivial solution is then, from eq. (\[Hamiltonian\_q3\]), $$\begin{gathered}
\Bigg[ \Bigg(- \frac{\partial_3^2}{2m} + V_\lambda - \varepsilon \Bigg) f \Bigg]^2
+ \Bigg[ \frac{Q \tilde{E}}{4 m^2 c^2} \Bigg( \partial_3 - \frac{\mathrm{Tr} \, \alpha}{2} \Bigg) f \Bigg]^2 = 0
\label{nontrivial}\end{gathered}$$ with $g_\uparrow$ and $g_\downarrow$ arbitrary, where $\tilde{E}_{(a)} \equiv \varepsilon_{(abc)} e_{(b)}^i e_{(c)}^3 E_i$ and $\tilde{E} \equiv \sqrt{\tilde{E}_{(x)}^2 + \tilde{E}_{(y)}^2 + \tilde{E}_{(z)}^2 }$. We try the form $f(q_1, q_2, q_3, t) = \exp (\pm \frac{iQ}{4 mc^2} \tilde{E} q_3 ) \tilde{f}(q_3)$ and substitute it into eq. (\[nontrivial\]). The condition then becomes $$\begin{gathered}
\Bigg[ - \frac{1}{2m} \partial_3^2
+ V_\lambda - \frac{ Q^2 \tilde{E}^2 }{32 m^3 c^4}
\mp \frac{i Q \tilde{E}}{8m^2c^2} \mathrm{Tr} \, \alpha
\Bigg]
\tilde{f}
= \varepsilon \tilde{f}
\label{diff_eq_f}
.\end{gathered}$$ It implies that the eigenvalue is expressed as $$\begin{gathered}
\varepsilon = \tilde{\varepsilon} - \frac{ Q^2 \tilde{E}^2}{32 m^3 c^4}
\mp \frac{i Q \tilde{E}}{8m^2c^2} \mathrm{Tr} \, \alpha
,
\label{evalue_H3}\end{gathered}$$ where $\tilde{\varepsilon}$ is an eigenvalue for $\tilde{E} = 0$, independent of $q_1, q_2$ and $t$. We neglect the second term on the right hand side of eq. (\[evalue\_H3\]) since it is on the order of $m^{-3}$. In addition, we take the average of the positive and negative signatures of the third term so that it vanishes. The first assumption needed for the present derivation is the validity of this treatment. The confinement potential is so strong that the wave function at any point on $S$ is, as the second assumption, expected to be written as $$\begin{gathered}
\chi(q_1, q_2, q_3, t) = \exp \Bigg(\pm \frac{iQ \tilde{E}}{4 mc^2} q_3 \Bigg) \tilde{f}_0(q_3) \chi_S(q_1, q_2, t)
,
\label{chi_f_chis}\end{gathered}$$ where $\tilde{f}_0$ is the nondegenerate solution of eq. (\[diff\_eq\_f\]) with the lowest eigenvalue $\tilde{\varepsilon}_0$. This assumption is feasible as long as low-energy excitation is discussed. The two assumptions introduced above are not needed for the Schrödinger case [@bib:1645] thanks to the absence of the relativistic term, for which the differential equation of the transverse dynamics remains unsolved after the variable separation. Hereafter we set $\tilde{\varepsilon}_0 \equiv 0$. Substitution of eq. (\[chi\_f\_chis\]) into eq. (\[diff\_eq\_chi\]) leads to the equation to be satisfied by the surface wave function $\chi_S$: $$\begin{gathered}
i \frac{\partial \chi_S}{\partial t} = - \frac{1}{2m}
\Bigg[
\frac{1}{\sqrt{g}} \partial_a (\sqrt{g} g^{ab} \partial_b \chi_S)
\nonumber \\
- \frac{iQ}{ c} \frac{1}{\sqrt{g}} \partial_a ( \sqrt{g} g^{ab} A_b) \chi_S
- \frac{2iQ}{ c} g^{ab} A_a \partial_b \chi_S
\nonumber \\
- \frac{Q^2}{ c^2} g^{ab} A_a A_b \chi_S
\Bigg]
+ ( V_S + V_{\mathrm{el}} ) \chi_S
+ H_{\mathrm{sp-rel}} \chi_S
\label{Pauli_eq_chi}
,\end{gathered}$$ where $$\begin{gathered}
H_{\mathrm{sp-rel}} \equiv
- \frac{Q}{2mc} \sigma_{(a)} \widetilde{B}_{(a)}
+ \frac{iQ}{4m^2c^2} h^{ia} E_i D_a
.\end{gathered}$$ It has been demonstrated that the variable separation is possible with the two assumptions for the Pauli equation on a curved surface under an electromagnetic field. The Hamiltonian of the resultant equation, eq. (\[Pauli\_eq\_chi\]), is simply that for the Schrödinger equation on a curved surface obtained by Ferrari [[*et al.*]{} ]{} [@bib:1645] plus $H_{\mathrm{sp-rel}}$. The new term acts on the two components of the surface wave function differently in general and hence physics which is absent in the Schrödinger case can emerge.
In what follows, energy spectra of a charged spin-$1/2$ particle on a sphere and a corrugated surface are provided as instructive applications of the surface Pauli equation. Analyses of their spinless and nonrelativistic electronic properties have been done using the Schrödinger equaiton for curved surfaces.[@bib:1645_2; @bib:1645_3; @bib:1680]
As the first example, let us calculate the energy spectrum of a charged spin-$1/2$ particle confined to a sphere of radius $R$. We here adopt the spherical coordinates as a new coordinate system. We assume that there is no magnetic field and a static electric field with a constant amplitude penetrates perpendicularly to the sphere: $\boldsymbol{E} = E \boldsymbol{e}_r$. In this case $V_S = V_{\mathrm{el}} = 0, E_r=E, E_\theta = E_\phi = 0$ and $$\begin{gathered}
h^{r \theta} = - \frac{\sin \phi}{R} \sigma_{(x)} + \frac{\cos \phi}{R} \sigma_{(y)} \nonumber \\
h^{r \phi} =
-\frac{\cot \theta \cos \phi}{R} \sigma_{(x)}
-\frac{\cot \theta \sin \phi}{R} \sigma_{(y)}
+\frac{1}{R} \sigma_{(z)} \nonumber \\
h^{\theta \phi} =
\frac{\cos \phi}{R^2} \sigma_{(x)}
+\frac{\sin \phi}{R^2} \sigma_{(y)}
+\frac{\cot \theta}{R^2} \sigma_{(z)} \end{gathered}$$ on the sphere. The Pauli Hamiltonian on the sphere is thus $$\begin{gathered}
H = - \frac{1}{2m R^2}
\Bigg[ \frac{1}{\sin \theta} \partial_\theta \sin \theta \partial_\theta
+ \frac{1}{\sin^2 \theta} \partial_\phi^2
\Bigg]
\nonumber \\
- \frac{i \alpha}{R}
\begin{pmatrix}
\partial_\phi & -i e^{-i \phi} ( \partial_\theta - i \cot \theta \partial_\phi) \\
i e^{i \phi} ( \partial_\theta + i \cot \theta \partial_\phi) & - \partial_\phi \\
\end{pmatrix}
\nonumber \\
= \frac{L^2}{2m R^2}
+ \frac{2 \alpha}{R} \boldsymbol{L} \cdot \boldsymbol{S}
,
\label{H_sphere}\end{gathered}$$ where $\alpha \equiv -\frac{QE}{4m^2c^2}$ and $\boldsymbol{L}$ is the orbital angular momentum operator. This Hamiltonian contains the spin-orbit interaction in the form well known in condensed matter physics, which is absent in the Schrödinger case. Spinor spherical harmonics [@bib:1042] $$\begin{gathered}
\mathcal{Y}_{jl}^{j_z} =
\begin{pmatrix}
\sqrt{\frac{l \pm j_z + 1/2}{2l+1}} Y_{l j_z-1/2} \\
\pm \sqrt{\frac{l \mp j_z + 1/2}{2l+1}} Y_{l j_z+1/2}
\end{pmatrix}
,\end{gathered}$$ is a simultaneous eigenfunction of $J^2, J_z, L^2$ and $S^2$ with the eigenvalues $j(j+1), j_z, l(l+1)$ and $3/4$, respectively. $\boldsymbol{J} \equiv \boldsymbol{L} + \boldsymbol{S}$ is the total angular momentum operator and $j$ can take only $l \pm 1/2$. It is easily confirmed that $\mathcal{Y}_{jl}^{j_z}/R$ is a normalized eigenfunction of the Hamiltonian, eq. (\[H\_sphere\]), with the eigenvalue $$\begin{gathered}
\varepsilon_{jl} = \frac{l(l+1)}{2m R^2}
+ \frac{\alpha}{R} [j(j+1) - l(l+1) - 3/4]
.
\label{evalue_sphere}\end{gathered}$$ The energy gap between two levels with common $l$ and different $j$ is $\frac{\alpha}{R}(2l + 1)$, which is a relativistic effect. Since $j \pm 1/2 = l \approx mvR = kR$, where $v$ is the velocity of the particle and $k$ is its wave number, in the limit of $R \to \infty$ the energy eigenvalue converges to $\varepsilon_{jl} \approx \frac{k^2}{2m} \pm \alpha k$. This energy dispersion is the same as that for the Rashba Hamiltonian [@bib:Rashba] $H_{\mathrm{R}} = \frac{k^2}{2m} + \alpha \boldsymbol{\sigma} \cdot \boldsymbol{e}_z \times \boldsymbol{k}$, which describes a charged particle moving freely on $xy$ plane under an electric field along $z$ axis, leading to the spin splitting as a relativistic effect due to the lack of inversion symmetry.
As the second example, let us calculate the energy spectrum of a charged spin-$1/2$ particle confined to a currugated surface. We assume here that the corrugation is along $x$ direction represented by $z = f(x)$ and there is no magnetic field and a static electric field with a constant amplitude penetrates perpendicularly to the surface: $\boldsymbol{E} = E (-f' \boldsymbol{e}_x + \boldsymbol{e}_z)/\sqrt{1 + f'^2}$ and $V_{\mathrm{el}} = 0$. We define the new coordinates on $S$ as $q_1 = \int_0^x \diff x' \sqrt{1 + f'(x')^2}, q_2 = y, q_3 = 0$. $q_1$ is the line length along $S$. $\boldsymbol{e}_1 = ( \boldsymbol{e}_x + f' \boldsymbol{e}_z )/\sqrt{ 1 + f'^2 }, \boldsymbol{e}_2 = \boldsymbol{e}_y$ and thus $g_{ab} = \delta_{ab}, \boldsymbol{n} = ( - f' \boldsymbol{e}_x + \boldsymbol{e}_z)/\sqrt{ 1 + f'^2 }$ and $V_S = - \frac{1}{8m} \frac{f''^2}{(1 + f'^2)^3}$. The nonzero components of the dreibein on $S$ are $e^1_{(x)} = e^3_{(z)} = \frac{1}{\sqrt{1 + f'^2}}, e^2_{(y)} = 1, e^1_{(z)} = -e^3_{(x)} = \frac{f'}{\sqrt{1 + f'^2}}$. Using $E_1 = E_2 = 0, E_3 = E, V_{\mathrm{el}} = 0$ and $$\begin{gathered}
h^{12} = \frac{\sigma_{(z)} - f' \sigma_{(x)}}{\sqrt{1 + f'^2}} \nonumber \\
h^{23} = \frac{\sigma_{(x)} + f' \sigma_{(z)}}{\sqrt{1 + f'^2}} \nonumber \\
h^{31} = \sigma_{(y)} \end{gathered}$$ on $S$, we obtain the following Pauli Hamiltonian for the surface wave function: $$\begin{gathered}
H = - \frac{1}{2m} ( \partial_1^2 + \partial_2^2 )
+ V_S
\nonumber \\
- i \alpha
\Bigg(
\sigma_{(y)} \partial_1
- \frac{\sigma_{(x)} + f' \sigma_{(z)}}{\sqrt{1 + f'^2}} \partial_2
\Bigg)
,
\label{H_corrugated}\end{gathered}$$ where the term proportional to $\alpha$ is responsible for relativistic effects. We examine periodic corrugation on $S$ in what follows, putting $f(x) = h g(x)$, where the constant $h$ measures the height of the corrugation and $g$ is a periodic function. When the corrugation is absent $(h = 0)$, the system reduces to the ordinary Rashba system [@bib:Rashba] and the time-independent Pauli equation $H u = \varepsilon u$ has two plane wave solutions: $$\begin{gathered}
\varepsilon^0_\pm = \frac{k^2}{2m} \pm \alpha k, \,
u^0_\pm = \frac{e^{i(k_1 q_1 + k_2 q_2)}}{\sqrt{2}}
\begin{pmatrix}
1 \\
\pm i e^{i \theta_k}
\end{pmatrix}
,\end{gathered}$$ where $k_1$ and $k_2$ are the wave numbers, $k \equiv \sqrt{k_1^2 + k_2^2}$ and $\theta_k$ is the argument of $k_1 + ik_2$ on the complex plane. From the perturbation theory, the perturbed energy spectrum to the lowest order for small corrugation is evaluated as the expectation value of the perturbed Hamiltonian with respect to the unperturbed wave functions. Expansion of the Pauli Hamiltonian, eq. (\[H\_corrugated\]), in terms of $h$ leads to the lowest-order perturbation $$\begin{gathered}
H^{(1)} = i \alpha h g' \sigma_{(z)} \partial_2\end{gathered}$$ and the second lowest-order perturbation $$\begin{gathered}
H^{(2)} = - \frac{h^2 g''^2}{8m} - i \frac{ \alpha h^2 g'^2}{2} \sigma_{(x)} \partial_2
.\end{gathered}$$ Since $\langle u^0_\pm | H^{(1)} | u^0_\pm \rangle = 0$ due to the periodicity of $g$ and $\langle u^0_\pm | H^{(2)} | u^0_\pm \rangle = -\frac{h^2 \langle g''^2 \rangle}{8m} \mp \frac{\alpha }{2} h^2 \langle g'^2 \rangle k \sin^2 \theta_k$, the perturbed energy dispersion is given by $$\begin{gathered}
\varepsilon_\pm = \frac{k^2}{2m}
\pm \alpha k \Bigg( 1 - \frac{h^2 \langle g'^2 \rangle}{2} \sin^2 \theta_k \Bigg)
-\frac{h^2 \langle g''^2 \rangle}{8m}
,\end{gathered}$$ which indicates that any small periodic corrugation decreases the Rashba splitting.
In conclusion, I have provided the Pauli equation for a charged spin-$1/2$ particle confined to a curved surface under an electromagnetic field by performing the variable separation on the manner similar to that for the Schrödinger equation. Energy spectra of a sphere and a corrugated surface to which a particle is confined were calculated. It was demonstrated that any small periodic corrugation decreases the Rashba splitting on a surface. The basic equation obtained will be a powerful tool for analyses of physics on curved surfaces and nanostructures where spin degree of freedom and/or spin-orbit interactions play important roles.
The author would like to thank Takashi Miyake and Shoji Ishibashi for useful discussions. This work was partly supported by the Next Generation Super Computing Project, Nanoscience Program, and by a Grant-in-Aid for Scientific Research on Innovative Areas, “Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics” (No. 22104010) from MEXT, Japan.
[99]{}
H. Aoki and H. Suezawa: Phys. Rev. A [**46**]{} (1992) R1163.
J. Kim, I. Vagner and B. Sundaram: Phys. Rev. B [**46**]{} (1992) 9501.
M. V. Entin and L. I. Magarill: Phys. Rev. B [**64**]{} (2001) 085330.
V. Atanasov and A. Saxena: Phys. Rev. B [**81**]{} (2010) 205409.
M. Encinosa: Phys. Rev. A [**73**]{} (2006) 012102.
S. Ono and H. Shima: Phys. Rev. B [**79**]{} (2009) 235407.
N. Fujita and O. Terasaki: Phys. Rev. B [**72**]{} (2005) 085459.
H. Taira and H. Shima: J. Phys.: Condens. Matter [**22**]{} (2010) 075301.
V. Y. Prinz, D. Grützmacher, A. Beyer, C. David, B. Ketterer and E. Deckardt: Nanotechnology [**12**]{} (2001) 399.
S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya and N. Hatakenaka: Nature (London) [**417**]{} (2002) 397.
J. Onoe, T. Nakayama, M. Aono and T. Hara: Appl. Phys. Lett. [**82**]{} (2003) 595.
M. Sano, A. Kamino, J. Okamura and S. Shinkai: Science [**293**]{} (2001) 1299.
Z. Gong, Z. Niu and Z. Fang: Nanotechnology [**17**]{} (2006) 1140.
L. Sainiemi, K. Grigoras and S. Franssila: Nanotechnology [**20**]{} (2009) 075306.
B. S. DeWitt: Rev. Mod. Phys. [**29**]{} (1957) 377.
R. C. T. da Costa: Phys. Rev. A [**23**]{} (1981) 1982.
S. Takagi and T. Tanzawa: Prog. Theor. Phys. [**87**]{} (1992) 561.
G. Ferrari and G. Cuoghi: Phys. Rev. Lett. [**100**]{} (2008) 230403.
C. J. Isham, in [*Relativity, Groups and Topology II*]{}, ed. B. S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984).
N. P. Landsman, N. Linden: Nucl. Phys. B [**371**]{} (1992) 415.
Y. Ohnuki and S. Kitakado: J. Math. Phys. [**34**]{} (1993) 2827.
D. Mcmullan and I. Tsutsui: Ann. Phys. [**237**]{} (1995) 269.
J. J. Sakurai, [*Advanced Quantum Mechanics*]{} (Addison Wesley, Menlo Park, CA, 1967).
L. L. Foldy and S. A. Wouthuysen: Phys. Rev. [**78**]{} (1950) 29.
G. B. Arfken and H. J. Weber, [*Mathematical Methods for Physicists*]{}.
R. Szmytkowski: J. Math. Chem. [**42**]{} (2007) 397.
E. I. Rashba: Sov. Phys. Solid State [**2**]{} (1960) 1109.
|
---
abstract: 'In this paper, blind signal classification and detection in a non-orthogonal multiple access (NOMA) system are explored. Since a NOMA scheme superposes the multiple-user (MU) signals within nonorthogonal resources, classical modulation classification methods used in orthogonal multiple access (OMA) systems are not sufficient to process the superposed NOMA signal. NOMA receivers require information about the multiple access schemes such as modulation order and need interference cancellation of the co-scheduled user’s signal; therefore, a NOMA system causes more high-layer signaling overheads than OMA during packet scheduling. Blind detection algorithms used for multiplexing information are considered to be possible solutions; however, they pose various challenges and could cause performance loss while performing blind modulation classification in the order of 1) OMA/NOMA classification, 2) co-scheduled user’s modulation classification, and 3) classification of the signal, due to the necessity for successive interference cancellation (SIC). To improve the performance of blind detection, we propose a NOMA transmission scheme that applies phase-rotation to data or pilot symbols depending on the NOMA multiplexing format, as an aid to the blind detection. The proposed classification algorithm can implicitly provide essential information on NOMA multiplexing without the need for any extra high layer signaling or resources. The performance improvement is verified through simulation studies, and it is found that the proposed algorithm provides a gain of more than 1 dB compared to the existing blind signal classification methods and shows almost equivalent performance as the genie information scheme.'
author:
- '[^1] [^2] [^3] [^4] [^5]'
title: 'Blind Signal Classification for Non-Orthogonal Multiple Access in Vehicular Networks'
---
[IEEE Transactions on Vehicular Technology (Submission)]{}
Non-orthogonal multiple access (NOMA), blind signal classification, signaling overhead, spectrum efficiency, 5G-enabled vehicular networks
Introduction {#sec:Introduction}
============
To utilize the radio spectrum efficiently for a massive number of user terminals (UTs), non-orthogonal multiple access (NOMA) based on power multiplexing has been widely studied [@OMA_vs_NOMA:Wang; @TVT:1; @TVT:2; @NOMA_basic:Docomo-Saito; @NOMA_basic:Higuchi]. In particular, NOMA has been actively researched as a promising technology to improve system performance in 5G networks[@NOMA_5G:Dai; @NOMA_5G:Ding; @NOMA_5G:Islam] and to provide robustness in high-mobility vehicular networks [@NOMA_VN:Di; @NOMA_VN:Qian]. The 3rd Generation Partnership Project (3GPP) has studied deployment scenarios and receiver designs for NOMA systems in Rel-14 in the context of a working item labeled multiple user superposition transmission (MuST)[@MuST:3GPP].
NOMA superposes the multiple-user (MU) signals within the same frequency, time, or spatial domain, and therefore successive interference cancellation (SIC) is generally considered to detect non-orthogonally multiplexed signals [@NOMA_basic:Docomo-Saito; @NOMA_basic:Higuchi]. Theoretically, NOMA is known to provide significant benefits in terms of improving the cell throughput[@Book:Tse]; nevertheless, such gains can be obtained only when the receiver is able to cancel or sufficiently suppress interference from co-scheduled users. NOMA has been extensively researched in conjunction with various technologies on the basis of the well-designed SIC. Studies have been conducted on a system that applies NOMA to multiple-input multiple-output (MIMO) systems [@NOMA-MIMO:TWC2016Ding; @NOMA-MIMO:TWC2016Ding2], and studies on NOMA in cooperative networks were reported in [@NOMA:CL2015Ding; @NOMA:TWC2018Choi]. In addition, a distributed NOMA scheme without the requirement of SIC for the Internet of Things (IoT) was studied in [@IOTJ2019Liu2]. In most existing studies on NOMA, an ideal SIC with a knowledge of the channel state information (CSI) was assumed, but in the recent studies in [@NOMA:TVT2018Gui; @IOTJ2019Liu], imperfect CSI for NOMA was handled using deep learning.
For orthogonal multiple access (OMA), there have been extensive research efforts on blind modulation classification (MC). MC was originally developed for military applications such as electronic warfare; therefore, most of the existing MC techniques were developed for systems having no knowledge of the signal amplitude, phase, channel fading characteristics, and noise distribution[@MC-ML-unknown1:Hameed; @MC-ML-unknown2:Erfan]. In [@MC-ML:Wei], a maximum likelihood (ML)-based classifier was presented to provide optimal performance in the presence of white Gaussian noise when candidate modulation schemes are equally probable. However, ML-based classification requires high computational complexity; therefore, the feature-based approaches for blind MC presented in [@MC-feature:ICCSPA2013Hazza] take advantage of the fact that good statistical features allow robust blind MC. Furthermore, convolutional neural network (CNN)-based feature extraction methods were recently addressed in [@MC-ML:TCCN2018Rajendran; @CNN-MC:DySPAN2017West; @CNN-MC:TVT2018Meng; @CNN-MC:TCCN2018Tian]. However, the existing studies on blind signal classification were primarily limited to MC techniques in an OMA system.
As NOMA has become popular and is being implemented in many applications, blind MC is also important in the NOMA system. Since the MU signals are superposed in the NOMA system, additional and necessary information of the signal modulation is required. Before attempting to handle interference, the NOMA receiver must first determine the presence of co-scheduled users. If SIC is to be used, the modulation order and power allocation ratio of the co-scheduled users should also be known to the receiver. In summary, the blind MC steps required for the NOMA receiver are as follows: 1) OMA/NOMA classification, 2) co-scheduled UTs’ MC, and 3) classification of the signal based on the necessity of SIC. Therefore, we refer to these classification steps in the NOMA system as blind signal classification, not to be confused with blind MC for OMA. Blind classification of something else, e.g., channel coding rate, can be considered, but this paper basically focuses on blind MC in the NOMA system; therefore, the classification steps for information not related to signal modulation are not considered in this paper.
The information required for signal classification and data detection can be transmitted to the receiver via a high layer; however, the required signaling overhead is a concern as more information of the signal is needed for decoding at the NOMA receiver. In particular, in vehicular networks with limited energy and resources (e.g., time limits due to the high mobility in vehicular networks) [@NOMA_VN:Di; @NOMA_VN:Qian], this motivates the use of blind signal classification at the receiver side followed by appropriate data detection. Moreover, vehicular networks must cope with periodic short burst communications related to safety information and alarm services [@CommMag2013Araniti; @TVT2016Kim]. Because the concern about signaling overheads becomes more critical in the case of short burst communications, blind signal classification could emerge as a promising technology to reduce these overheads in vehicular communications [@TVT2017Majhi].
While the existing MC techniques are aimed at determining the UT’s modulation, the NOMA receiver attempts to classify the co-scheduled user’s modulation to perform SIC. In addition, the NOMA system should perform OMA/NOMA classification, as well as determine whether SIC is required for the received signal. The blind MC technique in the context of NOMA has been actively studied in 3GPP, for example, [@MuST:3GPP] provides throughput analysis of ML-based blind MC in NOMA systems. However, the improved blind MC technique has not been well explored in academia. The recent study in [@ArXiv2019Choi] was focused on signal classification only with respect to the necessity of SIC, and both power allocation and the user scheduling scheme, which guarantee a reliable classification performance, were jointly optimized. In this paper, the performance effect of errors in blind NOMA signal classification is analyzed and the ensuing receiver challenges in practical MU cases are addressed. In addition, two transmission policies are proposed for improving the performances of ML-based blind NOMA signal classification.
The main contributions of this paper are summarized as follows:
- Signal to interference plus noise ratio (SINR) and user capacity analyses to determine the effect of errors on performance in the three blind NOMA signal classification steps are presented.
- A phase-rotated modulation is proposed for blind NOMA signal classification. The rotated data symbols implicitly render the constellations of modulation formats as blind detection aids. This method is based on the existing ML-based classification algorithm [@MC-ML:TCCN2018Rajendran].
- A pilot-rotation transmission method and the corresponding new signal classification algorithm are proposed. By using the algorithm, a receiver estimates the rotation value of pilots and utilizes it for blind signal classification. Since the proposed scheme depends only on the rotated phases of the pilots and not on the pilot values, it requires no extra pilot overhead.
- The presented numerical results verify the performance analysis of blind signal classification in the NOMA system. Moreover, the proposed phase-rotated modulation and the pilot-rotation transmission scheme are shown to provide better classification performances than the conventional ML-based method.
The rest of the paper is organized as follows. The NOMA system model and the blind classification steps for received NOMA signals are described in Section \[sec:Preliminaries\]. SINR analysis for the three steps of blind NOMA signal classification and the capacity of a NOMA UT in the presence of signal classification errors are provided in Sections \[sec:SINR\_analysis\] and \[sec:Capacity\_analysis\_for\_blind\_modulation\_Detection\], respectively. The proposed phase-rotated modulation is described in Section \[sec:phase\_rotated\_modulation\]. In Section \[sec:Pilot-Reuse\_based\_blind\_OMA/NOMA\_and\_Modualtion\_detection\], the proposed pilot-rotation transmission method and the corresponding new signal classification algorithm are presented. In Section \[sec:SimulationResults\], the performance improvements of the proposed algorithms are verified using the results of rigorous numerical simulations.
System Models {#sec:Preliminaries}
=============
Non-Orthogonal Multiple Access Signal Model and Receiver Structure
------------------------------------------------------------------
In downlink power-multiplexing NOMA, a base station (BS) intentionally superposes the signals for multiple UTs with different power weightings. However, when information about the transmitted NOMA signal is unknown at the receiver side, the computational complexity of the ML-based signal classification in the NOMA system grows significantly with the number of co-scheduled UTs [@ArXiv2019Choi]. Therefore, this paper considers a two-user NOMA system. The received signal in a two-user downlink NOMA transmission is given by $$y = h(s_f + s_n) + w,
\label{eq:received}$$ where $y, s, h,$ and $w$ correspond to the received signal, transmitted symbol, channel gain, and thermal noise, respectively, and the subscripts $f$ and $n$ denote far and near UTs. In addition, $\mathbb{E}[|s_f|^2] = P_f$ and $\mathbb{E}[|s_n|^2] = P_n$, where $P_f$ and $P_n$ are the power allocations to two UTs. A BS normally schedules UTs having a large channel gain difference and allocates larger power to a far (weak) UT to compensate its low channel gain, i.e., $P_f>P_n$. Suppose that there is a normalized power constraint, $P_f + P_n = 1$. When its power allocation is large, the far UT does not perform SIC and only detects its data while ignoring the near UT’s signal. Meanwhile, the near UT requires SIC to cancel the far UT’s signal; therefore, only the near UT is considered as a NOMA-serviced user in general. For this reason, all the statements in this paper are focused on the near UT of the NOMA system.
When the near UT performs SIC, interference, i.e., the far UT’s signal, is regenerated from the decoder or the detector, corresponding to the codeword-level interference cancellation (CWIC) or symbol-level interference cancellation (SLIC), respectively [@VTC2015Yan]. In this study, CWIC is mainly utilized to mitigate the intra-cell interference unless otherwise noted; signal classification is required before CWIC is applied.
Let $\mathcal{M}=\{M_0, M_{1}, \cdots, M_{L} \}$ be a set of modulation modes, including $L$ NOMA modes, $M_{l}$ for $l=1,\cdots,L$, and an OMA mode, $M_0$. The constellation set of the modulation mode $M_l$ is denoted by $\chi_l$ for all $l\in\{0,\cdots,L\}$. For $l\in \{1,\cdots,L\}$, $\chi_l$ is constructed by combinations of power-scaled near and far UTs’ constellation sets, $\chi_l = \chi_l^f \bigoplus \chi_l^n$, where $\chi_l^f$ and $\chi_l^n$ are power-scaled constellation sets of the near and far UTs, respectively. Therefore, the average powers of symbols in $\chi_l^f$ and $\chi_l^n$ are $P_f$ and $P_n$, respectively. In addition, let $\mathcal{N}$ be a set of the constellation points of all NOMA modes; i.e., $\mathcal{N} = \chi_1 \cup \cdots \cup \chi_L$.
Maximum Likelihood-based Signal Classification
----------------------------------------------
The existing ML-based MC algorithm [@MC-ML:Wei], which is optimal in OMA based on hypotheses testing, can be directly applied to NOMA signal classification. We define some hypotheses to identify the received signal information:
- $\mathcal{H}_l$: the hypothesis of the signal modulated by the $l$-th mode $M_l$ for all $l \in \{0,1,\cdots,L\}$
- $\mathcal{H}_N$: the hypothesis of the signal, i.e., $\mathcal{H}_N = \mathcal{H}_1 \cup \cdots \cup \mathcal{H}_L$.
- $\mathcal{H}^f$: the hypothesis of the signal that does not require SIC.
- $\mathcal{H}^n$: the hypothesis of the signal for which SIC is necessary.
- $\mathcal{H}_{l}^f$: the hypothesis of the signal that is modulated by the $l$-th NOMA mode and does not require SIC for all $l\in \{ 1,\cdots,L\}$.
- $\mathcal{H}_{l}^n$: the hypothesis of the signal that is modulated by the $l$-th NOMA mode and requires SIC for all $l \in \{1,\cdots,L\}$.
The ML-based hypothesis testing can classify the received signal according to whether it is modulated by OMA or NOMA, which modulation and power weightings are used, and whether SIC is required or not. For example, suppose that the transmitted signal is modulated by the $l$-th modulation mode for $l\in\{1,\cdots,L\}$ and that the target signal can be decoded after performing SIC, i.e., $\mathcal{H}_l^n$ is true. Then, the likelihood probability of hypothesis $\mathcal{H}_l^n$ is computed by $$p(y| \mathcal{H}_l^n) = \frac{1}{|\chi_l|} \sum_{s \in \chi_l}{\frac{1}{\pi \sigma^2}e^{-\frac{| y - hs|^2}{\sigma^2}}},$$ where $\sigma_n^2$ is the noise variance and $|\chi_l^n|$ is the number of symbols in $\chi_l^n$. If $K$ symbols are used for blind signal classification and are not correlated, the joint likelihood function of the $K$ symbols of $\mathbf{y}=[y_1, \cdots, y_K]$ is given by $$\Gamma(\mathbf{y}|\mathcal{H}_l^n) = \prod_{k=1}^K p(y_k| \mathcal{H}_l^n).$$ According to the ML criterion, the detected hypothesis $\hat{\mathcal{H}}$ can be determined by $$\hat{\mathcal{H}} = \underset{\xi \in \mathcal{H}}{\arg\max}~ \Gamma( \mathbf{y} | \xi),
\label{eq:ML-MC_whole}$$ where $\mathcal{H}=\{ \mathcal{H}_0, \mathcal{H}_{1}^f, \cdots, \mathcal{H}_{L}^f, \mathcal{H}_{1}^n, \cdots, \mathcal{H}_{L}^n \}$.
If $\hat{\mathcal{H}} = \mathcal{H}_0$, the receiver determines that the signal is modulated by OMA. However, if $\hat{\mathcal{H}} = \mathcal{H}_{l}^f$, then the received signal is classified as a NOMA signal modulated by $M_l$, which does not require SIC. In addition, $\hat{\mathcal{H}} = \mathcal{H}_{l}^n$ represents that the received signal is a NOMA signal modulated by $M_l$ for which SIC is necessary. However, the accuracy of hypothesis testing is significantly degraded as the number of hypotheses grows. Therefore, this paper considers the three-step classification framework for the NOMA signal to reduce the number of hypotheses in each classification step as follows: OMA/NOMA classification, MC (i.e., modulation order and power weightings), and near/far UT classification (i.e., the necessity for SIC). The relevant likelihood probabilities and the hypothesis testing results can be computed by
### OMA/NOMA Classification
\[t!\] ![Processes of maximum likelihood-based signal classification in non-orthogonal multiple access systems[]{data-label="Fig:MLPRM_scenario"}](Final_BSC_steps.pdf "fig:"){width="45.00000%"}
$$\begin{aligned}
p(y|\mathcal{H}_0) &= \frac{1}{|\chi_0|} \sum_{s \in \chi_0} {\frac{1}{\pi \sigma^2}e^{-\frac{|y-hs|^2}{\sigma^2}}}
\label{MLprobOMA}\\
p(y|\mathcal{H}_{N}) &= \frac{1}{|\mathcal{N}|} \sum_{s \in \mathcal{N}}{ \frac{1}{\pi \sigma^2}e^{-\frac{|y - hs|^2}{\sigma^2}} }
\label{MLprobNOMA}\\
\hat{\mathcal{H}} &= \underset{\xi \in \{\mathcal{H}_0, \mathcal{H}_N \}}{\arg\max} \Gamma (y|\xi);
\label{ML OMA/NOMA}
\end{aligned}$$
### Modulation Classification
$$\begin{aligned}
p(y|\mathcal{H}_{l}) &= \frac{1}{|\chi_l|} \sum_{s \in \chi_l}{ \frac{1}{\pi \sigma^2}e^{-\frac{|y - hs|^2}{\sigma^2}} }
\label{MLProb:SumExp}\\
\hat{\mathcal{H}} &= \underset{\xi \in\{ \mathcal{H}_1,\cdots,\mathcal{H}_L\}}{\arg\max} \Gamma (y|\xi);
\label{MLdetectedMod:SumExp}
\end{aligned}$$
### Near/Far UT Classification
$$\begin{aligned}
p(y|\mathcal{H}_{l}^n) &= \frac{1}{|\chi_{l}|} \sum_{s \in \chi_{l}}{\frac{1}{\pi \sigma^2}e^{-\frac{|y - hs|^2}{\sigma^2}}}
\label{ML_nearUT}\\
p(y|\mathcal{H}_{l}^f) &= \frac{1}{|\chi_{\hat{l}}^f|} \sum_{s \in \chi_{\hat{l}}^f}{ \frac{1}{\pi \sigma^2}e^{-\frac{|y - hs|^2}{\sigma^2}} } \\
\hat{\mathcal{H}} &= \underset{\xi \in\{\mathcal{H}_l^n, \mathcal{H}_l^f \}}{\arg\max} \Gamma (y|\xi).
\label{ML_farUT}
\end{aligned}$$
The overall steps of the ML-based signal classification in the NOMA system are shown in Fig. \[Fig:MLPRM\_scenario\]. In summary, OMA/NOMA classification should be performed first, followed by the classification of the modulation orders and power ratios of the UTs. Near/far UT classification is the last step because it requires $\chi_l$ and $\chi_l^f$, whose modulation mode is already given. This paper investigates the additional classification steps required for NOMA compared to OMA; therefore, we assume that a UT already knows its own modulation order, whose classification was extensively studied earlier. Then, if the far and near UTs’ modulation orders are different, the UT does not have to perform near/far UT classification; otherwise, near/far UT classification is necessary.
The hierarchical classification steps can reduce the computational dimension and increase the accuracy of hypothesis testing as compared to that performed with respect to the whole set of modulation formats in , in which $2L+1$ hypotheses are compared. In the proposed three-step classification framework, both OMA/NOMA and near/far UT classification compare only two hypotheses each, and $L$ hypotheses are compared in the MC step. In addition, a decrease in the number of compared hypotheses also reduces computational complexity.
SINR Analysis for Non-Orthogonal Multiple Access UT with Signal Classification Errors {#sec:SINR_analysis}
=====================================================================================
In this section, the effects of signal classification errors on the SINR are examined. Again, the near UT is considered only as the NOMA-serviced user.
OMA/NOMA and Near/Far UT Classification Errors
----------------------------------------------
When the BS transmits a NOMA signal, but the near UT incorrectly classifies it as an OMA signal, severe performance degradation is expected. Even though the transmitted signal contains the far UT’s signal component, incorrect OMA/NOMA classification causes the receiver to take no action to remedy the interference; i.e., it would not perform SIC. Similar results occur when an OMA signal is transmitted but the receiver classifies the signal as NOMA. In this case, the receiver performs SIC, but there is no interference in the OMA signal. Both cases do not guarantee a reliable classification performance. Therefore, in this paper, the throughput is reasonably considered to be zero when an OMA/NOMA classification error occurs.
Similarly, incorrect near/far UT classification significantly degrades the system performance. If incorrect near/far UT classification occurs, the far UT of the NOMA system cancels the target signal, and the near UT does not perform SIC. Therefore, an error in the near/far UT classification step is also assumed to yield no throughput. The classification results of the far UT’s modulation order and power ratio become meaningful only when the signal is classified as NOMA and near UT.
Power Ratio Classification Errors {#subsec:SINR_deg_PowerDetection}
---------------------------------
For the NOMA system, there are some modulation modes that have the same modulation orders but different power ratios for two UTs. The MC among these modes can be interpreted as power ratio classification. Although the receiver incorrectly classifies the power ratio as one of the competing modulation modes, the transmitted symbols can still be detected correctly if the incorrectly classified modulation mode has a constellation point indicating the same bit-labeling as the transmitted one. However, the SINR could be degraded because of erroneous power ratio classification.
For simplicity, consider a flat fading channel and two competing modulation modes, $M_1$ and $M_2$, having the same order but different power allocation ratios for two NOMA UTs. Suppose that the transmitted signal is modulated by $M_1$; then, the received signal is given by $$y = h( s_{f,1}(i) + s_{n,1}(k)) + w,
\label{eq:received_nearUT}$$ where $s_{f,1}(i)$ and $s_{n,1}(k)$ are the $i$-th and $k$-th symbols in $\chi_1^f$ and $\chi_1^n$ for the far and near UTs, respectively, and $\mathbb{E}_i[|s_{f,1}(i)|^2]=P_{f,1}$ and $\mathbb{E}_k[|s_{n,1}(k)|^2]=P_{n,1}$.
Assuming perfect SIC, correct MC yields an SINR of $$\eta_{1\rightarrow1} = \frac{P_{n,1}}{\tilde{\sigma}^2},
\label{eq:SINR_n,1->1}$$ where $\tilde{\sigma}^2 = \frac{\sigma^2}{|h|^2}$. The subscript $l\rightarrow m$ means that the transmitted mode is $M_l$, but $M_m$ is determined. Let $\hat{s}_f$ be the interference component for the near UT to be subtracted from the received signal , which is regenerated by SIC. Then, the signal after SIC is denoted by $$\begin{aligned}
y_{SIC} &= h(s_{f,1}(i)+ s_{n,1}(k) - \hat{s}_f) + w \\
&= h s_{n,2}(k) + h (s_{f,1}(i)- \hat{s}_f) \nonumber \\
&~~~+ h (s_{n,1}(k)-s_{n,2}(k)) + w.
\label{eq:y_SIC}
\end{aligned}$$ Although the classification of power ratio is incorrect, the data detection could still be correct if $s_{n,2}(k)$, whose bit-labeling is the same as that of the transmitted signal $s_{n,1}(k)$, is detected. Therefore, the SINR becomes $$\begin{aligned}
&\eta_{1\rightarrow2} = \nonumber \\ &~~~\frac{P_{n,2}}{\mathbb{E}_i[|s_{f,1}(i)-\hat{s}_f|^2]+\mathbb{E}_k[|s_{n,1}(k)-s_{n,2}(k)|^2]+\tilde{\sigma}^2}.
\label{eq:SINR_n,1->2}
\end{aligned}$$
Note that, when $P_{n,1} \geq P_{n,2}$, incorrect classification of power ratio obviously results in SINR degradation; i.e., $\eta_{1\rightarrow 1} - \eta_{1\rightarrow 2} \geq 0$. However, if $P_{n,1} < P_{n,2}$, $\eta_{1\rightarrow 1} - \eta_{1\rightarrow 2} \geq 0$ only when $$\begin{aligned}
P_{n,1}&(\mathbb{E}_i[|s_{f,1}(i)-\hat{s}_f|^2] + \mathbb{E}_k[|s_{n,1}(k)-s_{n,2}(k)|^2] + \tilde{\sigma}^2) \nonumber \\
& \geq P_{n,2}\tilde{\sigma}^2.
\label{eq:PowerDetection_condition}
\end{aligned}$$ If (\[eq:PowerDetection\_condition\]) is not satisfied, the SINR can increase even when the power ratio is incorrectly classified. Accordingly, power ratio classification becomes more important as the power allocated to the near UT, i.e., $P_{n,l}$, increases.
Suppose that the same index $i$ of $s_{f,1}(i)$ and $s_{f,2}(i)$ indicates the same bit-labeling. Then, when the power ratio classification is incorrect, it is highly likely that the interference regenerated by SIC is $\hat{s}_f = s_{f,2}(i)$ without a large $\tilde{\sigma}^2$. In this case, the inequality holds in the high SNR region.
Modulation Order Classification Errors
--------------------------------------
Consider two competing modes, $M_1$ and $M_2$, whose far UTs have different modulation orders. It is very important for the near UT to find the modulation order of the far UT in order to perform SIC appropriately. If the modulation order of the far UT is incorrectly classified, the interference estimated by SIC, i.e., $\hat{s}_f$, also becomes incorrect. Then, the signal after SIC and the SINR of $\eta_{1 \rightarrow 2}$ are the same as and , respectively.
Comparing with section \[subsec:SINR\_deg\_PowerDetection\], the only difference is that $\hat{s}_f$ is regenerated by SIC based on a competing mode whose modulation order is different from that of the transmitted mode, rather than power coefficient. In section \[subsec:SINR\_deg\_PowerDetection\], $\hat{s}_f$ can have the same bit-labeling as the transmitted symbol; however, $\hat{s}_f$ cannot because the modulation order of $M_2$ is different from that of the transmitted mode of $M_1$.
Similar to the power ratio classification error described in Section \[subsec:SINR\_deg\_PowerDetection\], a classification error of the far UT’s modulation order always causes SINR degradation at the near UT when $P_{n,1} \geq P_{n,2}$. However, when $P_{n,1} < P_{n,2}$, $\eta_{1\rightarrow 1}-\eta_{1\rightarrow 2} > 0$ only when the system satisfies the condition given by . Again, (\[eq:PowerDetection\_condition\]) can be satisfied in the high SNR region. If is not satisfied, the SINR is not degraded even when the far UT’s modulation order is incorrectly classified. However, this situation is considerably less likely to happen than when the power ratio is incorrectly classified but the SINR does not decrease. The reason is that $|s_{f,l}(i)-s_{f,u}(k)| > |s_{f,l}(i)-s_{f,v}(i)|$ in general, with the assumption that the far UT’s modulation orders of $M_l$ and $M_u$ are different, while its modulation orders of $M_l$ and $M_v$ are the same but with different power ratios.
An additional problem results from incorrect classification of a far UT’s modulation order. As mentioned in Section \[sec:Preliminaries\], the decision feedback for SIC can be generated at either the symbol or codeword level. However, incorrect classification of the far UT’s order does not allow the use of CWIC because of a mismatch in the codeword length. Because CWIC outperforms SLIC substantially, this is highly undesirable for the system-level performance.
Capacity of Non-Orthogonal Multiple Access UT with Signal Classification Errors {#sec:Capacity_analysis_for_blind_modulation_Detection}
===============================================================================
By performing SINR analysis, we can compute the capacity of a NOMA UT, which is denoted by $C$, including the effects of signal classification errors. With the assumption that the transmitted mode is $M_{l}$, let $p_{l \rightarrow m}$ be the probability that the classified modulation mode is $M_m$. The user capacity can then be computed as $$\begin{aligned}
C = \sum\limits_{l=0}^L \pi_l \mathbb{E}_h \Big[&\Big\{ p_{l \rightarrow l} q_l^n \log_2 (1+\eta_{l \rightarrow l}) \nonumber \\
&~ + \sum_{m \neq l} p_{l \rightarrow m} q_m^n \log_2(1+\eta_{l \rightarrow m}) \Big\} \Big],
\label{eq:capacity}
\end{aligned}$$ where $\pi_l$ is the probability that the signal is modulated by $M_l$ for all $l \in \{0,\cdots,L \}$ and $q_l^n$ is the probability that the signal of the modulation mode $M_l$ is determined to perform SIC. The equally probable modulation mode is assumed, i.e., $\pi_l=\frac{1}{L}$. Again, because only the near UT of the two-user NOMA system represents the NOMA user, the capacity in is achieved for the near UT case. In addition, $\eta_{l \rightarrow m}$ is achieved, as explained in Section \[sec:SINR\_analysis\]. Because incorrect OMA/NOMA and near/far classifications give rise to zero throughput, $\eta_{0\rightarrow l} = 0$, $\eta_{l\rightarrow 0} = 0$ for $l\neq0$, and only $q_l^n$ are considered, while the situation in which SIC is not necessary is not.
The performance of ML-based signal classification strongly depends on how well the constellations of the competing modulation modes can be distinguished from one another. To quantify this effect, we denote the minimum distance between constellation sets of two different modulation modes of $M_l$ and $M_m$ by $d_{\text{min}}(M_l, M_m)$, $l \neq m$. In general, $d_{\text{min}}$ can be defined for $L$ modulation modes as follows: $$\begin{aligned}
\forall s_1 \in \chi_1,~&\cdots,~\forall s_L \in \chi_L, \nonumber \\
&d_{\text{min}}(M_1, ..., M_L) = \min d(s_1,...,s_L).
\end{aligned}$$
\[t!\] ![Legacy constellations of two modulation modes[]{data-label="Fig:legacy_constellation"}](Final_legacy_constellation.pdf "fig:"){width="30.00000%"}
\[t!\] ![Rotated constellations of two modulation modes[]{data-label="Fig:rotated_constellation"}](Final_rotated_constellation.pdf "fig:"){width="30.00000%"}
Fig. \[Fig:legacy\_constellation\] shows the constellation sets of two competing modulation modes, $M_1$ and $M_2$. In this figure, $d_{\text{min}}(M_l, M_m)$ is the distance between two closest points from different modes, as marked by the dashed circles. These symbols are very close to each other; therefore, when ML-based signal classification is performed, these pairs are expected to be the main cause of incorrect classification. In this example, if all the symbol points are equally probable, the probability of a classification error can be computed as $$\begin{aligned}
p_{l\rightarrow m} &= \frac{1}{|\chi_l| |\chi_m| } \sum_{i=1}^{|\chi_l|} \sum_{k=1}^{|\chi_m|} Q\Big(\frac{|h(s_l(i)-s_m(k))|/2}{\sigma/\sqrt{2}}\Big)
\label{eq:modErrProb:allsymbol} \\
&\approx \frac{N_{\text{min}}}{|\chi_l| |\chi_m|} Q\Big(\frac{h\cdot d_{\text{min}}(M_l, M_m)/2}{\sigma/\sqrt{2}}\Big),
\label{eq:modErrProb:d_min}
\end{aligned}$$ where $s_l(i)$ is the $i$-th constellation point of $M_l$.
In (\[eq:modErrProb:allsymbol\]), let $s_m(k)$ be the closest to $s_l(i)$ among the constellation points of $M_k$. Then, the case where $s_l(i)$ is confused with $s_m(k)$ becomes one of the most frequent classification errors. In this context, (\[eq:modErrProb:d\_min\]) is approximated one step further by considering only the symbol pairs from different modes, providing the minimum distance $d_{\text{min}}(M_l,M_m)$. In addition, $N_{\text{min}}$ is the total number of symbol pairs providing the minimum distance $d_{\text{min}}(M_l, M_m)$, and $N_{\text{min}}=4$ in Fig. \[Fig:legacy\_constellation\].
We remark some conclusions of analysis here. First, we can determine user capacity analytically by substituting the expressions for the SINR and classification error probability in (\[eq:capacity\]). Second, the trade-off between the classification error probability of $p_{l\rightarrow m}$ and the SINR degradation term of $\eta_{l\rightarrow m}$ is observed in . According to , the classification error rate decreases with $d_{\text{min}}(M_l, M_m)$. However, if an MC error occurs, further demodulation process got meaningless, as the SINR is dramatically degraded with increasing $d_{\text{min}}(M_l, M_m)$. Third, selection of the appropriate power allocation ratios for each NOMA modulation mode which maximizes capacity in , can be beneficial. A maximum capacity is achieved with maximum power allocation on a near UT, but a far UT needs minimum power constrain in a reasonable problem setup.
Phase-Rotated Modulation\
based on ML Signal Classification {#sec:phase_rotated_modulation}
=================================
In this section, we propose a phase-rotated modulation method to increase the accuracy of signal classification. As explained in Section \[sec:Capacity\_analysis\_for\_blind\_modulation\_Detection\], the reliability of ML-based signal classification strongly depends on how well the competing modes are distinguished from one another. Based on this observation, different phase rotations are assigned to individual modulation modes so that their constellation points are more effectively separated.
A comparison of Fig. \[Fig:legacy\_constellation\] and Fig. \[Fig:rotated\_constellation\], which show the legacy and phase-rotated composite constellations of two different modulation modes, respectively, demonstrates this idea. In Fig. \[Fig:rotated\_constellation\], $M_1$ is rotated by $\theta$; therefore, it becomes $M_1e^{j\theta}$. The symbol pairs from different modes giving the minimum distance of $d_{\text{min}}(M_1e^{j\theta}, M_2)$ are marked by the dashed ellipses in Fig. \[Fig:rotated\_constellation\]. Because $d_{\text{min}}(M_1e^{j\theta}, M_2) > d_{\text{min}}(M_1, M_2)$, it is easily expected that the phase-rotated modulation provides a lower classification error probability than the legacy modulation method. If the same phase rotation is applied to every modulation mode, the minimum distance $d_{\text{min}}(M_1, M_2)$ between the rotated composite constellations of $M_1 e^{j\theta}$ and $M_2 e^{j\theta}$ remains unchanged. Accordingly, the application of different phase rotations to different modulation modes is a key point. We can make the phase list $\Theta =\{\theta_0, \theta_{1}, \cdots, \theta_{L} \}$, whose element $\theta_{l}$ corresponds to $M_{l}$ for all $l \in \{0,\cdots,L \}$. The modulation mode table is updated to include the phase rotations, as shown in Table \[Table:ModePhaseExample\].
Modulation Mode $M_0$ $M_{1}$ $\cdots$ $M_{L}$
-------------------- ------------ --------------------- ---------- ---------------------
Modulation orders $(m_O,-)$ $(m_{f,1},m_{n,1})$ $\cdots$ $(m_{f,L},m_{n,L})$
Power coefficients $(P,-)$ $(P_{f,1},P_{n,1})$ $\cdots$ $(P_{f,L},P_{n,L})$
Phase rotations $\theta_0$ $\theta_{1}$ $\cdots$ $\theta_{L}$
: Example of Modulation Mode Table for Phase-Rotated Modulation[]{data-label="Table:ModePhaseExample"}
However, a larger $d_{\text{min}}(M_1e^{j\theta},M_2)$ does not guarantee better user capacity, because the SINR terms in (\[eq:capacity\]) would be changed. We have mentioned the tradeoff between $p_{l \rightarrow m}$ and $\eta_{l \rightarrow m}$ in in Section \[sec:Capacity\_analysis\_for\_blind\_modulation\_Detection\]; although phase-rotated modulation can provide a lower classification error rate, the SINR degradation due to incorrect signal classification would be larger than that in legacy modulation. Therefore, we should carefully find the phase rotation values that maximize the user capacity by balancing the tradeoff between the classification error probability and the SINR degradation due to incorrect classification. We can formulate the optimization problem to find the phase rotation list $\Theta$ as follows:
$$\Theta = \arg \max_{\Theta=\{\theta_0 , \cdots,\theta_{L}\}}{C}.
\label{eq:optimization_Cmax}$$
The above optimization problem is difficult to solve theoretically because the expectation in is taken over random channel realizations; we have obtained the optimal phase rotations numerically in this paper. When there exist many modulation modes, however, numerical determination of all the rotation values requires excessively massive computations. In this study, phase-rotated modulation was applied for only OMA/NOMA classification, i.e., $\theta_0 \neq 0$ and $\theta_{l}=0$ for all $l \in \{1,\cdots,L\}$, because incorrect OMA/NOMA classification results in zero throughput, as shown in Section \[sec:SINR\_analysis\]. However, incorrect determination of the far UT’s modulation order or power allocation ratio is not as critical as OMA/NOMA misclassification.
On the other hand, near/far UT classification is not affected by phase-rotated modulation. According to (\[ML\_nearUT\]) and (\[ML\_farUT\]), near/far UT classification depends on the constellation structures of $\chi_l$ and $\chi_l^f$. Phase-rotated modulation changes the constellations to $\chi_l e^{j\theta_{l}}$ and $\chi_l^f e^{j \theta_{l}}$, but the minimum distance between them is not changed. Hence, phase-rotated modulation cannot influence the performance of ML-based near/far UT classification.
Phase-rotated Pilot reuse-based
----------------------- --------------- -------------------
OMA/NOMA
classification
Modulation
classification
Near/Far UT
classification
Additional overhead x x
Additional constraint x o
Complexity high low
: Characteristics of the proposed blind signal classification schemes[]{data-label="table:BSC_comp"}
Pilot Reuse-based Signal Classification {#sec:Pilot-Reuse_based_blind_OMA/NOMA_and_Modualtion_detection}
=======================================
In Section \[sec:phase\_rotated\_modulation\], the ML-based phase-rotated modulation scheme was proposed, which uses data symbols for blind signal classification, and we discussed the manner in which user capacity and classification accuracy change depending on the phase rotations. However, because a tradeoff exists between the classification accuracy and SINR degradation in the case of a classification error, the ML-based phase-rotated modulation scheme is limited in terms of improving the blind classification performance. Therefore, a pilot-based scheme that does not affect the SINR is presented in this section; however, this scheme requires an additional constraint, as explained below. This signal classification algorithm based on pilot reuse requires no changes in the modulation scheme. In addition, phase-rotated modulation cannot improve near/far UT classification, which means that phase rotations of data symbols do not influence the near/far UT classification performance. Therefore, the phase rotations assigned to pilots are considered.
The various processes of the pilot reuse-based signal classification algorithm are shown in Fig. \[Fig:PRBD\_scenario\]. The OMA/NOMA classification and modulation order and power ratio selection are conducted simultaneously. Then, near/far UT classification is executed. A comparison of the ML-based phase rotated modulation and the pilot reuse-based signal classification scheme is briefly shown in Table \[table:BSC\_comp\].
\[t!\] ![Processes of pilot reuse-based signal classification[]{data-label="Fig:PRBD_scenario"}](Final_PR_scenario.pdf "fig:"){width="47.00000%"}
Pilot Reuse-based OMA/NOMA and Modulation Classifications
---------------------------------------------------------
In contrast to phase-rotated modulation, phase rotations for existing pilots are introduced in this subsection. The data symbols are conventionally modulated; therefore, the SINR terms in (\[eq:capacity\]) do not change. This algorithm does not require additional pilots. It just rotates the existing pilot symbols already used for other purposes, e.g., carrier frequency offset estimation.
The proposed pilot reuse-based scheme requires some assumptions. First, the channel should be static for at least two pilot symbol durations. Second, an identical value should be used for two consecutive pilots that experience the static channel gain. In the proposed method, the BS transmits the legacy value during the first pilot symbol duration and rotates the identical pilot during the second pilot symbol duration. Different phase rotations are assigned to each modulation mode, as in phase-rotated modulation. The symbol $\phi$ is used to denote the pilot rotation in order to avoid confusion with $\theta$ in phase-rotated modulation.
Assuming $M_{l_0}$ is used to modulate the signal, the two received consecutive pilot symbols are given by $$\begin{aligned}
r_u &= hp_u + w_u
\label{eq:legacyPilot} \\
r_r &= hp_r+n_r = hp_u e^{j\phi_{l_0}} + w_r,
\label{eq:rotatedPilot}
\end{aligned}$$ where $p$ and $r$ are the transmitted and received pilots, respectively. The subscripts $\mathit{u}$ and $\mathit{r}$ denote unrotated and rotated symbols, respectively. In addition, $w_u, w_r \sim \mathcal{CN}(0,\sigma_n^2)$. The receiver can estimate the phase rotation of the pilot in the second symbol duration as $$\varphi = \measuredangle\{(r_{u})^* r_{r}\} \approx \measuredangle \{|r_{u}|^2 e^{j\phi_{l_0}}\}.
\label{eq:thetaEstimation1}$$ By comparing the estimated $\varphi$ with the exact rotations, the modulation mode, $M_{\hat{l}}$, can be easily classified as $$\hat{l} = \underset{l\in \{0,\cdots,L,\}}{\arg\min}~|\varphi-\phi_{l}|.
\label{eq:PRC_decision}$$
Since different phase rotation values are assigned to each modulation mode, the determination of the transmitted modulation mode in can be interpreted as finding the decision region of a particular modulation mode in which the estimated phase $\varphi$ is placed. In other words, we can make $L+1$ exclusive regions in $[0,2\pi)$ and assign separate regions to all modulation modes as their decision regions. First, let $\boldsymbol{\Phi}_O$ and $\boldsymbol{\Phi}_N$ be the phase ranges for OMA and NOMA with different intervals in $[0,2\pi)$, respectively. Then, obviously, $\phi_0 \in \boldsymbol{\Phi}_O$ and $\phi_{1},\cdots,\phi_{L} \in \boldsymbol{\Phi}_N$. It is also assumed that $\boldsymbol{\Phi}_O \cup \boldsymbol{\Phi}_N = [0,2\pi)$ and $\boldsymbol{\Phi}_O \cap \boldsymbol{\Phi}_N = \{\phi \}$, where $\{\phi\}$ is the empty set. The estimated $\varphi$ must be included in either $\boldsymbol{\Phi}_O$ or $\boldsymbol{\Phi}_N$, but not in both. If $\varphi \in \boldsymbol{\Phi}_O$, the signal is classified as OMA; otherwise, as NOMA. Accordingly, $\boldsymbol{\Phi}_O$ and $\boldsymbol{\Phi}_N$ are the decision regions of OMA and NOMA, respectively.
Similarly, the far UT’s modulation order can be classified by dividing $\boldsymbol{\Phi}_N$ into several nonoverlapping phase ranges corresponding to different far UT’s modulation order candidates. For example, let QAM, 16-QAM, and 64-QAM be the candidates of a far UT’s modulation. Then, we can generate $\boldsymbol{\Phi}_{N}^{QAM}$, $\boldsymbol{\Phi}_N^{16QAM}$, and $\boldsymbol{\Phi}_N^{64QAM}$, corresponding to the NOMA modes using QAM, 16-QAM, and 64-QAM as the far UT’s modulation, respectively. These ranges satisfy $\boldsymbol{\Phi}_{N}^{QAM} \cup \boldsymbol{\Phi}_N^{16QAM} \cup \boldsymbol{\Phi}_N^{64QAM} = \boldsymbol{\Phi}_N$. Therefore, if $\varphi \in \boldsymbol{\Phi}_N$, then $\varphi$ must be included in only one range among $\boldsymbol{\Phi}_{N}^{QAM}$, $\boldsymbol{\Phi}_N^{16QAM}$, and $\boldsymbol{\Phi}_N^{64QAM}$, and the far UT’s modulation can be determined.
Next, suppose that QAM is classified as the far UT’s modulation and there exist $L_1$ modulation modes with different power weightings. Again, we can generate decision regions for the classification of the power ratio by separating $\boldsymbol{\Phi}_N^{QAM}$ into $L_1$ nonoverlapping ranges denoted by $\boldsymbol{\Phi}_{N,l}^{QAM}$ for all $l\in \{1,\cdots,L_1\}$, satisfying $\bigcup_{l=1}^{L_1} \Phi_{N,l}^{QAM} = \Phi_N^{QAM}$, similar to the aforementioned classification steps. A series of the pilot reuse-based classification processes of OMA/NOMA and modulation are expressed in Algorithm \[alg:PRBD\]. In Algorithm \[alg:PRBD\], we assume that the far UT’s modulation of the first $L_1$ modes, $M_{1},...M_{L_1}$, is QAM. In addition, the next $L_2$ modes, $M_{L_1+1},...,M_{L_1+L_2}$, and the remaining $L_3$ modes, $M_{L_1+L_2+1},...M_{L}$, use 16-QAM and 64-QAM for the far UT’s modulation, respectively.
[$\hat{l}$ $\gets$ $\underset{l=1,...,L_1}{\arg\min} |\phi_{l} - \varphi|$]{} [$\hat{l}$ $\gets$ $\underset{l=L_1+1,...,L_1+L_2}{\arg\min} |\phi_{l} - \varphi|$]{} [$\hat{l}$ $\gets$ $\underset{l=L_1+L_2+1,...,L}{\arg\min} |\phi_{l} - \varphi|$]{}
Algorithm \[alg:PRBD\] performs OMA/NOMA and modulation classifications simultaneously; therefore, blind signal classification becomes simpler. In addition, this scheme does not require the rotation value to be accurately estimated. It is sufficient that the roughly estimated $\varphi$ is in the decision region of the transmitted modulation mode for correct signal classification. After MC, the pilot symbols should serve their original purposes; therefore, the rotated symbols must be de-rotated, $r_r e^{-j\phi_{\hat{l}}}$. In this case, even though the estimation of $\varphi$ is not accurate, if MC is correct, then we can find the accurate phase rotation value $\phi_{\hat{l}}$ according to the modulation mode table, which is shown in Table \[Table:ModePhaseExample\]. Thus, the proposed classification algorithm does not need to know the pilot value and the channel gain.
The performance of the proposed algorithm depends on the method of determining the phase rotation values and decision regions for each modulation mode. From a broad perspective, there are two phase assignment rules as follows.
### Uniform Assignment
The simplest rule is the uniform one, $\phi_l = \frac{2\pi \cdot l}{L+1}$ for $M_{l}$ and $\phi_O = 0$ for $M_O$. In this case, $\boldsymbol{\Phi}_N = [\frac{2\pi}{L+1},2\pi-\frac{2\pi}{L+1})$ and $\boldsymbol{\Phi}_O = [0,\frac{2\pi}{L+1}) \cup [2\pi-\frac{2\pi}{L+1},2\pi)$. The uniform assignment rule seems reasonable; however, $\boldsymbol{\Phi}_O$ becomes smaller as $L$ increases. Therefore, it is unfair when an OMA signal is transmitted.
### Non-uniform Assignment
We can give greater importance to OMA/NOMA classification than to the classification of modulation order or power ratio by using the non-uniform assignments of phase rotations. Because OMA/NOMA classification is more important than MC, the generation of $\boldsymbol{\Phi}_O$ and $\boldsymbol{\Phi}_N$ have the first priority, followed by $\boldsymbol{\Phi}_N^{QAM}$, $\boldsymbol{\Phi}_N^{16QAM}$, and $\boldsymbol{\Phi}_N^{64QAM}$. Finally, the phase rotations of the NOMA modes having the same far UT’s modulation order but different power ratios are determined.
\[t!\] ![Phase ranges of non-uniform phase rotation rule[]{data-label="Fig:PhaseRule"}](Final_PhaseRule.pdf "fig:"){width="35.00000%"}
The phase ranges of the non-uniform assignments are shown in Fig. \[Fig:PhaseRule\]. Although $\mathbf{\Phi}_N^{QAM}$, $\mathbf{\Phi}_N^{16QAM}$, and $\mathbf{\Phi}_N^{64QAM}$ consist of $L_1$, $L_2$, and $L_3$ modes, these ranges occupy the same amount of interval as that of $\mathbf{\Phi}_O$. After $\boldsymbol{\Phi}_O$ and $\boldsymbol{\Phi}_N$ are generated, $\boldsymbol{\Phi}_N$ is divided into $\boldsymbol{\Phi}_N^{QAM}$, $\boldsymbol{\Phi}_N^{16QAM}$, and $\boldsymbol{\Phi}_N^{64QAM}$, as shown in Fig. \[Fig:PhaseRule\]. The interval sizes of the phase ranges can be arbitrarily chosen depending on the parameters of the system environment, such as the number of modulation modes. The exact phase rotation values for the NOMA modes with different power weightings, i.e., $\phi_{1},\cdots,\phi_{L}$, can be uniformly chosen in the range in which each mode is included, because power ratio classification is less important than the other classification steps. For example, if $\boldsymbol{\Phi}_N^{QAM}=[\frac{\pi}{4}, \frac{3\pi}{4})$, then $\phi_{l} = \frac{\pi}{4} + \frac{\pi}{2L_1} \cdot (l-1)$, for all $l \in \{1,...,L_1\}$.
Pilot Reuse-based Near/Far UT Classification {#sec:Proposed_Blind_Near/Far_UT_Detection}
--------------------------------------------
Since the two signals with different power weightings are transmitted simultaneously, the pilot symbols targeting the two UTs are also superposed and transmitted. When the transmitted signal is modulated by $M_l$, the power-multiplexed legacy pilot becomes $$p_l = \sqrt{P_{f}}p_l^f + \sqrt{P_{n}}p_l^n,
\label{eq:powermultiplexed_pilot}$$ where $p_l^f$ and $p_l^n$ are legacy pilots for the far and near UTs, respectively. The rotated pilot symbols used for OMA/NOMA and modulation classifications can also be utilized for near/far UT classification after they have been de-rotated. Even though each UT knows only its own pilot values, the receiver cannot recognize which power ratio is weighted to its pilot symbol because it does not perform near/far UT classification yet. Let $p_l^0$ be a known pilot value. Then, if the receiver is far UT, $p_l^0 = p_l^f$, otherwise, $p_l^0 = p_l^n$.
The proposed near/far UT classification algorithm in a two-user NOMA system is summarized in Algorithm \[alg:Far/NearUT\]. This algorithm requires that channel estimation and MC should have been completed in advance. When the MC correctly determines $M_{l}$, the UT computes two hypotheses, $\Delta_l^f$ and $\Delta_l^n$, each of which is true when the receiver is the far or near UT, respectively.
To explain the algorithm clearly, an example is presented. From (\[eq:legacyPilot\]) and (\[eq:powermultiplexed\_pilot\]), the received pilot is given by $$y_{l} = h(\sqrt{P_f}p_l^f + \sqrt{P_n}p_l^n) + n_l.
\label{eq:ex:received}$$ Suppose that the channel estimation is perfect and the receiver is the near UT, i.e., $p_l^0 = p_l^n$; then, $$\begin{aligned}
a_f &= y_{l} - h\sqrt{P_f}p_{l}^0 = h(\sqrt{P_f}(p_{l}^f-p_l^n) + \sqrt{P_n}p_l^n) + n_{l}
\label{eq:ex:a_1^f}\\
a_n &= y_{l} - h\sqrt{P_n}p_{l}^0 = h\sqrt{P_f}p_{l}^f + n_l.
\label{eq:ex:a_1^n}
\end{aligned}$$ Here, $a_f$ and $a_n$ are obtained under the assumption that the receiver would be a far and near UT, respectively. The next step is to compute $\Delta_l^f$ and $\Delta_l^n$ as follows: $$\begin{aligned}
\Delta_l^f &= \min_{q \in \chi_l}|a_f - h\sqrt{P_n}q|
\label{eq:ex:Delta_1^f} \\
\Delta_l^n &= \min_{q \in \chi_l^f}|a_n - h\sqrt{P_f}q|.
\label{eq:ex:Delta_1^n}
\end{aligned}$$
Because MC has been completed, $P_f$, $P_n$, $\chi_l^f$, and $\chi_l$ are known. Note that $\Delta_l^n$ remains the only noise component when $q=p_{l}^f$. However, $a_f$ includes the non-zero term in (\[eq:ex:a\_1\^f\]), i.e., $h\sqrt{P_f}(p_{l}^f-p_l^n)$, as well as the noise component. Thus, mostly $\Delta_l^f \geq \Delta_l^n$ and the near UT is determined. Otherwise, the receiver does not require SIC, i.e., it is determined as the far UT.
Performance Evaluation {#sec:SimulationResults}
======================
Simulation Environments
-----------------------
This section describes a variety of performance comparisons of conventional ML signal classification with the proposed schemes. Acronyms are used for the signal classification methods in the related figures: “MLC" for ML classifier, “MLC-PRM" for phase-rotated modulation based on ML classification, and “PRC" for the pilot reuse-based classifier. For MLC and MLC-PRM, 10 data symbols were used to classify the received signals. PRC utilized only one pair of pilots, because the number of pilots is usually smaller than that of data symbols. Three example cases of the modulation modes are presented.
------------ ------------ ------------ -------------
Modulation Modulation Modulation Power ratio
mode (far UT) (near UT) (far UT)
$M_0$ QPSK - 1.0
$M_{1}$ QPSK QPSK 0.8
$M_{2}$ QPSK QPSK 0.8621
$M_{3}$ QPSK QPSK 0.9163
------------ ------------ ------------ -------------
: Case 1: Modulation Mode Table[]{data-label="Table:ModeExample1"}
------------ ------------ ------------ -------------
Modulation Modulation Modulation Power ratio
mode (far UT) (near UT) (far UT)
$M_0$ 16-QAM - 1.0
$M_{1}$ QPSK 16-QAM 0.8653
$M_{2}$ 16-QAM 16-QAM 0.95
------------ ------------ ------------ -------------
: Case 2: Modulation Mode Table[]{data-label="Table:ModeExample2"}
------------ ------------ ------------ -------------
Modulation Modulation Modulation Power ratio
mode (far UT) (near UT) (far UT)
$M_0$ 16-QAM - 1.0
$M_{1}$ QPSK 16-QAM 0.7619
$M_{2}$ QPSK 16-QAM 0.8653
$M_{3}$ QPSK 16-QAM 0.9275
$M_{4}$ 16-QAM 16-QAM 0.95
$M_{5}$ 16-QAM 16-QAM 0.97
------------ ------------ ------------ -------------
: Case 3: Modulation Mode Table[]{data-label="Table:ModeExample3"}
### Case 1
Case 1 is based on Table \[Table:ModeExample1\]. The far UT’s modulation is fixed; therefore, OMA/NOMA and power ratio classifications are considered. It is supposed that $M_{2}$ is transmitted.
### Case 2
Case 2 is based on Table \[Table:ModeExample2\]. A single power ratio is assigned to each mode; therefore, OMA/NOMA and the far UT’s modulation order classifications are considered. It is supposed that $M_{1}$ is transmitted.
### Case 3
Case 3 is based on Table \[Table:ModeExample3\]. It considers OMA/NOMA, the power ratio, and the far UT’s modulation order classifications. It is supposed that $M_{2}$ is transmitted.
The power ratios of the modulation modes, in which the far UT’s signal is modulated by QPSK in Tables \[Table:ModeExample1\]-\[Table:ModeExample3\], follow the MuST parameters of 3GPP [@MuST:3GPP]. Since MuST considers QPSK only for the far UT from this point onward, the power ratios of the far UT that uses 16-QAM were arbitrarily chosen.
We considered a two-user cellular NOMA system assuming a Rayleigh fading channel, $h \sim CN(0,1)$. An ML equalizer and a low-density parity check (LDPC) 11ad decoder [@ldpc_11ad] were used for word error rate (WER) simulations. CWIC was basically used for SIC, but the system occasionally chose SLIC when the decoder-feedback could not be obtained, i.e., the classification of the far UT’s modulation order was incorrect. The phase rotations of MLC-PRM were $\theta_0=0.6, 0.51$, and $0.69$ radians optimized at 13 dB, 20 dB, and 20 dB of SNR in Cases 1, 2, and 3, respectively. As explained previously, these phase rotations are applied for only OMA/NOMA classification. Additionally, the uniform phase assignment rule was used for PRC.
\[t!\] ![OMA/NOMA classification error rates in Case 1[]{data-label="Fig:DetOMANOMAErr_Ex1"}](Final_DetOMANOMAErr_Ex1.pdf "fig:"){width="30.00000%"}
\[t!\] ![Near/far UT and modulation classification error rates in Case 1[]{data-label="Fig:DetNOMANearErr_Ex1"}](Final_DetNOMANearErr_Ex1.pdf "fig:"){width="30.00000%"}
\[t!\] ![OMA/NOMA classification error rates in Case 2[]{data-label="Fig:DetOMANOMAErr_Ex2"}](Final_DetOMANOMAErr_Ex2.pdf "fig:"){width="30.00000%"}
\[t!\] ![Near/far UT and modulation classification error rates in Case 2[]{data-label="Fig:DetNOMANearErr_Ex2"}](Final_DetNOMANearErr_Ex2.pdf "fig:"){width="30.00000%"}
\[t!\] ![OMA/NOMA classification error rates in Case 3[]{data-label="Fig:DetOMANOMAErr_Ex3"}](Final_DetOMANOMAErr_Ex3.pdf "fig:"){width="30.00000%"}
\[t!\] ![Near/far UT and modulation classification error rates in Case 3[]{data-label="Fig:DetNOMANearErr_Ex3"}](Final_DetNOMANearErr_Ex3.pdf "fig:"){width="30.00000%"}
Signal Classification Error Rates and User Capacity
---------------------------------------------------
Figs. \[Fig:DetOMANOMAErr\_Ex1\], \[Fig:DetOMANOMAErr\_Ex2\], and \[Fig:DetOMANOMAErr\_Ex3\] show the OMA classification error rates in Cases 1, 2, and 3, respectively. We can easily see that PRC yields considerably better performances than MLC and MLC-PRM in each case. MLC-PRM is also clearly better than MLC, but not as good as PRC. Since an OMA/NOMA classification error hampers correct data restoration, the proposed MLC-PRM and PRC are expected to be favorable for data detection.
The near/far UT classification error rates in Cases 1, 2, and 3 are represented by the solid curves in Figs. \[Fig:DetNOMANearErr\_Ex1\], \[Fig:DetNOMANearErr\_Ex2\], and \[Fig:DetNOMANearErr\_Ex3\], respectively. Because the signals determined as OMA do not require near/far UT classification, these error rate curves include incorrect determinations of OMA, as well as of far UT of NOMA. As compared to OMA/NOMA classification, the performance improvements of MLC-PRM over MLC in near/far UT classification are reduced. It means that even though MLC-PRM correctly classifies the signals that are significantly contaminated by noise or channel fading as NOMA, they fail to be determined as signals for the near UT in the next step because MLC-PRM does not improve near/far UT classification, as explained in Section \[sec:phase\_rotated\_modulation\]. In MLC, these signals have been already classified as OMA in the first step (i.e., OMA/NOMA classification); therefore, they do not exacerbate the near/far UT classification error rate as much as MLC-PRM. A similar phenomenon can be seen in the PRC graphs, but the decrease in performance gain of near/far UT classification is not significant, because PRC is designed to improve both OMA/NOMA and near/far UT classification steps.
A comparison of the figures of the simulation cases reveals that there is not much difference in the near UT classification rates of the three methods shown in Fig. \[Fig:DetNOMANearErr\_Ex1\] as compared to those shown in Figs. \[Fig:DetNOMANearErr\_Ex2\] and \[Fig:DetNOMANearErr\_Ex3\]. In particular, the performances of MLC and MLC-PRM are similar to that of PRC in the high SNR region of Case 1. This is because Case 2 and Case 3 are less sensitive to near/far UT classification than Case 1. In Case 1, every NOMA mode has the same modulation order for both UTs in Case 1; therefore, even if the signal is classified as NOMA, near/far UT classification should always be performed for correct data detection. However, in Cases 2 and 3, there are some situations where near/far UT classification is not required; in other words, the modulation orders for near and far UTs are different for some modes. Note that the receiver already knows its own modulation order.
The dashed curves in Figs. \[Fig:DetNOMANearErr\_Ex1\], \[Fig:DetNOMANearErr\_Ex2\], and \[Fig:DetNOMANearErr\_Ex3\] represent the MC error rates among the signals correctly classified as near UT of NOMA. Incorrect OMA/NOMA or near/far UT classification almost always results in a packet error; therefore, only the MC error rates of the signals determined to be near UT of NOMA are meaningful. The MC error rates look better than the near/far UT classification error rates in the low SNR region, because most of the signals severely contaminated by noise or channel fading are already incorrectly classified as OMA or the far UT of NOMA; therefore, these signals do not influence the MC error rates. However, since the modulation classification step compares much more hypotheses than OMA/NOMA and near/far UT classification, the MC error rates become worse than the near/far UT classification error rates as SNR grows. In the case of MLC-PRM, a data symbol rotation is applied for only OMA/NOMA classification, and therefore, the MC error rates of MLC and MLC-PRM are almost the same in each case. However, PRC shows considerably better MC performances than both MLC and MLC-PRM.
\[t!\] ![Capacity decrease of MLC, MLC-PRM, and PRC as compared to that of Genie in Case 3[]{data-label="Fig:Capacity_Ex3"}](Final_CapacityDecrease_Ex3.pdf "fig:"){width="30.00000%"}
The capacity degradation due to incorrect signal classification, with respect to the Genie scheme with the assumption of ideal classification, is shown in Fig. \[Fig:Capacity\_Ex3\]. When the SNR is lower than 20 dB, MLC-PRM and PRC obviously provide better user capacity than MLC, and in particular, PRC shows almost the same capacity as Genie. In the high SNR region, the classification rates of all schemes achieve almost the same capacity as the Genie scheme.
\[t!\] ![Comparison of WER performances in Case 1[]{data-label="Fig:WER_Ex1"}](Final_WER_Ex1.pdf "fig:"){width="30.00000%"}
\[t!\] ![Comparison of WER performances in Case 2[]{data-label="Fig:WER_Ex2"}](Final_WER_Ex2.pdf "fig:"){width="30.00000%"}
\[t!\] ![Comparison of WER performances in Case 3[]{data-label="Fig:WER_Ex3"}](Final_WER_Ex3.pdf "fig:"){width="30.00000%"}
Word Error Rates
----------------
The word error rate (WER) performances of the signal classification schemes are obtained to verify the practical usefulness of the proposed schemes. Figs. \[Fig:WER\_Ex1\], \[Fig:WER\_Ex2\], and \[Fig:WER\_Ex3\] show the WER performances of the comparison classification schemes in Cases 1, 2, and 3, respectively.
In Case 1, the WER performance of every signal classification scheme is significantly worse than that of Genie. In Fig. \[Fig:DetNOMANearErr\_Ex1\], it can be seen that the classification error rates of none of the schemes are low enough to compete with the performance of Genie in the SNR region lower than 10 dB. As the SNR increases above 10 dB, the near/far UT classification rates obtained by MLC-PRM and PRC eventually become superior to those obtained by MLC, and this tendency is reflected in Fig. \[Fig:WER\_Ex1\].
The WER graphs of Cases 2 and 3 shown in Figs. \[Fig:WER\_Ex2\] and \[Fig:WER\_Ex3\], respectively, are quite similar. In these figures, the performance of PRC is nearly the same as that of the Genie scheme. It can be seen from Figs. \[Fig:DetNOMANearErr\_Ex2\] and \[Fig:DetNOMANearErr\_Ex3\] that PRC provides remarkably better classification error rates than MLC and MLC-PRM around 15 dB; In Figs \[Fig:WER\_Ex2\] and \[Fig:WER\_Ex3\], PRC appears almost always to classify the NOMA signals correctly in the operating SNR region.
MLC-PRM provides a 1 dB SNR gain at a WER of 0.1 as compared to MLC in Cases 2 and 3; however, the WER performances of PRC and Genie are still considerably better than those of MLC and MLC-PRM, even in the high SNR region. These results are consistent with the MC error rates shown in Figs. \[Fig:DetNOMANearErr\_Ex2\] and \[Fig:DetNOMANearErr\_Ex3\]. In the high SNR region, the near/far UT classification error rates of all methods are sufficiently improved, and the effect of the MC error becomes dominant. Thus, the WER performance of PRC is much better than that of MLC and MLC-PRM in the high SNR region. However, because the MC error rates of MLC and MLC-PRM are identical, their WER performances converge.
Conclusions
===========
In this paper, one of the key issues in NOMA systems, namely the blind signal classification problem of reducing high-layer signaling to provide information about the co-scheduled signal formats and to improve spectrum/resource efficiency in highly-mobile vehicular networks, was addressed. We considered the classification steps of OMA/NOMA, near/far UT, modulation orders, and power ratios for NOMA UTs. In this study, the effects of each type of classification error in terms of SINR were quantified, and the capacity of the NOMA user was derived considering the signal classification errors. This paper proposed a phase-rotated modulation scheme and a pilot reuse-based signal classification that rotate data symbols and pilot symbols, respectively, and utilize the estimated phase to classify the received signal. The proposed schemes yield better performances in terms of classification error rate, capacity, and WER than conventional ML classification in various environment settings. Hence, the proposed schemes can be helpful in vehicular networks where only limited energy and spectrum/resource/time are available because of high mobility.
[1]{}
P. Wang, J. Xiao, and L. Ping, “Comparison of Orthogonal and Nonorthogonal Approaches to Future Wireless Cellular Systems," [*[IEEE Vehicular Technology Magazine]{}*]{} vol. 1, no. 3, pp. 4–11, September 2006.
Y. Lin, S. Wang, X. Bu, C. Xing, and J. An, “NOMA-Based Calibration for Large-Scale Spaceborne Antenna Arrays," *IEEE Transactions on Vehicular Technology*, vol. 67, no. 3, pp. 2231–2242, March 2018.
Y. Gao, B. Xia, Y. Liu, Y. Yao, K. Xiao, and G. Lu, “Analysis of the Dynamic Ordered Decoding for Uplink NOMA Systems With Imperfect CSI," *IEEE Transactions on Vehicular Technology*, vol. 67, no. 7, pp. 6647–6651, July 2018.
Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Non-Orthogonal Multiple Access (NOMA) for Cellular Future Radio Access," in *Proc. of IEEE Vehicular Technology Conference (VTC)*, Dresden, Germany, June 2013.
K. Higuchi and Y. Kishiyama, “Non-Orthogonal Access with Random Beamforming and Intra-Beam SIC for Cellular MIMO Downlink," in *Proc. of IEEE Vehicular Technology Conference (VTC)*, Las Vegas, NV, USA, September 2013.
L. Dai, B. Wang, Y. Yuan, S. Han, C. l. I, and Z. Wang, “Non-Orthogonal Multiple Access for 5G: Solutions, Challenges, Opportunities, and Future Research Trends," *IEEE Communications Magazine*, vol. 53, no. 9, pp. 74–81, September 2015.
Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the Performance of Nonorthogonal Multiple Access in 5G Systems with Randomly Deployed Users," *IEEE Signal Processing Letters*, vol. 21, no. 12, pp. 1501–1505, December 2014.
S. M. R. Islam, N. Avazov, O. A. Dobre, and K. S. Kwak, “Power-Domain Non-Orthogonal Multiple Access (NOMA) in 5G Systems: Potentials and Challenges", *IEEE Communications Surveys & Tutorials*, vol. 19, no. 2, pp. 721–742, June 2017.
B. Di, L. Song, Y. Li, and Z. Han, “V2X Meets NOMA: Non-Orthogonal Multiple Access for 5G-Enabled Vehicular Networks," *IEEE Wireless Communications*, vol. 24, no. 6, pp. 14–21, Dec. 2017.
L. P. Qian, Y. Wu, H. Zhou, and X. Shen, “Non-Orthogonal Multiple Access Vehicular Small Cell Networks: Architecture and Solution," *IEEE Network*, vol. 31, no. 4, pp. 15–21, April 2017.
3GPP, “Study on Downlink Multiuser Superposition Transmission for LTE," *3GPP, TR 36.859*, January 2016.
D. Tse and P. Viswanath, *Fundamentals of Wireless Communication*, Cambridge University Press, 2005.
Z. Ding, F. Adachi, and H. V. Poor, “The application of MIMO to nonorthogonal multiple access,” *IEEE Transactions on Wireless Communications*, vol. 15, no. 1, pp. 537-–552, Jan. 2016.
Z. Ding, R. Schober, and H. V. Poor, “A general MIMO framework for NOMA downlink and uplink transmission based on signal alignment,” *IEEE Transactions on Wireless Communications*, vol. 15, no. 6, pp. 4438–-4454, Jun. 2016.
Z. Ding, M. Peng and H. V. Poor, “Cooperative Non-Orthogonal Multiple Access in 5G Systems," *IEEE Communications Letters*, vol. 19, no. 8, pp. 1462-1465, Aug. 2015.
M. Choi, D. Han and J. Moon, “Bi-Directional Cooperative NOMA Without Full CSIT," *IEEE Transactions on Wireless Communications*, vol. 17, no. 11, pp. 7515-7527, Nov. 2018.
M. Liu, J. Yang and G. Gui, “DSF-NOMA: UAV-Assisted Emergency Communication Technology in a Heterogeneous Internet of Things," Early Access, *IEEE Internet of Things Journal.*
G. Gui, H. Huang, Y. Song and H. Sari, “Deep Learning for an Effective Nonorthogonal Multiple Access Scheme," *IEEE Transactions on Vehicular Technology*, vol. 67, no. 9, pp. 8440–8450, Sept. 2018.
M. Liu, T. Song and G. Gui, “Deep Cognitive Perspective: Resource Allocation for NOMA based Heterogeneous IoT with Imperfect SIC," Early Access, *IEEE Internet of Things Journal.*
F. Hameed, O. A. Dobre, and D. C. Popescu, “On the Likelihood-based Approach to Modulation Classification," *IEEE Transactions on Wireless Communications*, vol. 8, no. 12, pp. 5884–5892, December 2009.
E. Soltanmohammadi and M. Naraghi-Pour, “Blind Modulation Classification over Fading Channels Using Expectation-Maximization," *IEEE Communications Letters*, vol. 17, no. 9, pp. 1692–1695, September 2013.
W. Wei and J. M. Mendel, “Maximum-Likelihood Classification for Digital Amplitude-Phase Modulations," *IEEE Transactions on Communications*, vol. 48, no. 2, pp. 189–193, February 2000.
A. Hazza, M. Shoaib, S. A. Alshebeili and A. Fahad, “An overview of feature-based methods for digital modulation classification," *2013 1st International Conference on Communications, Signal Processing, and their Applications (ICCSPA)*, Sharjah, 2013, pp. 1-6.
S. Rajendran, W. Meert, D. Giustiniano, V. Lenders and S. Pollin, “Deep Learning Models for Wireless Signal Classification With Distributed Low-Cost Spectrum Sensors," *IEEE Transactions on Cognitive Communications and Networking*, vol. 4, no. 3, pp. 433–445, Sept. 2018.
N. E. West and T. O’Shea, “Deep architectures for modulation recognition," *2017 IEEE International Symposium on Dynamic Spectrum Access Networks (DySPAN)*, Piscataway, NJ, 2017, pp. 1–6.
F. Meng, P. Chen, L. Wu and X. Wang, “Automatic Modulation Classification: A Deep Learning Enabled Approach," *IEEE Transactions on Vehicular Technology*, vol. 67, no. 11, pp. 10760–10772, Nov. 2018.
J. Tian, Y. Pei, Y. Huang and Y. Liang, “Modulation-Constrained Clustering Approach to Blind Modulation Classification for MIMO Systems," *IEEE Transactions on Cognitive Communications and Networking*, vol. 4, no. 4, pp. 894–907, Dec. 2018.
G. Araniti, C. Campolo, M. Condoluci, A. Iera, and A. Molinaro, “LTE for vehicular networking: A survey,” *IEEE Communications Magazine*, vol. 51, no. 5, pp. 148–-157, May 2013.
J. Kim, S.-C. Kwon, and G. Choi, “Performance of video streaming in infrastructure-to-vehicle telematic platforms with 60-GHz radiation and IEEE 802.11ad baseband,” *IEEE Trans. Vehicular Technology*, vol. 65, no. 12, pp. 10111–-10115, Dec. 2016.
S. Majhi, R. Gupta, W. Xiang and S. Glisic, “Hierarchical Hypothesis and Feature-Based Blind Modulation Classification for Linearly Modulated Signals," *IEEE Transactions on Vehicular Technology*, vol. 66, no. 12, pp. 11057–11069, Dec. 2017.
M. Choi and J. Kim, “Blind Signal Classification Analysis and Impact on User Scheduling and Power Allocation in Nonorthogonal Multiple Access," available at: arxiv.org/abs/1902.02090
C. Yan, *et. al.*, “Receiver Design for Downlink Non-Orthogonal Multiple Access (NOMA)," in [*[Proc. IEEE Vehicular Technology Conference (VTC),]{}*]{} Glasgow, UK, May 2015.
A. K. Gupta, *et. al.*, “BER Performance of IEEE 802.11ad for Single Carrier and Multi Carrier," [*[International Journal of Engineering Science and Technology]{}*]{}, vol. 1, no. 4, pp. 2180–2187, May 2012.
[Minseok Choi]{} is a postdoctoral researcher in the Department of Electrical and Computer Engineering at University of Southern California, Los Angeles, CA, USA, and also a postdoctoral associate in the School of Software at Chung-Ang University, Seoul, Korea. He received the B.S., M.S. and Ph.D. degrees in the School of Electrical Engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2011, 2013, and 2018, respectively. His research interests include wireless caching networks, stochastic network optimization, non-orthogonal multiple access and 5G networks.
[Daejung Yoon]{} received the B.S. and M.S. degrees in electrical engineering from Kyungpook National University, South Korea, in 2003 and 2005, respectively. He received the M.S./Ph.D. degrees in wireless communication signal processing from the Electrical Engineering Department, the University of Minnesota, Minneapolis, Minnesota, USA, in 2011. In 2009, he joined Samsung Information Systems America, San Jose, California, where he worked on advanced signal processing and performance optimization. Since 2011, he has worked at Intel Cooperation, Santa Clara, California, and is extending his research field to include wireless communication system developments. At Intel, he is contributing to Intel LTE/LTE-A handset modem design and products. Since 2014, he has served as an Intel 3GPP standardization delegate in the 3GPP working group RAN4, and now he is a senior research specialist in Bell Labs Nokia, France working on 5G communication systems and 3GPP NR system design in release 16 and beyond. He has published multiple technical papers and patents on 5G NR cellular network systems and usecase studies’ as well as LTE/LTE-A evolution.
[Joongheon Kim]{} (M’06–SM’18) has been an assistant professor with Korea University, Seoul, Korea, since 2019. He received his B.S. (2004) and M.S. (2006) in computer science and engineering from Korea University, Seoul, Korea; and his Ph.D. (2014) in computer science from the University of Southern California (USC), Los Angeles, CA, USA. Before joing Korea University as a faculty member, he was with Chung-Ang University (Seoul, Korea, 2016–2019), Intel Corporation (Santa Clara, CA, USA, 2013–2016), InterDigital (San Diego, CA, USA, 2012), and LG Electronics (Seoul, Korea, 2006–2009). He is a senior member of the IEEE; and a member of IEEE Communications Society. He was awarded Annenberg Graduate Fellowship with his Ph.D. admission from USC (2009).
[^1]: This research was supported by Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT) (No.2018-0-00170, Virtual Presence in Moving Objects through 5G).
[^2]: M. Choi is with University of Southern California, Los Angeles, CA 90089, USA (e-mail: choimins@usc.edu).
[^3]: D. Yoon is with Nokia Bell Labs, Nozay 91620, France (e-mail: yoondanny@hotmail.com).
[^4]: J. Kim is with the School of Electrical Engineering, Korea University, Seoul 02841, South Korea (e-mail: joongheon@korea.ac.kr).
[^5]: J. Kim is the corresponding author of this paper.
|
[10]{}
Niki Aifanti, Christos Papachristou, and Anastasios Delopoulos. The mug facial expression database. In [*Image analysis for multimedia interactive services (WIAMIS), 2010 11th international workshop on*]{}, pages 1–4. IEEE, 2010.
Martin Arjovsky, Soumith Chintala, and L[é]{}on Bottou. asserstein generative adversarial networks. In Doina Precup and Yee Whye Teh, editors, [*Proceedings of the 34th International Conference on Machine Learning*]{}, volume 70 of [ *Proceedings of Machine Learning Research*]{}, pages 214–223, International Convention Centre, Sydney, Australia, 06–11 Aug 2017. PMLR.
Yoshua Bengio, J[é]{}r[ô]{}me Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In [*Proceedings of the 26th annual international conference on machine learning*]{}, pages 41–48. ACM, 2009.
Nicolas Bonnotte. . PhD thesis, Universit[é]{} Paris Sud - Paris XI, 2013.
Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In [*2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*]{}, pages 4724–4733. IEEE, 2017.
Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. , 2017.
Ishan Deshpande, Ziyu Zhang, and Alexander Schwing. Generative modeling using the sliced wasserstein distance. , 2018.
Ishan Deshpande, Ziyu Zhang, and Alexander G. Schwing. Generative modeling using the sliced wasserstein distance. , abs/1803.11188, 2018.
Arnab Ghosh, Viveka Kulharia, Vinay Namboodiri, Philip HS Torr, and Puneet K Dokania. Multi-agent diverse generative adversarial networks. , 2017.
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, [*Advances in Neural Information Processing Systems 27*]{}, pages 2672–2680. Curran Associates, Inc., 2014.
Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Jimenez Rezende, and Daan Wierstra. Draw: A recurrent neural network for image generation. , 2015.
Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of wasserstein gans. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, [*Advances in Neural Information Processing Systems 30*]{}, pages 5767–5777. Curran Associates, Inc., 2017.
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In [*Advances in Neural Information Processing Systems*]{}, pages 6629–6640, 2017.
De-An Huang, Vignesh Ramanathan, Dhruv Mahajan, Lorenzo Torresani, Manohar Paluri, Li Fei-Fei, and Juan Carlos Niebles. What makes a video a video: Analyzing temporal information in video understanding models and datasets.
Xun Huang, Yixuan Li, Omid Poursaeed, John Hopcroft, and Serge Belongie. Stacked generative adversarial networks. In [*IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*]{}, volume 2, page 4, 2017.
Zhiwu Huang, Bernhard Kratzwald, Danda Pani Paudel, Jiqing Wu, and Luc Van Gool. Face translation between images and videos using identity-aware cyclegan. , 2017.
Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. , 2017.
Nal Kalchbrenner, A[ä]{}ron van den Oord, Karen Simonyan, Ivo Danihelka, Oriol Vinyals, Alex Graves, and Koray Kavukcuoglu. Video pixel networks. , abs/1610.00527, 2016.
Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of [GAN]{}s for improved quality, stability, and variation. In [*International Conference on Learning Representations*]{}, 2018.
H. Kim, P. Garrido, A. Tewari, W. Xu, J. Thies, N. Nie[ß]{}ner, P. P[é]{}rez, C. Richardt, M. Zollh[ö]{}fer, and C. Theobalt. Deep video portraits. , 2018.
Soheil Kolouri, Yang Zou, and Gustavo K Rohde. Sliced wasserstein kernels for probability distributions. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 5258–5267, 2016.
Marek Kowalski, Jacek Naruniec, and Tomasz Trzcinski. Deep alignment network: A convolutional neural network for robust face alignment. In [*Proceedings of the International Conference on Computer Vision & Pattern Recognition (CVPRW), Faces-in-the-wild Workshop/Challenge*]{}, volume 3, page 6, 2017.
Bernhard Kratzwald, Zhiwu Huang, Danda Pani Paudel, and Luc Van Gool. Towards an understanding of our world by ganing videos in the wild. , abs/1711.11453, 2017.
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In [*Advances in neural information processing systems*]{}, pages 1097–1105, 2012.
Christian Ledig, Lucas Theis, Ferenc Huszar, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, et al. Photo-realistic single image super-resolution using a generative adversarial network. In [*Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*]{}, pages 4681–4690, 2017.
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In [*Proceedings of International Conference on Computer Vision (ICCV)*]{}, 2015.
Alireza Makhzani, Jonathon Shlens, Navdeep Jaitly, and Ian Goodfellow. Adversarial autoencoders. In [*International Conference on Learning Representations*]{}, 2016.
Xudong Mao, Qing Li, Haoran Xie, Raymond YK Lau, Zhen Wang, and Stephen Paul Smolley. Least squares generative adversarial networks. In [*2017 IEEE International Conference on Computer Vision (ICCV)*]{}, pages 2813–2821. IEEE, 2017.
Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. , 2015.
Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. , 2014.
Katsunori Ohnishi, Shohei Yamamoto, Yoshitaka Ushiku, and Tatsuya Harada. Hierarchical video generation from orthogonal information: Optical flow and texture. In [*AAAI*]{}, 2018.
Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. , 2016.
Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. , 2015.
Masaki Saito, Eiichi Matsumoto, and Shunta Saito. Temporal generative adversarial nets with singular value clipping. In [*ICCV*]{}, 2017.
Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, and Xi Chen. Improved techniques for training gans. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, [*Advances in Neural Information Processing Systems 29*]{}, pages 2234–2242. Curran Associates, Inc., 2016.
Khurram Soomro, Amir Roshan Zamir, and Mubarak Shah. Ucf101: A dataset of 101 human actions classes from videos in the wild. , 2012.
Supasorn Suwajanakorn, Steven M Seitz, and Ira Kemelmacher-Shlizerman. Synthesizing obama: learning lip sync from audio. , 36(4):95, 2017.
Matthew Tesfaldet, Marcus A. Brubaker, and Konstantinos G. Derpanis. Two-stream convolutional networks for dynamic texture synthesis. In [*IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*]{}, 2018.
Du Tran, Heng Wang, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. , 2017.
Sergey Tulyakov, Ming[-]{}Yu Liu, Xiaodong Yang, and Jan Kautz. Mocogan: Decomposing motion and content for video generation. , abs/1707.04993, 2017.
Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Oriol Vinyals, Alex Graves, et al. Conditional image generation with pixelcnn decoders. In [*Advances in Neural Information Processing Systems*]{}, pages 4790–4798, 2016.
Ruben Villegas, Jimei Yang, Yuliang Zou, Sungryull Sohn, Xunyu Lin, and Honglak Lee. Learning to generate long-term future via hierarchical prediction. , 2017.
Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Generating videos with scene dynamics. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, [*Advances in Neural Information Processing Systems 29*]{}, pages 613–621. Curran Associates, Inc., 2016.
Chaoyue Wang, Chang Xu, Chaohui Wanga, and Dacheng Tao. Perceptual adversarial networks for image-to-image transformation. , 2018.
Lior Wolf, Tal Hassner, and Itay Maoz. Face recognition in unconstrained videos with matched background similarity. In [*Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on*]{}, pages 529–534. IEEE, 2011.
Jiajun Wu, Yifan Wang, Tianfan Xue, Xingyuan Sun, William T Freeman, and Joshua B Tenenbaum. . In [*Advances In Neural Information Processing Systems*]{}, 2017.
Jiajun Wu, Chengkai Zhang, Tianfan Xue, William T Freeman, and Joshua B Tenenbaum. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In [*Advances in Neural Information Processing Systems*]{}, pages 82–90, 2016.
Jiqing Wu, Zhiwu Huang, Wen Li, and Luc Van Gool. Generative autotransporters. , abs/1706.02631, 2017.
Saining Xie, Chen Sun, Jonathan Huang, Zhuowen Tu, and Kevin Murphy. Rethinking spatiotemporal feature learning for video understanding. , 2017.
Wei Xiong, Wenhan Luo, Lin Ma, Wei Liu, and Jiebo Luo. Learning to generate time-lapse videos using multi-stage dynamic generative adversarial networks. , abs/1709.07592, 2017.
Han Zhang, Tao Xu, Hongsheng Li, Shaoting Zhang, Xiaolei Huang, Xiaogang Wang, and Dimitris Metaxas. Stackgan: Text to photo-realistic image synthesis with stacked generative adversarial networks. In [*IEEE Int. Conf. Comput. Vision (ICCV)*]{}, pages 5907–5915, 2017.
Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. , 2017.
Jun-Yan Zhu, Richard Zhang, Deepak Pathak, Trevor Darrell, Alexei A Efros, Oliver Wang, and Eli Shechtman. Toward multimodal image-to-image translation. In [*Advances in Neural Information Processing Systems 30*]{}. 2017.
|
---
abstract: |
Multipartite entanglement has been widely regarded as key resources in distributed quantum computing, for instance, multi-party cryptography, measurement based quantum computing, quantum algorithms. It also plays a fundamental role in quantum phase transitions, even responsible for transport efficiency in biological systems.
Certifying multipartite entanglement is generally a fundamental task. Since an $N$ qubit state is parameterized by $4^N-1$ real numbers, one is interested to design a measurement setup that reveals multipartite entanglement with as little effort as possible, at least without fully revealing the whole information of the state, the so called “tomography", which requires exponential energy.
In this paper, we study this problem of certifying entanglement without tomography in the constrain that only single copy measurements can be applied. This task is formulate as a membership problem related to a dividing quantum state space, therefore, related to the geometric structure of state space. We show that universal entanglement detection among all states can never be accomplished without full state tomography. Moreover, we show that almost all multipartite correlation, include genuine entanglement detection, entanglement depth verification, requires full state tomography. However, universal entanglement detection among pure states can be much more efficient, even we only allow local measurements. Almost optimal local measurement scheme for detecting pure states entanglement is provided.
author:
-
title: 'Entanglement Verification, with or without tomography'
---
Introduction
============
Quantum computing has long seemed like one of those technologies that are 20 years away, and always will be. But 2017 could be the year that the field sheds its research-only image.
The world leading IT giants Google and Microsoft recently hired a host of leading lights, and have set challenging goals for this year. Their ambition reflects a broader transition taking place at start-ups and academic research labs alike: to move from pure science towards engineering.
Quantum computing offers the potential of considerable speedup over classical computing for some important problems such as prime factoring [@Sh94] and unsorted database search [@Gr97]. To take such advantage, entanglement, one striking feature of quantum many-body systems, must be provided. With shared entanglement, two or more parties can be correlated in the way that is much stronger than they can be in any classical way. Entanglement has been widely studied since it has been proven to be an asset to information processing and computational tasks. For instance, multipartite entanglement has been used as the central resource for quantum key distribution in multipartite cryptography [@MS08]; it is the initial resource in measurement based quantum computing [@RB01]; it is essential in understanding quantum phase transition [@SA11]; arguably, multipartite entanglement even should be responsible for transport efficiency in biological systems [@SIFW10]. However, entanglement is still mysterious to many people including experts due to its complex structure.
To understand multipartite entanglement, reliable techniques for characterising entanglement properties of general quantum states are required. Therefore, it is a fundamental problem to qualitatively test whether a given state is entangled or not. The pure state case has been extensively studied and fruitful result has been obtained. For instance, it is proved that almost all multi-qubit entangled states admit Hardy-type proofs of non-locality without inequalities or probabilities in [@ACY16]. In the setting of of multiple dishonest parties, it is showed how an agent of a quantum network can perform a distributed verification of a source creating multipartite Greenberger-Horne-Zeilinger (GHZ) states with minimal resources, which is, nevertheless, resistant against any number of dishonest parties in [@PCW+12]. However, a complete answer of entanglement detection for general mixed states is still missing so far. A considerable number of different separability criterions have been discovered, including the famous Positive Partial Transpose(PPT) criterion [@PER96], and Gurvits discovered it lies in the computational complexity class NP-Hard [@GUR04], a By borrowing idea from functional analysis, entanglement witnesses has been introduced to detect entanglement [@HHH96; @TER00]. A more challenging question is the detection of genuine multipartite entanglement, extensive study has not yielded satisfying results yet.
Entanglement detection problem is naturally fallen into the framework of quantum property testing, or quantum characterization, verification and validation, where one can test any interested property.
Now in the field of quantum computing, we are with a quantum version of the “big data" problem: the data collected from quantum systems generated in our labs are growing exponentially because the parameters is growing exponentially as the number of qubits grows. For instance, an $n$-qubit state has been created in our lad as the resource of measurement based quantum computing [@RB01], we want to see whether our preparation is correct.
Quantum property testing can be viewed in different settings: The first one is that the mathematical description of the quantum state is given, in other words, the complete information of the quantum object is known. Another one is that the quantum state is given as black box, where one can access its information by measuring it. Even in the latter setting, two very different scenarios should be considered, statistical fluctuations or accurate measurements. In the former one, the measurement outcomes are just bit strings distributed according to the outcome probability, see [@MW13] as an excellent survey. In the latter case, the measurements of experiments are accurate in the sense that the average of the measurement outcomes is exactly the probability distribution corresponding to the measurement. In this paper, we will focus on the latter one.
The general quantum property testing in our setting can be viewed as follows,
> [**Quantum property testing**]{}\
> Let $\mathcal{Q}$ be the set of quantum quantum states. A subset ${\mathcal{P}}\subseteq{\mathcal{Q}}$ is called a *property*. An *quantum property tester* for ${\mathcal{P}}$ is an algorithm (quantum procedure) that receives a black box as input $x\in \mathcal{Q}$. In the former case, the algorithm accepts; in the latter case, the algorithm rejects.
A property ${\mathcal{P}}$ is called trivial if ${\mathcal{P}}=\mathcal{Q}$ or ${\mathcal{P}}=\emptyset$.
Reconstructing the mathematical description of the given quantum states is called “quantum state tomography". Of course, one can obtain any information about this quantum state via quantum state tomography. However, an $N$ qubit state is parameterized by $4^N-1$ real numbers, therefore, informational complete measurements consist of exponential many observables, which is generally impossible. Formally, we regared the given quantum object as resource, and the goal of property testing is to test the property by accessing the object as less as possible. Therefore, we can define the sample complexity of the property ${\mathcal{P}}$ be the infimum on the number of access the object among all quantum property tester for ${\mathcal{P}}$. Notice that, the thing we care mostly is how many times do we need to access the quantum object to accomplish the property testing, not the post processing time of the algorithm. An optimal algorithm may heavily rely on collective measurements on many identical copies of given states. This is not friendly for current experimental technology, as collective measurements are usually much more difficult to implement than measuring single-copy ones. We will focus on measurements which only applies on single-copy of quantum state.
By noticing that these problems are decision problems with 1 bit outcome, one might hope to achieve the answers with very small number of measurements, or at least with something less than an informationally complete set-up. The bipartite version of this problem has been studied recently. Indeed, it was recently showed that testing whether a bipartite state is entangled or not requires an informationally-complete measurement [@CaHeKaScTo16; @Luetal16; @CaHeScTo17]. In [@CaHeScTo17], various sufficient criteria are given, under which the informationally-incomplete measurements can not reveal the property for unknown quantum state with certainty. Compare with bipartite entanglement, entanglement in the multipartite setting turns out much richer and more delicate to characterize.
In this paper, we are going to study two versions of the multipartite entanglement detection problem: We are giving multipartite quantum states, how do we universally detect entanglement through physical observables? In the first version, we do not have any further information of the state other than the state space it lives in. In other words, it can be any mixed state in that state space. We show that there is no such procedure which can detect multipartite entanglement without full state tomography among all mixed state. Actually, we prove the more stronger version: For any property that are invariant under stochastic local operations assisted by classical communication(SLOCC) requires full state tomography unless it is a trivial property if it contains some positive element but not all of them.
Due to the key role of multipartite entanglement in distributed quantum computing, our results can be interpreted as follows, in distributed quantum computation, one can not verify that whether the shared state is entangled or not without reconstructing the state using exponential measurement.
In the second version, we assume that state is pure, and then we provide an almost optimal quantum procedure to detect multipartite entanglement. Our algorithm only costs linear number of “local" measurements, where “local" means we only need to implement individual measurement on subsystems. This is extremely friendly for current available technology.
**Structure of the Paper.** In Section II, we provide technical preliminaries of the basic quantum mechanics. In Section III, we give the investigation on entanglement together with examples for illustration. In Section IV, we show that if we do not have any prior information on the given quantum state, then detecting its entanglement property requires full state tomography. Actually, almost all SLOCC equivalence property here required full state tomography. In Section V, we show that if we know that the given quantum state is pure, we provide one non-adaptive scheme and one adaptive scheme to detect entanglement which are exponential faster than doing full state tomography. Finally, in Section VII, we offer conclusions and some highlight open problems.
Preliminaries
=============
For convenience of the reader, we briefly recall some basic notions from linear algebra and quantum theory which are needed in this paper. For more details, we refer to [@NC00].
Basic linear algebra
--------------------
According to a basic postulate of quantum mechanics, the state space of an isolated quantum system is a Hilbert space. In this paper, we only consider finite-dimensional Hilbert spaces. We briefly recall some basic notions from Hilbert space theory. We write $\mathbb{C}$ for the set of complex numbers. For each complex number $c\in \mathbb{C}$, $c^*$ stands for the conjugate of $c$. An [*inner product space*]{} ${\mathcal{H}}$ is a vector space equipped with an inner product function $$\langle\cdot|\cdot\rangle:{\mathcal{H}}\times {\mathcal{H}}\rightarrow \mathbb{C}$$ such that
1. $\langle\psi|\psi\rangle\geq 0$ for any $|\psi\>\in{\mathcal{H}}$, with equality if and only if $|\psi\rangle =0$;
2. $\langle\phi|\psi\rangle=\langle\psi|\phi\rangle^{\ast}$;
3. $\langle\phi|\sum_i c_i|\psi_i\rangle=
\sum_i c_i\langle\phi|\psi_i\rangle$.
For any vector $|\psi\rangle\in{\mathcal{H}}$, its length $|||\psi\rangle||$ is defined to be $\sqrt{\langle\psi|\psi\rangle}$, and it is said to be [*normalized*]{} if $|||\psi\rangle||=1$. Two vectors $|\psi\>$ and $|\phi\>$ are [*orthogonal*]{} if $\<\psi|\phi\>=0$. An [*orthonormal basis*]{} of a Hilbert space ${\mathcal{H}}$ is a basis $\{|i\rangle\}$ where each $|i\>$ is normalized and any pair of them are orthogonal.
Let $\lh$ be the set of linear operators on ${\mathcal{H}}$. For any $A\in
\lh$, $A$ is [*Hermitian*]{} if $A^\dag=A$ where $A^\dag$ is the adjoint operator of $A$ such that $\<\psi|A^\dag|\phi\>=\<\phi|A|\psi\>^*$ for any $|\psi\>,|\phi\>\in{\mathcal{H}}$. The fundamental [*spectral theorem*]{} states that the set of all normalized eigenvectors of a Hermitian operator in $\lh$ constitutes an orthonormal basis for ${\mathcal{H}}$. That is, there exists a so-called spectral decomposition for each Hermitian $A$ such that $$A=\sum_i\lambda_i |i\>\<i|=\sum_{\lambda_{i}\in spec(A)}\lambda_i E_i,$$ where the set $\{|i\>\}$ constitutes an orthonormal basis of ${\mathcal{H}}$, $spec(A)$ denotes the set of eigenvalues of $A$, and $E_i$ is the projector to the corresponding eigenspace of $\lambda_i$. A linear operator $A\in \lh$ is [*unitary*]{} if $A^\dag A=A A^\dag=I_{\mathcal{H}}$ where $I_{\mathcal{H}}$ is the identity operator on ${\mathcal{H}}$.
The [*trace*]{} of $A\in\lh$ is defined as ${{\rm tr}}(A)=\sum_i \<i|A|i\>$ for some given orthonormal basis $\{|i\>\}$ of ${\mathcal{H}}$. It is worth noting that trace function is actually independent of the orthonormal basis selected. It is also easy to check that trace function is linear and ${{\rm tr}}(AB)={{\rm tr}}(BA)$ for any operators $A,B\in \lh$.
A matrix $A$ is called semi-definite positive if it is Hermitian and has no negative eigenvalues. A matrix $A$ is called positive if it is Hermitian and has positive eigenvalues only. We use $A\geq 0$ and $A>0$ to denote the semi-definite positivity and positivity of $A$, respectively.
$||A||$ stands for the 2-norm of $A\in\lh$ by definition $||A||=\sqrt{{{\rm tr}}(A^{\dag}A)}$.
We use $I_{{\mathcal{H}}}$ to denote the identity operator of $\lh$.
Basic quantum mechanics
-----------------------
According to von Neumann’s formalism of quantum mechanics [@vN55], an isolated physical system is associated with a Hilbert space which is called the [*state space*]{} of the system. A [*pure state*]{} of a quantum system is a normalized vector in its state space, and a [*mixed state*]{} is represented by a density operator on the state space. Here a density operator $\rho$ on Hilbert space ${\mathcal{H}}$ is a semi-definite positive linear operator such that ${{\rm tr}}(\rho)= 1$. Another equivalent representation of density operator is probabilistic ensemble of pure states. In particular, given an ensemble $\{(p_i,|\psi_i\rangle)\}$ where $p_i \geq 0$, $\sum_{i}p_i=1$, and $|\psi_i\rangle$ are pure states, then $\rho=\sum_{i}p_i{|\psi_i\rangle \langle \psi_i|}$ is a density operator. Conversely, each density operator can be generated by an ensemble of pure states in this way. The set of density operators on ${\mathcal{H}}$ is defined as $$\mathcal{D}=\{\ \rho\in\lh\ :\rho\ \mbox{is semi-definite positive and} {{\rm tr}}(\rho)=
\mbox{1}\}.$$
The general evolution of a quantum system is described by a trace-preserving super-operator on its state space: if the states of the system at times $t_1$ and $t_2$ are $\rho_1$ and $\rho_2$, respectively, then $\rho_2=\sum_k E_k\rho_1E_k^{\dag}$ for some $E_k$.
A (general) quantum [*measurement*]{} is described by a Hermitian operator $O$. If the system is in state $\rho$, then the measurement outcome is $${{\rm tr}}(O\rho),$$ in the accurate measurement setting of this paper.
Tensor Product of Hilbert Space
-------------------------------
The state space of a composed quantum system is the tensor product of the state spaces of its component systems. Let ${\mathcal{H}}_k$ be a Hilbert space with orthonormal basis $\{|\varphi_{i_k}\}$ for $1\leq k\leq n$. Then the tensor product $\bigotimes_{k=1}^{n}{\mathcal{H}}_k$ is defined to be the Hilbert space with $\{|\varphi_{i_1}\rangle...|\varphi_{i_n}\rangle\}$ as its orthonormal basis. Here the tensor product of two vectors is defined by a new vector such that $$\bigotimes_{k=1}^n\left(\sum_{i_k} \lambda_{i_k} |\psi_{i_k}\>\right)=\sum_{i_1,\cdots,i_n} \lambda_{i_1}\cdots\lambda_{i_n}
|\psi_{i_1}\>\otimes\cdots\otimes |\phi_{i_n}\>.$$ Then $\bigotimes_{k=1}^{n}{\mathcal{H}}_k$ is also a Hilbert space where the inner product is defined as the following: for any $|\psi_{k}\>,|\phi_{k}\>\in{\mathcal{H}}_k$ $$\<\psi_1\otimes\cdots\otimes \psi_n|\phi_1\otimes\cdots\otimes\phi_n\>=\<\psi_1|\phi_1\>_{{\mathcal{H}}_1}\cdots\<
\psi_n|\phi_n\>_{{\mathcal{H}}_n}$$ where $\<\cdot|\cdot\>_{{\mathcal{H}}_k}$ is the inner product of ${\mathcal{H}}_k$.
In the bipartite case, the [*partial trace*]{} of $A\in\mathcal{L}({\mathcal{H}}_1
\otimes {\mathcal{H}}_2)$ with respect to ${\mathcal{H}}_1$ is defined as ${{\rm tr}}_{{\mathcal{H}}_1}(A)=\sum_i \<i|A|i\>$ where $\{|i\>\}$ is an orthonormal basis of ${\mathcal{H}}_1$. Similarly, we can define the partial trace of $A$ with respect to ${\mathcal{H}}_2$. Partial trace functions are also independent of the orthonormal basis selected.
For a mixed state $\rho$ on ${\mathcal{H}}_1 \otimes {\mathcal{H}}_2$, partial traces of $\rho$ have explicit physical meanings: the density operators ${{\rm tr}}_{{\mathcal{H}}_1}\rho$ and ${{\rm tr}}_{{\mathcal{H}}_2}\rho$ are exactly the reduced quantum states of $\rho$ on the second and the first component system, respectively.
Entanglement
============
In this section, we introduce some basic facts about the most important quantum feature—Entanglement.
Note that in general, the state of a composite system cannot be decomposed into tensor product of the reduced states on its component systems. A well-known example is the 2-qubit state $$|\Psi\>=\frac{1}{\sqrt{2}}(|00\>+|11\>).$$ This kind of state is called [*entangled state*]{}. To see the strangeness of entanglement, suppose a measurement $M_{0}={|0\rangle \langle 0|}$ and $M_{1}={|1\rangle \langle 1|}$ are applied on the first qubit of $|\Psi\>$ (see the following for the definition of quantum measurements). Then after the measurement, the second qubit will definitely collapse into state $|0\>$ or $|1\>$ depending on whether the outcome $\lambda_0$ or $\lambda_1$ is observed. In other words, the measurement on the first qubit changes the state of the second qubit in some way. This is an outstanding feature of quantum mechanics which has no counterpart in classical world, and is the key to many quantum information processing tasks such as teleportation [@BB93] and superdense coding [@BW92].
In bipartite system, a pure state ${|\psi\rangle}$ is called product (or not entangled) if it is of form $${|\psi\rangle}={|\psi_1\rangle}{|\psi_2\rangle}.$$ A density matrix $\rho$ is called separable (or not entangled) if it can be written as some convex combination of the density of product pure states, that is $p_i>0$ and semi-definite positive $\rho_{i,1}$s and $\rho_{i,2}$s such that $$\rho=\sum_i p_i\rho_{i,1}\otimes \rho_{i,2}.$$ Otherwise, it is called entangled.
An $n$-particle pure state ${|\psi\rangle}$ is called product if it is of form $${|\psi\rangle}={|\psi_1\rangle}\cdots{|\psi_n\rangle}.$$ A density matrix $\rho$ is called separable if it can be written as some convex combination of the density of product pure states. Otherwise, it is called entangled.
Positive Partial Transpose
--------------------------
A bipartite quantum state $\rho\in \mathcal{L}({\mathcal{H}}_1\otimes {\mathcal{H}}_2)$ is called to have positive partial transpose (or simply PPT) if $\rho^{\Gamma_{{\mathcal{H}}_1}}\geq 0$, where ${\Gamma_{{\mathcal{H}}_1}}$ means the partial transpose with respect to the party ${\mathcal{H}}_1$, i.e., $$({|ij\rangle \langle kl|})^{\Gamma_{{\mathcal{H}}_1}}={|kj\rangle \langle il|}.$$
This definition can be seen more clearly if we write the state as a block matrix:
The result is independent of the party that was transposed, because $$\rho={\begin{bmatrix}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22} &A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\ \vdots &\vdots &\vdots &\cdots &\vdots\\ A_{n1}&A_{n2}&A_{n3}&\cdots &A_{nn}\\ \end{bmatrix}}$$ Where $n$ equals the dimension of ${\mathcal{H}}_a$, and each block is a square matrix of dimension equals the dimension of ${\mathcal{H}}_2$. Then the partial transpose is $$\rho^{\Gamma_2}={\begin{bmatrix}A^T_{11}&A^T_{12}&A^T_{13}&\cdots &A^T_{1n}\\A^T_{21}&A^T_{22} &A^T_{23}&\cdots &A^T_{2n}\\A^T_{31}&A^T_{32}&A^T_{33}&\cdots &A^T_{3n}\\ \vdots &\vdots &\vdots &\cdots &\vdots\\ A^T_{n1}&A^T_{n2}&A^T_{n3}&\cdots &A^T_{nn}\\ \end{bmatrix}}$$ It had been observed by Peres that any separable state has positive partial transpose [@PER96], $$\rho=\sum_i p_i\rho_{i,1}\otimes \rho_{i,2}\\
\Rightarrow \rho^{\Gamma_2}=\sum_i p_i\rho_{i,1}\otimes (\rho_{i,2})^T\geq 0.$$ The result is independent of the party that was transposed, because $\rho^{\Gamma_1}=(\rho^{\Gamma_2})^T$.
In [@KCKL00], it was proved that all $2\otimes n$ density operators that remain invariant after partial transposition with respect to the first system are separable.
Example
-------
Notice that, a multipartite pure state is product if and only if it is product under any bipartition. However, this is not true for mixed state.
Before going to further introduction on multipartite entanglement, we first give one example to illustrate the significant difference and complex of multipartite entanglement and bipartite entanglement.
Define three-qubit state as $$\rho=\frac{1}{4}(I-\sum_{i=1^4}{|\phi_i\rangle \langle \phi_i|}),$$ where ${|\phi_i\rangle}$ are defined as $$\begin{aligned}
{|\phi_1\rangle}={|0,1,+\rangle},\\
{|\phi_2\rangle}={|1,+,0\rangle},\\
{|\phi_3\rangle}={|+,0,1\rangle},\\
{|\phi_4\rangle}={|-,-,-\rangle},\end{aligned}$$ with ${|\pm\rangle}=\frac{{|0\rangle}\pm{|1\rangle}}{\sqrt{2}}$.
One can verify that:
i). $\rho$ is invariant under partial transpose of the any qubit.
Therefore, according to the result just been mentioned of [@KCKL00], $\rho$ is separable in any bipartition.
ii). There is no product state ${|\psi_1\rangle}{|\psi_2\rangle}{|\psi_3\rangle}$ which is orthogonal to all ${|\phi_i\rangle}$. That is, no product state lives in the orthogonal complement of the space spanned by ${|\phi_i\rangle}$.
Notice that $\rho$ is proportional to the projection on the orthogonal complement of the space spanned by ${|\phi_i\rangle}$. Therefore, $\rho$ is entangled as it can never be written as the convex combination of the density matrix of product states.
We have constructed three partite entangled state which has no bipartite entanglement.
In other words, multipartite entanglement enjoys much richer structure rather than union of bipartite entanglement.
Remark: The example we constructed here is called unextendable product bases (UPB) investigated in [@BDM+99].
Genuine entanglement
--------------------
An $n$-particle pure state ${|\psi\rangle}$ is called genuine entangled if it is not a product state of any bipartition. To defined the genuine entangle for mixed states, there are two inequivalent ways:
i). A density matrix $\rho$ is called genuine entangled if for any fixed bipartition, it can not be written as some convex combination of the density of pure states which is product in this bipartition.
ii). A density matrix $\rho$ is called genuine entangled if for it can not be written as some convex combination of the density of pure states which is product for any bipartition.
The second definition is stronger than the first one as the bipartition for different pure state can be different.
Entanglement depth
------------------
In [@SM01], entanglement depth is introduced to characterize the minimal number of particles in a system that are mutually entangled.
In an $n$-particle system ${\mathcal{H}}={\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes\cdots {\mathcal{H}}_n$, ${|\psi\rangle}$ is called $k$-product (separable) if it can be written as $${|\psi\rangle}={|\psi^{(1)}\rangle}\otimes{|\psi^{(2)}\rangle}\cdots\otimes{|\psi^{(k)}\rangle},$$ where decomposition corresponds to a partition of the $n$ particles, ${|\psi^{(i)}\rangle}$ is a genuine entangled state in ${\mathcal{H}}_{S_i}=\otimes_{j\in S_i} {\mathcal{H}}_j$ with $\bigcup S_i=\{1,2,\cdots,n\}$ and $S_i\bigcap S_k=\emptyset$ for $i\neq k$. The entanglement depth of ${|\psi\rangle}$, $\mathcal{D}(\psi)$, is defined as the largest cardinality of $S_i$.
An $n$-particle density matrix $\rho$ is called $k$-separable if it can be written as some convex combination of $k$-separable pure states. The entanglement depth of $\rho_N$ is defined as following, $$\mathcal{D}(\rho)=\min_{\rho_N=\sum p_i {|\psi^i\rangle}{\langle \psi^i|}}\max_i~~ \mathcal{D}(\psi^i),$$ where each ${|\psi^i\rangle}$ is an $N$-particle pure state and $\mathcal{D}(\psi^i)$ is the entanglement depth of ${|\psi^i\rangle}{\langle \psi^i|}$.
Mixed State Property Testing
============================
In this section, we study the possibility of detecting multipartite correlation without full state tomography by measuring only single-copy observables. For simplicity, we allow single-copy observables are only allowed to be measured nonaddaptively.
We assume that the state is mixed state, and the only know information about this state is the Hilbert space it lives in. We want to test properties of mixed states. In particular, we are interested in multipartite correlations, SLOCC invariant properties.
Let ${\mathcal{H}}=\bigotimes_{k=1}^{n}{\mathcal{H}}_k$ with $d_k$ being the dimension of ${\mathcal{H}}_k$. The set (state space) of density operators on ${\mathcal{H}}$ is defined as $$\mathcal{D}=\{\ \rho\in\lh\ :\rho\ \mbox{is semi-definite positive and} {{\rm tr}}(\rho)=
\mbox{1}\}.$$
The concept of stochastic local operations assisted by classical communication (SLOCC) has been used to study entanglement classification [@DVC00; @GW13] and entanglement transformation [@YCGD10; @YGD14]. Two $n$-partite quantum states $\rho$ and $\sigma$ are called SLOCC equivalent if $$\rho=(A_1\otimes A_2\otimes\cdots A_n)\sigma(A_1\otimes A_2\otimes\cdots A_n)^{\dag}$$ holds for some non-singular $A_i\in\lh_k$.
A property $\mathcal{P}\in \mathcal{D}$ is called SLOCC invariant if $\rho\in\mathcal{P}$ implies $\rho'\in\mathcal{P}$ for all $\rho'$ being SLOCC equivalent to $\rho$.
Our main result is given as follows,
For any stochastic local operations assisted by classical communication (SLOCC) invariant property $\mathcal{P}\subsetneq \mathcal{D}$, such that both $\mathcal{P}$ and $\mathcal{D}\setminus \mathcal{P}$ contain some positive elements respectively, it is impossible to determine with certainty of whether $\rho\in P$ or not without fully state tomography.
In other words, $t:=\Pi_{i=1}^n d_i^2-1$ measurements are necessary.
More precisely, for any set of informationally-incomplete measurements, there always exists two different states, $\rho\in\mathcal{P}$ and a $\sigma\notin \mathcal{P}$, which are not distinguishable according to the measurement results. That is, for any set of observables (Hermitian matrices) $\{O_1,O_2,\cdots,O_{s}\}$ of ${\mathcal{H}}$ with $s<t$, there always exist two different states, $\rho\in\mathcal{P}$ and a $\sigma\notin \mathcal{P}$, such that ${{\rm tr}}(O_i\rho)={{\rm tr}}(O_i\sigma)$ for all $i$.
Geometrically, any open and SLOCC invariant nontrivial $\mathcal{P}$ is not ‘cylinder-like’. In other words, the structural relation of $\mathcal{P}$ and $\mathcal{D}$ can not as (b).
\[fig:main\] ![[]{data-label="fig:theory"}](Figure-1Lu.pdf "fig:"){width="\columnwidth"}
Notice that, for quantum state in ${\mathcal{H}}=\bigotimes_{k=1}^{n}{\mathcal{H}}_k$ with $d_k$ being the dimension of ${\mathcal{H}}_k$, the informationally-complete measurements are set of linear independent Hermitian matrices $\{N_1,N_2,\cdots,N_{t}\}$ as quantum states are trace one which reduces one dimension.
To prove the validity of this theorem, we assume the existence of $\{O_1,O_2,\cdots,O_{s}\}$ of ${\mathcal{H}}$ with $s<t$ such that for any pairs of $\rho$ and $\sigma$, one can conclude that $\rho,\sigma\in\mathcal{P}$ or $\rho,\sigma\notin\mathcal{P}$ by giving ${{\rm tr}}(O_i\rho)={{\rm tr}}(O_i\sigma)$ for all $i$.
The proof is divided into two steps.
STEP 1: We transfer the problem into the existence of informationally-incomplete measurements in testing properties of semi-definite positive operators.
We first generalize the property into all semi-definite operators on ${\mathcal{H}}$ $$\widetilde{\mathcal{D}}=\{M\in\lh\ :\ M~\mbox{is semi-definite positive}.\}$$
For any property of $\mathcal{D}$, denoted by $\mathcal{P}$, satisfies that $\mathcal{P}\subsetneq \dh$, we first generalize it into property $\widetilde{\mathcal{P}}$ of $\widetilde{\mathcal{D}}$ as follows, $$\widetilde{\mathcal{P}}=\{M\in\lh :M/{{\rm tr}}(M)\in\mathcal{P}, A~\mbox{is semi-definite positive}\}.$$
We observe that $\mathcal{P}$ is SLOCC invariant if and only if $\widetilde{\mathcal{P}}$ is SLOCC invariant in the sense that for all non-singular matrices $A_i$s, $$M\in \widetilde{\mathcal{P}}\Leftrightarrow (A_1\otimes A_2\otimes\cdots A_n)M(A_1\otimes A_2\otimes\cdots A_n)^{\dag}\in \widetilde{\mathcal{P}}.$$
$\mathcal{P}$ contains some positive element if and only if $\widetilde{\mathcal{P}}$ is contains some positive element. $\mathcal{D}\setminus\mathcal{P}$ contains some positive element if and only if $\widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$ is contains some positive element.
More importantly, one can use the following set of observables $\{O_0,O_1,O_2,\cdots,O_{s}\}$ with $O_0=I_{{\mathcal{H}}}$ to test $\widetilde{\mathcal{P}}$ of of $\widetilde{\mathcal{D}}$. Notice that for any $\widetilde{\rho}\in \widetilde{D}$, we know that $\widetilde{\rho}\in \widetilde{\mathcal{P}}$ if and only if $\rho\in \mathcal{P}$ with $\rho=\widetilde{\rho}/{{\rm tr}}(O_0\widetilde{\rho})$. For $0<i$, ${{\rm tr}}(O_i\rho)={{\rm tr}}(O_i\widetilde{\rho})/{{\rm tr}}(O_0\widetilde{\rho})$. Thus, for an unknown $\widetilde{\rho}$ with $(o_0,o_1,\cdots,o_s)$ such that $o_i={{\rm tr}}(O_i\widetilde{\rho})$, one can conclude that $\widetilde{\rho}\in\widetilde{\mathcal{P}}$ if and only if the quantum states (trace 1) corresponding to $(o_1/o_0,\cdots,o_s/o_0)$ are in $\mathcal{P}$. On the other hand, $\{O_0,O_1,O_2,\cdots,O_{s}\}$ is not informationally-complete observalbes as $s<\Pi_{i=1}^n d_i^2-1$. This indicates informationally-incomplete measurements which can detect $\widetilde{\mathcal{P}}$ with certainty among all $\widetilde{\mathcal{D}}$.
STEP 2: We show there is no informationally-incomplete measurements, $\{O_0,O_1,O_2,\cdots,O_{s}\}$, which can detect properties of semi-definite positive operators with certainty.
Notice that, there exists an Hermitian $H\neq 0$ such that ${{\rm tr}}(O_iH)=0$ for all $0\leq i\leq s$. This $H$ enjoys the following property which we called “free". For any $\widetilde{\rho}\in \widetilde{\mathcal{D}}$, if $\widetilde{\rho}+rH\in \widetilde{\mathcal{D}}$ for some $r\in \mathbb{R}$, then $\widetilde{\rho}+rH\in\widetilde{\mathcal{P}}$ iff $\widetilde{\rho}\in \widetilde{\mathcal{P}}$.
Since $\widetilde{\mathcal{P}}$ is SLOCC invariant, we can conclude that for any “free" Hermitian $J$, any non-singular $A_i$s, $$(A_1\otimes A_2\otimes\cdots A_n)J(A_1\otimes A_2\otimes\cdots A_n)^{\dag}$$ is also a “free" Hermitian by the following observation. Notice that $\widetilde{\mathcal{D}}$ is SLOCC in variant, then for any $\widetilde{\rho}$ with $M=\widetilde{\rho}+r(A_1\otimes A_2\otimes\cdots A_n)J(A_1\otimes A_2\otimes\cdots A_n)^{\dag}\in \widetilde{\mathcal{D}}$, $M\in\widetilde{\mathcal{P}}$ if and only if $$(A_1^{-1}\otimes A_2^{-1}\otimes\cdots A_n^{-1})M(A_1^{-1}\otimes A_2^{-1}\otimes\cdots A_n^{-1})^{\dag}\in\widetilde{\mathcal{P}}.$$ That is $$(A_1^{-1}\otimes A_2^{-1}\otimes\cdots A_n^{-1})\widetilde{\rho}(A_1^{-1}\otimes A_2^{-1}\otimes\cdots A_n^{-1})^{\dag}+J\in\widetilde{\mathcal{P}}.$$ Invoking the fact that $J$ is “free", this is equivalent to $$(A_1^{-1}\otimes A_2^{-1}\otimes\cdots A_n^{-1})\widetilde{\rho}(A_1^{-1}\otimes A_2^{-1}\otimes\cdots A_n^{-1})^{\dag}\in\widetilde{\mathcal{P}}.$$ As $\widetilde{\mathcal{D}}$ being SLOCC invariant, this is true if and only if $$\widetilde{\rho}\in\widetilde{\mathcal{P}}.$$
This above argument leads us to the fact that $(A_1\otimes A_2\otimes\cdots A_n)J(A_1\otimes A_2\otimes\cdots A_n)^{\dag}$ is also a “free" Hermitian.
STEP 2 (a): In this part, we show that if the set of “free" Hermitian matrices is not empty, it contains elements which form a basis of the whole space $\lh$. In other words, there exist linear independent “free" Hermitian matrices $H_1,H_2,\cdots,H_t$.
For any nonzero Hermitian $J$, $S=\lh$ where $S$ is the matrix space spanning by all $$(A_1\otimes A_2\otimes\cdots A_n)J(A_1\otimes A_2\otimes\cdots A_n)^{\dag}$$ with $A_i$s being non-singular.
We actually prove a more general statment. The set of the following linear maps $$(A_1\otimes A_2\otimes\cdots A_n)\cdot(A_1\otimes A_2\otimes\cdots A_n)^{\dag}:\lh\rightarrow\lh,$$ spans the whole set of linear maps $\mathcal{L}(\lh,\lh):\lh\rightarrow\lh$.
We start from studying the case $n=1$. In this case, we are going to show that the following maps $$M\cdot I,I\cdot M$$ lie in the span of $$A\cdot A^{\dag}$$ with $A$ being non-singular.
Choose real $y\neq 0$ such that $M+yI,M-yI,M-iyI$ being non-singular. It is easy to verify that $A\cdot I$ lies in the span of $$\begin{aligned}
(M+yI)\cdot (M+yI)^{\dag},\\
(M-yI)\cdot (M-yI)^{\dag},\\
(M-iyI)\cdot (M-iyI)^{\dag}.\end{aligned}$$ That is $$\begin{aligned}
M\cdot I=\frac{1-i}{4y}(M+yI)\cdot (M+yI)^{\dag}\\-\frac{1+i}{4y}(M-yI)\cdot (M-yI)^{\dag}\\-\frac{i}{2y}(M-iyI)\cdot (M-iyI)^{\dag},\\
I\cdot M=\frac{1+i}{4y}(M+yI)\cdot (M+yI)^{\dag}\\-\frac{1-i}{4y}(M-yI)\cdot (M-yI)^{\dag}\\+\frac{i}{2y}(M-iyI)\cdot (M-iyI)^{\dag}.\\\end{aligned}$$
One crucial observation is that For non-singular $A,B$, $AB$ is still non-singular. Thus if any maps $\mathcal{E},\mathcal{F}$ lies in the span of non-singular $$A\cdot A^{\dag},$$ their composition $\mathcal{E}\circ\mathcal{F}$ also lies in that span.
Therefore, for all $M,N$, we can first implement $M\cdot I$, then apply $I\cdot N$. This observation indicates that $M\cdot N$ lie in the span of $A\cdot A^{\dag}$ with $A$ being non-singular.
Notice that any linear maps from $\lh$ to $\lh$ can be written as linear combination of form $M\cdot N$. Thus, for the case $n=1$, the following linear maps $A\cdot A^{\dag}$ spans the whole set of linear maps $\mathcal{L}(\lh,\lh):\lh\rightarrow\lh$.
Now back to the general $n$ case. Notice that any linear maps from $\lh$ to $\lh$ can be written as linear combination of form $M\cdot N$. $M\cdot N$ can be written into form $$\sum (A_{1p}\otimes A_{2p}\otimes\cdots A_{np})\cdot(B_{1p}\otimes B_{2p}\otimes\cdots B_{np})$$ with $A_{ip},B_{ip}$ being non-singular matrix of ${\mathcal{H}}_i$. We can first implement $A_{ip}\cdot B_{ip}$, then tensor them together. By linearity, we show that the set of the linear maps $$(A_1\otimes A_2\otimes\cdots A_n)\cdot(A_1\otimes A_2\otimes\cdots A_n)^{\dag}:\lh\rightarrow\lh,$$ spans the whole set of linear maps $\mathcal{L}(\lh,\lh):\lh\rightarrow\lh$.
Therefore, for any nonzero Hermitian $J$, $$(A_1\otimes A_2\otimes\cdots A_n)J(A_1\otimes A_2\otimes\cdots A_n)^{\dag}$$ forms a basis of $\lh$.
STEP 2 (b): In this part, we suppose $H_1,H_2,\cdots,H_t$ with $t=\Pi_{i=1}^n d_i^2-1$ be a set of linear independent “free" Hermitian matrices. We use the notation $||\cdot||$ to denote the two norm of the matrix.
We first let $\widetilde{H_1},\widetilde{H_2},\cdots,\widetilde{H_t}$ be the dual basis of $H_1,H_2,\cdots,H_t$. That is, ${{\rm tr}}(\widetilde{H_i}H_j)=\delta_{i,j}$ for $1\leq i,j\leq t$.
For Hermitian $Y=\sum_{i=1}^t \mu_i H_i$ such that $||Y||=1$, we have $\mu_i={{\rm tr}}(Y\widetilde{H_i})$. Therefore, $|\mu_i|\leq \sqrt{{{\rm tr}}{\widetilde{H_i}^2}}$.
Now we consider the $t$ matrices $Y_k=\sum_{i=1}^k \mu_i H_i)>0$ for $1\leq k\leq t$. Let $q=\max_{k=1}^t\{\lambda_k+\nu_k:1\leq k\leq t\}$ where $\lambda_k$ denotes the maximal eigenvalue of $Y_k$ and $\nu_k$ denotes the absolution of the minimal eigenvalue of $Y_k$.
For given $\widetilde{\rho}>0$ with $a>0$ being its minimal eigenvalue, we choose $r=\frac{a}{2q}$, then for any real number $r'$ with $|r'|<r$, and any real numbers $\mu_1,\mu_2,\cdots,\mu_t$ with $||\sum_{i=1}^t \mu_i H_i||=1$, we have $\widetilde{\rho}+r'(\sum_{i=1}^k \mu_i H_i)>0$ for all $1\leq k\leq t$.
Now we can see that if $\widetilde{\rho}\in \widetilde{\mathcal{P}}$, then for any $M$ with $||\widetilde{\rho}-M||<r$, we can have $M\in \widetilde{\mathcal{P}}$ by the following argument. Write $M=\widetilde{\rho}+r'Y$ with $Y=\sum_{i=1}^t \mu_i H_i$ and $||Y||=1$, then $r'<r$. Therefore, $\widetilde{\rho}+r'(\sum_{i=1}^k \mu_i H_i)>0$ for all $1\leq k\leq t$. As $H_1$ is “free", and $\widetilde{\rho}+r'\mu_1 H_1>0$, we have $\widetilde{\rho}+r'\mu_1 H_1\in\widetilde{\mathcal{P}}$,$\cdots$, $M=\widetilde{\rho}+r'Y \in \widetilde{\mathcal{P}}$.
If $\widetilde{\rho}\in \widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$, then for any $M$ with $||\widetilde{\rho}-M||<r$, we can have $M\in \widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$ by the following argument. Write $M=\widetilde{\rho}+r'Y$ with $Y=\sum_{i=1}^t \mu_i H_i$ and $||Y||=1$, then $r'<r$. Therefore, $\widetilde{\rho}+r'(\sum_{i=1}^k \mu_i H_i)>0$ for all $1\leq k\leq t$. As $H_1$ is “free", and $\widetilde{\rho}+r'\mu_1 H_1>0$, we have $\widetilde{\rho}+r'\mu_1 H_1\in\widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$,$\cdots$, $M=\widetilde{\rho}+r'Y \in \widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$.
Now suppose $0<\widetilde{\rho}\in \widetilde{\mathcal{P}}$ and $0<\widetilde{\sigma}\in \widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$. The for any $0\leq l\leq 1$, $M_l=l\widetilde{\rho}+(1-l)\widetilde{\sigma}>0$. Let $$l:=\sup\{l:M_x\in \widetilde{\mathcal{P}}~~\forall x\leq l\}.$$ Notice that there is a ball of center $\widetilde{\rho}$ lying in $\widetilde{\mathcal{P}}$, then $l>0$. Also there is a ball of center $\widetilde{\sigma}$ lying in $\widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$, then $l<1$. Now we consider the object $M_l$. If $M_l\in \widetilde{\mathcal{P}}$, then there is a ball of radius $r>0$ and center $M_l$ lying in $\widetilde{\mathcal{P}}$, then there is $\tilde{r}>0$ such that $M_x\in \widetilde{\mathcal{P}}~~\forall x\leq l+\tilde{r}$, contradict to the definition of $l$. Therefore, $M_l\in \widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$. Then there is a ball $B$ of center $M_l$ lying in $\widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$. Notice that $$\{M_x: x\leq l\}\bigcap B\neq \emptyset.$$ This is not possible as $\{M_x: x\leq l\}\subset \widetilde{\mathcal{P}}$ and $B\subset \widetilde{\mathcal{D}}\setminus\widetilde{\mathcal{P}}$.
Therefore, there is no informationally-incomplete measurements which can detect of property $\widetilde{\mathcal{P}}$ of $\widetilde{\mathcal{D}}$ with certainty.
Thus, there is no informationally-incomplete measurements which can detect of property $\mathcal{P}$ of $\mathcal{D}$ with certainty.
Almost all properties about multipartite correlations we are interested in are SLOCC invariant. Theorem 1 indicates that for detecting almost any multipartite correlations, fully state tomography is needed. In other words, exponential measurement resources are necessary.
In the following, we will applying our result on some examples.
$\mathcal{P}$ is the set of all PPT states, $i.e.$, states with positive partial transpose.
One can verify that $\mathcal{P}$ is SLOCC invariant. Obviously, $0<I/t\in \mathcal{P}$, and for sufficient small $x>0$, $xI/t+(1-x){|\Phi\rangle \langle \Phi|}\in\mathcal{D}\setminus \mathcal{P}$ with ${|\Phi\rangle}$ being an entangled pure states.
Applying Theorem 1, we know that fully state tomography is necessary to determine with certainty whether an unknown states is PPT or not.
$\mathcal{P}$ is the set of all entangled states.
Again, we can use the above arguments. One can verify that $P$ is SLOCC invariant. Also, $0<I/t\in \mathcal{D}\setminus \mathcal{P}$, and for sufficient small $x>0$, $xI/t+(1-x){|\Phi\rangle \langle \Phi|}\in\mathcal{P}$ with ${|\Phi\rangle}$ being an entangled pure states.
Applying Theorem 1, we know that fully state tomography is necessary to determine with certainty whether an unknown states is entangled or not.
$\mathcal{P}$ is the set of all states whose entanglement depth is $k$.
Clearly, $\mathcal{P}$ is SLOCC invariant.
If $k=1$, $0<I/t\in \mathcal{P}$, and for sufficient small $x>0$, $xI/t+(1-x){|\Phi\rangle \langle \Phi|}\in\mathcal{D}\setminus\mathcal{P}$. Applying Theorem 1, we know that fully state tomography is necessary to determine with certainty whether the entanglement depth of an unknown states is $k$ or not.
If $1<k\leq n$, $0<I/t\in \mathcal{D}\setminus\mathcal{P}$, and for sufficient small $x>0$, $xI/t+(1-x){|\Phi\rangle \langle \Phi|}\in\mathcal{P}$ with ${|\Phi\rangle}$ being an entangled pure states with depth $k$. Applying Theorem 1, we know that fully state tomography is necessary to determine with certainty whether the entanglement depth of an unknown states is $k$ or not.
If $k>n$, $\mathcal{P}=\emptyset$, no measurement is needed.
$\mathcal{P}$ is the set of all genuine entangled states (in any definition given in Section III.C).
One can verify that $\mathcal{P}$ is SLOCC invariant.
$0<I/t\in \mathcal{D}\setminus\mathcal{P}$, and for sufficient small $x>0$, $xI/t+(1-x){|\Phi\rangle \langle \Phi|}\in\mathcal{P}$ with ${|\Phi\rangle}$ being an entangled pure states.
Applying Theorem 1, we know that fully state tomography is necessary to determine with certainty whether an unknown state is genuine entangled or not.
Pure State Entanglement Testing
===============================
In this section, we study the possibility of detecting multipartite correlation without full state tomography by measuring only single-copy observables. For simplicity, we allow single-copy observables and we allow adaptive orocedures. We provide a lower bound together with an adaptive procedure with almost matching upper bound.
We assume that the state is a pure state, and the only know information about this state is the Hilbert space it lives in. We want to test whether the state is product or entangled.
Let ${\mathcal{H}}=\bigotimes_{k=1}^{n}{\mathcal{H}}_k$ with $d_k$ being the dimension of ${\mathcal{H}}_k$ and $d_1\geq d_2\geq\cdots\geq d_n$. The set (state space) of pure state on ${\mathcal{H}}$ is defined as $$\{\ {|\psi\rangle}: {\langle \psi|\psi\rangle}=\mbox{1}\}\subset{\mathcal{H}}.$$
Our problem can be formalized as following£º
Given a pure quantum ${|\psi\rangle}$, how many “local" measurements are needed to verify whether it is product, with in form $\otimes_{k=1}^n{|\psi_k\rangle}$, or not, where a measurement is called “local" if it applied only on one system nontrivially, say ${\mathcal{H}}_1$, or ${\mathcal{H}}_2$, or $\cdots$, or ${\mathcal{H}}_n$.
One can observe the following: ${|\psi\rangle}$ is product if and only if $\psi_{k}$ is a pure state for any $1\leq k\leq n$ with $\psi_k$ denoting the reduced density operator in ${\mathcal{H}}_k$. In other words, for any $k$, the resulting operator is pure (rank 1) after tracing out all other system except $k$.
We observe the following lower bound.
Any local “procedures" that can detect whether an $n$-partite pure state of ${\mathcal{H}}$ is product or not, must accomplish the pure state tomography of at least $n-1$ parties. Furthermore, at least $\sum_{k=2}^n 2(d_k-1)$ observables are necessary to detect product property.
As we observed, multipartite entanglement detecting corresponds to purity testing of each parties.
For a given $\sigma_k\in\mathcal{D}_k$, detect whether $\sigma_k$ is pure or not, where $\mathcal{D}_k$ denotes the mixed state space of ${\mathcal{H}}_k$, $$\mathcal{D}_k=\{\ \rho_k\in\lh_k\ :\rho_k\geq 0, {{\rm tr}}(\rho)=
\mbox{1}\}.$$ We first observe that purity testing must accomplish the task of pure state tomography. In other words, for different pure state ${|\psi_{k}\rangle},{|\phi_{k}\rangle}\in {\mathcal{H}}_k$, the purity testing should be able to distinguish them. Otherwise, by linearity, it can not distinguish ${|\psi_k\rangle \langle \psi_k|}$ and $1/2{|\psi_k\rangle \langle \psi_k|}+1/2{|\phi_k\rangle \langle \phi_k|}$, where the former one is pure and the later one is not a pure state. The procedure of testing purity can not determine to output $0$ (pure) or $1$ (not pure).
Suppose for parties ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, the procedure does not accomplish the pure state tomography. In other words, there exist ${|\psi_{1}\rangle},{|\phi_{1}\rangle}\in {\mathcal{H}}_1$ and ${|\psi_{2}\rangle},{|\phi_{2}\rangle}\in {\mathcal{H}}_2$ such that the procedure can not distinguish them. Then there exist a entangled pure bipartite state ${|\Omega\rangle}_{12}\in{\mathcal{H}}_1\otimes{\mathcal{H}}_2$ such that its reduced density matrices $\Omega_1=\lambda{|\psi_1\rangle \langle \psi_1|}+(1-\lambda){|\phi_1\rangle \langle \phi_1|}$, and $\Omega_2=\mu{|\psi_2\rangle \langle \psi_2|}+(1-\mu){|\phi_2\rangle \langle \phi_2|}$ for some $0<\lambda,\mu<1$. It is equivalent to find $0<\lambda,\mu<1$ such that $\lambda{|\psi_1\rangle \langle \psi_1|}+(1-\lambda){|\phi_1\rangle \langle \phi_1|}$ and $\mu{|\psi_2\rangle \langle \psi_2|}+(1-\mu){|\phi_2\rangle \langle \phi_2|}$ share the eigenvalues. We only need to choose $\lambda$ to be some very small positive number, then the corresponding $\mu$ does exist. Now, the procedure can not distinguish the following entangled state $${|\Omega_{12}\rangle}\otimes{|\psi_3\rangle}\otimes\cdots\otimes{|\psi_n\rangle}$$ and product state $${|\psi_1\rangle}\otimes{|\psi_2\rangle}\otimes{|\psi_3\rangle}\otimes\cdots\otimes{|\psi_n\rangle},$$ contradict to the assumption that the procedure can detect product property.
Therefore, the procedure must accomplish the pure state tomography of at least $n-1$ parties.
Notice that $d$-dimensional pure state tomography requires $2d-2$ observables as $d$-dimensional pure state has $2d-2$ free real parameters. Thus, at least $\sum_{k=2}^n 2(d_k-1)$ observables are necessary to detect product property.
For non-adaptive procedure, the lower bound becomes $\sum_{k=2}^n (4d_k-5)$ as the non-adaptive pure state tomography has lower bound [@HeMM12].
Notice that we do not need to accomplish the purity testing for each party since we have the constrain that the whole state is pure. In that sense $n-1$ parties are enough.
In the following, we provide an upper bound of detecting multipartite entanglement by presenting an algorithm. We suppose subsystem ${\mathcal{H}}_k$ with orthornormal basis ${|0\rangle},\cdots,{|d_k-1\rangle}$.
Let the unknown pure state ${|\psi\rangle}\in{\mathcal{H}}$ has $\psi_k$ be its reduced density matrix in subsystem ${\mathcal{H}}_k$ The algorithm output $0$ if ${|\psi\rangle}$ is product, $1$ if ${|\psi\rangle}$ is entangled $k\leftarrow 2$ $b\leftarrow 0$ Output $b$
We have the following result.
Algorithm 1 accomplishes the pure entanglement testing in ${\mathcal{H}}$ by using at most $\sum_{k=2}^n (2d_k-1)$ observables.
To prove Algorithm 1 accomplishes the pure entanglement testing in ${\mathcal{H}}$, we need to show two directions.
One direct is Algorithm 1 output $0$ if ${|\psi\rangle}$ is product. In other words, $\psi_k$ is pure for any $1\leq k\leq n$. As ${|\psi\rangle}$ is pure, we only need to prove that $\psi_k$ is pure for any $2\leq k\leq n$.
Assume $${|\psi_k\rangle}=\sum_{m=0}^{d_k-1}\beta_{m,k}{|m\rangle}.$$ According to the protocol, at Line 7, we measure ${|\psi_k\rangle}$ using measurements $E_l$ sequentially until ${{\rm tr}}({|\psi_k\rangle}{\langle \psi_k|} E_l)$ is non-zero, where $E_m={|m\rangle \langle m|}$. The goal is to find the smallest $l$ such that $\beta_l\neq 0$. The state becomes $${|\psi_k\rangle}=\sum_{m=l}^{d_k-1}\beta_{m,k}{|m\rangle},$$ where the summation starts from $m=l$ now. Now we know that $\alpha_{k,l}=\beta_{l,k}=\sqrt{{{\rm tr}}({|\psi\rangle}{\langle \psi|} E_k)}$ is positive since the global phase of a quantum state is ignorable.
The goal of Line 12 to Line 16 is to obtain $\beta_{m,j}$ for all $m\geq l$ by employing the coherence between ${|m\rangle}$ and ${|l\rangle}$. In terms of density matrix, our protocol actually provides the ($j+1$)-th row of ${|\psi\rangle \langle \psi|}$. Now we have $$\begin{aligned}
x=\langle\psi_k|(F_j+G_j){|\psi_k\rangle}&=&\beta_{l,k}\beta_{j,k}+\beta_{l,k}\beta_{j,k}^*, \\ \nonumber
y=\langle\psi_k|(F_j-G_j){|\psi_k\rangle}&=&i(\beta_{l,k}\beta_{j,k}-\beta_{l,k}\beta_{j,k})^*.\end{aligned}$$ As we have assumed that $\beta_{l,k}$ is real, it is obvious that $\beta_{l,k}^*\beta_{j,k}^*=\beta_{l,k}\beta_{j,k}^*$ for all $j>l$. Therefore, we can calculate the exact value of $\beta_{j,k}$ since we know the non-zero $\alpha_k$ and $\beta_{l,k}\beta_{j,k}$ from our measurements. $$\begin{aligned}
\beta_{j,k}=\frac{x+iy}{2\alpha_{l,k}}=\alpha_{j,k}.\end{aligned}$$ According to the fact that ${|\psi_k\rangle}$ is a normalized pure state, we have $$\begin{aligned}
\sum_{j=l}^{d_k-1}|\alpha_{j,k}|^2=\sum_{j=l}^{d_k-1}|\beta_{j,k}|^2={\langle \psi_k|\psi_k\rangle}=1.\end{aligned}$$ Therefore, if all $\psi_k$ are pure state, then Line 18-19 of Algorithm 1 will never be called. That means, the output $b$ is $0$.
In the next, we show the other direction. If ${|\psi\rangle}$ is entangled, then Algorithm 1 outputs $1$. To derive a contradiction, we assume that Algorithm 1 outputs $0$ for some entangled ${|\psi\rangle}$. We first notice that if ${|\psi\rangle}$ is entangled, there exist $k>1$ such that $\psi_k$ is not a pure state. In the next, we suppose there is the smallest $p>1$ such that $\psi_p$ is not a pure state. According to the previous argument, then the execution of Line 5-21 in Algorithm for such all $1<k<p$, would not change the value of $b$ as $\psi_k$ is pure state here. If the value of $b$ is not changed after the execution of Line 5-21 in Algorithm for $k=p$, we know that $\sum_{j=l}^{d_k-1}|\alpha_{j,k}|^2=1$. Therefore, we can define a pure state as follows $$\begin{aligned}
{|\phi_k\rangle}=\sum_{m=0}^{d_k-1}\alpha_{m,k}{|m\rangle}.\end{aligned}$$ We prove that $\psi_k=(r_{ka,kb})_{d_k\times d_k}$ is pure by showing $\psi_k={|\phi_k\rangle \langle \phi_k|}$.
For $m<l$, we have $$\begin{aligned}
r_{km,km}={{\rm tr}}(\psi_k{|m\rangle \langle m|})&={{\rm tr}}({|\phi_k\rangle \langle \phi_k|}{|m\rangle \langle m|})=0,\\
r_{kl,kl}={{\rm tr}}(\psi_k{|l\rangle \langle l|})&={{\rm tr}}({|\phi_k\rangle \langle \phi_k|}{|l\rangle \langle l|})=\alpha_{k,l}^2.\end{aligned}$$ For $l\leq m\leq d_k-1$, we have $$\begin{aligned}
r_{km,kl}={{\rm tr}}(\psi_k{|l\rangle \langle m|})&={{\rm tr}}({|\phi_k\rangle \langle \phi_k|}{|l\rangle \langle m|})=\alpha_{k,l}\alpha_{k,m}.\end{aligned}$$ As $\psi_k$ is semi-definite positive, we know that the first $l$ rows and columns of $\psi_k$ are all zero.
For any $l\leq m\leq d_k-1$, we choose the sub-matrix of $\psi_k$ of $\{{|l\rangle},{|m\rangle}\}\times\{{\langle l|},{\langle m|}\}$, $${\begin{bmatrix}\alpha_{k,l}^2&\alpha_{k,l}\alpha_{k,m}^*\\ \alpha_{k,l}\alpha_{k,m}&r_{km,km}\\ \end{bmatrix}}$$ This sub-matrix is also semi-definite positive. Thus, $$r_{km,km}\geq |\alpha_{k,m}|^2.$$ According to ${{\rm tr}}(\psi_k)=1$, we have $$1=\sum_m r_{km,km}\geq \sum_m |\alpha_{k,m}|^2=1.$$ Thus, $r_{km,km}=|\alpha_{k,m}|^2$.
Now for any $m,s>l$, we choose the sub-matrix of $\psi_k$ of $\{{|l\rangle},{|m\rangle},{|s\rangle}\}\times\{{\langle l|},{\langle m|},{\langle s|}\}$, $${\begin{bmatrix}\alpha_{k,l}^2&\alpha_{k,l}\alpha_{k,m}^*&\alpha_{k,l}\alpha_{k,s}^*\\\alpha_{k,l}\alpha_{k,m}&|\alpha_{k,m}|^2 &r_{km,ks}^*
\\ \alpha_{k,l}\alpha_{k,s}&r_{km,ks}&|\alpha_{k,s}|^2\\ \end{bmatrix}}.$$ According to its positivity of determinant, we have $r_{km,ks}=\alpha_{k,s}\alpha_{k,m}$. That is $\psi_k={|\phi_k\rangle \langle \phi_k|}$. This contradict to our assumption that $\psi_k$ is not pure. Therefore, if ${|\psi\rangle}$ is entangled, Algorithm 1 would output $1$.
For each $k>1$, the execution of testing $\psi_k$ uses at most $2d_k-1$ observables: Line 6-7 uses $l$ observables, Line 11-17 uses $2d_k-l$ observables. In total, ALgorithm 1 uses at most $\sum_{k=2}^n (2d_k-1)$ observables.
The gap between our upper bound and lower bound is at most $n-1$.
Conclusion
==========
In this paper, we study this problem of certifying entanglement without tomography in the constrain that only single copy measurements can be applied. We show that almost all multipartite correlation, include genuine entanglement detection, entanglement depth verification, requires full state tomography. However, universal entanglement detection among pure states can be much more efficient, even we only allow local measurements. Almost optimal adaptive local measurement scheme for detecting pure states entanglement is provided.
There are still many interesting open problems related to this topic. An immediate one is to generalize Theorem 1. There are two possible directions, about local unitary invariant property and adaptive measurements.
This work is supported by DE180100156.
[1]{} Algorithms for quantum computation: discrete log and factoring. In [*Proceedings of the 35th IEEE IEEE Symposium on Foundations of Computer Science*]{}. 124–134.
Quantum mechanics helps in searching for a needle in a haystack. [*78,*]{} 2, 325.
Graph states for quantum secret sharing. [*78,*]{} 042309.
A One-Way Quantum Computer. [*86,*]{} 5,188.
. Cambridge university press.
Quantum entanglement in photosynthetic light-harvesting complexes. [*6,*]{} 462-467.
. Hardy is (almost) everywhere: Nonlocality without inequalities for almost all entangled multipartite states , 250:3-14, 2016.
. Multipartite entanglement verification resistant against dishonest parties. , 108:260502, 2012.
Classical deterministic complexity of Edmonds’ Problem and quantum entanglement. , [*69,*]{} 3.
Separability Criterion for Density Matrices. , [*77,*]{} 1413.
1996\. Separability of Mixed States: Necessary and Sufficient Conditions. , [*222,*]{} 1.
Bell Inequalities and the Separability Criterion. , [*271,*]{} 319.
. A Survey of Quantum Property Testing. .
. Verifying the [Q]{}uantumness of [B]{}ipartite [C]{}orrelations. , 116:230403, 2016.
. Tomography is necessary for universal entanglement detection with single-copy observables. , 116:230501, 2016.
. Probing quantum state space: does one have to learn everything to learn something? , 473, 20160866 (2017).
. Quantum Tomography under Prior Information. , 318: 355–374, 2013.
. . Princeton University Press, Princeton, NJ.
. Cambridge university press.
, [Brassard, G.]{}, [Crepeau, C.]{}, [Jozsa, R.]{}, [Peres, A.]{}, [and]{} [Wootters, W.]{} Teleporting an unknown quantum state via dual classical and epr channels. [*70*]{}, 1895–1899.
Communication via one-and two-particle operators on einstein-podolsky-rosen states. [*69,*]{} 20, 2881–2884.
Separability in $2\times N$ composite quantum systems [*61,*]{} 062302.
. Unextendible Product Bases and Bound Entanglement. , 82:5385, 1999.
. Three qubits can be entangled in two inequivalent ways. , 62:062314, 2000.
. Classification of Multipartite Entanglement of All Finite Dimensionality. , 111:060502, 2013.
. Tensor rank of the tripartite state ${|W\rangle}^{\otimes n}$. , 81:014301, 2010.
. Obtaining a W State from a Greenberger-Horne-Zeilinger State via Stochastic Local Operations and Classical Communication with a Rate Approaching Unity. , 112:160401, 2014.
. [*86,*]{} 4431.
. [*89,*]{} 207901.
. [*89,*]{} 277906.
. [*70,*]{} 010302.
. [*53,*]{} 072203.
. [*86,*]{} 022339.
. [*88,*]{} 012109.
. .
. [*59*]{} [*4206–4216*]{} 1999.
. [*100,*]{} 070502.
. .
. [*57,*]{} 143–224.
. [*42,*]{} 504004.
. . [*1,*]{} 149.
. Springer Berlin.
. [*45,*]{} 7688.
|
---
abstract: |
We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval $1$$\;$$ < $$\;$$h$$\;$$ <$$\;$$ 2$, a fractal distribution function associated with a fractal von Neumann entropy. Fractons are charge-flux systems defined in two-dimensional multiply connected space and they carry rational or irrational values of spin. This formulation can be considered in the context of the fractional quantum Hall effect-FQHE and number theory.\
keywords: Fractal distribution function; fractal von Neumann entropy; fractional quantum Hall effect.\
Talk given at the 2nd International Londrina Winter School: Mathematical Methods in Physics, August, 26-30 (2002), Universidade Estadual de Londrina, Paraná, Brazil.
address: |
Departamento de Física,\
Universidade Estadual de Londrina, Caixa Postal 6001,\
Cep 86051-970 Londrina, PR, Brazil\
E-mail address: wdacruz@exatas.uel.br
author:
- Wellington da Cruz
title: 'A quantum-geometrical description of fracton statistics'
---
Introduction
============
We make out a review of some concepts introduced by us in the literature, such as[@R1; @R2; @R3; @R4; @R5; @R6]: fractons, universal classes $h$ of particles, fractal spectrum, duality symmetry betwenn classes $h$ of particles, fractal supersymmetry, fractal distribution function, fractal von Neumann entropy, fractal index etc. We apply these ideas in the context of the FQHE and number theory.
Fractons are charge-flux systems which carry rational or irrational values of spin. These objects are defined in two-dimensional multiply connected space and are classified in universal classes $h$ of particles or quasiparticles, with the fractal parameter or Hausdorff dimension $h$ , defined in the interval $1$$\;$$ < $$\;$$h$$\;$$ <$$\;$$ 2$. It is related to the quantum paths and can be extracted from the propagators of the particles in the momentum space[@R7]. The particles are collected in each class take into account the fractal spectrum
$$\begin{aligned}
&&h-1=1-\nu,\;\;\;\; 0 < \nu < 1;\;\;\;\;\;\;\;\;
h-1=\nu-1,\;
\;\;\;\;\;\; 1 <\nu < 2;\nonumber\\
&&h-1=3-\nu,\;\;\;\; 2 < \nu < 3;\;\;\;\;\;\;\;\;
h-1=\nu-3,\;
\;\;\;\;\;\; 3 <\nu < 4;etc.\end{aligned}$$
and the spin-statistics relation $\nu=2s$, valid for such fractons. The fractal spectrum establishes a connection between $h$ and the spin $s$ of the particles: $h=2-2s$, $0\leq s\leq \frac{1}{2}$. Thus, there exists a mirror symmetry behind this notion of fractal spectrum. Given the statistical weight for these classes of fractons
$$\label{e11}
{\cal W}[h,n]=\frac{\left[G+(nG-1)(h-1)\right]!}{[nG]!
\left[G+(nG-1)(h-1)-nG\right]!}$$
and from the condition of the entropy be a maximum, we obtain the fractal distribution function[@R2]
$$\begin{aligned}
\label{e.44}
n[h]=\frac{1}{{\cal{Y}}[\xi]-h}\end{aligned}$$
The function ${\cal{Y}}[\xi]$ satisfies the equation
$$\begin{aligned}
\label{e.4}
\xi=\biggl\{{\cal{Y}}[\xi]-1\biggr\}^{h-1}
\biggl\{{\cal{Y}}[\xi]-2\biggr\}^{2-h},\end{aligned}$$
with $\xi=\exp\left\{(\epsilon-\mu)/KT\right\}$. We understand the fractal distribution function as a quantum-geometrical description of the statistical laws of nature, since the quantum path is a fractal curve and this reflects the Heisenberg uncertainty principle.
We can obtain for any class its distribution function considering Eq.(\[e.44\]) and Eq.(\[e.4\]). For example, the universal class $h=\frac{3}{2}$ with distinct values of spin $\biggl\{\frac{1}{4},\frac{3}{4},\frac{5}{4},\cdots\biggr\}_{h=\frac{3}{2}}$, has a specific Fractal distribution
$$\begin{aligned}
n\left[\frac{3}{2}\right]=\frac{1}{\sqrt{\frac{1}{4}+\xi^2}}.\end{aligned}$$
We also have
$$\begin{aligned}
\xi^{-1}=\biggl\{\Theta[{\cal{Y}}]\biggr\}^{h-2}-
\biggl\{\Theta[{\cal{Y}}]\biggr\}^{h-1}\end{aligned}$$
where
$$\begin{aligned}
\Theta[{\cal{Y}}]=
\frac{{\cal{Y}}[\xi]-2}{{\cal{Y}}[\xi]-1}\end{aligned}$$
is the single-particle partition function. We verify that the classes $h$ satisfy a duality symmetry defined by ${\tilde{h}}=3-h$. So, fermions and bosons come as dual particles. As a consequence, we extract a fractal supersymmetry which defines pairs of particles $\left(s,s+\frac{1}{2}\right)$. In this way, the fractal distribution function appears as a natural generalization of the fermionic and bosonic distributions for particles with braiding properties. Therefore, our approach is a unified formulation in terms of the statistics which each universal class of particles satisfies: from a unique expression we can take out any distribution function. In some sense , we can say that fermions are fractons of the class $h=1$ and bosons are fractons of the class $h=2$.
The free energy for particles in a given quantum state is expressed as
$$\begin{aligned}
{\cal{F}}[h]=KT\ln\Theta[{\cal{Y}}].\end{aligned}$$
Hence, we find the average occupation number
$$\begin{aligned}
\label{e.h}
n[h]&=&\xi\frac{\partial}{\partial{\xi}}\ln\Theta[{\cal{Y}}].\end{aligned}$$
The fractal von Neumann entropy per state in terms of the average occupation number is given as[@R1; @R2]
$$\begin{aligned}
\label{e5}
{\cal{S}}_{G}[h,n]&=& K\left[\left[1+(h-1)n\right]\ln\left\{\frac{1+(h-1)n}{n}\right\}
-\left[1+(h-2)n\right]\ln\left\{\frac{1+(h-2)n}{n}\right\}\right]\end{aligned}$$
and it is associated with the fractal distribution function (Eq.\[e.44\]).
The entropies for fermions $\biggl\{\frac{1}{2},
\frac{3}{2},\frac{5}{2},\cdots\biggr\}_{h=1}$ and bosons $\biggl\{0,1,2,\cdots\biggr\}_{h=2}$, can be recovered promptly
$$\begin{aligned}
{\cal{S}}_{G}[1]=-K\biggl\{n\ln n +(1-n)\ln (1-n)\biggr\} \end{aligned}$$
and
$$\begin{aligned}
{\cal{S}}_{G}[2]=K\biggl\{(1+n)\ln (1+n)-n\ln n\biggr\}. \end{aligned}$$
Now, as we can check, each universal class $h$ of particles, within the interval of definition has its entropy defined by the Eq.(\[e5\]). Thus, for fractons of the self-dual class $\biggl\{\frac{1}{4},
\frac{3}{4},\frac{5}{4},\cdots\biggr\}_{h=\frac{3}{2}}$, we have
$$\begin{aligned}
{\cal{S}}_{G}\left[\frac{3}{2}\right]=K\left\{(2+n)\ln\sqrt{\frac{2+n}{2n}}
-(2-n)\ln\sqrt{\frac{2-n}{2n}}\right\} \end{aligned}$$
and for two more examples, the dual classes $\biggl\{\frac{1}{3},
\frac{2}{3},\frac{4}{3},\cdots\biggr\}_{h=\frac{4}{3}}$ and $\biggl\{\frac{1}{6},\frac{5}{6},\frac{7}{6},\cdots\biggr\}_{h=\frac{5}{3}}$,
the entropies read as
$$\begin{aligned}
{\cal{S}}_{G}\left[\frac{4}{3}\right]=K\left\{(3+n)\ln\sqrt[3]{\frac{3+n}{3n}}
-(3-2n)\ln\sqrt[3]{\frac{3-2n}{3n}}\right\} \end{aligned}$$
and
$$\begin{aligned}
{\cal{S}}_{G}\left[\frac{5}{3}\right]=K\left\{(3+2n)\ln\sqrt[3]{\frac{3+2n}{3n}}
-(3-n)\ln\sqrt[3]{\frac{3-n}{3n}}\right\}. \end{aligned}$$
We have also introduced the topological concept of fractal index, which is associated with each class. As we saw, $h$ is a geometrical parameter related to the quantum paths of the particles and so, we define[@R3]
$$\label{e.1}
i_{f}[h]=\frac{6}{\pi^2}\int_{\infty(T=0)}^{1(T=\infty)}
\frac{d\xi}{\xi}\ln\left\{\Theta[\cal{Y}(\xi)]\right\}.$$
We obtain for the bosonic class $i_{f}[2]=1$, for the fermionic class $i_{f}[1]=0.5$ and for some classes of fractons, we have $i_{f}[\frac{3}{2}]=0.6$, $i_{f}[\frac{4}{3}]=0.56$, $i_{f}[\frac{5}{3}]=0.656$. For the interval of the definition $ 1$$\;$$ \leq $$\;$$h$$\;$$ \leq $$\;$$ 2$, there exists the correspondence $0.5$$\;$$
\leq $$\;$$i_{f}[h]$$\;$$ \leq $$\;$$ 1$, which signalizes the connection between fractons and quasiparticles of the conformal field theories, in accordance with the unitary $c$$\;$$ <$$\;$$ 1$ representations of the central charge. For $\nu$ even it is defined by
$$\begin{aligned}
\label{e.11}
c[\nu]=i_{f}[h,\nu]-i_{f}\left[h,\frac{1}{\nu}\right]\end{aligned}$$
and for $\nu$ odd it is defined by
$$\begin{aligned}
\label{e.12}
c[\nu]=2\times i_{f}[h,\nu]-i_{f}\left[h,\frac{1}{\nu}\right],\end{aligned}$$
where $i_{f}[h,\nu]$ means the fractal index of the universal class $h$ which contains the statistical parameter $\nu=2s$ or the particles with distinct values of spin, which obey specific fractal distribution function. For example, we obtain the results
$$\begin{aligned}
&&c[0]=i_{f}[2,0]-i_{f}[h,\infty]=1;\nonumber\\
&&c[1]=2\times i_{f}[1,1]-i_{f}[1,1]=0.5;etc.\end{aligned}$$
We have noted in[@R3], for the first time, an unsuspected connection betwenn fractal geometry and conformal field theories, which second our considerations is expressed by Eqs.(\[e.1\],\[e.11\],\[e.12\]).
In another way, the central charge $c[\nu]$ can be obtained using the Rogers dilogarithm function, i.e.
$$\label{e.16}
c[\nu]=\frac{L[x^{\nu}]}{L[1]},$$
with $x^{\nu}=1-x$,$\;$ $\nu=0,1,2,3,etc.$ and
$$L[x]=-\frac{1}{2}\int_{0}^{x}\left\{\frac{\ln(1-y)}{y}
+\frac{\ln y}{1-y}\right\}dy,\; 0 < x < 1.$$
Thus, we have established a connection between fractal geometry and number theory, given that the dilogarithm function appears in this context, besides another branches of mathematics[@R8].
Fractional quantum Hall effect
==============================
Such ideas can be applied in the context of the FQHE. This phenomenon is characterized by the filling factor parameter $f$, and for each value of $f$ we have the quantization of Hall resistance and a superconducting state along the longitudinal direction of a planar system of electrons, which are manifested by semiconductor doped materials, i.e. heterojunctions, under intense perpendicular magnetic fields and lower temperatures[@R9].
The parameter $f$ is defined by $f=N\frac{\phi_{0}}{\phi}$, where $N$ is the electron number, $\phi_{0}$ is the quantum unit of flux and $\phi$ is the flux of the external magnetic field throughout the sample. The spin-statistics relation is given by $\nu=2s=2\frac{\phi\prime}{\phi_{0}}$, where $\phi\prime$ is the flux associated with the charge-flux system which defines the fracton $(h,\nu)$. According to our approach there is a correspondence between $f$ and $\nu$, numerically $f=\nu$. This way, we verify that the filling factors observed experimentally appear into the classes $h$ and from the definition of duality between the equivalence classes, we note that the FQHE occurs in pairs of these dual topological quantum numbers\
$(f,\tilde{f})=\left(\frac{1}{3},\frac{2}{3}\right),
\left(\frac{5}{3},\frac{4}{3}\right), \left(\frac{1}{5},\frac{4}{5}\right),
\left(\frac{2}{7},\frac{5}{7}\right),\left(\frac{2}{9},\frac{7}{9}\right),
\left(\frac{2}{5},\frac{3}{5}\right), \left(\frac{3}{7},\frac{4}{7}\right),
\left(\frac{4}{9},\frac{5}{9}\right) etc$.\
All the experimental data satisfy this symmetry principle. In this way, our formulatiom can predicting FQHE, that is, consider the duality symmetry discovered by us[@R2]. Thus, each Hall state is described by a system of quasiparticles(fractons) such that for a given value of filling factor $f$, the spin of the objects which constitute the physical system is $s=f/2$. We understand here fractons as modelling collective excitations of a two-dimensional electron gas under special conditions like FQHE.
We also observe that our approach, in terms of equivalence classes for the filling factors, embodies the structure of the modular group as discussed in the literature [@R2; @R10]. We have that the transitions allowed are those generated by the condition $\mid p_{2}q_{1}
-p_{1}q_{2}\mid=1$, with $h_{1}=\frac{p_{1}}{q_{1}}$ and $h_{2}=
\frac{p_{2}}{q_{2}}$. For example, we have the transitions between the classes
$$\biggl\{\frac{1}{3},\frac{5}{3},\frac{7}{3},\cdots\biggr\}_{h=\frac{5}{3}};
\biggl\{\frac{2}{5},\frac{8}{5},\frac{12}{5},\cdots\biggr\}_{h=\frac{8}{5}};
\biggl\{\frac{3}{7},\frac{11}{7},\frac{17}{7},\cdots\biggr\}_{h=\frac{11}{7}};$$ $$\biggl\{\frac{4}{9},\frac{14}{9},\frac{22}{9},\cdots\biggr\}_{h=\frac{14}{9}};
\biggl\{\frac{5}{11},\frac{17}{11},\frac{27}{11},\cdots\biggr\}_{h=\frac{17}{11}};
\biggl\{\frac{6}{13},\frac{20}{13},\frac{32}{13},\cdots\biggr\}_{h=\frac{20}{13}} etc.$$
This way, we define the universality classes of the quantum Hall transitions, which are labeled by the fractal parameter $h$. The topological character of these quantum numbers comes from the relation between $h$ and $f$, by the fractal spectrum.
Number theory
=============
We observe again that our formulation to the universal class $h$ of particles with any values of spin $s$ establishes a connection between Hausdorff dimension $h$ and the central charge $c[\nu]$. Besides this, we have obtained a relation between the fractal parameter and the Rogers dilogarithm function, through the concept of fractal index, which is defined in terms of the partition function associated with each universal class of particles. As a result, we have a connection between fractal geometry and number theory. Thus,
$$\begin{aligned}
c[\nu]&=&\frac{L[x^{\nu}]}{L[1]}=
i_{f}[h,\nu]-i_{f}\left[h,\frac{1}{\nu}\right],\;
\nu=0,2,4,etc.\\
c[\nu]&=&\frac{L[x^{\nu}]}{L[1]}=
2\times i_{f}[h,\nu]-i_{f}\left[h,\frac{1}{\nu}\right],\;
\nu=1,3,5,etc.\end{aligned}$$
Also we have established a connection between the fractal parameter $h$ and the Farey sequences of rational numbers. Now, the fractal curve is continuous and nowhere differentiable, it is self-similar, it does not depend on the scale and has fractal dimension just in the interval $1$$\;$$ < $$\;$$h$$\;$$ <$$\;$$ 2$. Given a closed path with length $L$ and resolution $R$, the fractal properties of this curve can be determined by $h-1=\lim_{R\rightarrow 0}\frac{\ln{L/l}}{\ln R}$, where $l$ is the usual length for the resolution $R$ and the curve is covering with $l/R$ spheres of diameter $R$.
Farey series $F_{n}$ of order $n$ is the increasing sequence of irreducible fractions in the range $0-1$ whose denominators do not exceed $n$. They satisfy the properties
P1. If $h_{1}=\frac{p_{1}}{q_{1}}$ and $h_{2}=\frac{p_{2}}{q_{2}}$ are two consecutive fractions $\frac{p_{1}}{q_{1}}$$ >$$ \frac{p_{2}}{q_{2}}$, then $|p_{2}q_{1}-q_{2}p_{1}|=1$.
P2. If $\frac{p_{1}}{q_{1}}$, $\frac{p_{2}}{q_{2}}$, $\frac{p_{3}}{q_{3}}$ are three consecutive fractions $\frac{p_{1}}{q_{1}}$$ >$$ \frac{p_{2}}{q_{2}}
$$>$$ \frac{p_{3}}{q_{3}}$, then $\frac{p_{2}}{q_{2}}=\frac{p_{1}+p_{3}}{q_{1}+q_{3}}$.
P3. If $\frac{p_{1}}{q_{1}}$ and $\frac{p_{2}}{q_{2}}$ are consecutive fractions in the same sequence, then among all fractions\
between the two, $\frac{p_{1}+p_{2}}{q_{1}+q_{2}}$ is the unique reduced fraction with the smallest denominator.
We have the following
[**Theorem**]{}[@R6]: [*The elements of the Farey series $F_{n}$ of the order $n$, belong to the fractal sets, whose Hausdorff dimensions are the second fractions of the fractal sets. The Hausdorff dimension has values within the interval $1$$\;$$ < $$\;$$h$$\;$$ <$$\;$$ 2$, which are associated with fractal curves.*]{}
We observe that the sets obtained are dual sets and, in particular, we have a fractal selfdual set, with Hausdorff dimension $h=\frac{3}{2}$. In this way, taking into account the fractal spectrum and the duality symmetry between sets, we can extract for any Farey series of rational numbers, fractal sets whose Hausdorff dimension is the second fraction of the set.
Conclusions
===========
We have introduced a unified description of particles with distinct values of spin in terms of their statistics. From a unique expression, the fractal distribution function, we can take out distribution functions for any universal class $h$ of particles. The Hausdorff dimension of the fractal quantum paths of the fractons are determined by the fractal spectrum. We have here a quantum-geometrical description of the statistical laws of nature.
We verify along these ideas that the FQHE occurs in pairs of dual filling factors. These quantum numbers get their topological character from the Hausdorff dimension, a geometrical parameter associated with the fractal curves of the particles. We can check that all experimental results satisfy the symmetry principle discovered by us, the duality symmetry betwenn universal classes $h$ of particles. The idea of supersymmetry, in some sense, appears in this context of the condensed matter and the universality classes of the quantum Hall transitions are established.
We emphasize that our formulation is supported by symmetry principles: mirror symmetry behind the fractal spectrum, duality symmetry betwenn classes $h$ of particles, fractal supersymmetry, modular group behind the quantum Hall transitions.
In another direction, we have established a connection betwenn Number Theory and Physics relating fractal geometry and dilogarithm function through the concept of fractal index. Also we have determined an algorithm for computation of the Hausdorff dimension of any fractal set related to the Farey sequences of rational numbers.
Finally, we are thinking about fracton quantum computing from the possible perspective of fractons qubits.
[99]{} W. da Cruz, Physica [**A313**]{} (2002), 446. W. da Cruz, Int. J. Mod. Phys. [**A15**]{} (2000), 3805. W. da Cruz and R. de Oliveira, Mod. Phys. Lett. [**A15**]{} (2000), 1931. W. da Cruz, J. Phys: Cond. Matter. [**12**]{} (2000), L673. W. da Cruz, Mod. Phys. Lett. [**A14**]{} (1999), 1933. W. da Cruz, Chaos, Solitons and Fractals [**17**]{} (2003), 975;\
W. da Cruz, cond-mat/0301587. A. M. Polyakov, in [*Proc. Les Houches Summer School [**vol. IL**]{}*]{}, ed. E. Brézin and J. Zinn-Justin (North Holland, 1990) 305. A. Kirillov, Prog. Theor. Phys. Suppl. [**118**]{} (1995), 61. R. B. Laughlin, Rev. Mod. Phys. [**71**]{}, (1999), 863;\
H. Stormer, Rev. Mod. Phys. [**71**]{}, (1999), 875;\
D. C. Tsui, Rev. Mod. Phys. [**71**]{}, (1999), 891;\
and references therein. B. P. Dolan, J. Phys. [**A32**]{} (1999), L243. Nucl. Phys. [**B554**]{} (1999), 487.
|
---
abstract: 'Several local geometric properties of Orlicz space $L_\phi$ are presented for an increasing Orlicz function $\phi$ which is not necessarily convex, and thus $L_\phi$ does not need to be a Banach space. In addition to monotonicity of $\phi$ it is supposed that $\phi(u^{1/p})$ is convex for some $p>0$ which is equivalent to that its lower Matuszewska-Orlicz index $\alpha_\phi>0$. Such spaces are locally bounded and are equipped with natural quasi-norms. Therefore many local geometric properties typical for Banach spaces can also be studied in those spaces. The techniques however have to be different, since duality theory cannot be applied in this case. In this article we present complete criteria, in terms of growth conditions of $\phi$, for $L_\phi$ to have type $0<p\le2$, cotype $q\ge 2$, to be (order) $p$-convex or $q$-concave, to have an upper $p$-estimate or a lower $q$-estimate, for $0<p,q<\infty$. We provide detailed proofs of most results, avoiding appealing to general not necessary theorems.'
address:
- 'Department of Mathematical Sciences, The University of Memphis, TN 38152-3240, U.S.A.'
- 'Department of Mathematical Sciences, The University of Memphis, TN 38152-3240, U.S.A.'
author:
- 'Anna Kamińska and Mariusz [Ż]{}yluk'
title: '**Local geometric properties in quasi-normed Orlicz spaces**'
---
,
We present here a number of results on local geometric properties of Orlicz spaces $L_\phi$, where $\phi:\mathbb{R}_+\to \mathbb{R}_+$, called an Orlicz function, is increasing, $\phi(0)=0$, continuous and $\lim_{u\to\infty} \phi(u) = \infty$. We will assume additionally that the space $L_\phi$ is locally bounded, which means that topology in $L_\phi$ is induced by a quasi-norm. We shall consider such notions like type, cotype, (order) convexity, concavity and upper, and lower estimates. In the case of Banach spaces (lattices) these properties are related, and in fact these relations have been thoroughly studied in the past decades [@LT2]. Although criteria for most of these properties have been known in the case of normed Orlicz space $L_\phi$ generated by a convex Orlicz function $\phi$, they have not been studied thoroughly in the case of quasi-Banach Orlicz spaces. On the other hand in paper [@K1998] the author studied Musielak-Orlicz spaces generated by the Orlicz functions (not convex) with parameter, and proved in this generality several results on (order) convexity, concavity, lower and upper estimates, though neither on type nor cotype. Theoretically from those results we can get corollaries for Orlicz spaces $L_\phi$ without parameters. However, both conditions on Orlicz functions with parameter and proofs in Musielak-Orlicz spaces, are far too complicated comparing with what can be done in just Orlicz spaces. In addition they can not be interpreted instantly in the case of Orlicz functions. Therefore it is desirable to provide direct and explicit statements and their proofs of these important properties in spaces $L_\phi$.
The paper is divided into four sections. The first section contains preliminaries comprising of notations, definitions, some general theorems, and a number of basic results on Orlicz spaces $L_\phi$. The first theorem states that $\alpha_\phi>0$ is necessary and sufficient for the Minkowski functional to be a quasi-norm in $L_\phi$. It justifies a general assumption made in the entire paper that $\alpha_\phi > 0$. We also show the relationship among Matuszewska-Orlicz indices $\alpha_\phi$, $\beta_\phi$ and the growth conditions $\Delta_2$, $\Delta^p$ and $\Delta^{*p}$. We give two regularization theorems on $\phi$, which allow us to assume that $\phi(u^{1/p})$ is convex or concave whenever $\phi\in\Delta^{*p}$ or $\phi\in \Delta^p$ respectively. It is also proved that if $\phi$ does not satisfy condition $\Delta_2$, then $L_\phi$ contains an order isomorphically isometric copy of $\ell_\infty$. All of these preliminary results are applied in the next sections.
Section 2 consists of two main theorems showing that for any $0<p,q<\infty$, $q$-convexity (resp., $p$-convexity) of $L_\phi$ is equivalent to lower $q$-estimate (resp. upper $p$-estimate) and this in turn is equivalent to $\phi\in \Delta^q$ (resp., $\phi\in \Delta^{*p}$). In the case of Banach lattices there exist duality among those properties [@LT2]. In that case it is possible to prove for instance the conditions on $q$-concavity, and by duality get the conditions on $p$-convexity. However in the case studied here when $L_\phi$ does not need to be a Banach space those theorems must be proved independently.
In section 3 we present results on type and cotype of $L_\phi$. Again we are able to characterize completely these notions in terms of conditions $\Delta_2$, $\Delta^q$ and $\Delta^{*p}$. If $1<p\le 2$ then $L_\phi$ has type $p$ if and only if $\phi\in\Delta_2$ and $\phi\in\Delta^{*p}$. For $0<p\le 1$ we show that $L_\phi$ has type $p$ if and only if $L_\Phi$ is $p$-normable which in turn is equivalent to the fact that $\phi(u^{1/p})$ is convex. In particular for $p=1$ we obtain that $1$-normability, that is the existence of a norm equivalent to the quasi-norm in $L_\phi$, is equivalent to type $1$. It is not a general fact because there exist spaces with type $1$ that are not normable [@KalKam; @Kam2018].
In the last section 4 there are theorems summarizing the conditions and their relationships for type, cotype, convexity, concavity and upper and lower estimates of $L_\phi$ for arbitrary Orlicz function with $\alpha_\phi >0$. Moreover in the case when $\phi$ is convex there are stated corollaries on conditions on $B$-convexity and uniform copies in $L_\phi$ of finite dimensional spaces $l_1^n$ and $l_\infty^n$, closely related to nontrivial type and finite cotype, respectively.
The paper is ended with some examples. We show that given $0<p\le 2 \le q<\infty$ there exist Orlicz spaces having type $p$ and cotype $q$. Moreover, for any $2\le q<\infty$ there are Orlicz spaces with the upper index $\beta_\phi = q$, which do not have cotype $q$, but they have cotype $q+\epsilon$ for any $\epsilon >0$.
All results on Orlicz spaces remain true in the case of three measure spaces, a non-atomic measure space with infinite or finite measure, or a discrete measure space identified with the set of natural numbers with counting measure. For each of these measure space, growth conditions of $\phi$ and indices are specified as follows. The growth conditions $\Delta_2$, $\Delta^q$, $\Delta^{*p}$ and indices $\alpha_\phi$, $\beta_\phi$ defined for all arguments, large arguments, and small arguments are corresponding to non-atomic infinite measure, non-atomic finite measure, and discrete counting measure, respectively. We will not repeat this convention while stating the results.
Preliminaries
=============
Let the symbols $\Bbb R$, $\Bbb R_+$ and $\Bbb N$ stand for reals, non-negative reals and natural numbers. Given a vector space $X$ the functional $x \mapsto
\|x\|$ is called a [*quasi-norm*]{} if for any $x\in
X$,
- $\|x\| = 0$ if and only if $x=0$,
- $\|ax\| = |a| \; \|x\|$, $a \in \mathbb{R}$,
- there exists $C > 0$ such that for all $x_1, x_2 \in X$, $$\|x_1 + x_2 \| \leq C (\|x_1\| + \|x_2\|).$$
We will say that $(X, \|\cdot\|)$ is a [*quasi-Banach space*]{} if it is complete [@KPR]. Notice that a quasi-Banach space is locally bounded [@KPR]. For $0 < p \leq 1$ the functional $x \mapsto ||x||$ is called a $p$-[*norm*]{} if it satisfies the previous conditions (1) and (2) and condition
- for any $x_1, x_2 \in X$, $$\|x_1 +x_2\| \leq (\|x_1\|^p + \|x_2\|^p)^{1/p}.$$
A quasi-Banach space equipped with a $p$-norm is called a $p$-Banach space.
A quasi-Banach space $(X, \|\cdot\|)$ is called [*$p$-normable*]{} for some $0<p\le 1$ if there exists $C>0$ such that $$\|x_1+ \dots + x_n\| \le C(\|x_1\|^p + \dots +\|x_n\|^p)^{1/p}$$ for all $x_1,\dots, x_n \in X$, $n\in \mathbb{N}$. If $X$ is $1$-normable then we say that $X$ is [*normable*]{}. For any $p$-normable space $(X,\|\cdot\|)$ there exists a $p$-norm equivalent to $\|\cdot\|$. In fact $$\|x\|_p = \inf\left\{\left(\sum_{i=1}^n \|x_i\|^p \right)^{1/p}:\ x=\sum_{i=1}^n x_i, \, n\in \mathbb{N}\right\}$$ is a $p$-norm and $\|x\|_p \le \|x\| \le C \|x\|_p$ (see [@KPR]).
[(Aoki-Rolewicz Theorem [@KPR; @R2])]{} \[th:aoki\]
For any quasi-Banach space $(X,\|\cdot\|)$ there exist $0<p \leq 1$ and a $p$-norm $\|\cdot\|_0$ which is equivalent to $\|\cdot\|$.
\[rem:aoki\]
If $(X, \|\cdot\|)$ is a quasi-normed lattice, then it is easy to modify the $p$-norm above to obtain an equivalent lattice $p$-norm. Indeed, setting $\|x\|_1 = \inf\{\|\,|y|\,\|_0 : |x| \le |y|\}$, the new functional $\|\cdot\|_1$ is a $p$-norm, which preserves the order and is equivalent to $\|\cdot\|$.
Let $r_n : [0,1]\rightarrow
\Bbb R, n\in\Bbb N$, be [*Rademacher functions*]{}, that is $r_n(t) =
\operatorname{sign}\,(\sin 2^n\pi t)$, $t\in [0,1]$. A quasi-Banach space $X$ has type $0 <
p\le 2$ if there is a constant $K>0$ such that, for any choice of finitely many vectors $x_1,\dots ,x_n$ from $X$, $$\int_0^1 \left\| \sum_{k=1}^n r_k(t)x_k \right\|dt \le K \left(\sum_{k=1}^n
\|x_k\|^p\right)^{1/p},$$ and it has cotype $q\ge 2$ if there is a constant $K>0$ such that for any finitely many elements $x_1,\dots ,x_n$ from $X$, $$\left(\sum_{k=1}^n \|x_k\|^q\right)^{1/q} \le K \int_0^1\left\|\sum_{k=1}^n r_k(t)
x_k\right\| dt.$$ Clearly if a quasi-Banach space has type $0<p\le 2$, respectively cotype $q\ge 2$, then it has type $r$ for any $0<r<p$, respectively cotype $r>q$. For these notions we refer to [@LT2] for Banach spaces and to [@Kal1981] for quasi-Banach spaces.
The following result by Kalton gives a connection between type $1<p\le 2$ and $1$-normability of quasi-Banach spaces.
\[th:kal1\] [@Kal1981 Theorem 4.1] Let $1<p\le 2$. If a quasi-Banach space has type $p$ then $X$ is normable.
A quasi-Banach space $(X,||\cdot||)$ which in addition is a vector lattice and $||x|| \leq ||y||$ whenever $|x|\leq |y|$ is called a [*quasi-Banach lattice*]{}. A quasi-Banach lattice $X=(X,\|\cdot \|)$ is said to be (order) $p$-*convex*, $0
< p < \infty$, respectively (order) $q$-*concave*, $0 < q< \infty,$ if there are positive constants $C_p$, respectively $D_q$, such that $$\left\|\left(\sum_{i=1}^n |x_i|^p \right)^{\frac{1}{p}}\right\| \leq C_p
\left(\sum_{i=1}^n \|x_i\|^p \right)^{\frac{1}{p}}$$ respectively, $$\left(\sum_{i=1}^n \|x_i\|^q \right)^{\frac{1}{q}} \leq D_q
\left\|\left(\sum_{i=1}^n |x_i|^q \right)^{\frac{1}{q}}\right\|$$ for every choice of vectors $x_1, \ldots, x_n \in X.$ When referring to these notions we skip the word “order”, and simply say $p$-convex or $q$-concave. We also say that $X$ satisfies an *upper $p$-estimate*, $0 < p < \infty$, respectively a *lower $q$-estimate*, $0<q<\infty$, if the inequalities in the definition of $p$-convexity, respectively $q$-concavity, hold true only for any choice of disjointly supported elements $x_1, \ldots, x_n$ in $X$ [@Kal1981; @LT2]. It is well known that if a quasi-Banach space is $p$-convex or has an upper $p$-convexity, then for any $0<r\le p$, $X$ is $r$-convex or has an upper $r$-convexity respectively. For $q$-concavity or lower $q$-estimate the similar property holds in reverse direction [@LT2].
The next two results on relationships among type, cotype, convexity and concavity hold true in Banach lattices.
\[th:LT1f18\] [@LT2 Theorem 1.f.18] A Banach lattice $X$ has type $p>1$ if and only if its dual space $X^*$ has cotype $p'$, $1/p + 1/p' =1$, and has a lower $q$-estimate for some $q<\infty$.
\[th:1f13\] A Banach lattice which has type $p>1$ is $q$-concave for some $q<\infty$.
The next result was proved in [@LT2], Theorem 1.d.6 for Banach lattices only, while it is shown here for quasi-normed lattices.
\[th:Khinchine\] Let $(X,\|\cdot\|)$ be a quasi-Banach lattice. If $X$ is $q$-concave for some $q<\infty$, then there exists $C>0$ such that for every $x_1,\dots,x_n \in X$, $n\in \mathbb{N}$, we have $$\int_0^1\left\|\sum_{i=1}^n r_i(t) x_i\right\|dt
\le C \left\|\left(\sum_{i=1}^n |x_i|^2\right)^{1/2}\right\|.$$ If $X$ is $p$-convex for some $p>0$, then there exists $C>0$ such that for every $x_1,\dots,x_n \in X$, $n\in \mathbb{N}$, we have $$\int_0^1\left\|\sum_{i=1}^n r_i(t) x_i\right\|dt
\ge C \left\|\left(\sum_{i=1}^n |x_i|^2\right)^{1/2}\right\|.$$
Recall first well known Khintchine’s inequality [@DJT; @LT1]. For every $1\le r < \infty$ there exist positive constants $A_r$ and $B_r$ such that $$A_r\left(\sum_{i=1}^n |a_i|^2 \right)^{1/2} \le \left(\int_0^1\left|\sum_{i=1}^n a_i r_i(t)\right|^rdt\right)^{1/r} \le B_r\left(\sum_{i=1}^n |a_i|^2\right)^{1/2}$$ for every choice of scalars $a_1,\dots, a_n$. We can assume that $0<p<1<q$. Let $x_1,\dots,x_n \in X$. Then by $q$-concavity of $X$ and Khintchine’s inequality, $$\begin{aligned}
\int_0^1\left\|\sum_{i=1}^n r_i(t) x_i\right\|dt &\le
\left(\int_0^1\left\|\sum_{i=1}^n r_i(t) x_i \right\|^q dt \right)^{1/q}
=\left(\frac{1}{2^n} \sum_{\theta_j =\pm 1} \left\|\sum_{j=1}^n \theta_j x_j \right\|^q\right)^{1/q}\\
&=\left(\sum_{\theta_j = \pm 1} \left\|\frac{1}{2^{n/q}}\sum_{j=1}^n \theta_j x_j \right\|^q\right)^{1/q}
\le D_q \left\|\left(\sum_{\theta_j=\pm 1}\left|\frac{1}{2^{n/q}} \sum_{j=1}^n \theta_j x_j \right|^q\right)^{1/q}\right\|\\
&= D_q \left\| \left(\frac{1}{2^n}\sum_{\theta_j =\pm 1} \left|\sum_{j=1}^n \theta_j x_j \right|^q \right)^{1/q} \right\|
= D_q \left\|\left(\int_0^1 \left|\sum_{i=1}^n r_i(t) x_i \right|^qdt \right)^{1/q}\right\| \\
&\le D_qB_q \left\|\left(\sum_{i=1}^n |x_i|^2\right)^{1/2}\right\|.\end{aligned}$$ On the other hand applying $p$-convexity of $X$ and Khintchine’s inequality, $$\begin{aligned}
\int_0^1 \left\|\sum_{i=1}^n r_i(t) x_i \right\| dt &\ge \left(\int_0^1 \left\|\sum_{i=1}^n r_i(t) x_i \right\|^pdt \right)^{1/p}
=\left(\sum_{\theta_j=\pm 1} \left\|\frac{1}{2^{n/p}} \sum_{j=1}^n \theta_j x_j \right\|^p\right)^{1/p}\\
&\ge C_p \left\| \left(\sum_{\theta_j=\pm 1} \frac{1}{2^n} \left| \sum_{j=1}^n \theta_j x_j \right|^p \right)^{1/p}\right\|
= C_p \left\|\left(\int_0^1 \left|\sum_{i=1}^n r_i(t) x_i \right|^p dt\right)^{1/p} \right\|\\
&\ge C_p A_p \left\|\left(\sum_{i=1}^n |x_i|^2\right)^{1/2}\right\|.\end{aligned}$$
In the sequel $(\Omega,\Sigma,\mu)$ denotes a $\sigma$-finite measure space. The space of all (equivalence classes of) $\Sigma$-measurable real functions defined on $\Omega$ is denoted by $L^0 = L^0(\mu)$. $L^0$ is a lattice with the pointwise order, that is $f\le g$ whenever $f(t)\le g(t)$ a.e.
In this article we will consider three types of measure spaces. $(\Omega, \Sigma,\mu)$ will be either nonatomic with $\mu(\Omega)
= \infty$ or nonatomic with $\mu(\Omega) < \infty$, or purely atomic with counting measure. In the last case we identify $\Omega$ with $\mathbb{N}$, where $\Sigma$ consists of all subsets of $\Bbb N$ and $\mu(\{n\}) = 1$ for every $n\in\Bbb N$.
A function $\phi : \Bbb R_+ \rightarrow \Bbb R_+$ is said to be an [*Orlicz function*]{} if $\phi(0) = 0$, $\phi$ is increasing (meaning strictly increasing), continuous, and $\lim_{u \rightarrow \infty} \phi
(u) = \infty$.
Several growth conditions of Orlicz functions will be considered. These conditions and also equivalence relations between Orlicz functions will be given in three different versions dependently on the measure space. Further, without additional comments (unless they are necessary), we will always associate the infinite non-atomic measure with conditions ”for all arguments”, the finite non-atomic measure with properties ”for large arguments” and finally the purely atomic measure with those defined ”for small arguments”.
In the sequel we will use the numbers called [*Matuszewska-Orlicz indices*]{}, which characterize growth conditions of real valued functions, in particular Orlicz functions $\phi$. There are three parallel definitions, for all, large or small arguments.
Let $\phi$ be an Orlicz function. The [*lower Matuszewska-Orlicz index*]{} $\alpha_\phi$ for all arguments (resp. large arguments; small arguments) is defined as follows
$\alpha_\phi = \sup\{ p\in \Bbb R: $ there exists $ c>0$ (resp. there exist $c>0$ and $v\ge 0$; there exist $c>0$ and $v>0$ such that $$\phi(au)\ge ca^p \phi(u)$$ for all $a\ge 1$ and $u\ge 0$ (resp. $u\ge v$; $0 <u\le au\le
v)\}$.
We define [*upper Matuszewska-Orlicz index*]{} $\beta_\phi$ for all arguments (resp. large arguments; small arguments) as
$\beta_\phi = \inf \{ p\in \Bbb R:$ there exists $c>0$ (resp. there exist $c>0$ and $v\ge 0$; there exist $c>0$ and $v>0$ such that $$\phi(au)\le ca^p \phi(u)$$ for all $a\ge 1$ and $u\ge 0$ (resp. $u\ge v$; $0 <u\le au\le
v)\}$.
The [*Orlicz space*]{} generated by an Orlicz function $\phi$ is denoted by $L_\phi$ and is defined as the set of all $f\in L^0$ such that $$I_\phi(\lambda f) = \int_\Omega \phi(\lambda |f(t)|)d\mu = \int
\phi (\lambda f) < \infty$$ for some $\lambda > 0$ dependent on $f$. The Minkowski functional of the set $\{f\in L^0: I_\phi(f) \le 1\}$, $$\| f \| = \| f \|_\phi = \inf \{\varepsilon > 0 :
I_\phi({f}/{\varepsilon}) \le 1\}$$ is finite on $L_\phi$. In the case of counting measure, usually the Orlicz space is denoted by $l_\phi$ and its elements are sequences $x = \{x(n)\}$.
The next result provides necessary and sufficient condition for $\|\cdot\|$ to be a quasi-norm [@MatOr].
\[th:quasinorm\] The Minkowski functional $\|\cdot\|$ is a quasi-norm in $L_\phi$ if and only if the upper index $\alpha_\phi > 0$.
If $\phi$ is a convex function then $\|\cdot\|$ is a norm in $L_\phi$, called the Luxemburg norm.
We will consider here only the case when the measure is non-atomic and infinite. Then according to our convention we use the index $\alpha_\phi$ for all arguments.
It is clear that $\|af\|= |a|\|f\|$ for $a\in \mathbb{R}$, and $\|f\|=0$ if and only if $f=0$ a.e.. If $\alpha_\phi >0$ then there exists $p>0$, such that $\phi(au) \ge C a^p\phi(u)$ for some $C>0$ and all $a\ge1$, $u\ge 0$. Setting then $a>1$ such that $C^{-1}a^{-p} = \frac12$ and $K=a$ we get $$\label{eq: 001}
\phi\left(\frac{u}{K}\right) \le \frac12 \phi(u), \ \ \ \ \text{for all} \ \ \ u\ge 0.$$ Observe also that in view of monotonicity of $\phi$ we have that $$\phi(\lambda t + (1 - \lambda)s) \le \phi(t) + \phi(s), \ \ \text{for all} \ \ \ t,s \ge 0,\ 0\le \lambda \le1.$$ Letting then $f,g\in L_\phi$ with $I_\phi\left(\frac{f}{\alpha}\right) \le 1$ and $I_\phi\left(\frac{g}{\beta}\right) \le 1$ we get $$\begin{aligned}
I_\phi\left(\frac{f+g}{K(\alpha + \beta)}\right) &\le \int\phi\left(\frac{\alpha}{\alpha+\beta} \frac{|f|}{K\alpha} + \frac{\beta}{\alpha+\beta} \frac{|g|}{K\beta}\right) \\
&\le I_\phi\left(\frac{f}{K\alpha}\right) + I_\phi\left(\frac{g}{K\beta}\right) \\
&\le \frac12 I_\phi\left(\frac{f}{\alpha}\right) + \frac12 I_\phi\left(\frac{g}{\beta}\right) \le 1. \end{aligned}$$ It follows $\|f+g\| \le K(\|f\| +\|g\|)$ for any $f,g \in L_\phi$.
If $\phi$ is convex then $$\phi(\lambda t + (1 - \lambda)s) \le \lambda\phi(t) + (1-\lambda)\phi(s), \ \ \text{for all} \ \ \ t,s \ge 0,\ 0\le \lambda \le1.$$ Then analogously as in general case letting $I_\phi\left(\frac{f}{\alpha}\right) \le 1$ and $I_\phi\left(\frac{g}{\beta}\right) \le 1$, and applying cnvexity of $\phi$, we get $$\begin{aligned}
I_\phi\left(\frac{f+g}{\alpha + \beta}\right) \
&\le \frac{\alpha}{\alpha+\beta} I_\phi\left(\frac{f}{\alpha}\right) + \frac{\beta}{\alpha+\beta} I_\phi\left(\frac{g}{\beta}\right) \le 1, \end{aligned}$$ which implies $\|f+g\| \le \|f\| + \|g\|$.
Let now $\|\cdot\|$ be a quasi-norm. Given a set $A$ with finite measure, clearly $$\|\chi_A\| = \frac{1}{\phi^{-1}\left(\frac{1}{\mu(A)}\right)}.$$ By non-atomicity of $\mu$ for any $t>0$ there exist disjoint sets $A, B$ such that $\mu(A) = \mu(B) =t$. By the assumption that $\|\cdot\|$ is a quasi-norm, there exists $M>1$ such that $$\frac{\|\chi_A + \chi_B\|}{\|\chi_A\| + \|\chi_B\|} = \frac{\phi^{-1}\left(\frac{1}{t}\right)}{2\phi^{-1}\left(\frac{1}{2t}\right)} \le M,$$ for all $t>0$. Therefore setting $K=2M$ we get $$2\phi(t) \le \phi(Kt),$$ for all $t>0$. Taking now $a\ge 1$ there exists $m\in\mathbb{N}$ such that $K^{m-1} \le a < K^m$. For $p= \frac{\ln 2}{\ln K}$ we have $2^{m-1} = (K^{m-1})^p$. It follows that for all $t\ge 0$ and $a\ge 1$, $$\phi(at) \ge \phi(K^{m-1} t) \ge 2^{m-1} \phi(t) = (K^{m-1})^p \phi(t) \ge a^p K^{-p} \phi(t).$$ Consequently $\alpha_\phi > 0$, and the proof is completed.
[**For the entire paper we assume that $\alpha_\phi > 0$. Then the space $L_\phi$ equipped with the Minkowski functional $\|\cdot\|$ is a quasi-Banach space.**]{}
Recall that Orlicz functions $\phi$ and $\psi$ are [*equivalent*]{} for all arguments (resp. large arguments; small arguments) if there exist positive constants $K_1, K_2$ (resp. positive constants $K_1, K_2$ and $v\ge 0$; positive constants $K_1, K_2$ and $v>0$) such that $$K_1^{-1}\psi(K_2^{-1}u)\le \phi(u)\le K_1\psi(K_2u)$$ for all $u\ge 0$ (resp. $u\ge v$; $u\le v$). Two equivalent Orlicz functions define the same Orlicz spaces with equivalent quasi-norms. In fact it is known more, two Orlicz spaces on non-atomic infinite measure (resp. non-atomic finite measure; purely atomic measure) are equal with equivalent quasi-norms if and only if the corresponding Orlicz functions are equivalent for all arguments (resp. large arguments; small arguments) [@MatOr; @Mus; @Lux].
We start with the most classical growth property of $\phi$, $\Delta_2$-condition. We say that an Orlicz function $\phi$ satisfies [*condition $\Delta_2$*]{} ($\phi\in \Delta_2$) for all arguments (resp. large arguments; small arguments) if there exists a positive constant $K$ (resp. there exist a positive constant $K$ and $v\ge 0$; a positive constant $K$ and $v>0$ ) such that $$\phi(2u)\le K\phi(u)$$ for all $u\ge
0$ (resp. $u\ge v$; $u\le v$).
The condition $\Delta_2$ is intimately connected to more subtle conditions $\Delta^p$ and $\Delta^{*p}$.
An Orlicz function $\phi$ satisfies [*condition $\Delta^q$*]{} ($\phi\in \Delta^q$), $q>0$, for all arguments (resp. large arguments; small arguments) if there exists a positive constant $K$ (resp. there exist a positive constant $K$ and $v\ge 0$; a positive constant $K$ and $v>0$) such that $$\phi(au)\le K a^q\phi(u)$$ for all $a\ge 1$ and $u\ge 0$ (resp. $u\ge v$; $au\le v$).
An Orlicz function $\phi$ satisfies condition $\Delta^{*p}$ ($\phi\in \Delta^{*p}$), $p> 0$, for all arguments (resp. large arguments; small arguments) if there exists a positive constant $K$ (resp. there exist a positive constant $K$ and $v\ge 0$; a positive constant $K$ and $v>0$) such that $$\phi(au)\ge K a^p \phi(u)$$ for all $a\ge 1$ and $u\ge 0$ (resp. $u\ge v$; $ au\le v$).
\[rem:delta\] If $\phi\in \Delta^q$, respectively $\phi\in \Delta^{*p}$, then $\phi\in \Delta^{q_1}$, respectively $\phi\in \Delta^{*p_1}$, for any $q_1 > q$, respectively for any $0<p<p_1$. Therefore $$\alpha_\phi = \sup\{p : \phi\in \Delta^{*p}\},\ \ \
\beta_\phi = \inf\{q : \phi\in\Delta^q\}.$$
Given an Orlicz function $\phi$, the function $\phi^*$ is defined as $$\phi^*(u) = \sup_{w\ge0}\{uw - \phi(w)\}.$$ It is called the Young conjugate or the compelmentary function, or the Legendre transformation of $\phi$. Let’s point out here that $\phi^*$ may assume infinite values although $\phi$ itself is finite. It is easy to check that $\phi^* : \Bbb R_+\rightarrow [0,+\infty]$ is convex, $\phi^*(0) = 0$, $\phi^*$ is left continuous and $\phi^*$ is not identically equal to infinity. Moreover, it is well known and easy to check that $\phi^{**}
\le \phi$ and $\phi^{**}$ is the largest convex minorant of $\phi$. Consequently $\phi = \phi^{**}$ whenever $\phi$ is convex [@IT Theorem 1 on p. 175].
Equivalent relations for Orlicz functions possibly assuming infinite values (like conjugate functions) are defined in the same manner like for the Orlicz finite valued functions. Observe that in the case when $\phi^*$ satisfies condition $\Delta_2$ for all and large arguments, it assumes only finite values, and when this condition is satisfied for small arguments, then $\phi^*$ may be replaced by an equivalent Orlicz convex finite valued function.
Let $$E_\phi= \{f\in L^0:\ I_\phi(\lambda f) < \infty\ \ \text{for every}\ \ \lambda >0\}.$$ $E_\phi$ is a closed subspace of $L_\phi$ and is usually called the subspace of finite elements. It is well known that $E_\phi$ is a closure in $L_\phi$ of the set of simple functions with finite measure supports. Moreover $L_\phi = E_\phi$ if and only if $\phi$ satisfies condition $\Delta_2$ [@Mus Theorem 8.14].
If $\phi$ is a convex Orlicz function satisfying condition $\Delta_2$ then the dual space $(E_\phi)^*$ is canonically isomorphic to $L_{\phi^*}$ via integral functionals. In fact $$(E_\phi, \|\cdot\|_\phi)^* \simeq (L_{\phi^*}, \|\cdot\|_{\phi^*}^0),$$ where the symbol $\simeq$ means the spaces are linearly isometric. Here $$\|f\|_{\phi}^0 = \|f\|^0 = \sup\left\{\int_I f\,g : I_{\phi^*}(g) \leq 1\right\}$$ is the Orlicz norm in $L_\phi$. It is well known that $\|f\| \leq \|f\|^0 \leq 2\|f\|$ for $f\in L_\phi$. [@KR; @Lux; @Mus].
The next result was proved for Musielak-Orlicz spaces in [@K1998].
\[prop:linfty\] If an Orlicz function $\phi$ does not satisfy the corresponding condition $\Delta_2$, then $L_\phi$ contains an order isomorphically isometric copy of $\ell_\infty$.
We shall conduct the proof only in the case of a non-atomic and infinite measure $\mu$.
Our first observation is that $\phi\in \Delta_2$ if and only if for every $a \ge 1$ there exists $K_a>0$ such that $$\label{eq:0000}
\phi(au) \le K_a \phi(u)$$ for all $u>0$. In fact for $a>1$ there exists $m \in\mathbb{N}$ such that $2^{m-1} \le a < 2^m$. Setting $q = \ln{K}/ln{2}$ and $K_a = 2^qa^q$, for every $u\ge 0$, $$\phi(au) \le \phi(2^mu) \le K^m\phi(u) = (2^m)^q\phi(u) \le K_a\phi(u).$$
We claim now that if $\phi$ does not satisfy $\Delta_2$, then there exists an infinite sequence $(f_i)$ of functions in $L_\phi$ of disjoint supports with $I_\phi (f_i) \leq \frac{1}{2^i}$ and $\|f_i\| = 1$, $i\in \mathbb{N}$. Indeed, if $\phi$ does not satisfy condition $\Delta_2$ then by (\[eq:0000\]) there exists a sequence $(u_n)\subset \mathbb{R}_+$ such that $$\label{eq:111}
\phi\left(\left(1 + \frac1n\right) u_n\right) \ge 2^n\phi(u_n), \ \ \ n\in\mathbb{N}.$$ Without loss of generality we can assume that $(u_n)$ is increasing. Since $\mu$ is non-atomic and infinite, for every $i\in\mathbb{N}$ there exists an infinite sequence $(A_n^i)$ such that $$\bigcup_{n=1}^\infty A_n^i \cap \bigcup_{n=1}^\infty A_n^j =\emptyset \ \ \ \text{if} \ \ i\ne j$$ and $$\label{eq:11113}
\phi(u_n) \mu(A_n^i) = \frac{1}{2^n} \ \ \ \text{ for every} \ \ \ i,n\in \mathbb{N}.$$ Define $$f_i = \sum_{n=i+1}^\infty u_n \chi_{A_n^i}, \ \ \ n\in\mathbb{N}.$$ Then $$\label{eq:114}
I_\phi(f_i) = I_\phi\left(\sum_{n=i+1}^\infty u_n \chi_{A_n^i}\right) = \sum_{n=i+1}^\infty \frac{1}{2^n} = \frac{1}{2^i},$$ and for any $\lambda >1$ there exists $n_0$ such that $\lambda > 1 + \frac{1}{n}$ for $n\ge n_0$. Thus by (\[eq:111\]) and (\[eq:11113\]) we get $$I_\phi(\lambda f_i) \ge \sum_{n=n_0}^\infty \phi\left(\left(1 + \frac{1}{n}\right) u_n\right)\mu(A_n^i)\ge \sum_{n=n_0}^\infty \frac{2^n \phi(u_n)}{2^n \phi(u_n)} = \infty.$$ Consequently $$\|f_i\| = 1,\ \ \text{and} \ \ \ f_i\wedge f_j =0, \ \ \text{for} \ \ \ i\ne j.$$ Hence and in view of (\[eq:114\]), $$I_\phi\left(\sum_{i=1}^\infty f_i \right) = \sum_{i=1}^\infty \frac{1}{2^i} = 1.$$ It follows $$\left\|\sum_{n=1}^\infty f_i\right\| =
\|f_i\| = 1, \ \ \ \text{ for} \ \ \ i\in\mathbb{N}.$$ Therefore for any $x = \{x(n)\} \in {\ell}_\infty$, $n\in\mathbb{N}$, $$|x(n)| = \|x_n f_n\| \le \left\|\sum_{n=1}^\infty x(n) f_n\right\| \ \ \
\text{and} \ \ \
\left\|\sum_{n=1}^\infty x(n) f_n\right\| \le \sup_n|x(n)| \left\|\sum_{n=1}^\infty f_n\right\| = \sup_n|x(n)|.$$ Hence $$\left\|\sum_{n=1}^\infty x(n) f_n\right\| = \sup_n|x(n)| = \|x\|_\infty,$$ and thus the closure of the linear span of $(f_n)$ in $L_\phi$ is lattice isomorphically isometric to $\ell_\infty$.
\[lem:equiv\] Conditions $\Delta^q$ and $\Delta^{*p}$, $p,q>0$, are preserved under equivalence relation of Orlicz functions.
We will show this only for $\Delta^{*p}$ when the equivalence relation and the condition are satisfied for all arguments. Let $\phi$ satisfy condition $\Delta^{*p}$ and $\psi$ be equivalent to $\phi$ that is $K_1^{-1}\psi(K_2^{-1}u)\le \phi(u) \le
K_1\psi(K_2u)$ for all $u\ge 0$ and some $K_1, K_2\ge 1$. Then for all $w=K_2^{-1}u\ge 0$ and $a\ge K_2^2$, $$K_1^{-1}\psi(aw) = K_1^{-2} K_1\psi(K_2aK_2^{-2}u)\ge K_1^{-2}
\phi(aK_2^{-2}u)\ge K_1^{-2}Ka^pK_2^{-2p}\phi(u)\ge K_1^{-3}Ka^pK_2^{-2p}\psi(w).$$ If $1\le a\le K_2^2$, then for any $w\ge 0$, $$\psi(aw)\ge \psi(w) \ge K_2^{-2p}a^p\psi(w).$$ Thus we showed that $\psi$ satisfies condition $\Delta^{*p}$.
The next two results demonstrates possible regularization of $\phi$ satisfying $\Delta^p$ or $\Delta^{*p}$. The earliest source of these theorems is in [@MatOr]. The thorough studies of these can be found in [@KMP].
\[prop:deltap\] Given $q > 0$ the following assertions are equivalent.
- An Orlicz function (resp. for $q\ge 1$, convex Orlicz function) $\phi$ satisfies condition $\Delta^q$.
- There exists an Orlicz function (resp. for $q\ge 1$, convex Orlicz function ) $\psi$ equivalent to $\phi$ such that for all $u\ge 0$, $a\ge 1$, $$\psi(au)\le a^q \psi(u).$$
- There exists an Orlicz function (resp. for $q\ge 1$, convex Orlicz function) $\psi$ equivalent to $\phi$ such that $\psi(u^{1/q})$ is concave.
\(a) $\Rightarrow$ (c) Let do it first for $\phi$ satisfying $\Delta^q$-condition for large arguments. Define $$r(u) =
\begin{cases}
\frac{\phi(v)}{v^q}, &\text{for $0\le u\le v$}\\
\inf_{v\le t\le u}\frac{\phi(t)}{t^q}, &\text{for $u>v$},
\end{cases}$$ where $v$ is a constant from condition $\Delta^q$ for large arguments. Clearly, $r(u)$ is decreasing and continuous and moreover for every $u\ge v$, $$\label{eq:11}
\frac1K\frac{\phi(u)}{u^q}\le r(u)\le\frac {\phi(u)}{u^q},$$ where $K$ is a constant in condition $\Delta^q$. Set $$\psi(u) =\int_0^u r(t)t^{q - 1}dt.$$ Clearly $\psi$ is an Orlicz function.
If $\phi$ is convex and $q\ge 1$, then the function $\psi$ is also convex since $r(t)t^{q - 1}$ is increasing. Indeed, it is constant on $(0,v]$, and for $v\le u_1 < u_2$ we have $$r(u_1)u_1^{q - 1} = u_1^{q - 1} \inf_{v\le t\le
u_1}\frac{\phi(t)}{t^q}\le u_2^{q - 1}\inf_{v\le t\le
u_1}\frac{\phi(t)}{t^q},$$ while by (\[eq:11\]) and in view of that $\phi(u)/u$ is increasing, $$r(u_1)u_1^{q - 1}\le \frac{\phi(u_1)}{u_1} = u_2^{q -
1}\inf_{u_1\le t\le u_2} \frac{\phi(u_1)}{t^{q - 1}u_1} \le u_2^{q
- 1}\inf_{u_1\le t\le u_2}\frac{\phi(t)}{t^q}.$$ Hence $$r(u_1)u_1^{q - 1} \le u_ 2^{q - 1}\min \left\{\inf_{v\le t\le
u_1}\frac{\phi(t)}{t^q} , \inf_{u_1\le t\le u_2}\frac{\phi(t)}{t^q}
\right\} = r(u_2)u_2^{q - 1}.$$
In order to check that $\psi$ is equivalent to $\phi$, observe by (\[eq:11\]) that for $u\ge 2v$, $$\psi(u)\ge r(u)\int_v^ut^{q - 1}dt \ge
\frac1q\left(1 - \frac{1}{2^q}\right)r(u)u^q \ge \frac1q\left(1 - \frac{1}{2^q}\right)\frac{1}{K}\phi(u),$$ which is half of what is needed. For the other inequality, note that in view of (\[eq:11\]) for $u\ge v$, $$\psi(u) \le \int_0^v\frac{\phi(v)}{v^q}t^{q - 1}dt +
\int_v^u\frac{\phi(u)}{u^q}t^{q - 1} dt\le \frac1q(\phi(v) +
\phi(u))\le \frac{2}{q} \phi(u).$$ Thus $\phi$ is equivalent to $\psi$. Next, an easy change of variables yields that $\psi(u^{1/q}) =
\frac1q\int_0^ur(t^{1/q})dt$. Since $r(t^{1/q})$ is decreasing, we conclude that the function $\psi(u^{1/q})$ is concave.
Since the case “for all arguments” is an easy repetition of the case “for large arguments” setting $v=0$, what remains to examine is the case when condition $\Delta^q$ is satisfied near zero. If we find an Orlicz function $\rho$ equivalent to $\phi$ and satisfying condition $\Delta^q$ for all arguments, then the proof can be done by the first step. Define $$\rho(u) =
\begin{cases}
\phi(u), &\text{for $0\le u\le v$}\\
cu^q + (\phi(v) - cv^q), &\text{for $u>v$},
\end{cases}$$ where $v$ is the number in condition $\Delta^q$ for small arguments. In case of non convex $\phi$, let $c>0$ be any constant, and in case of convex $\phi$, set $c= \frac{\phi'(v)}{qv^{q - 1}}$ where $\phi'(v)$ is a left derivative of $\phi$ at $v$. Clearly, $\rho$ is increasing and equivalent to $\phi$ for small arguments, so it is an Orlicz function. If $\phi$ is convex then $\rho$ is also convex.
Now it is enough to show that $\rho(au)\le Ka^q\rho(u)$ for all $a\ge 1$ , $u\ge 0$ and some positive constant $K$. From the assumption of $\Delta^q$ for small arguments this inequality is obvious if $au\le v$. In the case when $au>u>v$ and $\phi(v) - cv^q > 0$, $$\label{eq:001}
\rho(au) \le a^q [cu^q + (\phi(v) - cv^q)] = a^q \rho(u).$$ When $au>u>v$ and $b = -\phi(v) + cv^q > 0$, then $$\label{eq:002}
\frac{\rho(au)}{a^q\rho(u)} \le \frac{ca^qu^q}{a^q(cu^q - b)} =
\frac{cu^q}{cu^q - b} = 1 + \frac{b}{cu^q - b} \le 1 +
\frac{b}{cv^q - b} = 1 + \frac{b}{\phi(v)} = k_0,$$ where $k_0$ does not depend on $u$. Finally for $u\le v$ and $au>v$ we get the following estimation $$\rho(au)\le ca^pu^q + \phi(v)\le a^qu^q\left(c + \frac{\phi(v)}{v^q}\right) =
k_1a^qu^q,$$ with $k_1 = c + \frac{\phi(v)}{v^q}$ is not dependent on $u$. Applying condition $\Delta^q$ for $a = \frac{v}{u} \ge 1$ and $u\le v$ we get $$\phi(v) = \phi\left(\frac{v}{u} u\right)\le K\left(\frac{v}{u}\right)^q\phi(u),$$ which yields $u^q\le \frac{Kv^q}{\phi(v)} \phi(u)$. Hence $$\label{eq:003}
\rho(au)\le k_1a^qu^q\le k_2a^q \rho(u),$$ where $k_2 =\frac{k_1Kv^q}{\phi(v)} > 0$ does not depend on $u$. Combining (\[eq:001\]), (\[eq:002\]) and (\[eq:003\]) we proved that $\rho\in\Delta^q$.\
(c) $\Rightarrow$ (b) If $\psi(u^{1/q})$ is concave then the inequality in (b) is instant.\
(b) $\Rightarrow$ (a) The inequality in (b) means the $\Delta^q$-condition for all arguments of $\psi$. Since $\phi$ is equivalent with $\psi$, by Lemma \[lem:equiv\], $\phi$ also satisfies $\Delta^q$-condition.
\[prop:delta\*p\] Given $p > 0$ the following conditions are equivalent.
- The Orlicz (resp. for $p\ge 1$, convex Orlicz) function $\phi$ satisfies condition $\Delta^{*p}$.
- There exists an Orlicz (resp. for $p\ge 1$, convex Orlicz) function $\psi$ equivalent to $\phi$ such that for all $a\ge 1$, $u\ge 0$ $$\psi(au)\ge a^p\psi(u).$$
- There exists an Orlicz (resp. for $p\ge 1$, convex Orlicz) function $\psi$ equivalent to $\phi$ such that $\psi(u^{1/p})$ is convex.
Since this result is a dual part of Proposition \[prop:deltap\], the proof is analogous. We shall show only the essential construction part of implication (a) $\Rightarrow$ (c), in the case when the conditions are satisfied for large arguments. Define $$r(u) =
\begin{cases}
\frac{\phi(v)}{v^{p }}, &\text{for $0\le u\le
v$}\\
\sup_{v\le t\le u}\frac{\phi(t)}{t^p}, &\text{for $u>v$},
\end{cases}$$ where $v$ is a constant in condition $\Delta^{*p}$. The function $r(u)$ is increasing and satisfies the inequality $$\label{eq:115}
\frac{\phi(u)}{u^p}\le r(u)\le \frac1K \frac{\phi(u)}{u^p}$$ with $K$ from condition $\Delta^{*p}$. Letting $$\psi(u) = \int_0^u r(t)t^{p - 1}dt,$$ $\psi$ is an Orlicz function. If $\phi$ is convex and $p\ge 1$ then $\psi$ is also convex since $r(w) w^{p-1}$ is increasing. By changing variables $$\psi(u^{1/p})= \frac1p\int_0^ur(t^{1/p})dt,$$ and clearly $r(t^{1/p})$ is increasing, so $\psi(u^{1/p})$ is also convex. It remains to check that $\phi$ and $\psi$ are equivalent. By (\[eq:115\]) for $u\ge 2v$, $$\psi(u) \ge \int_{u/2}^{u}r(t)t^{p - 1}dt \ge
r\left(\frac{u}{2}\right)\int_{u/2}^u t^{p -1}dt \ge \frac
{\phi\left(\frac{u}{2}\right)}{(\frac{u}{2})^p} \left(\frac1p - \frac1{2^p p}\right)u^p = \phi\left(\frac u2\right)\frac{2^p - 1}{p}.$$ For the opposite inequality observe that by condition $\Delta^{*p}$ we have that $\frac1K \frac{\phi(u)}{u^p} \ge \frac{\phi(t)}{t^p}$ for $t\le u$. Letting $u\ge v$, $$\psi(u)\le \int_0^v \frac{\phi(v)}{v^{p}} t^{p-1} dt + \frac1K
\int_v^u \frac{\phi(t)}{t^p} t^{p - 1} dt
\le \frac{\phi(v)}{p} + \frac{1}{K^2} \int_v^u \frac{\phi(u)}{u^p} t^{p-1}\, dt \le \frac{1}{pK^2}\phi(u).$$
In the case of convex Orlicz function and $p\ge 1$, the method used in Propositions \[prop:deltap\], \[prop:delta\*p\] produces function $\psi$ which is not only convex but also smooth.
There exist several other constructions of $\psi$ (eg. [@BS; @KMP]). If for instance in the proof of Proposition \[prop:delta\*p\] we define the function $r(u)$ as $$r(u) =
\begin{cases}
\frac{K\phi(v)}{v^p}, &\text{for $0\le u\le v$}\\
\sup_{t\ge u}\frac{\phi(t)}{t^p}, &\text{for $u>v$},
\end{cases}$$ then $r(u)$ may not be continuous and thus $\psi$ will not be smooth.
\[prop:dual\] For $1<p<\infty$, a convex Orlicz function $\phi$ satisfies condition $\Delta^{*p}$ if and only if $\phi^*$ satisfies condition $\Delta^{q}$, where $\frac1{p} + \frac1q =
1$.
Since in view of Lemma \[lem:equiv\], both conditions $\Delta^{*p}$ and $\Delta^q$ are preserved under equivalence relation, assuming that $\phi$ satisfies $\Delta^{*p}$, by Proposition \[prop:delta\*p\] we suppose that $\phi(au)\ge a^p\phi(u)$ for every $a\ge 1, u\ge 0$. Hence $$\phi^*(au)= \sup_{w>0}\{auw - \phi(w)\} \ge \sup_{w>0}\{auw - a^{-p} \phi(aw)\} = \sup_{w>0}\{ uw - a^{-p}\phi(w)\}=
a^{-p}\phi^*(a^pu),$$ which implies that $\phi^*(a^{p}u)\le a^p \phi^*(au)$ or $\phi^*(a^{p - 1}u)\le a^p \phi^*(u)$ for every $a\ge 1 $ and $u\ge 0$. Letting $b = a^{p - 1}$, $b\ge 1$, we have $b^{q} = a^p$ and $$\phi^*(bu)\le b^{q}\phi^*(u)$$ for every $b\ge 1$ and $u\ge
0$, and so $\phi^*$ satisfies condition $\Delta^{q}$. The converse implication can be shown in the similar fashion.
\[prop:delta2\]
[**[(I)]{}** ]{} Let $\phi$ be an Orlicz function $\phi$. The following conditions are equivalent.
- $\phi$ satisfies condition $\Delta_2$.
- $\phi$ satisfies condition $\Delta^q$ for some $0<q<\infty$.
- $\beta_\phi<\infty$.
[**[(II)]{}**]{} Let $\phi$ be a convex Orlicz function. The following conditions are equivalent.
- $\phi^*$ satisfies condition $\Delta_2$.
- $\phi$ satisfies condition $\Delta^{*p}$ for some $p>1$.
- $\alpha_\phi>1$.
[**(I)**]{}. The equivalence of (2) and (3) follows from Remark \[rem:delta\].
The implication from (2) to (1) is obvious.
It is enough to show (1) implies (2). We shall prove this only in the case when the relations are satisfied for large arguments. The inequality $\phi(2u)\le K\phi(u)$ is valid for all $u\ge v$, where $v\ge 0$ and $K>1$ are some constants. Then for any $a \ge 1$ there exists $m \in\mathbb{N}$ such that $2^{m-1} \le a < 2^m$. Setting $q = \ln{K}/ln{2}$, for every $u \ge v$ and any $a \ge 1$, $$\phi(au) \le \phi(2^mu) \le K^m\phi(u) = (2^m)^q\phi(u) \le 2^qa^q\phi(u),$$ which shows that $\phi$ satisfies condition $\Delta^q$.
[**(II)**]{} From part (I), $\phi^*$ satisfies $\Delta_2$ if and only if $\phi^*$ satisfies $\Delta^q$ for some $1<q<\infty$. By the assumption that $\phi$ is convex $\phi = \phi^{**}$. Thus by Proposition \[prop:dual\], $\phi^*\in\Delta^q$ is equivalent to $\phi\in\Delta^{*p}$, where $1/p + 1/q =1$. Therefore (1’) is equivalent to (2’).
Finally (2’) is equivalent to (3’) by Remark \[rem:delta\].
Convexity, concavity, lower- and upper-estimates
================================================
\[th:concavity\] Let $0<q<\infty$ and $\phi$ be an Orlicz function. Then the following properties are equivalent.
- The Orlicz space $L_\phi$ is $q$-concave.
- $L_\phi$ satisfies a lower $q$-estimate.
- $\phi$ satisfies condition $\Delta^q$.
Since for disjointly supported functions $f_1,\dots,f_n \in L_\phi$, $$\left\|\sum_{i=1}^n f_i\right\|= \left\|\left(\sum_{i=1}^n |f_i|^q\right)^{\frac{1}{q}}\right\|,$$ clearly (i) implies (ii).
Assume (ii) and show (iii). Let $L_\phi$ satisfy a lower $q$-estimate.
Let us prove it first when the measure $\mu$ is non-atomic and finite. Without loss of generality assume that $\phi(1) = 1$ and $\mu(\Omega) = 1$. For arbitrary $u>w\ge 1$, set $$x =\frac1{\phi(u)} \ \ \ \text{and} \ \ \ y = \frac1{\phi(w)}$$ and $n = [\frac{y}{x}]$ is the entire part of $\frac{y}{x}$. Clearly $0<x<y\le 1$ and $n\le \frac{y}{x} \le
2n$. Since $nx\le y \le 1$, we are able to choose $n$ disjoint measurable sets $A_i$ with $\mu A_i = x$ for every $i = 1,\dots,
n$. Now, setting $$f_i = \phi^{-1} (1/x) \chi_{A_i} \ \ \ i = 1,\dots n,$$ where $\phi^{-1}$ is an inverse function of $\phi$ on $(0,\infty)$, we obtain that $\| f_i \| = 1$ and $$\left\|\sum_{i=1}^n f_i\right\| = \phi^{-1} (1/x) \| \chi_{\bigcup_{i=1}^n
A_i} \| = \frac {\phi^{-1}(1/x)}{\phi^{-1}(1/(nx))}.$$
If $D_q$ is a lower $q$-estimate constant then in view of $1/y \le 1/(nx)$, $$\frac{u}{w} =\frac {\phi^{-1}(1/x)}{\phi^{-1}(1/y)} \ge \frac
{\phi^{-1}(1/x)}{\phi^{-1}(1/(nx))} = \left\| \sum_{i=1}^n f_i
\right\| \ge D_q \left(\sum_{i=1}^n \|f_i\|^q\right)^{\frac1q} = D_q n^{\frac1q} \ge k
\left(\frac{y}{x}\right)^{\frac1q} = k \left(\frac{\phi(u)}{\phi(w)}\right)^{\frac1q},$$ with $k = D_q 2^{-\frac1q}$. This implies that for every $u>w\ge
1$, $$\phi(u) \le k^{-q} ({u}/{w})^q \phi(w),$$ and thus setting $s=w$ and $as=u$ where $a\ge 1$ and $s\ge 1$, $$\phi(as)\le k^{-q} a^q \phi(s),$$ which means $\Delta^q$ condition of $\phi$ for large arguments.
When the measure is non-atomic and infinite, then the proof is similar and in fact simpler.
Let’s sketch the proof in the case of discrete measure. Assume also that $\phi(1) = 1$. For any $0<w<u\le 1$, let $x = \frac1{\phi(u)}$ and $y = \frac1{\phi(w)}$ and $n = [\frac{y}{x}]$. There exists $m\in \Bbb N$ with $m\le x
< m+1$. Choose $n$ disjoint subsets $N_i$ of $\Bbb N$ with ${card}{N_i} = m$ and define $$x_{i}(j) =
\begin{cases}
\phi^{-1}(1/x),&\text{for $j\in N_{i}$} \\ 0,
&\text{for $ j\notin N_{i}$},
\end{cases}$$ for $i=1,\dots, n$. Then the vectors $x_i\in l_\phi$ have disjoint supports and $\|x_i\| = \frac{\phi^{-1}(1/x)}{\phi^{-1}(1/m)}$.
By the general assumption $\alpha_\phi > 0$, in view of Proposition \[prop:delta\*p\] there exists $p>0$ and equivalent Orlicz function $\psi$ to $\phi$ such that $\psi(u^{1/p})$ is convex. Without loss of generality we can assume that $\phi(u^{1/p})$ is convex. It follows that the function ${\phi^{-1}(u^p)}/{u}$ is decreasing. Consequently, $$\|x_i\| = \frac{\phi^{-1}(1/x)}{\phi^{-1}(1/m)} \ge
\frac{\phi^{-1}(1/(m+1))}{\phi^{-1}(1/m)} \ge
\left(\frac{m}{m+1}\right)^{1/p}.$$ By $$\frac{1}{mn} \ge \frac{1}{y}, \ \ \ \ n^{\frac{1}{q}}\ge \left(\frac{1}{2}\right)^{{1/q}}\left(\frac{y}{x}\right)^{1/q},$$ and by the lower $q$-estimate of $l_\phi$, $$\begin{aligned}
\frac{u}{w} &= \frac{\phi^{-1}(1/x)}{\phi^{-1}(1/y)} \ge \frac{\phi^{-1}(1/x)}{\phi^{-1}(1/(nm))} = \left\|\sum_{i=1}^n x_i\right\| \ge D_q\left(\sum_{i=1}^n \left(\frac{m}{m+1}\right)^{\frac{q}{p}} \right)^{\frac1q}\\
&= D_q n^{\frac{1}{q}}
\left(\frac{m}{m+1}\right)^{\frac{1}{p}}
\ge D_q \left(\frac{1}{2}\right)^{\frac1q+\frac1p } \left (\frac{y}{x}\right)^{1/q} = k \left(\frac{\phi(u)}{\phi(w)}\right)^{\frac1q},\end{aligned}$$ for every $0 < w < u \le 1$, where $D_q$ is the constant of lower $q$-estimate and $k = D_q (1/2)^{\frac1q +\frac1p}$ a constant not depending on $x$ and $y$. Similarly as in the first case it implies condition $\Delta^q$ for small arguments.
Below an alternative proof is provided in the case of finite non-atomic measure. Observe at first that $\phi$ must satisfy condition $\Delta_2$. Indeed, in the opposite situation in view of Proposition \[prop:linfty\], $l_\infty$ is an order isometric copy in $L_\phi$, and so $L_\phi$ can not satisfy any lower $q$-estimate. Note that it is easy to show that $l_\infty$ does not have any lower $q$-estimate for any $q<\infty$.
Assume now that $\phi$ does not satisfy condition $\Delta^q$ for large arguments. Assume without loss of generality that $\phi(1) = 1$. Since $\phi$ does not satisfy $\Delta^q$ for large arguments, there exist infinite sequences $\{a_n\}$ and $\{u_n\}$, such that $1\le
u_n \rightarrow \infty$ as $n\rightarrow \infty$, $a_n\ge 1$ and $$\label{eq:1222}
\phi(a_n^{1/q}u_n) \ge 2^n\, a_n\,\phi(u_n),$$ for every $n\in\Bbb N$. The sequence $\{a_n\}$ is not bounded. Indeed, if for some $M$, $a_n\le M$, then by $\Delta_2$ condition, $$2^na_n\phi(u_n)\le \phi(a_n^{1/q}u_n) \le \phi(M^{1/q}u_n) \le K
\phi(u_n)$$ for all $n\in\mathbb{N}$ and some $K>0$, which is a contradiction. Therefore we assume that $\sum_{n=1}^\infty \frac1{a_n} < \infty$, extracting a subsequence if necessary. Denoting by $[a_n]$ the entire part of $a_n$ with $[a_0] = 0$, let $$N_n = \left\{ 1 + \sum_{i=0}^{n-1} [a_i], 2 + \sum_{i=0}^{n-1} [a_i],
\dots, [a_n] + \sum_{i=0}^{n-1} [a_i] \right\},$$ for every $n\in \Bbb N$. The sequence $\{N_n\}$ is a disjoint partition of the natural numbers $\Bbb N$, and each $N_n$ contains exactly $[a_n]$ consecutive natural numbers. There will be no loss of generality if we suppose that $\mu (\Omega) = 1$. Observe that by (\[eq:1222\]), $$\sum_{n=1}^\infty \frac{[a_n]}{\phi(a_n^{1/q}u_n)} \le
\sum_{n=1}^\infty \frac1{2^n} = 1.$$ Therefore for every $n\in\Bbb
N$ there exists a finite sequence of sets $\{A_{in}\}_{i\in N_n}$ such that all $A_{in}$ are disjoint and $$\mu (A_{in}) = \frac1{\phi(a_n^{1/q}u_n)}$$ for each $i\in N_n$, $n\in \Bbb N$. Define $$g_{in} = u_n\chi_{A_{in}}$$ for $i\in N_n$ and $n\in \Bbb N$. For every $n$, $\{g_{in}\}_{i\in
N_n}$ is a finite sequence of functions with the same distributions and such that $$I_\phi (a_n^{1/q}g_{in}) = \phi (a_n^{1/q}u_n) \mu
(A_{in}) = 1.$$ Hence $\| g_{in} \| = 1/a_n^{1/q}$ for every $i\in
N_n , n\in\Bbb N$. It then implies that for sufficiently large $n\in
\Bbb N$, $$\label{eq:1333}
\sum_{i\in N_n} \|g_{in}\|^q = \sum_{i\in N_n} \frac1{a_n} = \frac
{[a_n]}{a_n} \ge \frac12.$$ On the other hand since $\phi$ satisfies condition $\Delta_2$ for large arguments, for every $\lambda > 1$ there exists a constant $K$ such that $\phi(\lambda u) \le K\phi (u)$ for all $u\ge 1$. Therefore by (\[eq:1222\]), $$\label{eq:1444}
I_\phi\left( \lambda \sum_{i\in N_n} g_{in}\right) =
[a_n]\phi(\lambda u_n)\mu(A_{in}) = \frac{[a_n]\phi(\lambda u_n)}{ \phi (a_n^{1/q}u_n)} \le \frac{[a_n] \phi (\lambda
u_n)}{2^n a_n \phi(u_n)}\le \frac{K}{2^n},$$ where $K$ depends only on $\lambda$. Hence for all $\lambda > 0$, $I_\phi (\lambda \sum_{i\in N_n} g_{in}) \rightarrow 0$, which means that $\|\sum_{i\in N_n} g_{in} \| \rightarrow 0$ as $n\rightarrow \infty$. Thus in view of (\[eq:1333\]), $L_\phi$ cannot satisfy any lower $q$-estimate.
We shall show now the implication from (iii) to (i). Let $\phi$ satisfy condition $\Delta^q$. In view of Proposition \[prop:deltap\] there exists a Young function $\psi$ equivalent to $\phi$ such that $\psi(u^{1/q})$ is concave. Since the norms determined by both functions $\psi$ and $\phi$ are equivalent, we assume not losing generality that $\phi(u^{1/q})$ is concave. By direct calculations we get
$$\|f\|_{\phi(u^{1/q})} = \| |f|^{1/q} \|_\phi^q.$$ Notice also that the Luxemburg functional defined by means of a concave function (e.g. $\|\cdot \|_{\phi(u^{1/q})}$) satisfies the reverse triangle inequality. Thus $$\left\|\left(\sum_{i=1}^n |f_i|^q\right)^{1/q} \right\|_\phi =\left\| \sum_{i=1}^n |f_i|^q
\right\|_{\phi(u^{1/q})}^{1/q}\ge \left(\sum_{i=1}^n \| \,|f_i|^q
\|_{\phi(u^{1/q})}\right)^{1/q}=\left(\sum_{i=1}^n \|f_i\|_\phi^q\right)^{1/q},$$ which shows that $L_\phi$ is $q$-concave.
\[th:convexity\] Let $0<p<\infty$ and $\phi$ be an Orlicz function. Then the following conditions are equivalent.
- The Orlicz space $L_\phi$ is $p$-convex.
- $L_\phi$ satisfies an upper $p$-estimate.
- $\phi$ satisfies condition $\Delta^{*p}$.
It is clear that (i) yields (ii).
\(ii) $\to$ (iii) Let now $L_\phi$ satisfy an upper $p$-estimate. We will show (iii) that is $\phi\in\Delta^{*p}$. We are giving the proof only in the case of finite and non-atomic measure. Assume without loss of generality that $\phi(1) = 1$ and $\mu(\Omega) = 1$. For the sake of convenience we will repeat several steps from the proof of the implication (ii) to (iii) of Theorem \[th:concavity\].
For arbitrary $u>w\ge 1$, set $$x =\frac1{\phi(u)} \ \ \ \text{and} \ \ \ y = \frac1{\phi(w)}$$ and $n = [\frac{y}{x}]$ is the entire part of $\frac{y}{x}$. Clearly $0<x<y\le 1$ and $n\le \frac{y}{x} \le
2n$. Since $nx\le y \le 1$, we are able to choose $n$ disjoint measurable sets $A_i$ with $\mu A_i = x$ for every $i = 1,\dots,
n$. Now, setting $$f_i = \phi^{-1} (1/x) \chi_{A_i} \ \ \ i = 1,\dots n,$$ where $\phi^{-1}$ is the inverse function on $(0,\infty)$ of $\phi$, we obtain that $\| f_i \| = 1$, $f_i\wedge f_j =0$ for $i\ne j$, and $$\left\|\sum_{i=1}^n f_i\right\| = \phi^{-1} (1/x) \| \chi_{\bigcup_{i=1}^n
A_i} \| = \frac {\phi^{-1}(1/x)}{\phi^{-1}(1/(nx))}.$$
The general assumption $\alpha_\phi >0$ implies that $\phi^{-1}$ satisfies condition $\Delta_2$. Indeed there exist a constant $1>C>0$, $r>0$ such that $\phi(at) \ge Ca^r\phi(t)$ for all $a\ge 1$ and $t\ge 0$. Hence for all $s =\phi(t)\ge 0$ we have $a\phi^{-1}(s) \ge \phi^{-1}(Ca^r s)$. Setting $K = a=(2/C)^r$, we get $$\phi^{-1}(2 s) \le (2/C)^r \phi^{-1}(s) = K \phi^{-1}(s).$$ If $C_p$ is an upper $p$-estimate constant then in view of $1/y\ge 1/(2nx)$ and the above inequality, $$\begin{aligned}
\frac{u}{w} &=\frac {\phi^{-1}(1/x)}{\phi^{-1}(1/y)} \le \frac
{\phi^{-1}(1/x)}{\phi^{-1}(1/(2nx))} \le K \frac
{\phi^{-1}(1/x)}{\phi^{-1}(1/(nx))} \\
& = K\left\| \sum_{i=1}^n f_i
\right\| \le C_p \left(\sum_{i=1}^n \|f_i\|^p\right)^{\frac1p} = KC_p n^{\frac1p} \le KC_p
\left(\frac{y}{x}\right)^{\frac1p} = KC_p \left(\frac{\phi(u)}{\phi(w)}\right)^{\frac1p}.\end{aligned}$$ This implies that for every $u>w\ge
1$, $$\phi(u) \ge (KC_p)^{-p} ({u}/{w})^p \phi(w),$$ and thus setting $s=w$ and $as=u$ where $a\ge 1$ and $s\ge 1$, $$\phi(as)\ge (KC_p)^{-p} a^p\phi(s),$$ which means $\Delta^{*p}$ condition of $\phi$ for large arguments.
[*Proof of (iii) $\to$ (i)*]{} [*if $\phi$ is convex.*]{}
In this case $L_\phi$ is a Banach space and we can use general duality between convexity and concavity. In view of Proposition \[prop:dual\], $\phi^*$ satisfies condition $\Delta^{p'}$. Thus by Theorem \[th:concavity\], $L_{\phi^*}$ must be $p'$-concave. Moreover $\phi^*\in \Delta^{p'}$ implies that $\phi^*\in\Delta_2$ in view of Proposition \[prop:delta2\]. Therefore $L_{\phi^*}= E_{\phi^*}$, and so $(L_{\phi^*})^* \simeq L_\phi$. Consequently, in view of Theorem \[th:LT1f18\], $ L_\phi$ is $p$-convex.
[*Proof of (iii) $\to$ (i)*]{} [*for arbitrary Orlicz function $\phi$.*]{}
We can give a proof analogous to the last part of the proof in Theorem \[th:concavity\] replacing concavity by convexity. However we can treat this more generally. Recall, given $0< p < \infty$ and a quasi-Banach lattice $(X, \|\cdot\|_X)$, the $p$-convexification of $X$ is the space $X^{(p)} = \{x\in X : |x|^p \in X\}$ equipped with the quasi-norm $\|x\|_{X^{(p)}} = \|\,|x|^p\,\|_X^{1/p}$. It is straightforward to show that $X^{(p)}$ is $p$-convex [@LT2 page 53].
Now if $\phi$ satisfies condition $\Delta^{*p}$, then in view of Proposition \[prop:delta\*p\], $\phi$ is equivalent to an Orlicz function $\psi$ such that $\psi(u^{1/p})$ is convex. Thus $L_{\psi(
u^{1/p})}$ is a Banach space. Moreover, $L_{\psi}= (L_{\psi(u^{1/p})})^{(p)}$ is a $p$-convexification of $L_{\psi(u^{1/p})}$. Hence $L_\psi$ and so $L_{\phi}$ is $p$-convex.
type and cotype
===============
\[th:cotype\] Let $q\ge 2$ and $\phi$ be an Orlicz function. Then the Orlicz space $L_\phi$ has cotype $q$ if and only if $\phi$ satisfies condition $\Delta^q$.
Under the assumption of convexity on $\phi$, the space $L_\phi$ is a Banach lattice. Therefore we can use general facts from Banach spaces and lattices. The summary of the relations among several local geometric properties is given in two diagrams in the monograph [@LT2]. By them, for $q>2$, lower $q$-estimate is equivalent to cotype $q$. Moreover, $2$-concavity and cotype $2$ are also equivalent. Thus in view of Theorem \[th:concavity\] the proof is done.
Assume now that $\phi$ is an arbitrary Orlicz function, not necessarily convex, and that $L_\phi$ has cotype $q<\infty$. We will show that $\phi$ satisfies condition $\Delta^q$.
Observe at first that $\phi$ must satisfy condition $\Delta_2$. Indeed, in the opposite case in view of Proposition \[prop:linfty\], $L_\phi$ contains an isomorphic copy of $l_\infty$. The space $l_\infty$ does not have any finite cotype. In fact let $e_i =(0,\dots,0,1,0,\dots)$, $i\in\mathbb{N}$, with $1$ on the $i$-th place be unit vectors in $l_\infty$. Then $$\label{eq:112}
\int_0^1\left\|\sum_{i=1}^n r_i(t)e_i\right\|_\infty\, dt = \left\|\sum_{i=1}^n e_i\right\|_\infty = 1 \ \ \ \ \text{and} \ \ \ \sum_{i=1}^n \|e_i\|_\infty^q = n,$$ for any $n\in\mathbb{N}$ and $q>0$, which implies that $l_\infty$ does not have a finite cotype. It follows that $L_\phi$ does not have it either, which contradicts the assumption.
We provide a further proof only in the case of finite non-atomic measure. Assume that $\phi$ does not satisfy condition $\Delta^q$ for large arguments. We employ now the functions $\{g_{in}\}_{i\in N_n}$ constructed in the proof of Theorem \[th:concavity\]. They satisfy the inequalities (\[eq:1333\]) and (\[eq:1444\]). Hence for all $\lambda > 0$, $I_\phi (\lambda \sum_{i\in N_n} g_{in}) \rightarrow 0$, equivalently that $\|\sum_{i\in N_n} g_{in} \| \rightarrow 0$ as $n\rightarrow \infty$. By this, (\[eq:1333\]) and the assumption that $L_\phi$ has cotype $q$, $$\frac12\le\sum_{i\in N_n} \|g_{in}\|^q \le K^q\left( \int_0^1\left\|\sum_{i\in N_n} r_i(t) g_{in}\right\| \, dt\right)^q = K^q \left\|\sum_{i\in N_n} g_{in} \right\|^q \to 0,$$ as $n\to\infty$, which concludes that $L_\phi$ has no cotype $q$, and thus $\phi\in \Delta^q$.
Let now $\phi$ satisfy condition $\Delta^q$. Then by Theorem \[th:concavity\], $L_\phi$ is $q$-concave.
By the general assumption $\alpha_\phi>0$ and Remark \[rem:delta\], $\phi\in\Delta^{*r}$ for some $r>0$. This yields that $L_\phi$ is $r$-convex by Theorem \[th:convexity\]. Therefore in view of the generalization of Khintchine’s inequality given by Theorem \[th:Khinchine\] and $2\le q<\infty$, $$\int_0^1\left\|\sum_{i=1}^n r(t) f_i\right\|_\phi\,dt \ge C \left\|\left(\sum_{i=1}^n |f_i|^2\right)^{1/2}\right\|_\phi \ge C \left\|\left(\sum_{i=1}^n |f_i|^q\right)^{1/q}\right\|_\phi \ge CD_q\left(\sum_{i=1}^n \|f_i\|_\phi^q\right)^{1/q},$$ where $D_q$ is a $q$-concavity constant of $L_\phi$. Hence $L_\phi$ has cotype $q$.
\[prop:type1\] Let $0<p\le 2$. If $L_\phi$ has type $p$ then $\phi$ satisfies condition $\Delta^{*p}$.
We provide a proof only in the case of finite non-atomic measure. Assume without loss of generality that $\phi(1) = 1$ and $\mu(\Omega) = 1$.
Assume that $\phi$ does not satisfy condition $\Delta^{*p}$ for large arguments. There exist infinite sequences $\{a_n\}$ and $\{u_n\}$, such that $1\le
u_n \rightarrow \infty$ as $n\rightarrow \infty$, $a_n\ge 1$ and $$\label{eq:16}
\phi(a_n^{1/p}u_n) \le a^{-n}\, a_n\,\phi(u_n),$$ for every $n\in\Bbb N$ and some $a \ge 2$. Observe that the sequence $\{a_n\}$ is not bounded. If for a contrary, for some $K>0$, $a_n\le K$ for all $n\in\mathbb{N}$, then $$\phi(u_n)\le \phi(a_n^{1/p} u_n)\le a^{-n} a_n \phi(u_n) \ \ \text{so}\ \ \ 1 \le \frac{a_n}{a^n} \le \frac{K}{a^n} \to 0 \ \ \text{as} \ \ \ n\to\infty,$$ which is a contradiction. Recall, given $a\in \mathbb{R}$, the ceiling function $\ceil*{a}$ is the smallest integer bigger than or equal to $a$. Let $\ceil*{a_0} = 0$ and $$N_n = \left\{ 1 + \sum_{i=0}^{n-1} \ceil*{a_i}, 2 + \sum_{i=0}^{n-1} \ceil*{a_i},
\dots, \ceil*{a_n} + \sum_{i=0}^{n-1} \ceil*{a_i} \right\},$$ for every $n\in \Bbb N$. The sequence $\{N_n\}$ is a disjoint partition of natural numbers $\Bbb N$, and each $N_n$ contains exactly $\ceil*{a_n}$ consecutive natural numbers. By the assumption $\alpha_\phi > 0$, in view of Remark \[rem:delta\], $\phi\in\Delta^{*s}$ for some $p>s>0$. Further by Proposition \[prop:delta\*p\], the function $\phi(u^{1/s})$ is equivalent to a convex function. Assume thus not loosing generality that $\phi(u^{1/s})$ is convex. Thus by (\[eq:16\]), $$\label{eq:166}
\ceil*{a_n} \phi(u_n) \ge 2^n \phi(a_n^{1/p} u_n) = 2^n \phi((a_n^{s/p} u_n^s)^{1/s}) \ge \phi(2^{n/s} a_n^{1/p} u_n).$$ Clearly $$\frac{\ceil*{a_n}}{\phi(2^{n/s} a_n ^{1/p} u_n)} \le \frac{a_n+1}{\phi(2^{n/s} a_n ^{1/p})}.$$ Consider now the convex function $\psi(u) = \phi(u^{1/s})$. It follows that $\psi(2^x x^{s/p}) \ge 2^x x^s \psi(1)$ for $x\ge 1$. Therefore $$\lim_{x\to\infty}\frac{x+1}{\phi(2^{x/s} x^{1/p})} = \lim_{x\to\infty}\frac{x+1}{\psi(2^x x^{s/p})} \le \lim_{x\to\infty}\frac{2x}{2^x x^{s/p} \psi(1)} = \lim_{x\to\infty}\frac{2x^{1-\frac{s}{p}}}{2^x \psi(1)} =0.$$ Hence choosing a subsequence if necessary we can assume that $$\sum_{n=1}^\infty \frac{\ceil*{a_n}}{\phi(2^{n/s} a_n ^{1/p} u_n)} < \mu(\Omega) = 1.$$ Therefore for every $n\in\Bbb
N$ there exists a finite sequence of disjoint sets $\{A_{in}\}_{i\in N_n}\subset \Omega$ such that $$\mu (A_{in}) = \frac1{\phi(2^{n/s}a_n^{1/p}u_n)}$$ for each $i\in N_n$, $n\in \Bbb N$. Defining for $i\in N_n$ and $n\in \Bbb N$, $$g_{in} = u_n\chi_{A_{in}},$$ it follows that $$I_\phi (2^{n/s} a_n^{1/p}g_{in}) = \phi (2^{n/s} a_n^{1/p}u_n) \mu
(A_{in}) = 1.$$ Consequently, $$\label{eq:13}
\| g_{in} \|^p = \frac{1}{2^{np/s} a_n} \ \ \ \text{and}\ \ \ \sum_{i\in N_n} \|g_{in}\|^p = \frac{\ceil*{a_n}}{2^{np/s}a_n} \le \frac
{2}{2^{np/s}} \to 0,$$ as $n\to \infty$. On the other hand by (\[eq:166\]), for every $n\in\mathbb{N}$, $$I_\phi\left( \sum_{i\in N_n} g_{in}\right) =
\ceil*{a_n}\phi(u_n)\mu(A_{in}) = \frac{\ceil*{a_n}\phi(u_n)}{ \phi (2^{n/s}a_n^{1/p}u_n)} \ge 1.$$ Now since $g_{in}$ are disjointly supported for $i\in N_n$, we get $$\label{eq:14}
K\left(\sum_{i\in N_n} \|g_{in}\|^p\right)^{1/p}\ge \int_0^1\left\|\sum_{i\in N_n} r_i(t) g_{in}\right\| \, dt= \left\|\sum_{i\in N_n} g_{in}\right\| \ge 1.$$ Finally combining (\[eq:13\]) and (\[eq:14\]), $L_\phi$ cannot have type $p$, which which finishes the proof.
\[prop:type2\] Let $1<p\le 2$. If $L_\phi$ has type $p$ then $\phi$ satisfies condition $\Delta_2$.
If $\phi$ does not satisfy $\Delta_2$ then by Proposition \[prop:linfty\], $L_\phi$ contains an isomorphic copy of $l_\infty$, which by reasoning as in (\[eq:112\]), does not have any finite cotype. Thus $L_\phi$ does not posses any finite cotype either. Now observe that $L_\phi$ can be treated as Banach space. Indeed if $L_\phi$ has type $p>1$ then by Theorem \[th:kal1\], the space $L_\phi$ is normable. Consequently in view of Theorem \[th:1f13\] it cannot have type bigger than $1$.
\[th:typecriterion\] Given $1<p\le 2$ and an Orlicz function $\phi$, the Orlicz space $L_\phi$ has type $p$ if and only if $\phi$ satisfies both conditions $\Delta_2$ and $\Delta^{*p}$.
[*Proof in the case $1<p\le2$ and a convex $\phi$.*]{}
Here we use general relations in Banach spaces between type and convexity, and duality between type and cotype.
Let $\phi$ satisfy $\Delta_2$ and $\Delta^{*p}$. It follows that $L_\phi$ is $p$-convex by Theorem \[th:convexity\]. Moreover, by Proposition \[prop:delta2\], $\phi\in\Delta^q$ for some $1<q<\infty$, and hence $L_\phi$ is $q$-concave in view of Theorem \[th:concavity\]. Applying now the diagram on page 101 in [@LT2], $L_\phi$ has type $p$.
If $L_\phi$ has type $p>1$, then due to Theorem \[th:1f13\], the space is $q$-concave for some $q<\infty$. Consequently, it cannot contain an order isomorphic copy of $l_\infty$ and so by Proposition \[prop:linfty\], $\phi$ must satisfy condition $\Delta_2$. It follows that $(L_\phi)^* \simeq L_{\phi^*}$. This in turn implies that $L_{\phi^*}$ has cotype $p'$ from Theorem \[th:LT1f18\]. Now by Theorem \[th:cotype\], $\phi^*\in\Delta^{p'}$ and then in view of Proposition \[prop:dual\], $\phi\in\Delta^{*p}$.
[*Proof in case $1< p\le2$ and an arbitrary Orlicz function $\phi$.*]{}
The [*necessity*]{} follows from Propositions \[prop:type1\] and \[prop:type2\].
[*Sufficiency.*]{} Let now $\phi\in \Delta_2$ and $\phi\in\Delta^{*p}$.
By Theorem \[th:convexity\], the condition $\Delta^{*p}$ implies that the space $L_\phi$ is $p$-convex.
In view of Proposition \[prop:delta2\] the assumption $\Delta_2$ for $\phi$ implies that $\phi\in \Delta^q$ for some $q<\infty$, and so $L_\phi$ is $q$-concave by Theorem \[th:concavity\]. Then applying Theorem \[th:Khinchine\] for $L_\phi$ and $0<p\le 2$, for any $f_1,\dots,f_n \in L_\phi$ we get $$\int_0^1\left\|\sum_{i=1}^n r(t) f_i\right\|_\phi\,dt \le C \left\|\left(\sum_{i=1}^n |f_i|^2\right)^{1/2}\right\|_\phi \le C \left\|\left(\sum_{i=1}^n |f_i|^p\right)^{1/p}\right\|_\phi \le C \left(\sum_{i=1}^n \|f_i\|_\phi^p\right)^{1/p}.$$ Hence $L_\phi$ has type $p$.
Observe that the proof of sufficiency in Theorem \[th:typecriterion\] works for any $0<p\le 2$.
Kalton [@Kal1981 Theorem 4.2] showed that for $0<p<1$, any quasi-Banach space is $p$-normable if and only if it has type $p$. For $p=1$ this is no longer true. Clearly any normable space has type $1$, however there exist quasi-Banach spaces that posses type $1$ but they are not normable [@Kal1981]. Natural examples of such spaces are weak-$L^1$ spaces considered in [@KalKam; @Kam2018]. In this respect Orlicz spaces behave more regularly. As we see in the next corollaries, normability is equivalent to type $1$ in $L_\phi$.
\[cor:pnormability\] Let $\phi$ be an Orlicz function and $0<p\le 1$. Then $\rm(i), (ii)$ and $\rm(iii)$ are equivalent.
- $\phi$ satisfies condition $\Delta^{*p}$.
- $L_\phi$ is $p$-normable.
- $L_\phi$ has type $p$.
\(i) $\Rightarrow$ (ii) By Theorem \[th:convexity\], $L_\phi$ is $p$-convex. Then by the inequality $|a+b|^p \le |a|^p + |b|^p$, $a,b\in \mathbb{R}$, we get $$\left\|\sum_{i=1}^n f_i\right\| \le \left\|\left(\sum_{i=1}^n |f_i|^p\right)^{1/p}\right\| \le C_p\left( \sum_{i=1}^n \|f_i\|^p\right)^{1/p}$$ for any $f_1,\dots,f_n\in L_\phi$, $n\in\mathbb{N}$. Thus $\|\cdot\|$ is $p$-normable.
\(ii) $\Rightarrow$ (iii) Let $f_1,\dots,f_n\in L_\phi$. By $p$-normability, there exists $C>0$ such that $$\int_0^1 \left\|\sum_{i=1}^n r_i(t) f_i \right\| dt \le \left\|\sum_{i=1}^n |f_i|\right\| \le C \left(\sum_{i=1}^n \|f_i\|^p\right)^{1/p}.$$
\(iii) $\Rightarrow $ (i) See Proposition \[prop:type1\].
In view of Proposition \[prop:delta\*p\], the immediate consequence of Corollary \[cor:pnormability\] for $p=1$ is the following result.
\[cor:1normability\] Let $\phi$ be an Orlicz function. Then $\rm(i), (ii)$ and $\rm(iii)$ are equivalent.
- $\phi$ is equivalent to a convex function.
- $L_\phi$ is normable.
- $L_\phi$ has type $1$.
Concluding results
==================
Summarizing the results from two previous sections, Propositions \[prop:deltap\], \[prop:delta\*p\], Theorems \[th:concavity\], \[th:convexity\], \[th:cotype\], \[th:typecriterion\] and Corollary \[cor:pnormability\], we obtain the next two theorems.
Let $\alpha_\phi> 0$ and $0< q < \infty$. Consider the following properties.
- $\phi$ satisfies condition $\Delta^{q}$.
- There exists an Orlicz function $\psi$ equivalent to $\phi$ such that $\psi(u^{\frac{1}{q}})$ is concave.
- $L_\phi$ is $q$-convex.
- $L_\phi$ satisfies a lower $q$-estimate.
- $L_\phi$ has cotype $q$.
The conditions [(i)-(iv)]{} are equivalent. If $q\ge 2$, then all conditions [(i)-(v)]{} are equivalent.
Let $\alpha_\phi > 0$ and $0<p < \infty$. Consider the conditions below.
- $\phi$ satisfies condition $\Delta^{*p}$.
- There exists an Orlicz function $\psi$ equivalent to $\phi$ such that $\psi(u^{\frac{1}{p}})$ is convex.
- $L_\phi$ satisfies an upper $p$-estimate.
- $L_\phi$ is $p$-convex.
- $L_\phi$ is $p$-normable.
- $\phi$ satisfies conditions $\Delta_2$ and $\Delta^{*p}$.
- $L_\phi$ has type $p$.
If $0<p<\infty$ then [(i)-(iv)]{} are equivalent.
If $1< p \le 2$ then [(vi)-(vii)]{} are equivalent.
If $0<p\le 1$, then [(i)-(v)]{} and [(vii)]{} are equivalent.
For Banach spaces the notions of type and cotype are related to geometric properties of subspaces uniformly isomorphic to finite dimensional spaces $l_1^n$ and $l_\infty^n$, $n\in\mathbb{N}$, respectively. Recall that $l_1^n$ and $l_\infty^n$ are $n$-dimensional spaces equipped with $l_1$- and $l_\infty$-norm respectively. A Banach space has always type $1\le p\le 2$. If it has type $p=1$, then we say that it has a trivial type.
Let $\phi$ be a convex function. The following conditions are equivalent.
- $L_\phi$ has nontrivial type.
- $L_\phi$ does not contain ${l_1^n}'s$ uniformly.
- $L_\phi$ is $B$-convex.
- $1<\alpha_\phi \le \beta_\phi< \infty$, that is $\phi$ and $\phi^*$ satisfy condition $\Delta_2$.
- $L_\phi$ is reflexive.
By the assumption of convexity of $\phi$, $L_\phi$ is a Banach space. The equivalence of (1) and (2) holds true for any Banach space by Pisier’s Theorem [@DJT Theorems 13.3], as well as the equivalence of (2) and (3) by [@DJT Theorem 13.6]. By Theorem \[th:typecriterion\], $L_\phi$ has type $1<p\le 2$ if and only if $\phi\in\Delta^{*p}$ and $\phi\in\Delta_2$. By Proposition \[prop:delta2\], the latter condition is equivalent to $\beta_\phi<\infty$, while the previous one is equivalent to $\alpha_\phi>1$. Moreover $L_\phi$ is reflexive if and only if $\phi$ and $\phi^*$ satisfy condition $\Delta_2$ [@BS; @KR; @Lux], so (4) is equivalent to (5).
Let $\phi$ be a convex function. All conditions below are equivalent.
- $L_\phi$ has a finite cotype.
- $L_\phi$ does not contain ${l_\infty^n}'s$ uniformly.
- $\beta_\phi< \infty$, that is $\phi$ satisfies condition $\Delta_2$.
For any Banach space, condtions (1) and (2) are equivalent [@DJT Theorem 14.1]. Moreover by Theorem \[th:cotype\], $L_\phi$ has cotype $2\le q < \infty$ if and only if $\phi\in \Delta^q$. In view of Proposition \[prop:delta2\], the existence of such a $q$, in turn, is equivalent to $\beta_\phi< \infty$.
Let $\phi$ be an Orlicz function with $\alpha_\phi > 0$.
[[**I**]{}]{}. The Orlicz space $L_\phi$ has cotype $\max (\beta_\phi + \varepsilon, 2)$ for every $\varepsilon > 0$. If $L_\phi$ has a finite cotype $q$ then $q\ge \max (\beta_\phi , 2)$.
[[**II**]{}]{}. The Orlicz space $L_\phi$ has type $\min (\alpha_\phi -
\varepsilon , 2)$, for every $\varepsilon > 0$ with $\alpha_\phi -
\varepsilon > 0$, whenever $ \beta_\phi < \infty$. If $L_\phi$ has a nontrivial type $p>0$, then $\beta_\phi < \infty$ and $p\le \min
(\alpha_\phi, 2)$.
Part [**I**]{} follows from Remark \[rem:delta\] and Theorem \[th:cotype\]. Part [**II**]{} is a result of Remark \[rem:delta\] and Theorem \[th:typecriterion\].
\(1) For any $p\in (0,2]$ and $q\in [2,\infty)$, there exists an Orlicz space of type $p$ and cotype $q$.
\(a) Let $L_\phi$ be induced by $\phi(u) = \max (u^p , u^q)$, $0<p\le 2\le q<\infty$. Then $\phi$ is an Orlicz function not necessarily convex with $\alpha_\phi = p$ and $\beta_\phi = q$. If $1\le p\le 2$ then $\phi$ is convex and $L_\phi$ is a Banach space.
Moreover $L_\phi =L_p\cap L_q$ where $\|\cdot\|_\phi$ is a quasi-norm equivalent to $\max\{\|\cdot\|_p, \|\cdot\|_q\}$. Here $\|f\|_r = \left(\int |f|^r\right)^{1/r}$, $r>0$. It is easy to check that $\phi\in\Delta_2$ and $\phi\in\Delta^{*p}$, and $\phi\in\Delta^q$. So by Theorem \[th:typecriterion\] it has type $p=\alpha_\phi$, and by Theorem \[th:cotype\] it has cotype $q=\beta_\phi$.
\(b) Let $L_\phi = L_p +
L_q $, where $\phi(u)= \min (u^p,
u^q)$, $0<p\le 2\le q<\infty$. Then $\alpha_\phi = p$ and $\beta_\phi = q$, and $\|f\|_\phi$ is equivalent to $ \inf\{\|g\|_p + \|h\|_q: f= g+h,\ g\in L_p, \ h\in L_q\}$. Again as above $\phi\in\Delta_2$ and $\phi\in\Delta^{*p}$, and $\phi\in\Delta^q$. Thus $L_\phi$ has type $p=\alpha_\phi$ and cotype $q=\beta_\phi$.
The function $\phi(u) = \min(u^p,u^q)$ is never convex. However if $1\le p \le 2$ then it is equivalent to a convex Orlicz function. In fact $\phi$ is equivalent to $$\Phi(u) =\int_0^u \min (t^{p-1},
t^{q-1})dt = \int_0^u \frac{\phi(t)}{t} dt.$$ Obviously $\frac{\phi(u)}{u}$ is increasing for $u>0$, so $\Phi$ is convex. Moreover, for every $u>0$, $$\phi\left(\frac{u}{2}\right) \le \Phi(u) \le \phi(u).$$ In this case the quasinorm $\|\cdot\|_\phi$ is equivalent to the Luxemburg norm $\|\cdot\|_\Phi$, that is $L_\phi$ is normable.
If $F:\Bbb R \rightarrow \Bbb R$ is any function then $F^*(v) = \sup_{u\in\Bbb R}\{uv - F(u)\}$, $v\in\mathbb{R}$, is said to be the [*Legendre transform*]{} $F$. It is well known that $F = F^{**}$ if and only if $F$ is convex [@IT Theorem 1 on p. 175]. Since $F^{**}\le F$ for any $F$, $F^{**}$ is a convex minorant of $F$.
In our case $\phi^{**}\le \phi$ is a convex minorant of $\phi$. Moreover $\Phi^{**} = \Phi$, and thus $$\phi^{**}\left(\frac{u}{2}\right) \le \Phi(u) \le \phi^{**}(u).$$ Therefore $\phi$ is equivalent to its convex minorant.
\(2) An Orlicz space need not have its upper index as a cotype. Let $q\ge 2$ and define $$\phi(u) =
\begin{cases}
0, &\text{for $u = 0$}\\
\frac{u^q}{|\ln u |}, &\text{for $0<u\le \frac1{e}$}\\
(\frac1q + 1)u^q - \frac1q e^{-q}, &\text{for
$u>\frac1{e}$.}
\end{cases}$$ Then $\beta_\phi = q$, but $\sup_{\lambda \ge 1 ,
u>0}\frac{\phi(\lambda u)}{\lambda^q\phi(u)} \ge \sup_{\lambda \ge 1} \frac{|\ln (1/e) |}{|\ln (\lambda /e)|}
= \infty$, which implies that $\phi$ does not satisfy condition $\Delta^q$. Thus $L_\phi$ has cotype $q + \varepsilon$ for every $\varepsilon > 0$, but does not have cotype $q$.
[99]{}
C. Bennet and R. Sharpley, *Interpolation of Operators*, Academic Press, 1988.
J. Diestel, H. Jarchov and A. Tonge, *Absolutely Summing Operators*, Cambridge University Press 1995.
A. D. Ioffe and V. M. Tihomirov,*Theory of Extremal Problems*, Translated from the Russian by Karol Makowski. Studies in Mathematics and its Applications, 6. North-Holland Publishing Co., Amsterdam-New York, 1979
N. J. Kalton, *Convexity, type and the three space problem*, Studia Math. **59** (1981), 247–287.
N. J. Kalton and A. Kamińska, *Type and order convexity of Marcinkiewicz and Lorentz spaces and applications*, Glasgow Math. J. **47** (2005) 123–137.
N. J .Kalton, N. T. Peck and J. W. Roberts, *An $F$ - Space Sampler*, London Math. Soc. Lecture Notes Series **89**, Cambridge University Press 1984.
A. Kamińska, *Convexity, concavity and indices in Musielak-Orlicz spaces*, Dedicated to Julian Musielak, Funct. Approx. Comment. Math. **26** (1998), 67–84.
A. Kamińska, *A note on type of weak-$L^1$ and weak $\ell^1$-spaces*, Banach Center Publications **** (2019),
A. Kamińska, L. Maligranda and L. E. Persson, *Indices and regularizations of measurable functions*, Proceedings of the Conference on Banach Function Spaces held in Poznań in 1998, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc; vol. **213** (2000), 231–246.
A. Kamińska and B. Turett, *Type and cotype in Musielak-Orlicz spaces*, Geometry of Banach Spaces, Cambridge University Press; London Mathematical Society, Lecture Note Series **158** (1990), 165–180.
M. A. Krasnoselskii and Ya. B. Rutickii, *Convex Functions and Orlicz Spaces*, Groningen 1961.
J. Lindenstrauss and L. Tzafriri, *Classical Banach Spaces I*, Springer-Verlag, 1977.
J. Lindenstrauss and L. Tzafriri, *Classical Banach Spaces II*, Springer-Verlag, 1979.
W. A. J. Luxemburg, *Banach Function Spaces*, Thesis, Delft Technical Univ., 1955.
W. Matuszewska and W. Orlicz, *On certain properties of $\phi$-functions*, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. **8** (1960), 439–443.
J. Musielak, *Orlicz Spaces and Modular Spaces*, Lecture Notes in Mathematics **1034**, Springer-Verlag, 1983.
S. Rolewicz, *Metric Linear Spaces*, PWN, Warsaw 1984.
|
---
abstract: 'With the quantum diffusion approach the behavior of capture cross sections and mean-square angular momenta of captured systems are revealed in the reactions with deformed nuclei at sub-barrier energies. The calculated results are in a good agreement with existing experimental data. With decreasing bombarding energy under the barrier the external turning point of the nucleus-nucleus potential leaves the region of short-range nuclear interaction and action of friction. Because of this change of the regime of interaction, an unexpected enhancement of the capture cross section is expected at bombarding energies far below the Coulomb barrier. This effect is shown its worth in the dependence of mean-square angular momentum of captured system on the bombarding energy. From the comparison of calculated and experimental capture cross sections, the importance of quasifission near the entrance channel is shown for the actinide-based reactions leading to superheavy nuclei.'
author:
- 'V.V.Sargsyan$^1$, G.G.Adamian$^{1,2}$, N.V.Antonenko$^1$, W.Scheid$^3$, and H.Q.Zhang$^4$'
title: 'Sub-barrier capture with quantum diffusion approach: actinide-based reactions'
---
Introduction
============
The measurement of excitation functions down to the extreme sub-barrier energy region is important for studying the nucleus-nucleus interaction as well as the coupling of relative motion with other degrees of freedom, and very little data exist on the fusion, fission and capture cross sections at extreme sub-barrier energies [@ZhangOth; @ZhangOU; @Og; @ZuhuaFTh; @Nadkarni; @trotta; @Ji1; @Tr1; @NishioOU; @Ji2; @NishioSiU; @Vino; @Dg; @NishioSU; @HindeSTh; @ItkisSU; @akn]. The experimental data obtained are of interest for solving astrophysical problems related to nuclear synthesis. Indications for an enhancement of the $S$-factor, $S=E_{\rm c.m.}\sigma \exp(2 \pi\eta)$ [@Zvezda; @Zvezda2], where $\eta(E_{\rm c.m.})=Z_1Z_2e^2\sqrt{\mu/(2\hbar^2E_{\rm c.m.})}$ is the Sommerfeld parameter, at energies $E_{\rm c.m.}$ below the Coulomb barrier have been found in Refs. [@Ji1; @Ji2; @Dg]. However, its origin is still under discussion.
To clarify the behavior of capture cross sections at sub-barrier energies, a further development of the theoretical methods is required [@Gomes]. The conventional coupled-channel approach with realistic set of parameters is not able to describe the capture cross sections either below or above the Coulomb barrier [@Dg]. The use of a quite shallow nucleus-nucleus potential [@Es] with an adjusted repulsive core considerably improves the agreement between the calculated and experimental data. Besides the coupling with collective excitations, the dissipation, which is simulated by an imaginary potential in Ref. [@Es] or by damping in each channel in Ref. [@Hag1], seems to be important.
The quantum diffusion approach based on the quantum master-equation for the reduced density matrix has been suggested in Ref. [@EPJSub]. This model takes into consideration the fluctuation and dissipation effects in collisions of heavy ions which model the coupling with various channels. As demonstrated in Ref. [@EPJSub], this approach is successful for describing the capture cross sections at energies near and below the Coulomb barrier for interacting spherical nuclei. An unexpected enhancement of the capture cross section at bombarding energies far below the Coulomb barrier has been predicted in [@EPJSub]. This effect is related to the switching off of the nuclear interaction at the external turning point $r_{ex}$ (Fig. 1). If the colliding nuclei approach the distance $R_{int}$ between their centers, the nuclear forces start to act in addition to the Coulomb interaction. Thus, at $R<R_{int}$ the relative motion is coupled strongly with other degrees of freedom. At $R>R_{int}$ the relative motion is almost independent of the internal degrees of freedom. Depending on whether the value of $r_{ex}$ is larger or smaller than the interaction radius $R_{int}$, the impact of coupling with other degrees of freedom upon the barrier passage seems to be different.
In the present paper we apply the approach of Ref. [@EPJSub] to the description of the capture process of deformed nuclei in a wide energy interval including the extreme sub-barrier region. The used formalism is presented in Sect. II. The results of our calculations for the reactions $^{16}$O,$^{19}$F,$^{32}$S,$^{48}$Ca+$^{232}$Th, $^{4}$He,$^{16}$O,$^{20}$Ne,$^{30}$Si,$^{36}$S,$^{48}$Ca+$^{238}$U, $^{36}$S,$^{48}$Ca,$^{50}$Ti+$^{244}$Pu, $^{48}$Ca+$^{246,248}$Cm, and $^{36}$S+$^{248}$Cm are discussed in Sect. III. The conclusions are given in Sect. IV.
Model
=====
The nucleus-nucleus potential
-----------------------------
![(Upper part) The nucleus-nucleus potentials calculated at $J$ = 0 (solid curve), 30 (dashed curve), 50 (dotted curve), and 65 (dash-dotted curve) for the $^{16}$O + $^{238}$U reaction. The interacting nuclei are assumed to be spherical in the calculation. (Lower part) The position $R_b$ of the Coulomb barrier, radius of interaction $R_{int}$, and external and internal turning points for some values of $E_{\rm c.m.}$ are indicated at the nucleus-nucleus potential for the same reaction at $J$=0.[]{data-label="1_fig"}](Fig1.eps){width="1.05\columnwidth"}
The potential describing the interaction of two nuclei can be written in the form [@poten] where $V_{N}$, $V_{C}$, and the last summand stand for the nuclear, the Coulomb, and the centrifugal potentials, respectively. The nuclei are proposed to be spherical or deformed. The potential depends on the distance $R$ between the center of mass of two interacting nuclei, mass $A_i$ and charge $Z_i$ of nuclei ($i=1,2$), the orientation angles $\theta_i$ of the deformed (with the quadrupole deformation parameters $\beta_{i}$) nuclei and the angular momentum $J$. The static quadrupole deformation parameters are taken from Ref. [@Ram] for the even-even deformed nuclei. For the nuclear part of the nucleus-nucleus potential, we use the double-folding formalism, in the form $$\begin{aligned}
V_N=\int\rho_1(\bold
{r_1})\rho_2(\bold{R}-\bold{r_2})F(\bold{r_1}-\bold{r_2})d\bold{r_1}d\bold{r_2},\end{aligned}$$ where $F(\bold {r_1}-\bold{r_2})=C_0[F_{\rm in}\frac{\rho_0(\bold{r_1})}{\rho_{00}}+F_{\rm
ex}(1-\frac{\rho_0(\bold{r_1})}{\rho_{00}})]\delta(\bold{r_1}-\bold{r_2})$ is the density-dependent effective nucleon-nucleon interaction and $\rho_0(\bold{r})=\rho_1(\bold{r})+\rho_2(\bold{R}-\bold{r})$, $F_{\rm in,ex}=f_{\rm in,ex}+f_{\rm in,ex}^{'}\frac{(N_1-Z_1)(N_2-Z_2)}{(N_1+Z_1)(N_2+Z_2)}$. Here, $\rho_i(\bold{r_i})$ and $N_i$ are the nucleon densities and neutron numbers of the light and the heavy nuclei of the dinuclear system, respectively. Our calculations are performed with the following set of parameters: $C_0=$ 300 MeV fm$^3$, $f_{\rm in}=$ 0.09, $f_{\rm ex}=$ -2.59, $f_{\rm in}^{'}=$ 0.42, $f_{\rm ex}^{'}=$ 0.54 and $\rho_{00}=$ 0.17 fm$^{-3}$ [@poten]. The densities of the nuclei are taken in the two-parameter symmetrized Woods-Saxon form with the nuclear radius parameter $r_0$=1.15 fm (for the nuclei with $A_i \ge 16$) and the diffuseness parameter $a$ depending on the charge and mass numbers of the nucleus [@poten]. We use $a$= 0.53 fm for the lighter nuclei $^{16}$O and $^{19}$F, $a$= 0.55 fm for the intermediate nuclei ($^{20}$Ne, $^{26}$Mg, $^{30}$Si, $^{32,34,36}$S, $^{40,48}$Ca, $^{50}$Ti), and $a$= 0.56 fm for the actinides. For the $^{4}$He nucleus $r_0$=1.02 fm and $a$=0.48 fm.
The Coulomb interaction of two deformed nuclei has the following form: $$\begin{aligned}
&&V_{C}(R,Z_i,A_i,\theta_{i},J)=
\frac{Z_1Z_2e^2}{R}\nonumber\\&+&\left(\frac{9}{20\pi}\right)^{1/2}
\frac{Z_1Z_2e^2}{R^3}\sum_{i=1,2}R_i^2\beta_{i}
\left[1+\frac{2}{7}\left(\frac{5}{\pi}\right)^{1/2}\beta_{i}\right]\nonumber\\&\times &P_2(\cos\theta_i),
\label{32ab_eq}\end{aligned}$$ where $P_2(\cos\theta_i)$ is the Legendre polynomial.
In Fig. 1 there is shown the nucleus-nucleus potential $V$ for the $^{16}$O + $^{238}$U reaction (for simplicity, $^{238}$U is assumed to be spherical) which has a pocket. With increasing centrifugal part of the potential the pocket depth becomes smaller, while the position of the pocket minimum moves towards the barrier at the position of the Coulomb barrier $R=R_b\approx R_1+R_2+2$ fm, where $R_i=1.15A_i^{1/3}$ are the radii of colliding nuclei. This pocket is washed out at large angular momenta $J>65$. Thus, only a limited part of angular momenta contributes to the capture process.
{width="2.0\columnwidth"}
For the reactions $^{36}$S + $^{238}$U and $^{16}$O + $^{238}$U (Fig. 2), the dependence of the potential energy on the orientation of the prolate deformed nucleus $^{238}$U is shown. The lowest Coulomb barriers are associated with collisions of the projectile nucleus with the tips of the target nucleus, while the highest barriers correspond to collisions with the sides of the target nucleus. The difference of the Coulomb barriers for the sphere-pole and sphere-side orientations is about 16 MeV (8 MeV) for the $^{36}$S + $^{238}$U ($^{16}$O + $^{238}$U) system.
Capture cross section
---------------------
The capture cross section is a sum of partial capture cross sections $$\begin{aligned}
\sigma_{cap}(E_{\rm c.m.})&=&\sum_{J}^{}\sigma_{\rm cap}(E_{\rm
c.m.},J)=\nonumber\\&=& \pi\lambdabar^2
\sum_{J}^{}(2J+1)\int_0^{\pi/2}d\theta_1\sin(\theta_1)\nonumber\\&\times &\int_0^{\pi/2}d\theta_2\sin(\theta_2) P_{\rm cap}(E_{\rm
c.m.},J,\theta_1,\theta_2),
\label{1a_eq}\end{aligned}$$ where $\lambdabar^2=\hbar^2/(2\mu E_{\rm c.m.})$ is the reduced de Broglie wavelength, $\mu=m_0A_1A_2/(A_1+A_2)$ is the reduced mass ($m_0$ is the nucleon mass), and the summation is over the possible values of angular momentum $J$ at a given bombarding energy $E_{\rm c.m.}$. Knowing the potential of the interacting nuclei for each orientation, one can obtain the partial capture probability $P_{\rm cap}$ which is defined by the passing probability of the potential barrier in the relative distance $R$ coordinate at a given $J$.
![ The calculated value $\langle V_b\rangle$ averaged over the orientations of the heavy deformed nucleus versus $E_{\rm c.m.}$ for the $^{36}$S + $^{238}$U reaction. The values of barriers $V^{orient}_b$(sphere-pole) for the sphere-pole configuration, $V_b=V_b$(sphere-sphere)=$V^{orient}_b$(sphere-sphere) for the sphere-sphere configuration and $V^{orient}_b$(sphere-side) for the sphere-side configuration are indicated by arrows. The static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286 and $\beta_{1}$($^{36}$S)=0. []{data-label="3_fig"}](Fig3.eps){width="1.0\columnwidth"}
![The calculated capture cross section (solid lines) versus $E_{\rm c.m.}$ for the reactions $^{16}$O + $^{232}$Th and $^{4}$He + $^{238}$U are compared with the available experimental data. The experimental data in the upper part are taken from Refs. [@BackOTh] (open triangles), [@ZhangOth] (closed triangles), [@MuakamiOTh] (open squares), [@KailasOTh] (closed squares), [@ZuhuaFTh] (open stars) and [@Nadkarni] (closed stars). The fission cross sections from Refs. [@trotta] and [@ViolaOU] are shown in the lower part by open circles and solid squares, respectively. The value of the Coulomb barrier $V_b$ for the spherical nuclei is indicated by arrow. The dashed curve represents the calculation by the Wong’s formula (\[wong1\_eq\]). The static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286, $\beta_{2}$($^{232}$Th)=0.261 and $\beta_{1}$($^{16}$O)=$\beta_{1}$( $^{4}$He)=0. []{data-label="4_fig"}](Fig4.eps){width="1.1\columnwidth"}
The value of $P_{\rm cap}$ is obtained by integrating the propagator $G$ from the initial state $(R_0,P_0)$ at time $t=0$ to the final state $(R,P)$ at time $t$ ($P$ is a momentum): $$\begin{aligned}
P_{\rm cap}&=&\lim_{t\to\infty}\int_{-\infty}^{r_{\rm in}}dR\int_{-\infty}^{\infty}dP\ G(R,P,t|R_0,P_0,0)\nonumber \\
&=&\lim_{t\to\infty}\frac{1}{2} {\rm erfc}\left[\frac{-r_{\rm in}+\overline{R(t)}}
{{\sqrt{\Sigma_{RR}(t)}}}\right].
\label{1ab_eq}\end{aligned}$$
![The same as in Fig. 4, but for the reactions $^{16}$O + $^{238}$U and $^{36}$S + $^{238}$U. The experimental cross sections are taken from Refs. [@NishioOU] (open triangles), [@TokeOU] (closed triangles), [@ZuhuaFTh] (open squares), [@ZhangOU] (closed squares), [@ViolaOU] (open stars), [@ItkisSU] (closed stars), and [@NishioSU] (rhombuses). The dashed curve represents the calculation by the Wong’s formula (\[wong1\_eq\]). The static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286 and $\beta_{1}$($^{16}$O)=$\beta_{1}$($^{36}$S)=0. []{data-label="5_fig"}](Fig5.eps){width="1.1\columnwidth"}
![The same as in Fig. 4, but for the reactions $^{32}$S + $^{232}$Th (solid line), $^{32}$S + $^{238}$U (dotted line) and $^{30}$Si + $^{238}$U. The experimental data are taken from Refs. [@HindeSTh] ($^{32}$S + $^{232}$Th, solid squares), [@NishioSiU] (solid circles) and [@Nishionew] (open squares). The static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286, $\beta_{2}$($^{232}$Th)=0.261, $\beta_{1}$($^{32}$S)=0.312 and $\beta_{1}$($^{30}$Si)=0.315. For the $^{30}$Si + $^{238}$U reaction, the results of calculations with $\beta_{1}$($^{30}$Si)=0 (the predictions of the mean-field and macroscopic-microscopic models) are presented by dashed line in the lower part of the figure. []{data-label="6_fig"}](Fig6.eps){width="1.1\columnwidth"}
![The same as in Fig. 4, but for the reactions $^{19}$F + $^{232}$Th and $^{20}$Ne + $^{238}$U. The experimental data are taken from Refs. [@Nadkarni] (open squares), [@ZuhuaFTh] (closed squares), and [@ViolaOU] (closed circles). The open circles in the lower part are the experimental data from Ref. [@ViolaOU] shifted by 2 MeV to the left. The results of calculations with the static quadrupole deformation parameters of $^{19}$F $\beta_{1}$($^{19}$F)=0.275 (as in Ref. [@Moel1]), 0.41, and 0.55 are shown by the dashed, solid, and dotted lines, respectively. The other static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286, $\beta_{2}$($^{232}$Th)=0.261 and $\beta_{1}$($^{20}$Ne)=0.335. []{data-label="7_fig"}](Fig7.eps){width="1.08\columnwidth"}
![The predicted capture cross sections for the reactions $^{36}$S + $^{244}$Pu,$^{248}$Cm. The static quadrupole deformation parameters are: $\beta_{2}$($^{244}$Pu)=0.293, $\beta_{2}$($^{248}$Cm)=0.297 and $\beta_{1}$($^{36}$S)=0. []{data-label="8_fig"}](Fig8.eps){width="1.1\columnwidth"}
![ The calculated mean-square angular momenta versus $E_{\rm c.m.}$ for the reactions $^{16}$O + $^{232}$Th,$^{238}$U are compared with experimental data [@ZuhuaFTh]. The dashed curve represents the calculation by the Eq. (\[Jwong\_eq\]). The static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286, $\beta_{2}$($^{232}$Th)=0.261 and $\beta_{1}$($^{16}$O)=0. The values of the Coulomb barriers $V_b$ corresponding to spherical interacting nuclei are indicated by arrows. []{data-label="9_fig"}](Fig9.eps){width="1.1\columnwidth"}
![The same as in Fig. 9, but for the indicated reactions $^{19}$F,$^{48}$Ca + $^{232}$Th. The experimental data are taken from Ref. [@ZuhuaFTh]. The results of calculations with quadrupole deformation parameters $\beta_{1}$($^{19}$F)=0.275, 0.41 and 0.55 are shown by the dashed, solid, and dotted lines, respectively. The other static quadrupole deformation parameters are: $\beta_{2}$($^{232}$Th)=0.261 and $\beta_{1}$($^{48}$Ca)=0. []{data-label="10_fig"}](Fig10.eps){width="1.1\columnwidth"}
![The calculated values of the astrophysical $S$-factor with $\eta_0=\eta(E_{\rm c.m.}=V_b)$ for the indicated reactions $^{16}$O+$^{232}$Th and $^{4}$He + $^{238}$U. The values of the Coulomb barriers $V_b$ corresponding to the spherical nuclei are 78.6 and 21.2 MeV. []{data-label="11_fig"}](Fig11.eps){width="1.1\columnwidth"}
![The calculated values of the astrophysical $S$-factor with $\eta_0=\eta(E_{\rm c.m.}=V_b)$ (middle part), the logarithmic derivative $L$ (upper part) and the fusion barrier distribution $d^2(E_{\rm c.m.}\sigma_{cap})/d E_{\rm c.m.}^2$ (lower part) for the $^{16}$O+$^{238}$U reaction. The value of $L$ calculated with the assumption of $\beta_1$($^{16}$O)=$\beta_2$($^{238}$U)=0 is shown by a dashed line. The solid and dotted lines show the values of $d^2(E_{\rm c.m.}\sigma_{cap})/d E_{\rm c.m.}^2$ calculated with the increments 0.2 and 1.2 MeV, respectively. The closed squares are the experimental data of Ref. [@DH]. The value of the Coulomb barrier $V_b$ corresponding to the spherical nuclei is 80 MeV. []{data-label="12_fig"}](Fig12.eps){width="1.1\columnwidth"}
![The same as in Fig. 4, but for the $^{48}$Ca + $^{232}$Th,$^{238}$U reactions. The excitation energies $E^*_{CN}$ of the corresponding nuclei are indicated. The experimental data are taken from Refs. [@Itkis1] (marked by squares) and [@Shen] (marked by circles). The static quadrupole deformation parameters are: $\beta_{2}$($^{238}$U)=0.286, $\beta_{2}$($^{232}$Th)=0.261 and $\beta_{1}$($^{48}$Ca)=0. []{data-label="13_fig"}](Fig13.eps){width="1.1\columnwidth"}
![The same as in Fig. 13, but for the reactions $^{48}$Ca + $^{246,248}$Cm. The experimental data are from Refs. [@Itkis1] (squares) and [@Itkis2] (circles). The static quadrupole deformation parameters are: $\beta_{2}$($^{246}$Cm)=0.298, $\beta_{2}$($^{248}$Cm)=0.297 and $\beta_{1}$($^{48}$Ca)=0. []{data-label="14_fig"}](Fig14.eps){width="1.1\columnwidth"}
The second line in (\[1ab\_eq\]) is obtained by using the propagator $G=\pi^{-1}|\det {\bf \Sigma}^{-1}|^{1/2}
\exp(-{\bf q}^{T}{\bf \Sigma}^{-1}{\bm q})$ (${\bf q}^{T}=[q_R,q_P]$, $q_R(t)=R-\overline{R(t)}$, $q_P(t)=P-\overline{P(t)}$, $\overline{R(t=0)}=R_0$, $\overline{P(t=0)}=P_0$, $\Sigma_{kk'}(t)=2\overline{q_k(t)q_{k'}(t)}$, $\Sigma_{kk'}(t=0)=0$, $k,k'=R,P$) calculated in Ref. [@DMDadonov] for an inverted oscillator which approximates the nucleus-nucleus potential $V$ in the variable $R$. The frequency $\omega$ of this oscillator with an internal turning point $r_{\rm in}$ is defined from the condition of equality of the classical actions of approximated and realistic potential barriers of the same hight at given $J$. It should be noted that the passage through the Coulomb barrier approximated by a parabola has been previously studied in Refs. [@Hofman; @VAZ; @Ayik; @Hupin; @our]. This approximation is well justified for the reactions and energy range, which are here considered. Finally, one can find the expression for the capture probability: $$\begin{aligned}
P_{\rm cap}&=&
\frac{1}{2} {\rm erfc}\left[\left(\frac{\pi s_1(\gamma-s_1)}{2\mu(\omega_0^2-s_1^2)}\right)^{1/2}
\frac{\mu\omega_0^2 R_0/s_1+P_0}
{\left[\gamma \ln(\gamma/s_1)\right]^{1/2}}\right],
\label{PC_eq}\end{aligned}$$ where $\gamma$ is the internal-excitation width, $\omega_0^2=\omega^2\{1-\hbar\tilde\lambda\gamma/
[\mu(s_1+\gamma)(s_2+\gamma)]\}$ is the renormalized frequency in the Markovian limit, the value of $\tilde\lambda$ is related to the strength of linear coupling in coordinates between collective and internal subsystems. The $s_i$ are the real roots ($s_1\ge 0> s_2 \ge s_3$) of the following equation: $$\begin{aligned}
(s+\gamma)(s^2- \omega_0^2)+\hbar\tilde\lambda\gamma s/\mu=0.
\label{Root_eq}\end{aligned}$$ The details of the used formalism are presented in [@EPJSub]. We have to mention that most of the quantum-mechanical, dissipative effects and non-Markovian effects accompanying the passage through the potential barrier are taken into consideration in our formalism [@EPJSub; @our]. For example, the non-Markovian effects appear in the calculations through the internal-excitation width $\gamma$.
As shown in [@EPJSub], the nuclear forces start to play a role at $R_{int}=R_b+1.1$ fm where the nucleon density of colliding nuclei approximately reaches 10% of the saturation density. If the value of $r_{\rm ex}$ corresponding to the external turning point is larger than the interaction radius $R_{int}$, we take $R_0=r_{\rm ex}$ and $P_0=0$ in Eq. (\[PC\_eq\]). For $r_{\rm ex}< R_{int}$, it is naturally to start our treatment with $R_0=R_{int}$ and $P_0$ defined by the kinetic energy at $R=R_0$. In this case the friction hinders the classical motion to proceed towards smaller values of $R$. If $P_0=0$ at $R_0>R_{int}$, the friction almost does not play a role in the transition through the barrier. Thus, two regimes of interaction at sub-barrier energies differ by the action of the nuclear forces and the role of friction at $R=r_{\rm ex}$.
Calculated results
==================
Besides the parameters related to the nucleus-nucleus potential, two parameters $\hbar\gamma$=32 MeV and the friction coefficient $\hbar\lambda=-\hbar (s_1+s_2)$=2 MeV are used for calculating the capture probability in reactions with deformed actinides. The value of $\tilde\lambda$ is set to obtain this value of $\hbar\lambda$. The most realistic friction coefficients in the range of $\hbar \lambda\approx 1-2$ MeV are suggested from the study of deep inelastic and fusion reactions [@Obzor]. These values are close to those calculated within the mean field approach [@Den]. All calculated results presented are obtained with the same set of parameters and are rather insensitive to a reasonable variation of them [@EPJSub; @VAZ; @our].
Effect of orientation
---------------------
The influence of orientation of the deformed heavy nucleus on the capture process in the reactions $^{36}$S + $^{238}$U and $^{16}$O + $^{238}$U is studied in Fig. 2. We demonstrate that the capture cross section $\sigma_{cap}$ at fixed orientation as a function of $E_{\rm c.m.}-V_b^{orient}$, where $V_b^{orient}$ is the Coulomb barrier for this orientation, is almost independent of the orientation angle $\theta_2$.
In Fig. 3 the value of the Coulomb barrier $$\begin{aligned}
\nonumber
<V_b>&=&\frac{\pi\lambdabar^2}{\sigma_{\rm cap}(E_{\rm c.m.})}
\sum_{J}^{}(2J+1)\int_0^{\pi/2}d\theta_2\sin(\theta_2)\\\nonumber &\times & P_{\rm cap}(E_{\rm
c.m.},J,\theta_1,\theta_2) V(R_b,Z_i,A_i,\theta_{i},J)\end{aligned}$$
averaged over all possible orientations of the heavy nucleus versus $E_{\rm c.m.}$ is shown for the $^{36}$S + $^{238}$U reaction. With increasing (decreasing) $E_{\rm c.m.}$ the value of $<V_b>$ approaches the value of the Coulomb barrier for the sphere-sphere configuration (for the sphere-pole configuration). The influence of deformation on the capture cross section is very weak already at bombarding energies about 15 MeV above the Coulomb barrier corresponding to spherical nuclei.
Comparison with experimental data and predictions
-------------------------------------------------
In Figs. 4–6 the calculated capture cross sections for the reactions $^{16}$O,$^{19}$F,$^{32}$S+$^{232}$Th and $^{4}$He,$^{16}$O,$^{30}$Si,$^{32,36}$S+$^{238}$U are in a rather good agreement with the available experimental data [@NishioOU; @TokeOU; @ZuhuaFTh; @ZhangOU; @ViolaOU; @ItkisSU; @NishioSU; @BackOTh; @ZhangOth; @MuakamiOTh; @KailasOTh; @Nadkarni; @trotta; @HindeSTh; @NishioSiU; @Nishionew]. Because of the uncertainties in the definition of the deformation of the light nucleus and in the experimental data [@NishioSiU; @Nishionew] in Fig. 6, we show the calculated results for the $^{30}$Si+$^{238}$U reaction with $\beta_1$($^{30}$Si) from Ref. [@Ram] as well as with $\beta_1$($^{30}$Si)=0 (lower part of Fig. 6). Note that $\beta_1$($^{30}$Si)=0 for the ground state were predicted within the mean-field and macroscopic-microscopic models.
In Fig. 7 (upper part) we are not able to describe well the data of Ref. [@Nadkarni] for the $^{19}$F+$^{232}$Th reaction at $E_{\rm c.m.}< 74$ MeV, even by varying the static quadrupole deformation parameters $\beta_1$ of $^{19}$F. However, the deviations of the solid curve in the upper part of Fig. 7 from the experimental data are within the uncertainty of these data. Note that the value of $\beta_1$ mainly influences the slope of curve at $E_{\rm c.m.}< V_b$ and one can extract the ground state deformation of nucleus from the experimental capture cross section data. For the $^{20}$Ne + $^{238}$U reaction, the calculated capture cross sections in Fig. 7 are consistent with the experimental data [@ViolaOU] if the latter ones are shifted by 2 MeV to lower energies. For the $^{20}$Ne nucleus, the experimental quadrupole deformation parameter $\beta_1$=0.727 related in Ref. [@Ram] to the first 2$^+$ state seems to be unrealistically large and we take $\beta_1$=0.335 as predicted in Ref. [@Moel1]. The capture cross sections for the reactions $^{32}$S + $^{238}$U and $^{36}$S + $^{244}$Pu,$^{248}$Cm are shown in Figs. 6 and 8, respectively.
One can see in Figs. 4–8 that there is a sharp fall-off of the cross sections just under the Coulomb barrier corresponding to undeformed nuclei. With decreasing $E_{\rm c.m.}$ up to about 8–10 MeV (when the projectile is spherical) and 15–20 MeV (when both projectile and target are deformed nuclei) below the Coulomb barrier the regime of interaction is changed because at the external turning point the colliding nuclei do not reach the region of nuclear interaction where the friction plays a role. As result, at smaller $E_{\rm c.m.}$ the cross sections fall with a smaller rate. With larger values of $R_{int}$ the change of fall rate occurs at smaller $E_{\rm c.m.}$. However, the uncertainty in the definition of $R_{int}$ is rather small. Therefore, an effect of the change of fall rate of sub-barrier capture cross section should be in the data if we assume that the friction starts to act only when the colliding nuclei approach the barrier. Note that at energies of 10–20 MeV below the barrier the experimental data have still large uncertainties to make a firm experimental conclusion about this effect. The effect seems to be more pronounced in collisions of spherical nuclei, where the regime of interaction is changed at $E_{\rm c.m.}$ up to about 3–5 MeV below the Coulomb barrier [@EPJSub]. The collisions of deformed nuclei occur at various mutual orientations affecting the value of $R_{int}$.
The well-known Wong formula for the capture cross section is $$\begin{aligned}
\nonumber
\sigma(E_{\rm c.m.})&=&
\frac{R_b^2\hbar\omega}{2E_{\rm c.m.}}\int_0^{\pi/2}d\theta_1 \sin\theta_1\int_0^{\pi/2}d\theta_2 \sin\theta_2
\\ &\times &\ln(1+\exp[2\pi(E_{\rm c.m.}-E_b(\theta_1,\theta_2))/\hbar\omega]),
\label{wong1_eq}\end{aligned}$$ where $E_b(\theta_1,\theta_2)$ is value of the Coulomb barrier which depends on the orientations of the deformed nuclei [@Wong]. As seen from Figs. 4 and 5 (dashed lines) the Wong formula (\[wong1\_eq\]) does not reproduce the capture cross section at $E_{\rm c.m.}< V_b$ even taking into consideration the static quadrupole deformation of target-nucleus.
The calculated mean-square angular momenta $$\begin{aligned}
\nonumber
\langle J^2\rangle & =&\frac{\pi\lambdabar^2
\sum_{J}^{} J(J+1)(2J+1)}{\sigma_{cap}(E_{\rm c.m.})}\\ \nonumber &\times &\int_0^{\pi/2}d\theta_1\sin(\theta_1)\int_0^{\pi/2}d\theta_2\sin(\theta_2) P_{\rm cap}(E_{\rm
c.m.},J,\theta_1,\theta_2)\\
\label{J_eq}\end{aligned}$$ of captured systems versus $E_{\rm c.m.}$ are presented in Figs. 9–10 for the reactions mentioned above. At energies below the barrier the value of $\langle J^2\rangle$ has a minimum. This minimum depends on the deformations of nuclei and on the factor $Z_1\times Z_2$. For the reactions $^{16}$O + $^{232}$Th, $^{16}$O + $^{238}$U, $^{19}$F + $^{232}$Th and $^{48}$Ca + $^{232}$Th, these minima are about 7, 8, 12 and 15 MeV below the corresponding Coulomb barriers, respectively. The experimental data [@Vand] indicate the presence of the minimum as well. On the left-hand side of this minimum the dependence of $\langle J^2\rangle$ on $E_{\rm c.m.}$ is rather weak. A similar weak dependence has been found in Refs. [@Bala] in the extreme sub-barrier region. Note that the found behavior of $\langle J^2\rangle$, which is related to the change of the regime of interaction between the colliding nuclei, would affect the angular anisotropy of the products of fission-like fragments following capture. Indeed, the values of $\langle J^2\rangle$ are extracted from data on angular distribution of fission-like fragments [@akn].
In the Wong model [@Wong] the value of the mean-square angular momentum is determined as $$\begin{aligned}
\nonumber
\langle J^2\rangle &=&\frac{\mu R_b^2\hbar\omega}{\pi\hbar^2}
\int_0^{\pi/2}d\theta_1 \sin\theta_1\int_0^{\pi/2}d\theta_2 \sin\theta_2\\
& \times &\frac{-Li_2(-\exp[2\pi(E_{\rm c.m.}-E_b(\theta_1,\theta_2))/\hbar\omega])}{\ln(1+\exp[2\pi(E_{\rm c.m.}-E_b(\theta_1,\theta_2))/\hbar\omega])}.
\label{Jwong_eq}\end{aligned}$$ Here, the $Li_2(z)$ is the polylogarithm function. At $\exp[2\pi(E_{\rm c.m.}-E_b)/\hbar\omega])\ll$1 (much below the Coulomb barrier), $\frac{-Li_2(-\exp[2\pi(E_{\rm c.m.}-E_b)/\hbar\omega])}{\ln(1+\exp[2\pi(E_{\rm c.m.}-E_b)/\hbar\omega])}\approx 1$ and one can obtain the saturation value of the mean-square angular momentum [@Gomes]: $$\begin{aligned}
\langle J^2\rangle=\frac{\mu R_b^2\hbar\omega}{\pi\hbar^2}.
\label{7abc_eq}\end{aligned}$$ The agreement between $\langle J^2\rangle$ calculated with Eq. (\[Jwong\_eq\]) and experimental $\langle J^2\rangle$ is not good. At energies below the barrier $\langle J^2\rangle$ has no a minimum (see Fig. 9). However, for the considered reactions the saturation values of $\langle J^2\rangle$ are close to those obtained with our formalism.
Astrophysical factor, L-factor and barrier distribution
-------------------------------------------------------
In Figs. 11 and 12 the calculated astrophysical $S$–factors versus $E_{\rm c.m.}$ are shown for the reactions $^{4}$He,$^{16}$O+$^{238}$U and $^{16}$O+$^{232}$Th. The $S$-factor has a maximum for which there are experimental indications in Refs. [@Ji1; @Ji2; @Es]. After this maximum $S$-factor slightly decreases with decreasing $E_{\rm c.m.}$ and then starts to increase. This effect seems to be more pronounced in collisions of spherical nuclei [@EPJSub]. The same behavior has been revealed in Refs. [@LANG] by extracting the $S$-factor from the experimental data.
In Fig. 12, the so-called logarithmic derivative, $L(E_{\rm c.m.})=d(\ln (E_{\rm c.m.}\sigma_{cap}))/dE_{\rm c.m.},$ and the barrier distribution $d^2(E_{\rm c.m.}\sigma_{cap})/d E_{\rm c.m.}^2$ are presented for the $^{16}$O+$^{238}$U reaction. The logarithmic derivative strongly increases below the barrier and then has a maximum at $E_{\rm c.m.}\approx V_b^{orient}$(sphere-pole)-3 MeV (at $E_{\rm c.m.}\approx V_b$-3 MeV for the case of spherical nuclei). The maximum of $L$ corresponds to the minimum of the $S$-factor.
The barrier distributions calculated with an energy increment 0.2 MeV have only one maximum at $E_{\rm c.m.}\approx V_b^{orient}$(sphere-sphere)$=V_b$ as in the experiment [@DH]. With increasing increment the barrier distribution is shifted to lower energies. Assuming a spherical target nucleus in the calculations, we obtain a more narrow barrier distribution (see Fig. 12).
Capture cross sections in reactions with large fraction of quasifission
-----------------------------------------------------------------------
In the case of large values of $Z_1\times Z_2$ the quasifission process competes with complete fusion at energies near barrier and can lead to a large hindrance for fusion, thus ruling the probability for producing superheavy elements in the actinide-based reactions [@trota; @nasha]. Since the sum of the fusion cross section $\sigma_{fus}$, and the quasifission cross section $\sigma_{qf}$ gives the capture cross section, $$\sigma_{cap}=\sigma_{fus}+\sigma_{qf},$$ and $\sigma_{fus}\ll \sigma_{qf}$ in the actinide-based reactions $^{48}$Ca + $^{232}$Th,$^{238}$U,$^{244}$Pu,$^{246,248}$Cm and $^{50}$Ti + $^{244}$Pu [@nasha], we have $$\sigma_{cap}\approx\sigma_{qf}.$$
In a wide mass-range near the entrance channel, the quasifission events overlap with the products of deep-inelastic collisions and can not be firmly distinguished. Because of this the mass region near the entrance channel is taken out in the experimental analyses of Refs. [@Itkis1; @Itkis2]. Thus, by comparing the calculated and experimental capture cross sections one can study the importance of quasifission near the entrance channel for the actinide-based reactions leading to superheavy nuclei.
The capture cross sections for the quasifission reactions [@Shen; @Itkis1; @Itkis2] are shown in Figs. 13-15. One can observe a large deviations of the experimental data of Refs. [@Itkis1; @Itkis2] from the the calculated results. The possible reason is an underestimation of the quasifission yields measured in these reactions. Thus, the quasifission yields near the entrance channel are important. Note that there are the experimental uncertainties in the bombarding energies.
![The same as in Fig. 13, but for the indicated $^{48}$Ca,$^{50}$Ti + $^{244}$Pu reactions. The experimental data are from Refs. [@Itkis2] (squares) and [@Itkis1] (circles). The static quadrupole deformation parameters are: $\beta_{2}$($^{244}$Pu)=0.293, and $\beta_{1}$($^{48}$Ca)=$\beta_{1}$($^{50}$Ti)=0. []{data-label="15_fig"}](Fig15.eps){width="0.85\columnwidth"}
![The ratio of theoretical and experimental capture cross sections versus the excitation energy $E_{\rm c.m.}$ of the compound nucleus for the reactions $^{48}$Ca+$^{238}$U (closed stars), $^{48}$Ca+$^{244}$Pu (closed triangles), $^{48}$Ca+$^{246}$Cm (closed squares), $^{48}$Ca+$^{248}$Cm (closed circles), and $^{50}$Ti+$^{244}$Pu (closed rhombuses). []{data-label="16_fig"}](Fig16.eps){width="0.8\columnwidth"}
One can see in Fig. 16 that the experimental and the theoretical cross sections become closer with increasing bombarding energy. This means that with increasing bombarding energy the quasifission yields near the entrance channel mass-region decrease with respect to the quasifission yields in other mass-regions. The quasifission yields near the entrance channel increase with $Z_1\times Z_2$.
Summary
=======
The quantum diffusion approach is applied to study the capture process in the reactions with deformed nuclei at sub-barrier energies. The available experimental data at energies above and below the Coulomb barrier are well described, showing that the static quadrupole deformations of the interacting nuclei are the main reasons for the capture cross section enhancement at sub-barrier energies. Since the deformations of the interacting nuclei mainly influence the slope of curve at $E_{\rm c.m.}< V_b$ and one can extract the ground state deformation of projectile or target from the experimental capture cross section data.
Due to a change of the regime of interaction (the turning-off of the nuclear forces and friction) at sub-barrier energies, the curve related to the capture cross section as a function of bombarding energy has smaller slope $E_{\rm c.m.}-V_b <$ – 5 MeV. This change is also reflected in the functions $\langle J^2\rangle$, $L(E_{\rm c.m.})$, and $S(E_{\rm c.m.})$. The mean-square angular momentum of captured system versus $E_{\rm c.m.}$ has a minimum and then saturates at sub-barrier energies. This behavior of $\langle J^2\rangle$ would increase the expected anisotropy of the angular distribution of the products of fission and quasifission following capture. The astrophysical factor has a maximum and a minimum at energies below the barrier. The maximum of $L$-factor corresponds to the minimum of the $S$-factor. One can suggest the experiments to check these predictions.
The importance of quasifission near the entrance channel is shown for the actinide-based reactions leading to superheavy nuclei.
acknowledgements
================
This work was supported by DFG, NSFC, and RFBR. The IN2P3-JINR, MTA-JINR and Polish-JINR Cooperation programs are gratefully acknowledged.
[99]{} H.Q. Zhang [*et al.*]{}, Phys. Rev. C [**42**]{}, 1086 (1990). H.Q. Zhang [*et al.*]{}, Phys. Rev. C [**49**]{}, 926 (1994). Yu.Ts. Oganessian [*et al.*]{}, JINR Rapid Communications [**75**]{}, 123 (1996). Z. Liu [*et al.*]{}, Phys. Rev. C [**54**]{}, 761 (1996). D.M. Nadkarni [*et al.*]{}, Phys. Rev. C [**59**]{}, R580 (1999). M. Trotta [*et al.*]{}, Phys. Rev. Lett. [**84**]{}, 2342 (2000). C.L. Jiang [*et al.*]{}, Phys. Rev. Lett. [**89**]{}, 052701 (2002). S.P. Tretyakova, A.A. Ogloblin, R.N. Sagaidak, S.V. Khlebnikov, and W. Trzaska, Nucl. Phys. [**A734**]{}, E33 (2004); S.P. Tretyakova, A.A. Ogloblin, R.N. Sagaidak, W. Trzaska, S.V. Khlebnikov, R. Julin, and J. Petrowski, Nucl. Phys. [**A738**]{}, 487 (2004). K. Nishio [*et al.*]{}, Phys. Rev. Lett. [**93**]{}, 162701 (2004). C.L. Jiang [*et al.*]{}, Phys. Rev. C [**71**]{}, 044613 (2005). K. Nishio [*et al.*]{}, Eur. Phys. J. A [**29**]{} (2006). W. Loveland et al., Phys. Rev. C 74 (2006) 044604; A.M. Vinodkumar et al., Phys. Rev. C 74 (2006) 064612; A.M. Vinodkumar et al., Phys. Rev. C 78 (2008) 054608. M. Dasgupta [*et al.*]{}, Phys. Rev. Lett. [**99**]{}, 192701 (2007). K. Nishio [*et al.*]{}, Phys. Rev. C. [**77**]{}, 064607 (2008). D.J. Hinde [*et al.*]{}, Phys. Rev. Lett. [**101**]{}, 092701 (2008). M.G. Itkis [*et al.*]{}, Nucl. Phys. A [**834**]{}, 374c (2010). H.Q. Zhang [*et al.*]{}, Phys. Rev. C. [**81**]{}, 034611 (2010). K. Langanke and C.A. Barnes, Adv. Nucl. Phys. [**22**]{}, 173 (1996). A. Aprahamian, K. Langanke, and M. Wiescher, Prog. Part. Nucl. Phys. [**54**]{}, 535 (2005). L.E. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. [**424**]{}, 1 (2006). H. Esbensen and C.L. Jiang, Phys. Rev. C [**79**]{}, 064619 (2009); S. Misicu and H. Esbensen, Phys. Rev. C [**75**]{}, 034606 (2007); H. Esbensen and S. Misicu, Phys. Rev. C [**76**]{}, 054609 (2007). K. Hagino and N. Rowley, AIP Conf. Proc. [**1098**]{}, 18 (2009). V.V.Sargsyan, G.G. Adamian, N.V. Antonenko, and W. Scheid, Eur. Phys. J. A [**45**]{}, 125 (2010). G.G. Adamian [*et al.*]{}, Int. J. Mod. Phys. E [**5**]{}, 191 (1996). S. Raman, C.W. Nestor, Jr, and P. Tikkanen, At. Data Nucl. Data Tables [**78**]{}, 1 (2001). V.V. Dodonov and V.I. Man’ko, Trudy Fiz. Inst. AN [**167**]{}, 7 (1986). H. Hofmann, Phys. Rep. [**284**]{}, 137 (1997); C. Rummel and H. Hofmann, Nucl. Phys. A [**727**]{}, 24 (2003). G.G. Adamian, N.V. Antonenko, Z. Kanokov, and V.V. Sargsyan, Teor. Mat. Fiz. [**145**]{}, 87 (2005) \[Theor. Math. Phys. [**145**]{}, 1443 (2006)\]; Z. Kanokov, Yu.V. Palchikov, G.G. Adamian, N.V. Antonenko, and W. Scheid, Phys. Rev. E [**71**]{}, 016121 (2005); Yu.V. Palchikov, Z. Kanokov, G.G. Adamian, N.V. Antonenko, and W. Scheid, Phys. Rev. E [**71**]{}, 016122 (2005). N. Takigawa, S. Ayik, K. Washiyama, and S. Kimura, Phys. Rev. C [**69**]{}, 054605 (2004). G. Hupin and D. Lacroix, Phys. Rev. C [**81**]{}, 014609 (2010). V.V. Sargsyan, Z. Kanokov, G.G. Adamian, N.V. Antonenko, and W. Scheid, Phys. Rev. C [**80**]{}, 034606 (2009); Phys. Rev. C [**80**]{}, 047603 (2009). G.G. Adamian, A.K. Nasirov, N.V. Antonenko, and R.V. Jolos, Phys. Part. Nucl. [**25**]{}, 583 (1994). K. Washiyama, D. Lacroix, and S. Ayik, Phys. Rev. C [**79**]{}, 024609 (2009); S. Ayik, K. Washiyama, and D. Lacroix, Phys. Rev. C [**79**]{}, 054606 (2009). K. Langanke and S.E. Koonin, Nucl. Phys. [**A410**]{}, 334 (1983); A. Redder, H.W. Becker, C. Rolfs, H.P. Trautvetter, T.R. Donoghue, T.C. Rinckel, J.W. Hammer, and K. Langanke, Nucl. Phys. [**A462**]{}, 385 (1987). J. Toke [*et al.*]{}, Nucl. Phys. A [**440**]{}, 327 (1985). V.E. Viola, Jr., and T. Sikkeland [*et al.*]{}, Phys. Rev. [**128**]{}, 767 (1962). B.B. Back [*et al.*]{}, Phys. Rev. C. [**32**]{}, 195 (1985). T. Murakami [*et al.*]{}, Phys. Rev. C [**34**]{}, 1353 (1986). S. Kailas [*et al.*]{}, Phys. Rev. C [**59**]{}, 2580 (1999). K. Nishio [*et al.*]{}, Phys. Rev. C [**82**]{}, 044604 (2010). P. Möller [*et al.*]{}, At. Data Nucl. Data Tables [**59**]{}, 185 (1995). R. Vandenbosch, Annu. Rev. Nucl. Part. Sci. [**42**]{}, 447 (1992). A.B. Balantekin, J.R. Bennett, and S. Kuyucak, Phys. Lett. B [**335**]{}, 295 (1994). C.Y. Wong, Phys. Rev. Lett. [**31**]{}, 766 (1973). M. Trotta [*et al.*]{}, Eur. Phys. J. A [**25**]{}, 615 (2005). G.G. Adamian, N.V. Antonenko, and W. Scheid, Phys. Rev. C [**68**]{}, 034601 (2003). W.Q. Shen [*et al.*]{}, Phys. Rev. C [**36**]{}, 115 (1987). M.G. Itkis [*et al.*]{}, Nucl. Phys. A [**787**]{}, 150c (2007). M.G. Itkis [*et al.*]{}, Nucl. Phys. A [**734**]{}, 136c (2004). D.J. Hinde [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 1295 (1995).
|
---
abstract: 'We explore the ground-state phase diagram of a Heisenberg spin chain coupled locally to optical phonons (bond coupling), using large-scale density matrix renormalization group calculations and an extended perturbative analysis. For the quantum phase transition from the spin liquid to the dimerized phase, we find deviations from previous quantum Monte Carlo and flow equation results.'
author:
- 'Alexander Wei[ß]{}e'
- Georg Hager
- 'Alan R. Bishop'
- Holger Fehske
title: 'Phase diagram of the spin-Peierls chain with local coupling'
---
The interaction of electronic and lattice degrees of freedom in combination with reduced dimensionality can lead to a variety of interesting effects, one of which is the instability of a one-dimensional metal towards lattice distortion and the opening of a gap at the Fermi surface that was first described by Peierls [@Pe55]. A similar effect is observed in quantum spin chains, where the coupling to the lattice can cause a transition from a spin liquid with gap-less excitations to a dimerized phase with an excitation gap. Experimentally such behavior was first observed in the 1970s for organic compounds of the TTF and TCNQ family [@Brea75], but the topic regained attention after the discovery of the first inorganic spin-Peierls compound [CuGeO$_3$]{} in 1993 by @HTU93. In this material Cu$^{2+}$ ions form well separated spin-$1/2$ chains with an exchange interaction that couples to high-frequency *optical* phonons ($\omega\approx J$), and the phonon dynamics at the phase transition is governed by a central peak rather than a soft-mode behavior. [@BHRDR98; @FHW00] These features distinguish [CuGeO$_3$]{} from other spin-Peierls systems and sparked the interest in a non-adiabatic modelling.
A good starting point is the study of simplified microscopic models, which can be build from three ingredients, $$H = H_s + H_p + H_{sp}\,.$$ Here $H_s = \sum_i {\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}}$ and $H_p = \omega \sum_i
b_i^{\dagger} b_i^{}$ describe a Heisenberg spin-$1/2$ chain and a set of harmonic (Einstein) oscillators which are coupled by an interaction term $H_{sp}$. For this interaction we can consider two simple forms, $$\begin{aligned}
H_{sp}^{\text{diff}} & = \smash[b]{g\omega \sum_i (b_i^{\dagger} + b_i^{})
({\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}}-{\vec{S}_{i-1}\!\cdot\!\vec{S}_{i}})\,,}\\
\intertext{and}
H_{sp}^{\text{loc}} & = \smash[t]{g\omega \sum_i (b_i^{\dagger} + b_i^{}) {\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}}\,.}\end{aligned}$$
The first type of spin-phonon interaction, $H_{sp}^{\text{diff}}$, has been studied with a number of methods, including perturbation theory [@KF87; @WWF99], flow equations [@Uh98; @RBU01], exact diagonalization [@WWF99] and DMRG [@BMH99]. The latter approach identified the ground-state phase diagram, but also analytically the quantum phase transition from the gap-less to the dimerized phase is rather well understood: For finite phonon frequency $\omega$ the spin-phonon coupling $g$ leads to effective spin interactions beyond nearest-neighbor exchange, i.e., the low energy physics is governed by a frustrated Heisenberg model. As we know from the spin model $$\label{frust}
H = \sum_i ({\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}} + \alpha{\vec{S}_{i}\!\cdot\!\vec{S}_{i+2}})\,,$$ frustration can lead to dimerization if the parameter $\alpha$ exceeds a certain critical value ($\alpha_c = 0.241167$ in this case [@ON92; @CCE95; @Eg96]), and similarly we obtain a finite $g_c(\omega)$.[@Uh98; @BMH99; @WWF99; @HWWJF06p]
![(Color online) Ground-state phase diagram of the spin-Peierls chain with local coupling: QMC and flow equation (FEq) results of @RLUK02 compared to DMRG data and results of 12th order Schrieffer-Wolff transformation (SWT).[]{data-label="fig:pd"}](phasedia.eps){width="\linewidth"}
For the second type of spin-phonon coupling, $H_{sp}^{\text{loc}}$, which applies to [CuGeO$_3$]{}, to date the precise location of the phase boundary was arguable. In previous studies [@WWF99; @WFK98] we calculated $g_c$ using perturbation theory, a variational ansatz, as well as exact diagonalization of small systems. This was challenged by results of the flow equation method and, in a limited parameter range, by quantum Monte Carlo.[@RLUK02] In this article we present unbiased results of large-scale density matrix renormalization group (DMRG) calculations, and extend our perturbation theory by several orders in $g$, high enough to ensure convergence. The new results are summarized in Figure \[fig:pd\] and compared to other approaches. More details now follow.
#### Numerical results: {#numerical-results .unnumbered}
DMRG calculations are certainly the most precise numerical tool for studying the low energy properties of one-dimensional models, but as yet have only been performed for the spin-Peierls model with difference coupling [@BMH99] $H_{sp}^{\text{diff}}$, models with acoustic phonons [@BB05], or with $XY$-type spin interaction [@CM96]. We therefore implemented the local spin-phonon interaction $H_{sp}^{\text{loc}}$, using a high-performance, parallel version of the usual two-block finite-lattice algorithm with up to 1000 states per block. To detect the quantum phase transition from the gap-less to the dimerized phase we use the established criterion of the level-crossing between the first singlet and the first triplet excitation, which was derived for the frustrated spin chain [@AGSZ89], Eq. , and has successfully been applied [@BMH99; @HWWJF06p] in the case of $H_{sp}^{\text{diff}}$. For finite systems in the gap-less phase the lowest singlet excitation is above the lowest triplet, both becoming degenerate with the singlet ground state for system size $N\to\infty$. In the gapped phase, for $N\to\infty$ the lowest singlet becomes degenerate with the ground state to form the symmetry-broken dimerized state, whereas the lowest triplet maintains a finite gap. Consequently, the two excitations will cross at the critical point. Note however, that for small phonon frequency $\omega$ the relevant singlet excitation can be confused with a copy of the ground state plus an excited phonon. This frequency range therefore requires rather large $N$ for the correct crossing to be detectable in the low energy spectrum of the spin-Peierls models.
![(Color online) Crossing of the singlet and triplet gaps for large (right) and small (left) phonon frequency. Note the different finite-size scalings.[]{data-label="fig:cr"}](crossing.eps){width="\linewidth"}
In Figure \[fig:cr\] we show the level difference as a function of the spin-phonon coupling for various system sizes and two typical phonon frequencies. For $\omega \gtrsim 0.5$ there is almost no finite-size dependence of the critical coupling $g_c$, whereas for smaller frequencies the data scales noticeably. It is tempting to attribute this different behavior to the cross-over from the anti-adiabatic to the adiabatic regime, naïvely expected for $\omega\sim J \equiv 1$. However, as was pointed out by @COG05 using bosonization techniques, the relevant scale for the adiabatic to anti-adiabatic cross-over is given by the excitation gap, $\omega\sim\Delta$. The larger $N$-dependence of $g_c$ observed in Figure \[fig:cr\] for $\omega=0.1$ is therefore related to the finite-size gap still being close $\omega$, and will disappear for $N\to\infty$. Note also, that the bosonization results support the analytical approach we present in the following, which is based on the assumption of anti-adiabaticity.
![(Color online) Convergence of the Schrieffer-Wolff approach with increasing expansion order $O(k)$. Main panel: phase diagram showing DMRG and expansion data for order $4$–$12$ without and with oscillator shifts. Inset: $\alpha_{\text{eff}}$ as a function of $g$ for $\omega=50$ and orders $4$–$12$.[]{data-label="fig:cv"}](converge.eps){width="\linewidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$ J_0 $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
-------------------------- ----------------------------- ------------------------------------ ------------------------------------- ---------------------------------- ------------------------------------- ---------------------------------- ------------------------------------- --------------------------
$ g^0 $ $ \xi^2 \omega $ $ 1 $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^2 $ $ \cdot $ $ -\frac{1}{2} $ $ \frac{1}{2} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^2 \omega $ $ -\frac{3}{16} $ $ \frac{1}{2} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^3 \xi \omega $ $ \cdot $ $ -\frac{2}{3} $ $ \frac{2}{3} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^4 $ $ \cdot $ $ \frac{7}{24} $ $ -\frac{37}{96} $ $ \frac{3}{32} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^4 \omega $ $ \frac{3}{64} $ $ -\frac{3}{8} $ $ \frac{3}{16} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^5 \xi \omega $ $ \cdot $ $ \frac{7}{15} $ $ -\frac{37}{60} $ $ \frac{3}{20} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^6 $ $ \cdot $ $ -\frac{697}{5760} $ $ \frac{541}{2880} $ $ -\frac{29}{384} $ $ \frac{5}{576} $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^6 \omega $ $ -\frac{1}{64} $ $ \frac{29}{144} $ $ -\frac{91}{576} $ $ \frac{5}{192} $ $ \cdot $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^7 \xi \omega $ $ \cdot $ $ -\frac{697}{3360} $ $ \frac{541}{1680} $ $ -\frac{29}{224} $ $ \frac{5}{336} $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^8 $ $ \cdot $ $ \frac{5119}{129024} $ $ -\frac{88339}{1290240} $ $ \frac{45749}{1290240} $ $ -\frac{3685}{516096} $ $ \frac{35}{73728} $ $ \cdot $ $ \cdot $
$ g^8 \omega $ $ \frac{107}{24576} $ $ -\frac{3611}{46080} $ $ \frac{3569}{46080} $ $ -\frac{683}{30720} $ $ \frac{35}{18432} $ $ \cdot $ $ \cdot $ $ \cdot $
$ g^9 \xi \omega $ $ \cdot $ $ \frac{5119}{72576} $ $ -\frac{88339}{725760} $ $ \frac{45749}{725760} $ $ -\frac{3685}{290304} $ $ \frac{35}{41472} $ $ \cdot $ $ \cdot $
$ g^{10} $ $ \cdot $ $ -\frac{2516207}{232243200} $ $ \frac{9363217}{464486400} $ $ -\frac{105569}{8601600} $ $ \frac{171601}{51609600} $ $ -\frac{7601}{19353600} $ $ \frac{7}{409600} $ $ \cdot $
$ g^{10} \omega $ $ -\frac{18101}{17203200} $ $ \frac{35017}{1433600} $ $ -\frac{72439}{2580480} $ $ \frac{1243}{115200} $ $ -\frac{42433}{25804800} $ $ \frac{7}{81920} $ $ \cdot $ $ \cdot $
$ g^{11} \xi \omega $ $ \cdot $ $ -\frac{2516207}{127733760} $ $ \frac{9363217}{255467520} $ $ -\frac{105569}{4730880} $ $ \frac{171601}{28385280} $ $ -\frac{691}{967680} $ $ \frac{7}{225280} $ $ \cdot $
$ g^{12} $ $ \cdot $ $ \frac{624432139}{245248819200} $ $ -\frac{820409053}{163499212800} $ $ \frac{62408713}{18166579200} $ $ -\frac{138813341}{122624409600} $ $ \frac{12753401}{70071091200} $ $ $ \frac{77}{176947200} $
-\frac{6906257}{490497638400} $
$ g^{12} \omega $ $ \frac{69371}{309657600} $ $ -\frac{441857}{68812800} $ $ \frac{1701589}{206438400} $ $ -\frac{7128059}{1857945600} $ $ \frac{280187}{348364800} $ $ -\frac{4939}{66355200} $ $ $ \cdot $
\frac{77}{29491200} $
$ g^{13} \xi \omega $ $ \cdot $ $ \frac{624432139}{122624409600} $ $ -\frac{820409053}{81749606400} $ $ \frac{62408713}{9083289600} $ $ -\frac{138813341}{61312204800} $ $ \frac{12753401}{35035545600} $ $ $ \frac{77}{88473600} $
-\frac{6906257}{245248819200} $
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Analytical results: {#analytical-results .unnumbered}
Already in our earlier work we suggested construction of an effective spin model for the low energy physics of the spin-Peierls problem by integrating out the phonons with a Schrieffer-Wolff transformation [@SW66] (which removes direct spin-phonon interactions) followed by an average over the phonon vacuum. In more detail, we apply the unitary transformation $\tilde H
= \exp(S) H \exp(-S)$, where $$S = g \sum_i (b_i^\dagger - b_i){\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}}\,.$$ Unfortunately, this transformation cannot be evaluated exactly, but needs to be approximated by an expansion in iterated commutators: $$\tilde H = \sum_k [S,H]_k / k!,$$ where $[S,H]_{k+1} = [S,[S,H]_k]$ and $[S,H]_0 = H$. For increasing expansion order these commutators quickly become very complicated and easily involve millions of terms. Using FORM [@Ver00; @Ver06], an algebra tool popular in high energy physics, we are now able to push the limit of the expansion to order $k=12$, a tremendous advantage over our previous results with $k=4$. For the last step, we decided to be more general by averaging the transformed Hamiltonian over coherent states with $b_i|\xi\rangle = \xi|\xi\rangle$ instead of just the phonon vacuum: $$\label{heff}
\begin{aligned}
H_{\text{eff}} & = \langle \xi|\tilde H|\xi\rangle \\
& = J_0 N + \sum_i\sum_{n=1}^{7} J_n {\vec{S}_{i}\!\cdot\!\vec{S}_{i+n}} + \text{multi-spin terms}
\end{aligned}$$ This allows for a direct comparison with the flow equation result of @RLUK02, which is equivalent to a 2nd order expansion ($k=2$) with phonon shift $\xi = - \langle {\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}} \rangle g
\approx (1/4 - \ln 2) g \approx 0.44 g$. Here the spin correlator is approximated by its value in the isotropic Heisenberg chain. For a comparison see the small squares and the thin dashed line in Figure \[fig:pd\].
In Table \[tab:jn\] we list the expansion coefficients of the resulting long-ranged exchange interactions $J_n$ that contribute to our effective Hamiltonian Eq. . We neglect interactions that involve more than two spin operators. The phase transition line in Figure \[fig:pd\] is obtained by equating the effective frustration $\alpha_{\text{eff}} := J_2/J_1$ with the critical value $\alpha_c$ of the next-nearest-neighbor spin chain (see Eq. ), where for the phonon shift we use the “neutral” value $\xi=0$, i.e., the vacuum. Except for a constant offset, the analytical result matches the DMRG data quite well. In particular, the slope of the critical line is captured correctly.
The convergence of our analytical approach is illustrated in Figure \[fig:cv\], depicting the phase transition lines obtained for increasing expansion order. We observe an oscillatory behavior, such that reliable data results only beyond order $k\ge 8$. For lower orders, $\alpha_{\text{eff}}$ may not even reach the critical value, as is evident from the order $6$ data shown in the inset. Nevertheless, for order $6$ the phase transition can be estimated using a Shanks transformation [@BO99].
We also checked, if the phonon shift $\xi\approx 0.44 g$ proposed by @RLUK02 leads to an improved effective description, but clearly the dot-dashed line in Figure \[fig:cv\] deviates qualitatively from the DMRG data. Within the flow equation approach this shift was motivated by the technical requirement for normal ordering such that the phonons couple to $({\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}} -
\langle{\vec{S}_{i}\!\cdot\!\vec{S}_{i+1}}\rangle)$, but not by, e.g., some variational principle. Hence, the discrepancy is not too surprising. Interestingly, for $\omega\lesssim 3$ we obtain an almost perfect description of the phase transition, if instead we assume a phonon shift of $\xi\approx 0.44 g/\omega$. As yet we did not find a reasonable physical motivation for this $\omega$-dependence, and the good agreement might be accidental.
To summarize, using DMRG we obtained the, to date, most precise numerical result for the location of the quantum phase transition from the spin liquid to the dimerized phase in the one-dimensional Heisenberg model with local coupling to optical phonons. In addition, we proved the convergence of the unitary transformation approach that maps the full spin-phonon model to an effective frustrated spin model and allows an analytical calculation of the phase boundary in good agreement with the numerical data.
We thank E. Jeckelmann and G. Wellein for many helpful comments. In addition, we acknowledge the generous grant of resources by HLRN and NERSC, and financial support by DFG through SPP 1073. Work at Los Alamos was performed under the auspices of the US DOE.
[24]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ** (, , ).
, , , , , , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , in **, edited by (, , ), pp. .
, ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , , (), , <http://arXiv.org/abs/cond-mat/0606360>.
, , , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
(), , <http://arXiv.org/abs/math-ph/0010025>.
, ****, ().
, ** (, , ).
|
---
abstract: |
The $q$-oscillator representation for the Borel subalgebra of the affine symmetry $U_q'(\widehat{sl_N})$ is presented. By means of this $q$-oscillator representation, we give the free field realizations of the Baxter’s $Q$-operator ${\bf Q}_j(\lambda),
\overline{\bf Q}_j(\lambda), (j=1,2,\cdots,N)$ for the $W$-algebra $W_N$. We give the functional relations of the $T$-$Q$ operators, including the higher-rank generaliztion of the Baxter’s $T$-$Q$ relation.
title: |
**The Baxter’s $Q$-operator\
for the $W$-algebra $W_N$\
**
---
=eufm10 =eufm7 =eufm5 ===\#1[[\#1]{}]{} addtoreset[equation]{}[section]{}
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Fact]{} \[section\] \[thm\][Definition]{}
[ Takeo KOJIMA]{}\
\
[*Department of Mathematics, College of Science and Technology, Nihon University,\
Surugadai, Chiyoda-ku, Tokyo 101-0062, JAPAN* ]{}
\
\
Key Words : CFT, $q$-oscillator, $Q$-operator, functional relation, free field realization
Introduction
============
The Baxter’s $T$-$Q$-operator have various exceptional properties and play an important role in many aspect of the theory of integrable systems. Originally the $Q$-operator was introduced by R.Baxter [@Baxter1], in terms of some special transfer matrix of the 8-vertex model. Over the last three decades, this method of the $Q$-operator has been developed by many literatures. We would like to refer some of these literatures, written by R.Baxter [@Baxter2; @Baxter3; @Baxter4; @Baxter5], by L.Takhtadzhan and L.Faddeev [@TF], by K.Fabricius and B.McCoy [@FM1; @FM2; @FM3], by K.Fabricius [@Fab], by V.Bazhanov and V.Mangazeev [@BM], by B.Feigin, T.Kojima, J.Shiraishi and H.Watanabe [@FKSW2], by T.Kojima and J.Shiraishi [@KS]. However a full theory of the $Q$-operator for the 8-vertex model is not yet developed. For the simpler models associated with the quantum group $U_q(g)$, there have been many papers which extend, generalize, and comment on the $T$-$Q$ relation. We would like to refer some of these literatures, including Sklyanin’s separation variable method, written by E.Sklyanin [@Skl1; @Skl2; @Skl3], by V.Kuzunetsov, V.Mangazeev and E.Sklyanin [@KMS], by V.Pasquier and M.Gaudin [@PG], by S.Derkachov [@Der1] by S.Derkachov, G.Karakhanyan and A.Mansahov [@DKM1; @DKM2] by S.Derkachov, G.Karakhanyan and R.Kirschner, [@DKK] by S.Derkachov and A.Mansahov [@DM], by A.Belisty, S.Derkachov, G.Korchemesky and A.Manasahov [@BDKM], by C.Korff [@Kor1; @Kor2], by A.Bytsko and J.Teschner [@BT], by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [@BLZ1; @BLZ2; @BLZ3; @BLZ4], by M.Rossi and R.Weston [@RW], by P.Dorey and R.Tateo [@DT], by V.Bazhanov, A.Hibberd and S.Khoroshkin [@BHK], by P.Kulish and Z.Zeitlin [@KZ], by A.Antonov and B.Feigin [@AF], by I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin [@KLWZ], by V.Bazhanov and N.Reshetikhin [@BR], by A.Kuniba, T.Nakanishi and J.Suzuki [@KNS], by H.Boos, M.Jimbo, T.Miwa, F.Smirnov and Y.Takeyama [@BJMST1; @BJMST2], by A.Chervov and G.Falqui [@CF]. Each paper added to our understanding of the great Baxter’s original paper [@Baxter1]. Especially for example the $T$-$Q$-operators acting on the Fock space of the Virasoro algebra $Vir$ were introduced by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov [@BLZ1; @BLZ2; @BLZ3]. They derived various functional relations of the $T$-$Q$ operators and gave the asymptotic behavior of the eigen-value of the $T$-$Q$ operators. P.Dorey and R.Tateo [@DT] revealed the hidden connection between the vacuum expectation value of the $Q$-operator and the spectral determinant for Schrödinger equation. V.Bazhanov, A.Hibberd and S.Khoroshkin [@BHK] achieved the $W_3$-algebraic generalization of [@BLZ1; @BLZ2; @BLZ3; @BLZ4; @DT]. In this paper we study the higher-rank $W_N$-generalization of [@BHK]. We study the $T$-$Q$-operators acting on the Fock space of the $W$-algebra $W_N$. We give the free field realization of the $Q$-operator and functional relations of the $T$-$Q$-operators for the $W$-algebra $W_N$, including the higher-rank generalization of the Baxter’s $T$-$Q$ relation, $$\begin{aligned}
&&
{\bf Q}_i(tq^N)+\sum_{s=1}^{N-1}(-1)^{s}
{\bf T}_{\Lambda_1+\cdots+
\Lambda_s}(tq^{-1}){\bf Q}_i(tq^{N-2s})+(-1)^{N}{\bf Q}_i(tq^{-N})=0,
\nonumber
\\
&&
\overline{\bf Q}_i(tq^{-N})+\sum_{s=1}^{N-1}(-1)^{s}
\overline{\bf T}_{\Lambda_1+\cdots+\Lambda_s}(tq)
\overline{\bf Q}_i(tq^{-N+2s})+(-1)^{N}
\overline{\bf Q}_i(tq^{N})=0,\nonumber\end{aligned}$$ where $i=1,2,\cdots,N$. The organization of this paper is as following. In section 2, we give basic definitions, including $q$-oscillator representation of the Borel subalgebra of the affine symmetry $U_q'(\widehat{sl_N})$, which play an essential role in construction of the $Q$-operator. In section 3, we give the definition of the $T$-operator and the $Q$-operator. In section 4 we give conjecturous funtional relations between the $T$-opeartor and the $Q$-operator, including Baxter’s $T$-$Q$ relation. In appendix, we give supporting arguments on conjecturous formulae stated in section 4.
Basic Definition
================
In this section we give the different realizations of the Borel subalgebra of the affine quantum algebra $U_q'(\widehat{sl_N})$, which will play an important role in construction of the Baxter’s $T$-$Q$ operator. Let us fix the integer $N \geqq 3$. Let us fix a complex number $1<r<N$. In this paper, upon this setting, we construct the Baxter’s $T$-$Q$ operators on the space of the $W$-algebra $W_N$ with the central charge $-\infty<C_{CFT}<-2$, where $$\begin{aligned}
C_{CFT}=(N-1)\left(1-\frac{N(N+1)}{r (r-1)}\right).\nonumber\end{aligned}$$ Becuse $C_{CFT} \to -\infty$ represents the classical limit, we call $-\infty<C_{CFT}<-2$ “quasi-classical domain”. By anlytic continuation, it is possible to extend our theory to the CFT with central charge $C_{CFT}<1$. We would like to note that the unitary minimal CFT is described by the central charge $C_{CFT}=
(N-1)\left(1-\frac{N(N+1)}{r (r-1)}\right)$ for $N,r \in {\mathbb Z}$, $(N \geqq 2, r\geqq N+2)$ [@FL]. We set parameters $r^*=r-1$ and $q=e^{2\pi i \frac{r^*}{r}}$. In what follows we use the $q$-integer $[n]_q=\frac{q^n-q^{-n}}{q-q^{-1}}$.
The $q$-oscillator representation
---------------------------------
Let $\{\epsilon_j\}$ be an orthonormal basis of ${\mathbb R}^N$, relative to the standard inner product $(\epsilon_i|\epsilon_j)=\delta_{i,j}$. Let us set $\bar{\epsilon}_j=\epsilon_j-\epsilon$ where $\epsilon=\frac{1}{N}\sum_{j=1}^N \epsilon_j$. We have $(\bar{\epsilon}_i|\bar{\epsilon}_j)
=\delta_{i,j}-\frac{1}{N}$. Let us set the simple roots $\alpha_j=\bar{\epsilon}_j-\bar{\epsilon}_{j+1},
(1\leqq j \leqq N-1)$ and $\alpha_N=-\sum_{j=1}^{N-1}\alpha_j$. Let us set the fundamental weights $\omega_j$ as the dual vector of $\alpha_j$, i.e. $(\alpha_i|\omega_j)=\delta_{i,j}$. Explicitly we have $\omega_j=\bar{\epsilon_1}+\cdots+
\bar{\epsilon_j}$. Let us set the weight lattice $P=\oplus_{j=1}^N {\mathbb Z}
\bar{\epsilon}_j$. We consider the quantum affine algebra $U_q'(\widehat{sl_N})$, which is generated by $e_1,\cdots,e_{N}$, $f_1, \cdots, f_{N}$, and $h_1,\cdots, h_N$, with the defining relations, $$\begin{aligned}
&&~[h_i,h_j]=0,
~[h_i,e_j]=(\alpha_i|\alpha_j)e_j,
~[h_i,f_j]=-(\alpha_i|\alpha_j)f_j,
~[e_i,f_j]=\delta_{i,j}\frac{q^{h_i}-q^{-h_i}}
{q-q^{-1}},\nonumber\\
&&e_i^2 e_j-[2]_qe_i e_j e_i+e_j e_i^2=0,
~~
f_i^2 f_j-[2]_qf_i f_j f_i+f_j f_i^2=0,~~{\rm for}~~(\alpha_i|\alpha_j)=-1.\nonumber\end{aligned}$$ Here $((\alpha_j|\alpha_k))_{1 \leqq j,k \leqq N}$ is the Cartan matrix of type $\widehat{sl_N}$. Let us introduce the Borel subalgebra of $U_q'(\widehat{sl_N})$. The Borel subalgebra $U_q'(\widehat{\bf b}^+)$ is generated by $e_1,\cdots,e_{N}$, $h_1,\cdots, h_{N}$, and $U_q'(\widehat{\bf b}^-)$ by $f_1,\cdots,f_{N}$, $h_1,\cdots,h_{N}$. In this paper we consider the level $c=0$ case, with the central element $c=h_1+\cdots+h_{N}$. Let us introduce the $q$-oscillator representation $o_t$ of the Borel subalgebra $U_q'(\widehat{\bf b}^+)$. The $q$-oscillator algebra $Osc_j$, $(1\leqq j \leqq N-1)$, is generated by elements ${\cal E}_j, {\cal E}_j^*, {\cal H}_j$, with the defining relations, $$\begin{aligned}
~[{\cal H}_j,{\cal E}_j]={\cal E}_j,
~[{\cal H}_j,{\cal E}_j^*]=-{\cal E}_j^*,~
q{\cal E}_j{\cal E}_j^*-q^{-1}{\cal E}_j^*{\cal E}_j
=\frac{1}{q-q^{-1}}.\end{aligned}$$ Let us set $Osc=Osc_1 \otimes_{{\mathbb C}} \cdots
\otimes_{{\mathbb C}} Osc_{N-1}$. We have $[{\cal E}_j,{\cal E}_k]=0$, $[{\cal E}_j^*,{\cal E}_k^*]=0$, $[{\cal E}_j,{\cal E}_k^*]=0$, $[{\cal H}_j,{\cal H}_k]=0$ for $j \neq k$. Let us set the auxiliarry operator ${\cal H}_N=-{\cal H}_1
-{\cal H}_2-\cdots-{\cal H}_{N-1}$. We define homomorphism $o_t: U_q'(\widehat{\bf b}^+) \to
Osc$ by $$\begin{aligned}
o_t(e_1)&=&t q^{\frac{1}{2}}(q-q^{-1})
q^{-{\cal H}_2}{\cal E}_1^*{\cal E}_2,
\nonumber\\
o_t(e_2)&=&q^{\frac{1}{2}}(q-q^{-1})
q^{-{\cal H}_3}{\cal E}_2^*{\cal E}_3,
\nonumber\\
\cdots \nonumber\\
o_t(e_{N-2})&=&q^{\frac{1}{2}}(q-q^{-1})q^{-{\cal H}_{N-1}}
{\cal E}_{N-2}^*{\cal E}_{N-1},\nonumber\\
o_t(e_{N-1})&=&{\cal E}_{N-1}^*,\nonumber\\
o_t(e_N)&=& q^{-{\cal H}_1-{\cal H}_N}{\cal E}_1,
\label{def:q-osc}\end{aligned}$$ $$\begin{aligned}
o_t(h_1)=-{\cal H}_1+{\cal H}_2,~
o_t(h_2)=-{\cal H}_2+{\cal H}_3,\cdots,
o_t(h_N)=-{\cal H}_N+{\cal H}_1.\nonumber\end{aligned}$$ This $q$-oscillator representation $o_t$ satisfies level zero condition $o_t(h_1+h_2+\cdots+h_N)=0$. This $q$-oscillator representation give a higher-rank generalization of those in [@BHK]. By means of the Dynkin-diagram automorphism $\tau, \sigma$, we construct a family of the $q$-oscillator representation $o_{t,j}, \bar{o}_{t,j}$. Let us set the Dynkin-diagram automorphism $\tau$ of the affine algebra $U_q'(\widehat{sl_N})$. $$\begin{aligned}
\tau(e_1)=e_2, \cdots, \tau(e_j)=e_{j+1},\cdots, \tau(e_N)=e_1,\nonumber\\
\tau(h_1)=h_2, \cdots, \tau(h_j)=h_{j+1},\cdots, \tau(h_N)=h_1,\nonumber\\
\tau(f_1)=f_2, \cdots, \tau(f_j)=f_{j+1},\cdots, \tau(f_N)=f_1.\nonumber\end{aligned}$$ Let us set the Dynkin-diagram automorphism $\sigma$ of the finite simple algebra $U_q({sl_N})$, generated by $e_2,\cdots,e_{N}, h_2,\cdots,h_{N}, f_2,\cdots, f_{N}$. $$\begin{aligned}
\sigma(e_2)=e_{N},\cdots, \sigma(e_j)=e_{N+2-j},\cdots, \sigma(e_{N})=e_2, \nonumber\\
\sigma(h_2)=h_{N},\cdots, \sigma(h_j)=h_{N+2-j},\cdots, \sigma(h_{N})=h_2, \nonumber\\
\sigma(f_2)=f_{N},\cdots, \sigma(f_j)=f_{N+2-j},\cdots, \sigma(f_{N})=f_2, \nonumber\end{aligned}$$ and $\sigma$ is esxtended to the affine vertex as $\sigma(e_1)=e_1,
\sigma(h_1)=h_1,
\sigma(f_1)=f_1$. We have the action of $\tau^{j}\cdot \sigma \cdot
\tau^{-1}$, $$\begin{aligned}
\tau^{j}\cdot \sigma \cdot \tau^{-1}(e_{i})
=e_{j-1-i},\nonumber\\
\tau^{j}\cdot \sigma \cdot \tau^{-1}(h_{i})=
h_{j-1+i},\nonumber\\
\tau^{j}\cdot \sigma \cdot \tau^{-1}
(f_{i})=f_{j-1-i},\nonumber\end{aligned}$$ with $s, j \in {\mathbb Z}$. We set homomorphism $o_{t,j},\bar{o}_{t,j}:
U_q'(\widehat{\bf b}^+) \to Osc$, $(1\leqq j \leqq N)$, $$\begin{aligned}
o_{t,j}=o_t \cdot \tau^{-j},~~
\bar{o}_{t,j}=o_{(-1)^N t} \cdot \tau^{j}
\cdot \sigma \cdot \tau^{-1},\end{aligned}$$ These $q$-oscillator representations $o_{t,j},\bar{o}_{t,j}$ will play an important role in construction of the Baxter’s $Q$-operator.
Evaluation highest weight representation
----------------------------------------
Let us consider the quantum simple algebra $U_q(gl_N)$, which is generated by $E_{\alpha_1},\cdots,E_{\alpha_{N-1}}$, $H_1,\cdots, H_{N}$, and $F_{\alpha_1},\cdots, F_{\alpha_{N-1}}$, with the defining relations, $$\begin{aligned}
&&~[H_i,H_j]=0,
~[H_i,E_{\alpha_j}]=(\delta_{i,j}-\delta_{i,j+1})E_{\alpha_j},~
~[H_i,F_{\alpha_j}]=(-\delta_{i,j}+\delta_{i,j+1})F_{\alpha_j},
\nonumber\\
&&~[E_{\alpha_i},F_{\alpha_j}]=\delta_{i,j}\frac{q^{H_i-H_{i+1}}-q^{-H_i+H_{i+1}}}
{q-q^{-1}},\nonumber\\
&&E_{\alpha_i}^2 E_{\alpha_j}-[2]_qE_{\alpha_i} E_{\alpha_j} E_{\alpha_i}
+E_{\alpha_j} E_{\alpha_i}^2=0,
~~
F_{\alpha_i}^2 F_{\alpha_j}-[2]_qF_{\alpha_i} F_{\alpha_j} F_{\alpha_i}
+F_{\alpha_j} F_{\alpha_i}^2=0.\nonumber\end{aligned}$$ Let us set the root vectors, $$\begin{aligned}
&&F_{\alpha_1+\alpha_2}=[F_{\alpha_2},F_{\alpha_1}]_{\sqrt{q}}
=\sqrt{q}F_{\alpha_2}F_{\alpha_1}-\frac{1}{\sqrt{q}}
F_{\alpha_1}F_{\alpha_2},\nonumber\\
&&\bar{F}_{\alpha_1+\alpha_2}=[F_{\alpha_2},F_{\alpha_1}]_{
\frac{1}{\sqrt{q}}}
=\frac{1}{\sqrt{q}}
F_{\alpha_2}F_{\alpha_1}
-\sqrt{q}F_{\alpha_1}F_{\alpha_2},\nonumber\\
&&F_{\alpha_1+\cdots+\alpha_{N-1}}=[F_{\alpha_{N-1}},[F_{\alpha_{N-2}},
\cdots,[F_{\alpha_2},F_{\alpha_1}]_{\sqrt{q}}\cdots ]_{\sqrt{q}}]_{\sqrt{q}},\nonumber\\
&&\overline{F}_{\alpha_1+\cdots+\alpha_{N-1}}
=[[
\cdots,[F_{\alpha_{N-1}},F_{\alpha_{N-2}}]_{\frac
{1}{\sqrt{q}}}\cdots ,F_{\alpha_2}]_{\frac{1}{\sqrt{q}}},
F_{\alpha_1}]_{\frac{1}{\sqrt{q}}}.\nonumber\end{aligned}$$ Let us set the automorhism $\sigma$ by $$\begin{aligned}
&&\sigma(E_{\alpha_1})=E_{\alpha_{N-1}},\cdots,
\sigma(E_{\alpha_j})=E_{\alpha_{N-j}},\cdots,
\sigma(E_{\alpha_{N-1}})=E_{\alpha_{1}},\nonumber\\
&&\sigma(H_1)=-H_{N},\cdots,
\sigma(H_j)=-H_{N-j+1},\cdots,
\sigma(H_N)=-H_1,\nonumber\\
&&\sigma(F_{\alpha_1})=F_{\alpha_{N-1}},\cdots,
\sigma(F_{\alpha_j})=F_{\alpha_{N-j}},\cdots,
\sigma(F_{\alpha_{N-1}})=F_{\alpha_{1}}.\nonumber\end{aligned}$$ We have the evaluation representation $ev_t$, $\overline{ev}_t$: $U_q'(\widehat{sl_N})\to U_q(gl_N)$, given by $$\begin{aligned}
&&
ev_t(e_2)=E_{\alpha_1},\cdots,
ev_t(e_{j+1})=E_{\alpha_j},\cdots,
ev_t(e_{N})=E_{\alpha_{N-1}},\nonumber\\
&&
ev_t(h_2)=H_1-H_2,\cdots,
ev_t(h_{j+1})=H_j-H_{j+1},\cdots,
ev_t(h_{N})=H_{N-1}-H_N,\nonumber\\
&&
ev_t(f_2)=F_{\alpha_1},\cdots,
ev_t(f_{j+1})=F_{\alpha_j},\cdots,
ev_t(f_{N})=F_{\alpha_{N-1}},\nonumber
\\
&&
ev_t(e_1)=t
F_{\alpha_1+\alpha_2+\cdots+\alpha_{N-1}}q^{H_1+H_{N}},~
ev_t(f_1)=t^{-1}
E_{\alpha_1+\alpha_2+\cdots+\alpha_{N-1}}
q^{-H_1-H_{N}},\nonumber\\
&&ev_t(h_1)=H_N-H_1.\nonumber\end{aligned}$$ $$\begin{aligned}
&&
\overline{ev}_t(e_2)=E_{\alpha_1},\cdots,
\overline{ev}_t(e_{j+1})=E_{\alpha_j},\cdots,
\overline{ev}_t(e_{N})=E_{\alpha_{N-1}},\nonumber\\
&&
\overline{ev}_t(h_2)=H_1-H_2,\cdots,
\overline{ev}_t(h_{j+1})=H_j-H_{j+1},\cdots,
\overline{ev}_t(h_{N})=H_{N-1}-H_N,\nonumber\\
&&
\overline{ev}_t(f_2)=F_{\alpha_1},\cdots,
\overline{ev}_t(f_{j+1})=F_{\alpha_j},\cdots,
\overline{ev}_t(f_{N})=F_{\alpha_{N-1}},\nonumber\\
&&
\overline{ev}_t(e_1)=t
\overline{F}_{\alpha_1+\alpha_2+\cdots+\alpha_{N-1}}
q^{-H_1-H_{N}},~
\overline{ev}_t(f_1)=t^{-1}
\overline{E}_{\alpha_1+\alpha_2+\cdots+\alpha_{N-1}}
q^{H_1+H_{N}},\nonumber\\
&&
\overline{ev}_t(h_{1})=H_{N}-H_1.\nonumber\end{aligned}$$ We have the conjugation $\overline{ev}_t=
\sigma \cdot ev_{(-)^N t} \cdot \sigma^{-1}$. We set the irreducible highest representation of $U_q(gl_N)$ with the highest weight $\lambda=m_1 \Lambda_1+\cdots+m_{N}\Lambda_{N}$, the highest weight vector $|\lambda \rangle$ of $U_q(sl_N)$. $$\begin{aligned}
&&
\pi^{(\lambda)}(E_{\alpha_j})|\lambda \rangle=0,~~
\pi^{(\lambda)}(H_{j})|\lambda \rangle=m_j|\lambda \rangle,
~~(1\leqq j \leqq N).\nonumber\end{aligned}$$ In what follows we consider the case $m_j-m_{j+1} \in {\mathbb N}$, $(1\leqq j \leqq N-2)$. In this case the representation $\pi^{(\lambda)}$ is finite dimension. Let us set the evaluation highest weight representation $\pi_t^{(\lambda)}$ of the affine symmetry $U_q'(\widehat{sl_N})$, as $$\begin{aligned}
\pi^{(\lambda)}_t=\pi^{(\lambda)}\cdot ev_t,~~~
\overline{\pi}^{(\lambda)}_t=\pi^{(\lambda)}\cdot
\overline{ev}_t.\nonumber\end{aligned}$$ These evaluation highest weight representation will play an important role in construction of the $T$-operator ${\bf T}_\lambda(t),
\overline{\bf T}_\lambda(t)$.
Screening current
-----------------
Let us introduce bosons $B_m^i$, $(m \in {\mathbb Z}_{\neq 0}; i=1,2,\cdots,N-1)$ by $$\begin{aligned}
[B_m^i,B_n^j]=m \delta_{m+n,0}(\alpha_i|\alpha_j)\frac{r-1}{r},~~
(1\leqq i,j \leqq N-1).\end{aligned}$$ Let us set $B_m^N=-\sum_{j=1}^{N-1}B_m^j$. We have the commutation relation $
[B_m^i,B_n^j]=m \delta_{m+n,0}(\alpha_i|\alpha_j)\frac{r-1}{r}$, for $1\leqq i,j \leqq N$. Let us set the zero-mode operators $P_\lambda$ and $Q_\lambda$, $(\lambda \in P=\oplus_j {\mathbb Z}\bar{\epsilon}_j)$ by $$\begin{aligned}
[P_\lambda, iQ_\mu]=(\lambda |\mu).\end{aligned}$$ Let us set the Heisenberg algebra ${\cal B}$ generated by $B_m^1,\cdots,B_m^{N-1}$, $P_\lambda, Q_\lambda$, $(\lambda \in P)$ and its completion $\widehat{\cal B}$. Let us set the Fock space ${\cal F}_{l,k}$ by $$\begin{aligned}
B_m^j|l,k\rangle&=&0,~~(m>0)\\
P_\alpha|l,k\rangle&=&
\left(
\alpha \left|
\sqrt{\frac{r}{r-1}}l-
\sqrt{\frac{r-1}{r}}k
\right.
\right)
|l,k\rangle,\\
|l,k\rangle&=&e^{i\sqrt{\frac{r}{r-1}}Q_l-
i\sqrt{\frac{r-1}{r}}Q_k}|0,0\rangle.\end{aligned}$$ Let us set the screening currents of the $W$-algebra $W_N$ by $$\begin{aligned}
V_{\alpha_j}(u)&=&
\exp\left(i\sqrt{\frac{r^*}{r}}Q_{\alpha_j}\right)
\exp\left(\sqrt{\frac{r^*}{r}}P_{\alpha_j}i u\right)\nonumber\\
&\times&
\exp\left(\sum_{m>0}\frac{1}{m}B_{-m}^j e^{imu}\right)
\exp\left(-\sum_{m>0}\frac{1}{m}B_m^j e^{-imu}\right),~
(1\leqq j \leqq N).\end{aligned}$$ Here we have added one operator $V_{\alpha_N}(u)$, which looks like affinization of the classical $A_{N-1}$. We can find the elliptic deformation of $V_{\alpha_j}(u)$ for $j \neq N$ in [@FKSW2; @KS]. For ${\rm Re}(u_1)>{\rm Re}(u_2)$, we have $$\begin{aligned}
V_{\alpha_j}(u_1)V_{\alpha_j}(u_2)&=&
:V_{\alpha_j}(u_1)V_{\alpha_j}(u_2):(e^{iu_1}-e^{iu_2})^{\frac{2r^*}{r}},~~
(1\leqq j \leqq N),
\nonumber
\\
V_{\alpha_j}(u_1)V_{\alpha_{j+1}}(u_2)&=&
:V_{\alpha_j}(u_1)V_{\alpha_{j+1}}(u_2):
(e^{iu_1}-e^{iu_2})^{-\frac{r^*}{r}},~~
(1\leqq j \leqq N),\nonumber
\\
V_{\alpha_{j+1}}(u_1)V_{\alpha_j}(u_2)&=&
:V_{\alpha_{j+1}}(u_1)V_{\alpha_{j}}(u_2):
(e^{iu_1}-e^{iu_2})^{-\frac{r^*}{r}},~~
(1\leqq j \leqq N).
\nonumber\end{aligned}$$ By analytic continuation, we have $$\begin{aligned}
V_{\alpha_i}(u_1)V_{\alpha_j}(u_2)=q^{(\alpha_i|\alpha_j)}
V_{\alpha_j}(u_2)V_{\alpha_i}(u_1),
~~(1 \leqq i,j \leqq N).\end{aligned}$$ Let us set $$\begin{aligned}
z_{j}=\exp\left(-2 \pi i \sqrt{\frac{r^*}{r}}
P_{\bar{\epsilon}_{j}}\right),~~(1 \leqq j \leqq N).\end{aligned}$$ We have $z_1 z_2\cdots z_{N}=1$ and $$\begin{aligned}
V_{\alpha_i}(u+2\pi)
=
z_{i}^{-1}z_{i+1}V_{\alpha_i}(u),~~
z_i V_{\alpha_j}(u)&=&q^{\delta_{i,j+1}-\delta_{i,j}}
V_{\alpha_j}(u) z_i.\nonumber\end{aligned}$$ Let us set the nilpotent subalgebra $U_q'(\widehat{\bf n}^-)$ generated by $f_1,f_2,\cdots,f_N$. We have homomorphism $sc : U_q'(\widehat{\bf n}^-) \to \widehat{\cal B}$ given by $$\begin{aligned}
sc(f_j)=\frac{1}{q-q^{-1}}\int_0^{2\pi}
V_{\alpha_j}(u)du,~~(1\leqq j \leqq N).\nonumber\end{aligned}$$
Baxter’s $Q$-operator
=====================
In this section we define the Baxter’s $T$-$Q$ operator by means of the trace of the universal $R$, and present conjecturous functional relations of the $T$-$Q$ operator, which include the higher-rank generalization of the Baxter’s $T$-$Q$ relation.
${\cal L}$-operator
-------------------
Let us set the universal $L$-operator ${\cal L}
\in \widehat{\cal B}\otimes U_q(\widehat{\bf n}^-)$ by $$\begin{aligned}
{\cal L}=\exp\left(-\pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^{N}
P_{\omega_j} \otimes h_j
\right){\cal P}\exp\left(
\int_0^{2\pi}K(u)du\right).\end{aligned}$$ Here we have set $$\begin{aligned}
K(u)=\sum_{j=1}^{N}
V_{\alpha_j}(u)\otimes e_j.\nonumber\end{aligned}$$ Here ${\cal P}\exp
\left(\int_0^{2\pi}K(u)du\right)$ represents the path ordered exponential $$\begin{aligned}
{\cal P}\exp
\left(\int_0^{2\pi}K(u)du\right)=
\sum_{n=0}^\infty
\int \cdots \int_{2\pi \geqq u_1 \geqq u_2 \geqq
\cdots \geqq u_n \geqq 0}
K(u_1)K(u_2)\cdots K_n(u_n)du_1 du_2 \cdots du_n.
\nonumber\end{aligned}$$ The above integrals converge in “quasi-classical domain” $-\infty < C_{CFT} <-2$. For the value of $C_{CFT}$ outside the quasi-classical domain, the integrals should be understood as analytic continuation. Let us set $U_q(\widehat{sl_N})$ the extension of $U_q'(\widehat{sl_N})$ by the degree operator $d$. Let us set $U_q(\widehat{\bf n}^\pm)$ the extension of $U_q'(\widehat{\bf n}^\pm)$ by the degree operator $d$. There exists the unique universal $R$-matrix ${\cal R} \in U_q(\widehat{\bf n}^+)\otimes
U_q(\widehat{\bf n}^-)$ satisfying the Yang-Baxter equation. $$\begin{aligned}
{\cal R}_{12}{\cal R}_{13}{\cal R}_{23}=
{\cal R}_{23}{\cal R}_{13}{\cal R}_{12}.\nonumber\end{aligned}$$ The universal-$R$’s Cartan elements ${\bf t}$ is factored as $$\begin{aligned}
{\cal R}=q^{\bf t} \overline{\cal R},
~~
{\bf t}=
\sum_{j=1}^{N-1}h_j \otimes h^j+c \otimes d+d \otimes c,
\nonumber\end{aligned}$$ where $(h^i|h_j)=\delta_{i,j}$. We call the element $\overline{\cal R}
\in U_q'(\widehat{\bf n}^+)\otimes
U_q'(\widehat{\bf n}^-)$ the reduced universal $R$-matrix. The ${\cal L}$-operator is an image of the reduced $R$-matrix [@BHK], $$\begin{aligned}
{\cal L}=(sc \otimes id )(\overline{R}).\nonumber\end{aligned}$$ The ${\cal L}$-operator will play an important role in trace construction of the $T$-$Q$ operator.
T-operator
----------
Let us set the $T$-operator ${\bf T}_\lambda(t)$ and $\overline{\bf T}_\lambda(t)$ by $$\begin{aligned}
&&{\bf T}_\lambda(t)={\rm Tr}_{
\pi_t^{(\lambda)}}
\left(
\exp\left(-\pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^N
P_{\omega_j} \otimes h_j
\right){\cal L}
\right),\\
&&\overline{\bf T}_\lambda(t)={\rm Tr}_{
\overline{\pi}_t^{(\lambda)}}
\left(
\exp\left(-\pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^N
P_{\omega_j} \otimes h_j
\right){\cal L}
\right).\end{aligned}$$ Let us set an image of ${\cal L}$ as ${\bf L}_\lambda(t)=(id \otimes \pi_t^{(\lambda)})
\left(
{\cal L}
\right)$, and the $R$-matrix $R_{\lambda_1,\lambda_2}(t_1/t_2)=
\pi_{t_1}^{(\lambda_1)}\otimes \pi_{t_2}^{(\lambda_2)}({\cal R})
$. We have so-called $RLL$ relation, $$\begin{aligned}
{R}_{\lambda_1, \lambda_2}(t_1/t_2)
{\bf L}_{\lambda_1}(t_1)
{\bf L}_{\lambda_2}(t_2)=
{\bf L}_{\lambda_2}(t_2)
{\bf L}_{\lambda_1}(t_1)
{R}_{\lambda_1, \lambda_2}(t_1/t_2).\nonumber\end{aligned}$$ Multiplying the $R$-matrix ${R}_{\lambda_1, \lambda_2}(t_1/t_2)^{-1}$ from the right, and taking trace, we have the commutation relation, $$\begin{aligned}
~[{\bf T}_{\lambda_1}(t_1), {\bf T}_{\lambda_2}(t_2)]=
~[\overline{\bf T}_{\lambda_1}(t_1),
\overline{\bf T}_{\lambda_2}(t_2)]=
~[{\bf T}_{\lambda_1}(t_1), \overline{\bf T}_{\lambda_2}(t_2)]=0.
\nonumber\end{aligned}$$ The coefficients of the Taylor expansion of ${\bf T}_\lambda(t)$ commute with each other. Hence we have infinitly many commutative operators, which give quantum deformation of the conservation laws of the $N$-th KdV equation.
Q-operator
----------
Let us set the Fock representation $\pi_j^\pm$: $Osc_j \to W^\pm$ with $j=1,2,\cdots,N-1$, $$\begin{aligned}
W^+=\oplus_{k \geqq 0}{\mathbb C}|k\rangle_+,
~~
W^-=\oplus_{k \geqq 0}{\mathbb C}|k\rangle_-.\nonumber\end{aligned}$$ The action is given by $$\begin{aligned}
\pi_j^+({\cal H}_j)|k\rangle_+=-k|k\rangle_+,~
\pi_j^+({\cal E}_j)|k\rangle_+=\frac{1-q^{-2k}}{(q-q^{-1})^2}
|k-1\rangle_+,~
\pi_j^+({\cal E}_j^*)|k\rangle=|k+1\rangle_+,\nonumber\\
\pi_j^-({\cal H}_j)|k\rangle_-=k|k\rangle_-,~
\pi_j^-({\cal E}_j)|k\rangle_-
=\frac{1-q^{2k}}{(q-q^{-1})^2}|k-1\rangle_-,~
\pi_j^-({\cal E}_j)|k\rangle_-=|k+1\rangle_-.\nonumber\end{aligned}$$ Let $\pi_j$ and $\overline{\pi}_j$ be any representation of the $q$-oscillator $Osc=Osc_1 \otimes_{\mathbb C}\cdots \otimes_{\mathbb C}
Osc_{N-1}$ such that the partition $Z_j(t), \overline{Z}_j(t)$ converge. $$\begin{aligned}
Z_j(t)&=&{\rm Tr}_{\pi_j o_{t,j}}
\left(
\exp\left(-2 \pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^{N}
P_{\omega_j} \otimes h_j
\right)
\right),\nonumber\\
\overline{Z}_j(t)&=&
{\rm Tr}_{\overline{\pi}_j \overline{o}_{t,j}}
\left(
\exp\left(-2 \pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^{N}
P_{\omega_j} \otimes h_j
\right)
\right).\nonumber\end{aligned}$$ Let us set the operators ${\bf A}_j(t)$ and $\overline{\bf A}_j(t)$ with $j=1,2,\cdots,N$ $$\begin{aligned}
&&{\bf A}_j(t)=\frac{1}{Z_j(t)}
{\rm Tr}_{
{\pi}_j {o}_{t,j}
}
\left(
\exp\left(-\pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^{N}
P_{\omega_j} \otimes h_j
\right){\cal L}
\right),\label{def:A1}
\\
&&\overline{\bf A}_j(t)=\frac{1}{\overline{Z}_j(t)}
{\rm Tr}_{
\overline{\pi}_j \overline{o}_{t,j}}
\left(
\exp\left(-\pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^{N}
P_{\omega_j} \otimes h_j
\right){\cal L}
\right).\label{def:A2}\end{aligned}$$ Let us set the Baxter’s $Q$-operator ${\bf Q}_j(t)$ and $\overline{\bf Q}_j(t)$ with $j=1,2,\cdots,N,$ $$\begin{aligned}
{\bf Q}_j(t)=t^{-\frac{1}{2}\sqrt{\frac{r}{r^*}}
P_{\bar{\epsilon}_{j}}}
{\bf A}_j(t),~~
\overline{\bf Q}_j(t)=
t^{\frac{1}{2}\sqrt{\frac{r}{r^*}}
P_{\bar{\epsilon}_{j}}}
\overline{\bf A}_j(t).\label{def:Q}\end{aligned}$$ We would like to note convenient relation, $$\begin{aligned}
\sum_{k=1}^{N}P_{\omega_k}\otimes o_{t,j}(h_k)&=&
\sum_{k=1}^{N-1}(P_{\bar{\epsilon}_{j}}
-P_{\bar{\epsilon}_{j+k}})
\otimes {\cal H}_k,\nonumber\\
\sum_{k=1}^{N}P_{\omega_k}\otimes \overline{o}_{t,j}(h_k)
&=&
\sum_{k=1}^{N-1}(P_{\bar{\epsilon}_{j-k}}
-P_{\bar{\epsilon}_{j}})
\otimes {\cal H}_k.\nonumber\end{aligned}$$ Here we should understand the surfix number as modulus $N$, i.e. $\bar{\epsilon}_{j+N}=\bar{\epsilon}_j$.
From the Yang-Baxter equation, we have the commutation relations $$\begin{aligned}
~[{\bf Q}_{j_1}(t_1),{\bf Q}_{j_2}(t_2)]
=[\overline{\bf Q}_{j_1}(t_1),\overline{\bf Q}_{j_2}(t_2)]
=[{\bf Q}_{j_1}(t_1),\overline{\bf Q}_{j_2}(t_2)]=0,\nonumber\end{aligned}$$ and $$\begin{aligned}
~[{\bf Q}_{j}(t_1),{\bf T}_{\lambda}(t_2)]
=[{\bf Q}_{j}(t_1),\overline{\bf T}_{\lambda}(t_2)]
=[\overline{\bf Q}_{j}(t_1),{\bf T}_{\lambda}(t_2)]
=[\overline{\bf Q}_{j}(t_1),\overline{\bf T}_{\lambda}(t_2)]=0.\nonumber\end{aligned}$$ The operators ${\bf A}_j(t)$ can be written as power series. $$\begin{aligned}
{\bf A}_j(t)&=&1+\sum_{n=1}^\infty
\sum_{\sigma_1,\cdots,\sigma_{Nn} \in {\mathbb Z}_N}
a_{Nn}^{(j)}(\sigma_1,\cdots,\sigma_{Nn})\nonumber\\
&\times&
\int \cdots \int_{2\pi \geqq u_1 \geqq u_2 \geqq \cdots
\geqq u_{Nn} \geqq 0}
V_{\alpha_{\sigma_1}}(u_1)
\cdots V_{\alpha_{\sigma_{Nn}}}(u_{Nn})
du_1 \cdots du_{Nn}.\nonumber\end{aligned}$$ Here we have set $$\begin{aligned}
a_{Nn}^{(j)}(\sigma_1,\cdots,\sigma_{Nn})=
\frac{1}{Z_j(t)}{\rm Tr}_{
{\pi}_j {o}_{t,j}}
\left(
\exp\left(-2\pi i \sqrt{\frac{r^*}{r}}
\sum_{j=1}^N
P_{\omega_j} \otimes h_j
\right)e_{\sigma_1}e_{\sigma_2}\cdots e_{\sigma_{Nn}}
\right).\nonumber\end{aligned}$$ The coefficients $a_{Nn}^{(j)}$ vanishes unless $
n=|\{ j |\sigma_j=s \}|$ for $s \in {\mathbb Z}_N$, and behaves like $a_{Nn}^{(j)} \sim O(t^n)$. The coefficients $a_{Nn}^{(j)}$ are determined by the commutation relations of the Borel subalgebra $U_q(\widehat{\bf n}^-)$ and the cyclic property of the trace, hence the specific choice of representation $\pi_j, \overline{\pi}_j$ is not significant as long as it converges. In [@FKSW2; @KS] we have constructed the elliptic version of the integral of the currents, $$\int \cdots \int_{2\pi \geqq u_1 \geqq u_2 \geqq \cdots
\geqq u_{Nn} \geqq 0}
V_{\alpha_{\sigma_1}}(u_1)
V_{\alpha_{\sigma_2}}(u_2)
\cdots V_{\alpha_{\sigma_{Nn}}}(u_{Nn})
du_1 du_2 \cdots du_{Nn}.$$
Functional relations
====================
In the previous section, we show that the $T$-$Q$ operators commute with each other. In this section we give conjecturous functional relations of the $T$-$Q$ operators, which coincide with the previous work [@BHK] upon $N=3$ specialization. We have checked those functional relations up to the order $O(t^2)$ in appendix. Some of similar formulae have been obtained in the context of the solvable lattice models associated with $U_q(\widehat{sl_N})$ [@KLWZ; @BR; @KNS]. At the end of this section we summarize conclusion.
Functional relations
--------------------
The $T$-operator is written by determinant of the $Q$-operators. Let us set the Young diagram $\mu=(\mu_1,\mu_2,\cdots,\mu_N)$, $(\mu_j\geqq \mu_{j+1}; \mu_j \in {\mathbb N})$. Using the same character as the Young diagram $\mu$, we represent the highest weight $\mu=\mu_1 \Lambda_1+\cdots+\mu_{N}\Lambda_{N}$. We set $c_0=\prod_{1\leqq j<k \leqq N}
\left(
\sqrt{\frac{z_j}{z_k}}-
\sqrt{\frac{z_k}{z_j}}
\right)$. We have the following determinant formulae of the $T$-operator, $$\begin{aligned}
{\bf T}_\mu(t)&=&
\frac{1}{c_0}\left|
\begin{array}{cccc}
{\bf Q}_1(tq^{2\tilde{\mu}_1})&
{\bf Q}_1(tq^{2\tilde{\mu}_2})&\cdots&
{\bf Q}_1(tq^{2\tilde{\mu}_N})\\
{\bf Q}_2(tq^{2\tilde{\mu}_1})&
{\bf Q}_2(tq^{2\tilde{\mu}_2})&\cdots&
{\bf Q}_2(tq^{2\tilde{\mu}_N})\\
\cdots&\cdots&\cdots&\cdots
\\
{\bf Q}_N(tq^{2\tilde{\mu}_1})&
{\bf Q}_N(tq^{2\tilde{\mu}_2})&\cdots&
{\bf Q}_N(tq^{2\tilde{\mu}_N})
\end{array}
\right|,
\label{rel:det1}\\
\nonumber\\
\nonumber\\
\overline{\bf T}_\mu(t)&=&
\frac{1}{c_0}
\left|
\begin{array}{cccc}
\overline{\bf Q}_1(tq^{-2\tilde{\mu}_1})&
\overline{\bf Q}_1(tq^{-2\tilde{\mu}_2})&\cdots&
\overline{\bf Q}_1(tq^{-2\tilde{\mu}_N})\\
\overline{\bf Q}_2(tq^{-2\tilde{\mu}_1})&
\overline{\bf Q}_2(tq^{-2\tilde{\mu}_2})&\cdots&
\overline{\bf Q}_2(tq^{-2\tilde{\mu}_N})\\
\cdots&\cdots&\cdots&\cdots
\\
\overline{\bf Q}_N(tq^{-2\tilde{\mu}_1})&
\overline{\bf Q}_N(tq^{-2\tilde{\mu}_2})&\cdots&
\overline{\bf Q}_N(tq^{-2\tilde{\mu}_N})
\end{array}
\right|.\label{rel:det2}\end{aligned}$$ Here we have used the auxiliarry parameters $2\tilde{\mu}_j=2\mu_j+N-2j+1, (1\leqq j \leqq N)$. We have checked the above formulae (\[rel:det1\]) and (\[rel:det2\]) for $\mu=\Lambda_1$ and $\mu=
\Lambda_1+\cdots+\Lambda_{N-1}$, up to the order $O(t^2)$. See appendix. As the special case $\mu_j=0, (1\leqq j \leqq N)$, we have the quantum Wronskian condition. $$\begin{aligned}
c_0&=&
\left|
\begin{array}{cccc}
{\bf Q}_1(tq^{N-1})&{\bf Q}_1(tq^{N-3})&\cdots&{\bf Q}_1(tq^{-N+1})\\
{\bf Q}_2(tq^{N-1})&{\bf Q}_2(tq^{N-3})&\cdots&{\bf Q}_2(tq^{-N+1})\\
\cdots&\cdots&\cdots&\cdots
\\
{\bf Q}_N(tq^{N-1})&{\bf Q}_N(tq^{N-3})&\cdots&{\bf Q}_N(tq^{-N+1})
\end{array}
\right|,
\label{rel:det1'}
\\
\nonumber
\\
\nonumber\\
c_0&=&
\left|
\begin{array}{cccc}
\overline{\bf Q}_1(tq^{-N+1})&
\overline{\bf Q}_1(tq^{-N+3})&\cdots&\overline{\bf Q}_1(tq^{N-1})\\
\overline{\bf Q}_2(tq^{-N+1})&\overline{\bf Q}_2(tq^{-N+3})&\cdots&
\overline{\bf Q}_2(tq^{N-1})\\
\cdots&\cdots&\cdots&\cdots
\\
\overline{\bf Q}_N(tq^{-N+1})&\overline{\bf Q}_N(tq^{-N+3})&\cdots&
\overline{\bf Q}_N(tq^{N-1})
\end{array}
\right|.\label{rel:det2'}\end{aligned}$$ We have checked the above formulae (\[rel:det1’\]), (\[rel:det2’\]), up to the order $O(t^2)$. See appendix. Let us set $c_i=\prod_{1\leqq j<k \leqq N
\atop{j,k \neq i}}
\left(
\sqrt{\frac{z_j}{z_k}}-
\sqrt{\frac{z_k}{z_j}}
\right)$ for $1\leqq i \leqq N$. The two kind of $Q$-operator, ${\bf Q}_j(t)$ and $\overline{\bf Q}_j(t)$, are functionally dependent. The $Q$-operator ${\bf Q}_i(t)$ is written by the determinant of the $Q$-operator $\overline{\bf Q}_j(t)$, $$\begin{aligned}
c_i {\bf Q}_i(t)&=&
\left|
\begin{array}{cccc}
\overline{\bf Q}_1(tq^{N-2})&\overline{\bf Q}_1(tq^{N-4})&\cdots&
\overline{\bf Q}_1(tq^{-N+2})\\
\cdots&\cdots&\cdots&\cdots
\\
\overline{\bf Q}_{i-1}(tq^{N-2})&
\overline{\bf Q}_{i-1}(tq^{N-4})&\cdots&\overline{\bf Q}_{i-1}(tq^{-N+2})\\
\overline{\bf Q}_{i+1}(tq^{N-2})&\overline{\bf Q}_{i+1}(tq^{N-4})&\cdots&
\overline{\bf Q}_{i+1}(tq^{-N+2})\\
\cdots&\cdots&\cdots&\cdots
\\
\overline{\bf Q}_N(tq^{N-2})&
\overline{\bf Q}_N(tq^{N-4})&\cdots&\overline{\bf Q}_N(tq^{-N+2})
\end{array}
\right|,
\label{rel:det3}
\\
\nonumber
\\
\nonumber
\\
c_i
\overline{\bf Q}_i(t)&=&
\left|
\begin{array}{cccc}
{\bf Q}_1(tq^{-N+2})&{\bf Q}_1(tq^{-N+4})&\cdots&{\bf Q}_1(tq^{N-2})\\
\cdots&\cdots&\cdots&\cdots
\\
{\bf Q}_{i-1}(tq^{-N+2})&{\bf Q}_{i-1}(tq^{-N+4})&\cdots&{\bf Q}_{i-1}(tq^{N-2})\\
{\bf Q}_{i+1}(tq^{-N+2})&{\bf Q}_{i+1}(tq^{-N+4})&\cdots&{\bf Q}_{i+1}(tq^{N-2})\\
\cdots&\cdots&\cdots&\cdots
\\
{\bf Q}_N(tq^{-N+2})&{\bf Q}_N(tq^{-N+4})&\cdots&{\bf Q}_N(tq^{N-2})
\end{array}
\right|,
\label{rel:det4}\end{aligned}$$ with $i=1,2,\cdots,N$. We have checked the determinat formulae (\[rel:det3\]) and (\[rel:det3\]) up to the order $O(t^2)$. See appendix. We derive the following (\[rel:BaxTQ1\]), (\[rel:BaxTQ2\]), (\[rel:qua1\]), (\[rel:qua2\]), (\[rel:qua3\]), (\[rel:qua4\]), and (\[rel:JT\]) from the above formulae (\[rel:det1\]), (\[rel:det2\]), (\[rel:det3\]) and (\[rel:det4\]). We have the higher-rank generalization of the Baxter’s $T$-$Q$ relation (\[rel:BaxTQ1\]) and (\[rel:BaxTQ2\]), as the consequence of (\[rel:det1\]) and (\[rel:det2\]), $$\begin{aligned}
&&
{\bf Q}_i(tq^N)+\sum_{s=1}^{N-1}(-1)^{s}
{\bf T}_{\Lambda_1+\cdots+
\Lambda_s}(tq^{-1}){\bf Q}_i(tq^{N-2s})+(-1)^{N}{\bf Q}_i(tq^{-N})=0,
\label{rel:BaxTQ1}\\
&&
\overline{\bf Q}_i(tq^{-N})+\sum_{s=1}^{N-1}(-1)^{s}
\overline{\bf T}_{\Lambda_1+\cdots+\Lambda_s}(tq)
\overline{\bf Q}_i(tq^{-N+2s})+(-1)^{N}
\overline{\bf Q}_i(tq^{N})=0,
\label{rel:BaxTQ2}\end{aligned}$$ with $i=1,2,\cdots,N$. This Baxter’s $T$-$Q$ relation, (\[rel:BaxTQ1\]) and (\[rel:BaxTQ2\]), coincides with those in [@BHK] upon $N=3$ specialization. Note that the specialization to $N=2$ does not yield the formulae in [@BLZ1; @BLZ2; @BLZ3], because the Dynkin-diagram for $N=2$ is different from those for $N \geqq 3$. We have to give separate definitions of the bosons, the $q$-oscillator and the screening currents for $N=2$, [@BLZ1; @BLZ2; @BLZ3]. This Baxter’s $T$-$Q$ relation (\[rel:BaxTQ1\]), (\[rel:BaxTQ2\]) coincides with those of [@KLWZ] for $N \geqq 3$. In [@KLWZ], I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin gave the conjecture that the standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. For simplest example they showed that the fusion rules for quantum transfer matrices coincide with the Hirota-Miwa’s bilinear difference equation [@H; @M] (the discrete KP). They derived higher-rank generalization of Baxter’s $T$-$Q$ relation by analysing the Hirota-Miwa’s bilinear difference equation (classical nonlinear integrable difference equation), too. In this paper, we derive the same Baxter’s $T$-$Q$ relation by analysing the quantum field theory of the KP (quantum integrable model). Hence this paper give a supporting argument of the conjecture on quantum and classical-discrete integrable models, by I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin [@KLWZ]. As the consequence of (\[rel:det3\]) and (\[rel:det4\]), we have the bilinear formulae of the $T$-operator (\[rel:qua1\]) and (\[rel:qua2\]). $$\begin{aligned}
(-1)^{\frac{(N-1)(N-2)}{2}}c_0
{\bf T}_{m\Lambda_1}(t)&=&\sum_{s=1}^N
(-1)^{s+1}c_s
{\bf Q}_s(tq^{2m+N-1})\overline{\bf Q}_s(tq^{-1}),
\label{rel:qua1}\\
(-1)^{\frac{(N-1)(N-2)}{2}}c_0
\overline{\bf T}_{m\Lambda_1}(t)&=&
\sum_{s=1}^N (-1)^{s+1}c_s
\overline{\bf Q}_s(tq^{-2m-N+1}){\bf Q}_s(tq),
\label{rel:qua2}\end{aligned}$$ and $$\begin{aligned}
(-1)^{\frac{(N-1)(N-2)}{2}}c_0
{\bf T}_{m(\Lambda_1+\cdots+\Lambda_{N-1})}(t)&=&
\sum_{s=1}^N (-1)^{N+s}c_s
\overline{\bf Q}_s(tq^{-2m-1})
\overline{\bf Q}_s(tq^{N-1}),
\label{rel:qua3}\\
(-1)^{\frac{(N-1)(N-2)}{2}}c_0
\overline{\bf T}_{m(\Lambda_1+\cdots+\Lambda_{N-1})}
(t)&=&\sum_{s=1}^N (-1)^{N+s}
c_s {\bf Q}_s(tq^{2m+1})
\overline{\bf Q}_s(tq^{-N+1}).
\label{rel:qua4}\end{aligned}$$ As a consequence of the determinant formulae (\[rel:det1\]) and (\[rel:det2\]), we have the Jacobi-Trudi formulae of the $T$-operator. For the Young-diagram $\mu=(\mu_1,\mu_1,\cdots,\mu_{N-1},0)$, we have $$\begin{aligned}
{\bf T}_\mu(t)=\left|\begin{array}{ccccc}
{\bf \tau}^{(\mu_1')}(t)&\cdots&
{\bf \tau}^{(\mu_1'+j-1)}(tq^{2(j-1)})&\cdots&
{\bf \tau}^{(\mu_1'+l(\mu')-1)}(tq^{2(l(\mu')-1)})\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
{\bf \tau}^{(\mu_i'-i+1)}(t)&\cdots&
{\bf \tau}^{(\mu_i'-i+j)}(tq^{2(j-1)})&\cdots&
{\bf \tau}^{(\mu_i'-i+l(\mu'))}(tq^{2(l(\mu')-1)})\\
\cdots&\cdots&\cdots&\cdots&\cdots\\
{\bf \tau}^{(\mu_{l(\mu')}'-l(\mu')+1)}(t)&\cdots&
{\bf \tau}^{(\mu_{l(\mu')}'-l(\mu')+j)}(tq^{2(j-1)})&\cdots&
{\bf \tau}^{(\mu_{l(\mu')}')}(tq^{2(l(\mu')-1)})
\end{array}\right|.\nonumber\\
\label{rel:JT}\end{aligned}$$ Here we have set $\mu'=(\mu_1',\mu_2',\cdots,\mu_N')$ the transpose Young-diagram of $\mu$, and $l(\mu')=\mu_1$. We have set $\tau^{(s)}(t)={\bf T}_{\Lambda_1+\cdots+\Lambda_s}(t)$. We have $\tau^{(0)}(t)=\tau^{(N)}(t)=1$. The above conjecturous functional relations of the $T$-$Q$ operators, (\[rel:det1\]), (\[rel:det2\]), (\[rel:det1’\]), (\[rel:det2’\]), (\[rel:det3\]), (\[rel:det4\]), (\[rel:BaxTQ1\]), (\[rel:BaxTQ2\]), (\[rel:qua1\]), (\[rel:qua2\]), (\[rel:qua3\]), (\[rel:qua4\]), (\[rel:JT\]), coincide with the previous work [@BHK] upon $N=3$ specialization.
Conclusion
----------
In this paper we present $q$-oscillator representation of the Borel subalgebra $U_q'(\widehat{sl_N})$, (\[def:q-osc\]). By using this $q$-oscillator representation, we give the free field realization of the Baxter’s $Q$-operator ${\bf Q}_j(t), \overline{\bf Q}_j(t)$ with $j=1,2,\cdots,N$, for the $W_N$-algebra, (\[def:A1\]), (\[def:A2\]), (\[def:Q\]). The commutativity of the $Q$-operator is direct consequence of the Yang-Baxter equation. We give conjecturous determinant formulae of the $T$-$Q$ operator for the $W_N$-algebra, (\[rel:det1\]), (\[rel:det2\]), (\[rel:det3\]), (\[rel:det4\]), which produce the higher-rank $W_N$-generalization of the Baxter’s $T$-$Q$ relation, (\[rel:BaxTQ1\]), (\[rel:BaxTQ2\]). We have checked these determinant formulae of the $T$-$Q$ operator, (\[rel:det1\]), (\[rel:det2\]), (\[rel:det3\]), (\[rel:det4\]) up to the order $O(t^2)$ in appendix. Because the scheme of funtional relations works well, we conclude that the number of the $Q$-operators for the $W_N$-algebra, is just $2N$, $(N \geqq 3)$. In this paper we didn’t give complete proof of the determinant formulae for the $W_N$-algebra. V.Bazhanov, A.Hibberd and S.Khoroshkin [@BHK] gave proof of the determinant formulae for the $W_3$-algebra. Their proof is based on the trace of the universal ${\cal L}$-operator over Verma module, and the Bernstein-Gel’fand-Gel’fand (BGG) resolution. Because we have already established conjecturous determinant formulae, higher-rank generalization of complete proof seems calculation problem. However it is not so easy.
\
\
[**Acknowledgements**]{}\
The author would like to thank Prof.V.Bazhanov and Prof.M.Jimbo for useful communications. The author would like to thank Institute of Advanced Studies, Australian National University for the hospitality during his visit to Canberra in March 2008. The author would like to thank Prof.P.Bouwknegt, Prof.A.Chervov, Prof.V.Gerdjikov, Prof.K.Hasegawa and Prof.V.Mangazeev for their interests in this work. This work is partly supported by Grant-in Aid for Young Scientist [**B**]{} (18740092) from JSPS.
Supporting Arguments
====================
In this appendix we give some supporting arguments on conjecturous formulae of the determinant formulae (\[rel:det1\]), (\[rel:det2\]), (\[rel:det1’\]), (\[rel:det2’\]), (\[rel:det3\]), (\[rel:det4\]). We check those determinant formulae up to the order $O(t^2)$. At first we prepare the Taylor expansion of ${\bf A}_j(t), \overline{\bf A}_j(t)$. Let us set $\pi_j=\overline{\pi}_j=
\pi_1^+\otimes \cdots \otimes \pi_{N-1}^+$. Taking the trace for the basis $\{|n_1,n_2,\cdots, n_{N-1}\rangle
=({\cal H}_1^*)^{n_1}({\cal H}_2^*)^{n_2}\cdots
({\cal H}_{N-1}^*)^{n_{N-1}}|0\rangle_+ \otimes \cdots \otimes |0\rangle_+\}_{n_1,n_2,
\cdots, n_{N-1} \in {\mathbb N}}$, we have $$\begin{aligned}
Z_j(t)={\rm Tr}_{\pi_j}\left(
\exp\left(-2\pi i \sqrt{\frac{r^*}{r}}
\sum_{k=1}^{N-1}(P_{\bar{\epsilon}_{j}}-P_{\bar{\epsilon}_{j+k}})
\otimes {\cal H}_k\right)
\right)=
\prod_{k=1
\atop{k \neq j}}^N
\left(1-\frac{z_k}{z_{j}}\right)^{-1},\nonumber\end{aligned}$$ with $j=1,2,\cdots, N$. As the same manner as the above, we have $$\begin{aligned}
\overline{Z}_j(t)
=
\prod_{k=1
\atop{k \neq j}}^N\left(1-\frac{z_{j}}{z_{k}}\right)^{-1}.
\nonumber\end{aligned}$$ Let us set $a_i, \overline{a}_i$ by $$\begin{aligned}
{\bf A}_i(t)=1+a_it+O(t^2),~~
\overline{\bf A}_i(t)=1+\overline{a}_i t+O(t^2).\nonumber\end{aligned}$$ Let us set $$\begin{aligned}
{\it J}^{(n)}_{k_1,k_2,\cdots,k_N}
&=&\int \cdots \int_{2\pi \geqq u_1 \geqq u_2 \geqq \cdots \geqq u_{N n}
\geqq 0}
V_{k_1}(u_1)V_{k_2}(u_2)\cdots V_{k_N}(u_N)\nonumber\\
&\times&
V_{k_1}(u_{N+1})V_{k_2}(u_{N+2})\cdots V_{k_N}(u_{2N})\cdots
\nonumber\\
&\times&
V_{k_1}(u_{N(n-1)+1})V_{k_2}(u_{N(n-1)+2})\cdots V_{k_{Nn}}(u_{N n})
du_1 du_2 \cdots du_{N n}.\nonumber\end{aligned}$$ Let us calculate coefficient of ${\cal J}_{1,2,\cdots,N}^{(1)}$ in $a_i$. We have $$\begin{aligned}
&&
{\rm Tr}_{\pi_i}\left(
o_{t,i}\left(
\exp\left(-2\pi i \sqrt{\frac{r^*}{r}}\sum_{k=1}^{N-1}P_{\omega_k}\otimes h_k \right)
e_{1} e_2 e_{3} \cdots e_{N}\right)\right)\times
{\cal J}_{1,2,\cdots,N}^{(1)}\nonumber\\
&=&t(q-q^{-1})^{N-2}q^{\frac{N-2}{2}}
\nonumber\\
&\times&\prod_{k=1}^{N-1}
{\rm Tr}_{\pi^+_k}
\left(\exp\left(-2\pi i \sqrt{\frac{r^*}{r}}
(P_{\bar{\epsilon}_{i}}-P_{\bar{\epsilon}_{i+k}})
\otimes {\cal H}_k\right) {\cal E}_k {\cal E}_k^*\right)
\times
{\cal J}_{1,2,\cdots,N}^{(1)}.\nonumber\end{aligned}$$ Taking the trace and dividing $Z_i(t)$, we have $$\begin{aligned}
a_{i}&=&
\frac{
q^{\frac{3}{2}N-2}z_i^{N-2}z_1
}{\displaystyle (q-q^{-1})
\prod_{k=1
\atop{k \neq i}}^N
(q^2z_i-z_k)}\times
{\cal J}_{1,2,\cdots,N}^{(1)}+\cdots,\nonumber\end{aligned}$$ with $i=1,2,\cdots,N$. As the same manner as the above, we have $$\begin{aligned}
\overline{a}_{i}&=&
(-1)^N
\frac{
q^{\frac{1}{2}N}z_i^{N-2}z_1
}{\displaystyle (q-q^{-1})
\prod_{k=1
\atop{k \neq i}}^N
(q^2z_k-z_i)}\times
{\cal J}_{1,2,\cdots,N}^{(1)}+\cdots,\nonumber\end{aligned}$$ with $i=1,2,\cdots,N$. Let us check the determinant relations between ${\bf Q}_i(t)$ and $\overline{\bf Q}_i(t)$, (\[rel:det3\]) and (\[rel:det4\]). We have $$\begin{aligned}
&&\left|
\begin{array}{cccc}
{\bf Q}_1(tq^{N-2})&{\bf Q}_1(tq^{N-4})&\cdots&
{\bf Q}_1(tq^{-N+2})\\
\cdots&\cdots&\cdots&\cdots\\
{\bf Q}_{i-1}(tq^{N-2})&
{\bf Q}_{i-1}(tq^{N-3})&\cdots&
{\bf Q}_{i-1}(tq^{-N+2})\\
{\bf Q}_{i+1}(tq^{N-2})&
{\bf Q}_{i+1}(tq^{N-3})&\cdots&
{\bf Q}_{i+1}(tq^{-N+2})\\
\cdots&\cdots&\cdots&\cdots
\\
{\bf Q}_N(tq^{N-2})&{\bf Q}_N(tq^{N-4})&\cdots&
{\bf Q}_N(tq^{-N+2})
\end{array}
\right|\nonumber\\
&=&
t^{-\sqrt{\frac{r}{r^*}}P_{\bar{\epsilon}_i}}
\prod_{1\leqq j<k \leqq N
\atop{j,k\neq i}}
\left(\sqrt{\frac{z_j}{z_k}}-
\sqrt{\frac{z_k}{z_j}}\right)
\left(
1+\left(
\sum_{j=1
\atop{j \neq i}}^N
a_j
\prod_{k=1\atop{k \neq i,j}}^N
\frac{(q^2z_j-z_k)}
{(z_j-z_k)}q^{-N+2}\right)t+O(t^2)\right).\nonumber\end{aligned}$$ Inserting the formulae of $a_i$ into RHS and using the following identity, $$\begin{aligned}
\sum_{j=1
\atop{j \neq i}}^N
\frac{z_j^{N-2}}{
\displaystyle
(z_jq^2-z_i)
\prod_{k=1\atop{k\neq i,j}}^N
(z_j-z_k)}=(-1)^N \frac{z_i^{N-2}}{
\displaystyle
\prod_{k=1
\atop{k \neq i}}^N
(z_kq^2-z_i)}.\nonumber\end{aligned}$$ we have $$\begin{aligned}
t^{-\sqrt{\frac{r}{r^*}}P_{\bar{\epsilon}_i}}
c_i
\left(
1+(-1)^N \frac{q^{\frac{1}{2}N}z_i^{N-2}z_1}{
\displaystyle
(q-q^{-1})
\prod_{k=1\atop{k \neq i}}^N
(z_kq^2-z_i)
}\times
{\cal J}_{1,2,\cdots,N}^{(1)}\times t+\cdots
\right),\nonumber\end{aligned}$$ which coincides with leading terms of $\overline{\bf Q}_i(t)$. As the same argument as the above, the coefficients of ${\cal J}_{k_1,k_2,\cdots,k_N}^{(1)}$ coincide with each other up to the order $O(t^2)$. Now we have checked the determinant formulae (\[rel:det3\]) and (\[rel:det4\]) up to the order $O(t^2)$. For the second we check the quantum Wronskian condition (\[rel:det1’\]) and (\[rel:det2’\]) up to the order $O(t^2)$. We have Taylor expansion of determinant of ${\bf Q}_j(t)$, $$\begin{aligned}
&&\left|
\begin{array}{cccc}
{\bf Q}_1(tq^{N-1})&{\bf Q}_1(tq^{N-3})&\cdots&{\bf Q}_1(tq^{-N+1})\\
{\bf Q}_2(tq^{N-1})&{\bf Q}_2(tq^{N-3})&\cdots&{\bf Q}_2(tq^{-N+1})\\
\cdots&\cdots&\cdots&\cdots
\\
{\bf Q}_N(tq^{N-1})&{\bf Q}_N(tq^{N-3})&\cdots&{\bf Q}_N(tq^{-N+1})
\end{array}
\right|\nonumber\\
&=&
\prod_{1\leqq j<k \leqq N}
\left(\sqrt{\frac{z_j}{z_k}}-
\sqrt{\frac{z_k}{z_j}}\right)
\left(
1+\left(
\sum_{i=1}^N
a_i \prod_{k=1
\atop{k \neq i}}^N
\frac{(q^2z_i-z_k)}{(z_i-z_k)}q^{-N+1}
\right)t+O(t^2)\right).\nonumber\end{aligned}$$ Insrting the explicit formulae of $a_i$ into RHS and using the following identity, $$\begin{aligned}
\sum_{i=1}^N
(-1)^{i+1}z_i^{N-2}
\prod_{1\leqq j<k \leqq N
\atop{j,k\neq i}}(z_j-z_k)=0,\nonumber\end{aligned}$$ we have $$\begin{aligned}
\prod_{1\leqq j<k \leqq N}
\left(\sqrt{\frac{z_j}{z_k}}-
\sqrt{\frac{z_k}{z_j}}\right)
\left(
1+O(t^2)\right).\nonumber\end{aligned}$$ Now we have checked the quantum Wronskian condition (\[rel:det1’\]) and (\[rel:det2’\]) up to the order $O(t^2)$. Next we consider the determinant formulae (\[rel:det1\]) and (\[rel:det2\]) for the special cases $\mu=\Lambda_1$ and $\mu=\Lambda_1+\cdots+\Lambda_{N-1}$. Because we have checked the formulae (\[rel:det3\]) and (\[rel:det4\]) up to the order $O(t^2)$, it is enough to show (\[rel:qua1\]), (\[rel:qua2\]), (\[rel:qua3\]) and (\[rel:qua4\]) in order to show (\[rel:det1\]) and (\[rel:det2\]) up to the order $O(t^2)$. We have $$\begin{aligned}
&&(-1)^{\frac{(N-1)(N-2)}{2}}
\sum_{s=1}^N (-1)^{s+1}c_s
{\bf Q}_s(tq^{N+1})\overline{\bf Q}_s(tq^{-1})\nonumber\\
&=&
\sum_{s=1}^N
(-1)^{s+1}
z_s^N
\prod_{1\leqq j<k\leqq N
\atop{j,k \neq s}}(z_j-z_k)
\left(1+t(
q^{N+1}a_s+q^{-1}\overline{a}_s)+O(t^2)\right).\nonumber\end{aligned}$$ Using the following relation, $$\begin{aligned}
\sum_{s=1}^N
(-1)^{s+1}
z_s^N
\prod_{1\leqq j<k\leqq N
\atop{j,k \neq s}}(z_j-z_k)=
\prod_{1\leqq j<k\leqq N}(z_j-z_k)
(z_1+z_2+\cdots+z_N),\nonumber\end{aligned}$$ we show that the first leading term becomes $c_0 \sum_{j=1}^Nz_j.$ Inserting the explicit formulae of $a_s,
\overline{a}_s$ and using the following relation, $$\begin{aligned}
\sum_{s=1}^N\frac{z_s^{2N-2}}{
\displaystyle
\prod_{k=1\atop{k\neq s}}^N (z_s-z_k)}
\left(\frac{q}{
\displaystyle
\prod_{k=1\atop{k \neq s}}^N(z_s-q^{2}z_k)}-
\frac{q^{-1}}{
\displaystyle
\prod_{k=1\atop{k \neq s}}^N(z_s-q^{-2}z_k)}
\right)=(q-q^{-1})\nonumber\end{aligned}$$ we have the second leading term, $$\begin{aligned}
t q^{\frac{Nn}{2}} \prod_{1\leqq j<k \leqq N}(z_k-z_j)
\left(
z_1 {\cal J}_{1,2,\cdots,N}^{
(1)}+z_2 {\cal J}_{2,3,\cdots,N,1}^{(1)}+\cdots \right).\nonumber\end{aligned}$$ Now we need explicit formulae of ${\bf T}_{\Lambda_1}(t)$. Let us fix a basis of the irreducible highest representation of $U_q(gl_N)$ with $\Lambda_1$ by $$\begin{aligned}
|\Lambda_1\rangle,
\pi^{(\Lambda_1)}(E_{\alpha_1})|\Lambda_1\rangle,
\pi^{(\Lambda_1)}(E_{\alpha_2}E_{\alpha_1})|\Lambda_1\rangle,
\cdots,
\pi^{(\Lambda_1)}(E_{\alpha_{N-1}}\cdots
E_{\alpha_1})|\Lambda_1\rangle,\nonumber\end{aligned}$$ The matrix representation of $\pi^{(\Lambda_1)}$ are written upon this basis by $$\begin{aligned}
&&\pi^{(\Lambda_1)}(E_{\alpha_i})=(
\delta_{j,i}
\delta_{k,i+1})_{1\leqq j,k \leqq N},~~(1\leqq i \leqq N-1),\nonumber\\
&&
\pi^{(\Lambda_1)}(F_{\alpha_i})=(\delta_{j,i+1}
\delta_{k,i})_{1\leqq j,k \leqq N},~~(1\leqq i \leqq N-1),
\nonumber\\
&&\pi^{(\Lambda_1)}(H_{i})=
(\delta_{j,i}\delta_{k,i})
_{1\leqq j,k \leqq N},~(1\leqq i \leqq N).\nonumber\end{aligned}$$ Using this matrix representation, we have $$\begin{aligned}
{\bf T}_{\Lambda_1}(t)&=&
\sum_{j=1}^N z_j
+\sum_{n=1}^\infty
t^n q^{\frac{N n}{2}}\sum_{j=1}^N z_{j}
{\it J}_{j,j+1,j+2,\cdots,j+N-1}^{(n)}.\nonumber\end{aligned}$$ Now we have checked the determinant formulae (\[rel:det1\]) for $\mu=\Lambda_1$ up to the order $O(t^2)$. As the same manner we checked the determinant formulae (\[rel:det1\]) for $\mu=\Lambda_1+\cdots+\Lambda_{N-1}$ and (\[rel:det2\]) for $\mu=\Lambda_1,
\Lambda_1+\cdots+\Lambda_{N-1}$, up to the order $O(t^2)$. For reader’s convenience we sumarize the explicit formulae of ${\bf T}_{\Lambda_1+\cdots+\Lambda_{N-1}}(t)$, $\overline{\bf T}_{\Lambda_1}(t)$ and $\overline{\bf T}_{\Lambda_1+\cdots+\Lambda_{N-1}}(t)$. The matrix representation of $\pi^{(\Lambda_1+\cdots+\Lambda_{N-1})}$ are written by $$\begin{aligned}
&&\pi^{(\Lambda_1+\cdots+\Lambda_{N-1})}(E_{\alpha_i})=(
\delta_{j,N-i}
\delta_{k,N-i+1})_{1\leqq j,k \leqq N},~~(1\leqq i \leqq N-1),\nonumber\\
&&
\pi^{(\Lambda_1+\cdots+\Lambda_{N-1})}(F_{\alpha_i})=(\delta_{j,N-i+1}
\delta_{k,N-i})_{1\leqq j,k \leqq N},~~(1\leqq i \leqq N-1),
\nonumber\\
&&\pi^{(\Lambda_1+\cdots+\Lambda_{N-1})}(H_{i}-H_{i+1})=
(\delta_{j,N-i}\delta_{k,N-i}-\delta_{j,N-i+1}\delta_{k,N-i+1})
_{1\leqq j,k \leqq N},~(1\leqq i \leqq N-1).\nonumber\end{aligned}$$ We have $$\begin{aligned}
{\bf T}_{\Lambda_1+\cdots+\Lambda_{N-1}}(t)&=&
\sum_{j=1}^N \frac{1}{z_j}
+\sum_{n=1}^\infty
t^n q^{(2-\frac{N}{2})n}
(-1)^{Nn}\sum_{j=1}^N \frac{1}{z_{j}}
{\it J}_{j+N-1,\cdots ,j+1,j}^{(n)},\nonumber\\
\overline{\bf T}_{\Lambda_1}(t)&=&
\sum_{j=1}^N z_{j}
+\sum_{n=0}^\infty
t^n q^{-\frac{nN}{2}}\sum_{j=1}^N z_{j}
{\it J}_{j,j+1,j+2,\cdots,j+N-1}^{(n)},\nonumber\\
\overline{\bf T}_{\Lambda_1+\cdots+\Lambda_{N-1}}(t)&=&
\sum_{j=1}^N \frac{1}{z_j}
+\sum_{n=1}^\infty t^n q^{(\frac{N}{2}-2)n}
\sum_{j=1}^N \frac{1}{z_{j}}
{\it J}_{j+N-1,\cdots ,j+1,j}^{(n)}.\nonumber\end{aligned}$$ Using these explicit formulae, we have $$\begin{aligned}
{\bf T}_{\Lambda_1}(q^{-\frac{N}{2}}t)&=&
\overline{\bf T}_{\Lambda_{1}}(q^{\frac{N}{2}}t),\nonumber\\
{\bf T}_{\Lambda_1+\cdots+\Lambda_{N-1}}(q^{\frac{N-4}{2}}t)&=&
\overline{\bf T}_{\Lambda_1+\cdots+\Lambda_{N-1}}(q^{\frac{4-N}{2}}t),\nonumber\\
{\bf T}_0(t)&=&\overline{\bf T}_0(t)=1.\nonumber
\nonumber\end{aligned}$$
[99]{} R.Baxter : Partition function of the eight vertex model, [*Ann.Phys.*]{}[**70**]{} 193-228, (1972). R.Baxter : Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain I: Some fundamental eigenvectors, [*Ann.Phys.*]{}[**76**]{} 1-24, (1973). R.Baxter : Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II: Equivalence to a generalized Ice-type lattice model, [*Ann.Phys.*]{}[**76**]{} 25-47, (1973). R.Baxter : Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain III: Eigenvectors of the transfer matrix and the Hamiltonian, [*Ann.Phys.*]{}[**76**]{} 48-71, (1973). R.Baxter : Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982. L.Takhtadzhan and L.Faddeev: The Quantum Method of the Inverse Problem and the Heisenberg XYZ model, [*Russian.Math.Surveys*]{} [**34**]{}:5, 11-68,(1979). K.Fabricius and B.McCoy: New Development in the eight vertex model, [*J.Stat.Phys.*]{}[**111**]{}, 323-337, (2003). K.Fabricius and B.McCoy: Functional equations and fusion matrices for the eight vertex model, [*Pub.Res.Inst.Math.Sci.*]{}[**40**]{}, 905-932, (2004). K.Fabricius and B.McCoy : An elliptic current operator for the eight vertex model, [*J.Phys.*]{} [**A39**]{}, 14869-14886, (2006). K.Fabricius : A new $Q$-matrix in the eight vertex model, [*J.Phys.*]{} [**A40**]{}, 4075-4086, (2007). V.Bazhanov and V.Mangazeev: Analytic theory of the eight-vertex model, [*Nucl.Phys*]{} [**B775**]{} \[FS\] 225-282, (2007). B.Feigin, T.Kojima, J.Shiraishi and H.Watanabe: The Integrals of Motion for the Deformed $W$-Algebra $W_{q,t}(\widehat{sl_N})$, [*Proceeding for Representation Theory 2006*]{}, Atami, Japan (2006). T.Kojima and J.Shiraishi: The Integrals of Motion for the Deformed $W$-Algebra $W_{q,t}(\widehat{sl_N})$ II: Proof of commutation relations, \[arXiv:0709.2305\], to appear in [*Commun.Math.Phys.*]{} E.Sklyanin : The Quantum Toda Chain, [*Lec.Notes.Phys.*]{}[**226**]{}, 196-223, (1985). E.Sklyanin : Functional Bethe Ansatz, Integrable and Superintegrable systems, 8-33, World Sci.Publ.Teaneck,N, 1990. E.Sklyanin : Separation of Variables - New trends. Quantum field theory, integrable models and beyond (Kyoto), [*Prog.Theoret.Phys.Suppl.*]{}[**118**]{}, 35-60, (1995). V.Kuznetsov V.Mangazeev and E.Sklyanin : $Q$-operator and factorized separation chain for Jack polynomials, [*Indag.Math.*]{}[**14**]{}, 451-482, (2003). V.Pasquier and M.Gaudin : The periodic Toda chain and matrix generalization of the Bessel function recursion relations, [*J.Phys.*]{}[**A25**]{}, 5243-5252, (1992). S.Derkachov: Baxter’s $Q$-operator for the homogeneous XXX spin chain, [*J.Phys.*]{} [**A32**]{}, 5299-5316, (1999). S.Derkachov, G.Korchemsky and A.Mansahov : Noncompact Heisenberg spin magnets from high-energy QCD I: [*Nucl.Phys.*]{}[**B617**]{}, 375-440, (2001). Baxter $Q$-operator and Separation of Variables, S.Derkachov, G.Korchemsky and A.Mansahov : Baxter $Q$-operator and Separation of Variables for the open $SL(2,{\mathbb R})$ spin chain, [*J.High.Energy.Phys.*]{}[**2003**]{}, no.10, paper 053, 31pp (electronic), (2003). S.Derkachov, D.Karakhanyan and R.Kirschner : Baxter $Q$-operators of the XXZ chain and $R$-matrix factorization, [*Nucl.Phys.*]{}[**B738**]{}, 368-390, (2006). S.Derkachov, A.Manashov : $R$-matrix and Baxter operators for the noncompact quantum $sl(N,{\mathbb C})$ invariant spin chain, [*SIGMA Symmetry Integrability Geom.Mthods Appl.*]{}, paper 084, 20pp (electronic), (2006). A.Belisty, S. Derkachov, G.Korchemsky and A.Manashov: The Baxter $Q$-operator for the graded $SL(2|1)$ spin chain, [*J.Stat.Mech.Theory Exp.*]{}[**2007**]{}, paper1005, 63pp (electronic), (2007). C.Korff: A $Q$-OperatorIdentity for the Correlation Functions of the infinite XXZ spin-chain, [*J.Phys.*]{}[**A39**]{}, 3203-3219, (2006). C.Korff: A $Q$-operator for the quantum transfer matrix, [*J.Phys.*]{}[**A40**]{}, 3749-3774, (2007). A.Bytsko and J.Teschner : Quantization of models with non-compact quantum group symmetry, Modular XXZ magnet and lattice sinhGordon model, [*J.Phys.*]{}[**A39**]{}, 12927-12981, (2006). V.Bazhanov, S.Lukyanov and Al.Zamolodchikov: Integrable structure of conformal field theory : Quantum KdV Theory and Thermodynamic Bethe Ansatz, [*Commun.Math.Phys.*]{}[**177**]{} 381-398, (1996). V.Bazhanov, S.Lukyanov and Al.Zamolodchikov: Integrable structure of conformal field theory II: Q-operator and DDV equation, [*Commun.Math.Phys.*]{}[**190**]{} 247-278, (1997). V.Bazhanov, S.Lukyanov and Al.Zamolodchikov: Integrable structure of conformal field theory III: The Yang-Baxter Relation, [*Commun.Math.Phys.*]{}[**200**]{} 297-324, (1999). V.Bazhanov, S.Lukyanov and Al.Zamolodchikov: Spectral determinant for Schödinger equation and $Q$-operator of Conformal Field Theory, [*J.Stat.Phys.*]{} [**102**]{}, 567-576, (2001). M.Rossi and R.Weston : A generalized $Q$-operator for $U_q(\widehat{sl_2})$-vertex model, [*J.Phys.*]{}[**A35**]{}, 10015-10032,(2002). P.Dorey and R.Tateo: Anharmonic oscillators, the thermodynamic Bethe Ansatz and nonlinear integral equation, [*J.Phys.*]{}[**A32**]{} L419-l425, (1999). V.Bazhanov, A.Hibberd and S.Khoroshkin: Integrable structure of $W_3$ Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory, [*Nucl.Phys.*]{}[**B622**]{} 475-547,(2002). P.Kulish and Z.Zeitlin : Superconformal field theory and SUSY $N=1$ KdV hierarchy II: The $Q$-operator, [*Nucl.Phys.*]{} [**B709**]{}, 578-591, (2005). A.Antonov and B.Feigin : Quantum Group Representation and Baxter Equation, [*Phys. Lett.*]{}[**B392**]{}, 115-122, (1997). Y.Asai, M.Jimbo, T.Miwa,A and Ya.Pugai, Bosonization of vertex operators for $A_{n-1}^{(1)}$ face model, [*J.Phys.*]{}[**A29**]{}, 6595-6616, (1996). I.Krichever, O.Lipan, P.Wiegmann and A.Zabrodin, Quantum integrable models and discrete classical Hirota equations, [*Commun.Math.Phys.*]{}[**188**]{} 267-304, (1997). V.Bazhanov and N.Reshetikhin, Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, [*J.Phys.*]{}[**A23**]{} 1477-1492, (1990). A.Kuniba, T.Nakanishi and J.Suzuki : Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory, [*Int.J.Mod.Phys.*]{}[**A9**]{} 5215-5266, (1994). H.Boos, M.Jimbo, T.Miwa, F.Smirnov and Y.Takeyama, Hidden Grassmann Structure in the XXZ model, [*Commun.Math.Phys.*]{}[**272**]{}, 263-281, (2007). H.Boos, M.Jimbo, T.Miwa, F.Smirnov and Y.Takeyama, Hidden Grassmann Structure in the XXZ model II: Creation Operators, \[arXiv:0801.1176\]. A.Chervov and G.Falqui : Manin matrices and Talalaev’s formula, \[arXiv:0711.2236\] V.Fateev and S.Lukyanov : The Models of Two-Dimensional Conformal Quantum Field Theory with ${\mathbb Z}_n$ Symmetry, [*Int.J.Mod.Phys.*]{}[**A3**]{}, 507-520, (1988). R.Hirota : Discrete analogue of a generalized Toda equation, [*J.Phys.Soc.Japan*]{} [**50**]{}, 3785-3791, (1981). T. Miwa : On Hirota’s difference equations, [*Proc. Japan Acad.*]{} [**58**]{}, 9-12, (1982).
|
---
abstract: 'We present a basic building block of a quantum network consisting of a quantum dot coupled to a source cavity, which in turn is coupled to a target cavity via a waveguide. The single photon emission from the high-$Q/V$ source cavity is characterized by a twelve-fold spontaneous emission (SE) rate enhancement that results in a SE coupling efficiency $\beta\sim0.98$ into the source cavity mode. Single photons are efficiently transferred into the target cavity through the waveguide, with a source/target field intensity ratio of 0.12 (up to 0.49 observed in other structures without coupled quantum dots). This system shows great promise as a building block of future on-chip quantum information processing systems.'
author:
- Dirk Englund
- Andrei Faraon
- Bingyang Zhang
- Yoshihisa Yamamoto
- Jelena Vučković
bibliography:
- '0603\_CcSPS\_Arxiv\_V3.bib'
title: Generation and transfer of single photons on a photonic crystal chip
---
Recent years have witnessed dramatic practical and theoretical advancements towards creating the basic components of quantum information processing (QIP) devices. One essential element is a source of single indistinguishable photons, which is required in quantum teleportation[@Bouwmeester97nature], linear-optics quantum computation[@KLM01], and several schemes for quantum cryptography [@NC00]. Sources have been demonstrated from a variety of systems[@SVG2005Europhysics] including semiconductor quantum dots (QDs)[@Michler03book], whose efficiency and indistinguishability can be dramatically improved by placing it inside a microcavity[@Santori02]. A second major component is a quantum channel for efficiently transferring information between spatially separated nodes of a quantum network[@CZKM1997PRL]. This network would combine the ease of storing and manipulating quantum information in quantum dots[@PhysRevLett.83.4204], atoms or ions[@PhysRevLett.75.4714; @Wineland2004Nature], with the advantages of transferring information between nodes via photons, using coherent interfaces[@FattalThesis2005; @Sham2005PRL; @Monroe2004Nature]. Here we demonstrate a basic building block of such a quantum network by the generation and transfer of single photons on a photonic crystal (PC) chip. A cavity-coupled QD single photon source is connected through a $25$ channel to an otherwise identical target cavity so that different cavities may be interrogated and manipulated independently (Fig. \[fig:Fig1\]). This system provides a source of single photons with a high degree of indistinguishability (mean wavepacket overlap of $\sim 67$%), 12-fold spontaneous emission (SE) rate enhancement, SE coupling factor $\beta \sim 0.98$ into the cavity mode, and high-efficiency coupling into a waveguide. These photons are transferred into the target cavity with a target/source field intensity ratio of 0.12, showing the system’s potential as a fundamental component of a scalable quantum network for building on-chip quantum information processing devices.
The structure consists of two linear 3-hole defect cavities[@Noda2003Nature], butt-coupled and connected via a $25$-long closed portion of a waveguide (Fig. \[fig:Fig2\](a)). It was designed by component-wise Finite Difference Time Domain (FDTD) simulations in three dimensions. The waveguide design shown here supports four modes; we picked one of these modes, designated as $B_{oe}$ in Fig. \[fig:Fig2\](b), to transfer light between the end-cavities. This mode offers a wide, relatively flat spectral region of guided modes for coupling and can furthermore be confined in the high-$Q$ end-cavities. Thanks to their near-minimum mode volume $V_{mode} \equiv (\int_V \varepsilon(\vec{r}) |\vec{E}(\vec{r})|^{2} d^{3}\vec{r})/\max(\varepsilon(\vec{r}) |\vec{E}(\vec{r})|^{2})\approx 0.74 (\lambda/n)^{3}$, these end-cavities allow a large SE rate enhancement $\propto Q/V_{mode}$.
The cavity and waveguide field decay rates can be expressed as a sum of vertical, in-plane, and material loss, respectively: $\kappa=\kappa_{\perp}+\kappa_{\|}+\gamma$. Removed from the waveguide, the ‘bare’ outer cavities radiate predominantly in the vertical direction at rate $\kappa_{\perp}$, as in-plane losses can be suppressed with enough PC confinement layers. Introducing a waveguide coupled to the cavity creates additional loss $\kappa_{\|}$. In designing the cavity/waveguide system, we therefore optimized for the ratio of the coupling rate into the waveguide versus other losses, $\kappa_{\|}/(\kappa_{\perp}+\gamma)$, while retaining a high cavity $Q$ value for enhancing the SE rate of QDs coupled to the cavity.
In a waveguide of finite extent, the continuum of modes in the waveguide band breaks up into discrete resonances. For photon transfer, one of these must be coupled to the outer cavities. Assuming spectral matching and negligible material losses[@Englund05_OptExp], the field amplitudes in the source and target cavities (S,T) and waveguide (W) evolve according to: $$\begin{aligned}
\label{eq:rates}
\nonumber \dot{c_s}(t) &=& -i \kappa_{\|} c_w(t) - \kappa_{\perp} c_s(t) + p(t) \\
\dot{c_t}(t) &=& -i \kappa_{\|} c_w(t) - \kappa_{\perp} c_t(t) \\
\dot{c_w}(t) &=& -i\kappa_{\|} c_s(t) -i\kappa_{\|} c_t(t) - \kappa_W c_w(t) \nonumber\end{aligned}$$ Here we assume equal coupling rates for the outer cavities, based on their near-identical SEM images and on their $Q$ values, which fall within a linewidth of each other in the great majority of structures. The constant $\kappa_W$ denotes the waveguide loss rate (other than loss into the end-cavities), and $p(t)$ represents a dipole driving the source cavity. It will suffice to analyze this system in steady-state since excitation of the modes, which happens on the order of the exciton lifetime $\tau\sim 100$, occurs slowly compared to relaxation time of the photonic network, which is of order $\tau_{cav}=\omega/Q \sim 1$ for the cavity and waveguide resonances involved. Then the amplitude ratio between the S and T fields is easily solved to be $|c_s/c_t|=1+\kappa_{\perp}\kappa_W/\kappa_{\|}^{2}$.
In the present application, where high photon transfer probability and reasonably high output coupling is desired, we optimized for a design with a two-hole separation between cavity and waveguide, giving $Q_{\perp} \equiv 2 \omega/\kappa_{\perp}=23000, Q_{\|}\equiv 2 \omega/\kappa_{\|}=5200$, and $\kappa_{\|}/\kappa_{\perp}=4.4$. In this design, the cavity resonances were targeted to a linear region of the waveguide dispersion, slightly above the lower waveguide cut-off frequency (see Fig.\[fig:Fig2\]). In this way, we ensured coupling that is tolerant to slight fabrication inaccuracies.
The structures were fabricated on a 160 GaAs membrane, grown by molecular-beam epitaxy with a central layer of self-assembled InAs QDs whose photoluminesce peaks at 932 with an inhomogeneous linewidth of $\sim 60$ and a density of 200. The structures were fabricated by electron beam lithography and reactive ion etching, followed by a wet chemical etch to remove a sacrificial layer underneath the PC membrane[@Englund05PRL]. Fabrication inaccuracies decreased $Q$ values to about $1000-5000$. We will now focus on a particular system that showed high coupling among cavities, while simultaneously evincing large QD coupling in the source cavity. Measurements were done with the sample at 5K in a continuous-flow cryostat and probed with the confocal microscope setup connected to the various instruments shown in Fig. \[fig:Fig3\]. A movable aperture is located the image plane of the microscope so that the pump and observation regions can be adjusted independently. The structures were characterized by measuring the combinations of pump/probe regions (waveguide ‘WG’, source cavity ‘S’, or target cavity ‘T’). Here the broad-band photoluminescence (PL) of high-intensity, above-band pumped QDs was used as an internal illumination source. Fig. \[fig:Fig4\](a) shows good spectral match between direct measurements of the source/target cavities (plots ‘SS’ and ‘TT’), together with the coupled emission (plots ‘ST’ and ‘TS’). A comparison of the emission intensities from the S and T cavities gives the transfer efficiency $|c_t/c_s|^{2}=0.12$. To understand how this transmission occurs through the terminated waveguide, we illuminate the waveguide resonances by pumping it near the center. Photoluminescence that is resonant inside the waveguide, but off-resonant from the cavities, is scattered primarily at the waveguide/cavities interfaces. On the other hand, PL that is resonant with the cavities is dropped into them and can be spatially separated with the pinhole. This drop-filtering is shown in \[fig:Fig4\](b), where the bottom plot (‘WG-all’) shows all modes resonant in the coupled system, while the top two plots ‘WG-WG’ and ‘WG-T’ show collection from only the waveguide terminus and the cavity, respectively. Fitting the measurements of panels (a,b) to the frequency-domain model in Eq.\[eq:rates\] gives the coupling coefficients $\kappa_{\perp}=455$, $\kappa_{W}=322$, $\kappa_{\|}=283$. Other instances of a slightly modified cavity/waveguide design, in which the waveguide and cavity were separated by only a single hole, yielded photon transfer ratios as large as $|c_t/c_s|^{2}=0.49$ (see supplemental material), though we did not produce this system in large enough numbers to find high cavity/QD coupling. The cavities/waveguide system strongly isolates transmission to the cavity linewidth, as seen from the transmission spectrum (‘ST’) (Fig. \[fig:Fig4\] inset) when S is pumped above the GaAs bandgap.
We now consider the problem of coupling a QD to the source cavity, which requires a high degree of spatial and spectral matching. Though it primarily relies on chance, the spectral coupling can be fine-tuned by shifting the QD transitions with changing sample temperature. In Fig. \[fig:Fig4\](d), we show a single-excition transition coupled to cavity S at 897. The transition is driven resonantly through a higher-order excited QD state with a 878 pump from a Ti-Saph laser. The SE rate enhancement is measured from the modified emitter lifetime, which a direct streak camera measurement puts at 116 (Fig. \[fig:Fig5\](b)). Compared to the average lifetime of 1.4 for QDs in the bulk semiconductor of this wafer, this corresponds to a Purcell enhancement of $F=12$. The SE coupling factor into the cavity mode is then $\beta=F/(F+F_{PC})\sim0.98$, where $F_{PC}\sim 0.3$ reflects the SE rate suppression into other modes due to the bandgap of the surrounding PC[@Englund05PRL]. We characterized the exciton emission by measurements of the second-order coherence and indistinguishability of consecutive photons. The second-order coherence $g^{(2)}(t')=\braket{I(t) I(t+t')}/\braket{I(t)}^{2}$ is measured with a Hanbury-Brown and Twiss interferometer, as described earlier [@Englund05PRL]. When the QD in cavity S is pumped resonantly, then photons observed from S shows clear antibunching (Fig. \[fig:Fig5\](a)), with $g^{2}(0)=0.35\pm0.01$. This value is larger than what we observe for resonantly pumped cavity-detuned QDs, where typically $g^{(2)}(0)<0.05$, similar to previous reports[@Santori02]. The main contributor to the larger $g^{(2)}(0)$ for the coupled QD is enhanced background emission from nearby transitions and the wetting layer emission tail, which decays at $\sim 100$ time scales (see Fig. \[fig:Fig5\](b)) and is not completely filtered by our grating setup. The background emission is rather large in this study because of the high QD density of the sample (e.g., four times larger than that in the experiment by *Santori et al.*[@Santori02]). Because of the shortened lifetime of the cavity-coupled QD exciton, the coherence time of emitted photons becomes dominated by radiative effects and results in high photon indistinguishability[@Kiraz04]. We measured the indistinguishability using the Hong-Ou-Mandel (HOM) type setup sketched in Fig. \[fig:Fig3\], similar to a recent experiment on PC-cavity coupled QDs[@Abram2005APL]. Following the analysis of *Santori et al.*[@Santori02], the data (inset Fig.\[fig:Fig5\](b)) indicate a mean wavefunction overlap of $I=0.67\pm0.18$, where we adjusted for the imperfect visibility (88%) of our setup and subtracted dark counts in the calculation. Even with higher SE rate enhancement, we expect that $I \lesssim 0.80$ for resonantly excited QDs[@Vuckovic2005PRE] because of the finite relaxation time, measured here at $ 23$ by the streak camera.
We will now consider the transfer of single photons to the target cavity T. The single photon transfer is described by Eqs. \[eq:rates\], where cavity S is now pumped by the QD single exciton with state $e(t) \ket{e}+ g(t)\ket{g}$: $$\begin{aligned}
p(t)&=&-i g_0 e(t) \\ \nonumber
\dot{e}(t)&=&-{\frac{{\Gamma}}{{2}}} e(t)-i g_0 g(t)\end{aligned}$$ Here, $g_0$ is the QD-field coupling strength and $\Gamma$ the QD spontaneous emission rate. In the present situation, where coupling rates greatly exceed the exciton decay rate, the previous steady-state results apply. Thus, the signal from cavity T mirrors the SE of the single exciton coupled to cavity S. Experimentally, we verified photon transfer from S to T by spectral measurements as in Fig. \[fig:Fig4\](d): the exciton line is observed from T only if S is pumped. It is not visible if the waveguide or cavity T itself are pumped, indicating that this line originates from the QD coupled to cavity S and that a fraction of the emission is transferred to T. This emission has the same polarization and temperature-tuned wavelength dependence as emission from S. Photon autocorrelation measurements on the signal from T indicate the antibunching characteristic of a single emitter when S is pumped (Fig. \[fig:Fig5\](c)). The signal-to-noise ratio is rather low because autocorrelation count rates are $(|c_t/c_s|^{2})^{2} \sim 0.014$ times lower than for collection from S. Nevertheless, the observed antibunching does appear higher, in large part because the background emission from cavity S is additionally filtered in the transfer to T, as shown in Fig.\[fig:Fig4\](b). Indeed, this filtering through the waveguide/cavity system suffices to bypass the spectrometer in the HBT setup (a 10-nm bandpass filter was used to eliminate room lights). Slightly more background is permitted, but the count rate is about three times higher while antibunching, $g^{2}(0)=0.50\pm0.11$, is still clearly evident (Fig. \[fig:Fig5\](d)). The largest contribution to $g^{2}(0)$ comes from imperfectly filtered photoluminesce near the QD distribution peak seen in Fig. \[fig:Fig4\](c). This transmission appears to occur through the top of the dielectric band near $k=0.7 \pi/a$, and could easily be eliminated in the future by increasing the waveguide frequency with slightly larger bounding holes around the PC cavity. This on-chip filtering will be essential in future QIP applications and should also find uses in optical communications as a set of cascaded drop filters. Table 1\[tab:Tab1\] summarizes the relevant parameters of this system.
\[tab:Tab1\]
$g_0$ (GHz) $\kappa$ (GHz) $\kappa_{\|}/\kappa_{\perp}$ $\Gamma $(GHz) $I$
--------------- ------------- ---------------- ------------------------------ --------------------- ------ --
system 1 50 800 0.62 10 0.67
best observed 210 1.9
theoretical 230 46 4.4 $\sim180^{\dagger}$
: Paramaters of the structure described here (‘system 1’), best observed values from other structures on the same sample (if different), and theoretical predictions of the design. $^{\dagger}$ *Limited by the onset of strong coupling.*
In conclusion, we have demonstrated a basic building block of a quantum network consisting of a quantum dot with large coupling to a high-$Q/V$ cavity, which in turn is coupled to a target cavity via a waveguide. This system functions as an efficient on-demand source of single photons with mean wavepacket overlap of $\sim 67$%, SE coupling efficiency $\beta\sim0.98$ into the cavity mode, and high out-coupling efficiency into the waveguide. These single photons from cavity S are channeled to the target cavity, as confirmed by localized spectroscopic measurements. We measured a high photon transfer with a field intensity ratio of $0.12$ for this system. In other structures we measured field ratios up to 49%, though without coupled QDs. These efficiencies greatly exceed what is possible in off-chip transfer and demonstrate the great potential of this system as a building block of future on-chip quantum information processing systems.
Financial support was provided by the MURI Center for photonic quantum information systems (ARO/DTO Program DAAD19-03-1-0199) and NSF grants ECS-0424080 and ECS-0421483. Dirk Englund was also supported by the NDSEG fellowship. Dr.Zhang is supported by JST. We thank David Fattal and Edo Waks for helpful discussions.
|
---
abstract: 'A displacement operator $\hat d_\zeta$ is introduced, verifying commutation relations $[\hat d_\zeta, a_f^\dagger]=[\hat d_\zeta, a_f]=\zeta(f)\hat d_\zeta$ with field creation and annihilation operators that verify $[a_f,a_g]=0$, $[a_f,a_g^\dagger]=(g,f)$, as usual. $f$ and $g$ are test functions, $\zeta$ is a Poincaré invariant real-valued function on the test function space, and $(g,f)$ is a Poincaré invariant Hermitian inner product. The $\star$-algebra generated by all these operators, and a state defined on it, nontrivially extends the $\star$-algebra of creation and annihilation operators and its Fock space representation. If the usual requirement for linearity is weakened, as suggested in , we obtain a deformation of the free quantum field.'
address: 'Physics Department, Yale University, CT 06520.'
author:
- Peter Morgan
title: Displacement deformed quantum fields
---
Introduction
============
In an earlier paper, I introduced a weakening of the axioms of quantum field theory that allows a nonlinear inner product structure [@MorganWLQF]. I refer to that paper for notation, motivation, and an introduction to the approach that is further pursued here. There, I mentioned that I had investigated deformations of the Heisenberg algebra of the Arik-Coons type [@Quesne], but had found no way to apply deformations of a comparable type to quantum fields. Here, I briefly describe the failure, and move on to introduce a displacement operator $\hat d_\zeta$, verifying $[\hat d_\zeta, a_f^\dagger]=[\hat d_\zeta, a_f]=\zeta(f)\hat d_\zeta$, where $\zeta$ is an arbitrary real-valued scalar function on the test function space (taken to be a Schwartz space [@Haag §II.1]), which will allow us to construct an extension of Fock space, generated by the action of displacement operators on a vacuum state as well as by the action of creation operators $a_f^\dagger$. Note that the “displacement” is not a space-time displacement, but will shortly be seen to “displace” creation and annihilation operators in the sense of adding a scalar. What follows will show some of the uses to which such operators can be put.
A comparable (but Hermitian) number operator $\hat n_\zeta$ would verify the very different commutation relation $[\hat n_\zeta, a_f^\dagger]=\zeta(f)a_f^\dagger$. Number operators are important for a uniform presentation of algebras of the Arik-Coons type[@Quesne], but we cannot in general construct an associative algebra if we use the operator $\hat n_\zeta$ to extend the free quantum field algebra; it is straightforward to verify, for example, that for the undeformed commutation relation $[a_f,a_g^\dagger]=(g,f)$, $\hat n_\zeta a_f a_g^\dagger$ becomes either $(a_g^\dagger a_f+(g,f))(\hat n_\zeta-\zeta(f)+\zeta(g))$ or $a_g^\dagger a_f(\hat n_\zeta-\zeta(f)+\zeta(g))+(g,f)\hat n_\zeta$, depending on the order in which the commutation relations are applied, which is incompatible with associativity unless $\zeta$ is a constant function on the test function space. We will here take the constant function number operator to be relatively uninteresting, particularly because we cannot generate an associative algebra using both a number operator $\hat n_1$ (with the constant function $1$) and a displacement operator $\hat d_\zeta$; $\hat d_\zeta\hat n_1 a_f^\dagger$, for example, becomes different values depending on the order in which commutation relations are applied. Equally, every attempt I have made at deforming the commutation relations $[a_f,a_g^\dagger]=(g,f)$ and $[a_f,a_g]=0$ using number operators or displacement operators have failed to be associative, with $a_f(a_h a_g^\dagger)\not=a_h(a_f a_g^\dagger)$.
We will work with a $\star$-algebra $\mathcal{A}_1$ that is generated by creation and annihilation operators that verify $[a_f,a_g^\dagger]=(g,f)$ and $[a_f,a_g]=0$, together with a single displacement operator pair $\hat d_\zeta$ and $\hat d_\zeta^\dagger$. We will take $\hat d_\zeta^\dagger$ to be equivalent to $\hat d_{-\zeta}$; $\hat d_\zeta^k$ to be equivalent to $\hat d_{k\zeta}$; and $\hat d_{0\zeta}$ to be equivalent to $1$. The commutation relations above and the state we will define in a moment are consistent with these equivalences. $\hat d_{0\zeta}$ is central in $\mathcal{A}_1$, for example. In general, we will take $\hat d_{m\zeta}\hat d_{n\zeta}$ to be equivalent to $\hat d_{(m+n)\zeta}$.
$\mathcal{A}_1$ has the familiar subalgebra $\mathcal{A}_0$ that is generated by the creation and annihilation operators alone. A basis for $\mathcal{A}_1$ is $a_{g_1}^\dagger a_{g_2}^\dagger...a_{g_m}^\dagger \hat d_{k\zeta} a_{f_1} a_{f_2}...a_{f_n}$, $k\in{{\mathrm{Z\hspace{-.4em}Z}}}$, for some set of test functions $\{f_i\}$. We construct a linear state $\varphi_0$ on this basis as $$\begin{aligned}
&&\varphi_0(1)=1,\\
&&\varphi_0(a_{g_1}^\dagger a_{g_2}^\dagger...a_{g_m}^\dagger
\hat d_{k\zeta} a_{f_1} a_{f_2}...a_{f_n})=0
\quad\mathrm{if}\ m>0\ \mathrm{or}\ n>0\ \mathrm{or}\ k\not=0.\end{aligned}$$ If $k$ is always zero, this is exactly the vacuum state for the conventional free quantum field. To establish that $\varphi_0$ is a state on $\mathcal{A}_1$, we have to show that $\varphi_0(\hat A^\dagger \hat A)\ge 0$ for every element of the algebra. A general element of the algebra can be written as $\hat A=\sum_k\sum_r\lambda_{kr}\hat X^\dagger_{kr}\hat d_{k\zeta} \hat Y_{kr}$, where $\hat X_{kr}$ and $\hat Y_{kr}$ are products of annihilation operators, so that $$\begin{aligned}
\varphi_0(\hat A^\dagger \hat A)&=&\varphi_0(
(\sum_j\sum_s\lambda_{js}^*\hat Y^\dagger_{js}\hat d_{-j\zeta} \hat X_{js})
(\sum_k\sum_r\lambda_{kr}\hat X^\dagger_{kr}\hat d_{k\zeta} \hat Y_{kr}))\cr
&=&\sum_k\varphi_0(
(\sum_s\lambda_{js}^*\hat Y^\dagger_{js}{\hat X}'_{js})
(\sum_r\lambda_{kr}\hat{X'}^\dagger_{kr}\hat Y_{kr}))\cr
&=&\sum_k\varphi_0(\hat A_k^\dagger\hat A_k)\ge 0,\end{aligned}$$ because only terms for which $j=k$ contribute, and $\hat A_k=\sum_r\lambda_{kr}\hat{X'}^\dagger_{kr}\hat Y_{kr}$ is an operator in the free quantum field algebra $\mathcal{A}_0$ for each $k$. The critical observation is that ${\hat X}'_{kr}=\hat d_{-k\zeta}\hat X_{kr}\hat d_{k\zeta}$ is a sum of products of annihilation operators only.
Given the state $\varphi_0$, we can use the GNS construction to construct a Hilbert space $\mathcal{H}_0$ (see, for example, [@Haag §III.2]), then we can use the $C^\star$-algebra of bounded operators $\mathcal{B}(\mathcal{H}_0)$ that act on $\mathcal{H}_0$ as an algebra of observables, but this or a similar construction is not strictly needed for *Physics*. From the point of view established in [@MorganWLQF], we can be content to use a finite number of creation operators and annihilation operators to generate a $\star$-algebra of operators. This is not enough to support a continuous representation of the Poincaré group, but the formalism is Poincaré invariant, adequate (if we take *enough* generators) to construct complex enough models to be as empirically adequate as a continuum limit, and is much simpler, more constructive, and more appropriate for general use than Type $\mathrm{III}_1$ von Neumann algebras. This paper broadly follows the general practice in physics of fairly freely employing unbounded creation and annihilation operators. Completion of a $\star$-algebra in a norm to give at least a Banach $\star$-algebra structure, which would allow us to construct an action on the GNS Hilbert space directly, is a useful nicety for mathematics, but it is not essential for constructing physical models.
For future reference, I list some of the simplest identities that are entailed by the commutation relation of the displacement operator with the creation and annihilation operators (using a Baker-Campbell-Hausdorff (BCH) formula for the exponentials): $$\begin{aligned}
&&[\hat d_\zeta^k,a_f^\dagger]=[\hat d_\zeta^k,a_f]=k\zeta(f)\hat d_\zeta^k,\\
\label{DAnnihilation}
&&\hat d_\zeta^k a_f^\dagger=(a_f^\dagger+k\zeta(f)) \hat d_\zeta^k,\qquad
\hat d_\zeta^k e^{i\lambda a_f^\dagger}=e^{i\lambda(a_f^\dagger+k\zeta(f))} \hat d_\zeta^k,\\
\label{DCreation}
&&\hat d_\zeta^k a_f=(a_f+k\zeta(f)) \hat d_\zeta^k,\qquad
\hat d_\zeta^k e^{i\lambda a_f}=e^{i\lambda(a_f+k\zeta(f))} \hat d_\zeta^k,\\
&&e^{\alpha\hat d_\zeta-\alpha^*\hat d_\zeta^\dagger\,}a_f=
\left[a_f+\zeta(f)(\alpha\hat d_\zeta+\alpha^*\hat d_\zeta^\dagger)\right]
e^{\alpha\hat d_\zeta-\alpha^*\hat d_\zeta^\dagger}.\end{aligned}$$ From these it should begin to be clear why I have called $\hat d_\zeta$ a “displacement” operator. Equations (\[DAnnihilation\]) and (\[DCreation\]) make apparent the useful practical consequence that it is sufficient to sum the powers of displacement operators in a term to be sure whether the term contributes to $\varphi_0(\hat A)$ — if the sum of powers is zero — because displacement operators are not modified if they are moved to left or right in the term.
We can introduce as many displacement operators as needed, all mutually commuting, $[\hat d_{\zeta_1},\hat d_{\zeta_2}]=0$, without changing any essentials of the above, but probably not as far as a continuum of such operators without significant extra care. It is most straightforward to introduce linear dependency between products of the displacement operators immediately, $\hat d_{\zeta_1}\hat d_{\zeta_2}=\hat d_{\zeta_1+\zeta_2}$, which is consistent with the commutation relations, although we could also proceed by considering equivalence relations later in the development. The only other comment that seems necessary is that the action of the state $\varphi_0$ on a basis constructed as above is zero unless there are no displacement operators present, so that $$\begin{aligned}
\varphi_0(1)=1,\quad\varphi_0(a_{g_1}^\dagger a_{g_2}^\dagger...a_{g_m}^\dagger
\hat d_{\zeta_1}^{k_1}\hat d_{\zeta_2}^{k_2}...\hat d_{\zeta_l}^{k_l}
a_{f_1} a_{f_2}...a_{f_n})&=&0,\cr
&&\hspace{-10em}\quad\mathrm{if}\ m>0\ \mathrm{or}\ n>0\ \mathrm{or\ any}\ k_i\not=0.\end{aligned}$$ $\hat d_{\zeta_1}^{k_1}\hat d_{\zeta_2}^{k_2}...\hat d_{\zeta_l}^{k_l}$ should be taken to be equal to $\hat d_{k_1\zeta_1+k_2\zeta_2+\cdots+k_l\zeta_l}$.
The basic algebra is adequately defined above, the rest of this paper develops some of the consequences for modelling correlations. Three ways in which the displacement operators can be used are described below. In particular, probability densities are calculated for various models, as far as possible. All three ways can be combined freely with the two ways of constructing nonlinear quantum fields that are described in [@MorganWLQF], so the comment made there must be emphasized, that the approach discussed here should at this point be considered essentially empirical, because there is an embarrassing number of models. The reason for pursuing this approach nonetheless — from a high theoretical point of view the lack of constraints on models might be seen as a serious failing — is that it brings much better mathematical control to discussions of renormalization, and might lead to new and hopefully useful conceptualizations and phenomenological models of physical processes. Even if the nonlinear quantum field theoretic models discussed here and in [@MorganWLQF] do not turn out to be empirically useful, they nonetheless give an approach that can be compared in detail with standard renormalization approaches, and an understanding of precisely why these nonlinear models and others like them cannot be made to work should give some insight into both approaches.
Displaced vacuum states {#VacuumDisplacement}
=======================
The way to use displacement operators that is discussed in this section in effect constructs representations of the subalgebra $\mathcal{A}_0$, because the commutation relation $[\hat\phi_f,\hat\phi_g]=(g,f)-(f,g)$ is unchanged. However, we will be able to construct vacuum states in which the 1-measurement probability density in the Poincaré invariant vacuum state can be any probability density in convolution with the conventional Gaussian probability density, which seems useful regardless, particularly if used in conjunction with the methods of [@MorganWLQF]. The vacuum probability density may depend on any set of nonlinear Poincaré invariants of the test function that describes a 1-measurement.
Let $\hat\phi_f=a_f+a_f^\dagger$ be the quantum field, for which the conventional vacuum state generates a characteristic function $\chi_0(\lambda|f)$ of the 1-measurement probability density; using a BCH formula, we obtain $$\begin{aligned}
\chi_0(\lambda|f)&=&\varphi_0(e^{i\lambda\hat\phi_f})=
e^{-{{\frac{1}{2}}}\lambda^2(f,f)}\varphi_0(e^{i\lambda a_f^\dagger}e^{i\lambda a_f})\\
&=&e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\end{aligned}$$ so that the probability density associated with single measurements in the vacuum state is the Gaussian $\rho_0(x|f):=\exp{(-x^2/2(f,f))/\sqrt{2\pi(f,f)}}$.
Consider first the elementary alternative vacuum state, $\varphi_d(\hat A)=\varphi_0(\hat d_\zeta \hat A \hat d_\zeta^\dagger)$. For a vacuum state, $\zeta$ should be Poincaré invariant; this is a physical requirement on vacuum states to which the mathematics here is largely indifferent. Using this modified vacuum state, we can generate a characteristic function for single measurements, $$\begin{aligned}
\chi_d(\lambda|f)&=&\varphi_0(\hat d_\zeta e^{i\lambda\hat\phi_f} \hat d_\zeta^\dagger)=
e^{-{{\frac{1}{2}}}\lambda^2(f,f)}\varphi_0(\hat d_\zeta e^{i\lambda a_f^\dagger}e^{i\lambda a_f} \hat d_\zeta^\dagger)\\
&=&e^{-{{\frac{1}{2}}}\lambda^2(f,f)+2i\lambda\zeta(f)},\end{aligned}$$ so that the probability density associated with single measurements in the modified vacuum state is still Gaussian, but “displaced”, $$\rho_d(x|f):=\frac{1}{\sqrt{2\pi(f,f)}}\exp{\left(-\frac{(x-2\zeta(f))^2}{2(f,f)}\right)}.$$ As $\zeta(f)$ varies with some Poincaré invariant scale of $f$, the expected displacement of the Gaussian varies accordingly. $\zeta(f)$ might be large for “small” $f$, small at intermediate scale, and large again for “large” $f$; any function of multiple Poincaré invariant scales of the test functions may be used.
Introducing a linear combination $\hat\Xi=\sum_k \xi_k\hat d_\zeta^k/\sqrt{N}$ of higher powers of $\hat d_\zeta$, with normalization constant $N=\sum_k\left|\xi_k\right|^2$, we can construct another modified vacuum state, $\varphi_c(\hat A)=\varphi_0(\hat\Xi \hat A \hat\Xi^\dagger)$, which generates a characteristic function $$\begin{aligned}
\chi_c(\lambda|f)&=&\varphi_0(\hat\Xi e^{i\lambda\hat\phi_f} \Xi^\dagger)=
e^{-{{\frac{1}{2}}}\lambda^2(f,f)}\varphi_0(\hat\Xi e^{i\lambda a_f^\dagger}e^{i\lambda a_f} \hat\Xi^\dagger)\\
&=&\frac{1}{N}\sum_k \left|\xi_k\right|^2 e^{-{{\frac{1}{2}}}\lambda^2(f,f)+2ik\lambda\zeta(f)},\end{aligned}$$ so that we obtain a probability density $$\rho_c(x|f)=\frac{1}{N}\sum_k \frac{\left|\xi_k\right|^2}{\sqrt{2\pi(f,f)}}
\exp{\left(-\frac{(x-2k\zeta(f))^2}{2(f,f)}\right)}.$$ If we are prepared to introduce a continuum of displacement operators, this probability density can be any probability density in convolution with the conventional Gaussian probability density. A finite number of displacement operators will generally be as empirically adequate as a continuum of displacement operators.
Finally, we can explicitly generate the $n$-measurement probability density in the state $\varphi_C(\hat A)=\varphi_0(\hat\Xi' \hat A \hat{\Xi'}^\dagger)$, where $\hat\Xi'=\sum_m \xi'_m\hat d_{\zeta_m}/\sqrt{N'}$, with normalization constant $N'=\sum_m\left|\xi'_m\right|^2$. The characteristic function is $$\begin{aligned}
\chi_C(\lambda_1,\lambda_2,...,\lambda_n|f_1,f_2,...,f_n)&=&
\varphi_0(\hat\Xi' e^{i\sum_j\lambda_j\hat\phi_{f_j}} {\Xi'}^\dagger)\\
&=&\frac{1}{N'}\sum_m \left|\xi'_m\right|^2
e^{-{{\frac{1}{2}}}\underline{\lambda}^T F\underline{\lambda}+2i\sum_j\lambda_j\zeta_m(f_j)},\end{aligned}$$ where $F$ is the gram matrix $(f_i,f_j)$ and $\underline{\lambda}$ is a vector of the variables $\lambda_i$. $\chi_C(\lambda_1,\lambda_2,...,\lambda_n|f_1,f_2,...,f_n)$ generates the probability density $$\rho_C(x_1,x_2,...,x_n|f_1,f_2,...,f_n)=
\frac{1}{N'}\sum_m \frac{\left|\xi'_m\right|^2}{\sqrt{2\pi \mathrm{det}(F)}}
e^{-{{\frac{1}{2}}}\underline{x}(m)^T F^{-1}\underline{x}(m) },$$ where the set of vectors $\underline{x}(m)$ is given by $x(m)_j=x_j-2\zeta_m(f_j)$. With a suitable choice of $\zeta_m$ and $|\xi'_m|^2$, we can make the probability density vary with multiple Poincaré invariant scales of the individual measurements. Note, however, that in the approach of this paper only the gram matrix $F$ describes the relationships *between* the measurements described by the test functions $f_i$, and all such relationships are pairwise.
Displacements of the field observable-I {#FieldDeformationI}
=======================================
This and the following section introduce deformations of the field instead of deformations of the ground state. As above, the quantum field discussed in this section still satisfies the commutation relation $[\hat\phi_f,\hat\phi_g]=(g,f)-(f,g)$, so the states we can construct again effectively generate many representations of the free field algebra of observables (the *next* section modifies the commutation relations satisfied by the observable field). If we think of ourselves as constructing empirically effective models for physical situations, it is worth considering different models for the different intuitions they present, while of course also presenting, as clearly as possible, isomorphisms between models, or – less restrictively – empirical equivalences between models.
The simplest deformation discussed in this section is $$\hat\phi_f=i(a_f-a_f^\dagger)+\alpha(f)\hat d_\zeta+\alpha^*(f)\hat d_\zeta^\dagger,$$ This deformed field satisfies microcausality because $\hat d_\zeta$ commutes with $i(a_f-a_f^\dagger)$[^1]. Note that in this section and in the next we take $a_f+a_f^\dagger$ not to be an observable of the theory, because $[(a_f+a_f^\dagger),i(a_g-a_g^\dagger)]\not=0$ when $f$ and $g$ have space-like separated supports.
We can straightforwardly calculate the vacuum state 1-measurement characteristic function for $\hat\phi_f$, $$\begin{aligned}
\chi_J(\lambda|f)&=&\varphi_0(e^{i\lambda\hat\phi_f})=
e^{-{{\frac{1}{2}}}\lambda^2(f,f)}\varphi_0(e^{\lambda a_f^\dagger}e^{-\lambda a_f}
e^{i\lambda(\alpha(f)\hat d_\zeta+\alpha^*(f)\hat d_\zeta^\dagger)})\cr
\rule[-3ex]{0pt}{7ex}
&=&e^{-{{\frac{1}{2}}}\lambda^2(f,f)}\sum_{j=0}^\infty \frac{(i\lambda|\alpha(f)|)^{2j}}{(2j)!}\frac{(2j)!}{j!^2}
\varphi_0(e^{\lambda a_f^\dagger}e^{-\lambda a_f})\cr
&=&e^{-{{\frac{1}{2}}}\lambda^2(f,f)}J_0(2\lambda|\alpha(f)|),\end{aligned}$$ where the Bessel function emerges because the only contributions to the result are those for which $\hat d_\zeta$ and $\hat d_\zeta^\dagger$ cancel, which gives the contribution $\frac{(2j)!}{j!^2}$. This results in a probability density that is the convolution of the conventional Gaussian and the probability density $\frac{1}{\sqrt{|2\alpha(f)|^2-x^2}}$ (when $|x|<|2\alpha(f)|$, otherwise $0$). The probability density we have just calculated is independent of $\zeta$, because $\hat d_\zeta$ commutes with $i(a_f-a_f^\dagger)$, but $\zeta$ will turn up in expressions for non-vacuum state probability densities. The scales of $(f,f)$ and $|\alpha(f)|$ determine the “shape” of the convolution. The convolution is displayed in figure \[PhiDeformationOne\] for $(f,f)=1$ and $|\alpha(f)|=0$, $\frac{1}{3}$, $1$, and $3$.
![The probability densities that result from the deformation\
$\hat\phi_f=i(a_f-a_f^\dagger)+\alpha(f)\hat d_\zeta+\alpha^*(f)\hat d_\zeta^\dagger$, with $(f,f)=1$ and $|\alpha(f)|=0$ (blue,\
highest function at zero), $\frac{1}{3}$ (red, second highest), $1$ (green, third\
highest), $3$ (cyan, lowest function at zero) \[colour on the web\].[]{data-label="PhiDeformationOne"}](PhiDeformationOne-resized.eps){width="20em" height="20em"}
We can also compute characteristic functions for higher powers such as $\hat \phi_f=i(a_f-a_f^\dagger)+\alpha(f)(\hat d_\zeta+\hat d_\zeta^\dagger)^k$, $$\begin{aligned}
k=1&\longrightarrow&{}_0F_1(;1;-(\lambda\alpha(f))^2)e^{-{{\frac{1}{2}}}\lambda^2(f,f)}
=J_0(2\lambda\alpha(f))e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
k=3&\longrightarrow&{}_2F_3({{\scriptstyle\frac{1}{6}}}, {{\scriptstyle\frac{5}{6}}};
{{\scriptstyle\frac{1}{3}}}, {{\scriptstyle\frac{2}{3}}}, 1;-16(\lambda\alpha(f))^2)
e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
k=5&\longrightarrow&{}_4F_5({{\scriptstyle\frac{1}{10}}}, {{\scriptstyle\frac{3}{10}}}, {{\scriptstyle\frac{7}{10}}}, {{\scriptstyle\frac{9}{10}}};
{{\scriptstyle\frac{1}{5}}}, {{\scriptstyle\frac{2}{5}}}, {{\scriptstyle\frac{3}{5}}}, {{\scriptstyle\frac{4}{5}}}, 1;-256(\lambda\alpha(f))^2)
e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
&\mathit{etc.},&\cr
k=0&\longrightarrow&{}_0F_0(;;2i\lambda\alpha(f))e^{-{{\frac{1}{2}}}\lambda^2(f,f)}
=e^{2i\lambda\alpha(f)}e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
k=2&\longrightarrow&{}_1F_1({{\scriptstyle\frac{1}{2}}};1;4i\lambda\alpha(f))e^{-{{\frac{1}{2}}}\lambda^2(f,f)}
=J_0(2\lambda|\alpha(f)|)e^{2i\lambda\alpha(f)}e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
k=4&\longrightarrow&{}_2F_2({{\scriptstyle\frac{1}{4}}},{{\scriptstyle\frac{3}{4}}};{{\scriptstyle\frac{1}{2}}},1;16i\lambda\alpha(f))e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
k=6&\longrightarrow&{}_3F_3({{\scriptstyle\frac{1}{6}}},{{\scriptstyle\frac{3}{6}}},{{\scriptstyle\frac{5}{6}}};
{{\scriptstyle\frac{1}{3}}},{{\scriptstyle\frac{2}{3}}},1;64i\lambda\alpha(f))e^{-{{\frac{1}{2}}}\lambda^2(f,f)},\\
&\mathit{etc.}& \end{aligned}$$ The $k=0$ entry is trivially tractable, indeed trivial; otherwise only the $k=2$ entry is immediately tractable, being just a trivially displaced version of the $k=1$ entry we have just discussed, because $(d_\zeta+d_\zeta^\dagger)^2=(d_{2\zeta}+d_{2\zeta}^\dagger)+2$. The combinatorics for arbitrary Hermitian functions of $\hat d_\zeta$ and $\hat d_\zeta^\dagger$ added to $i(a_f-a_f^\dagger)$, potentially using multiple Poincaré invariant displacement functions $\zeta_i$, can be as complicated as we care to consider.
Further possibilities that must be considered, because $\hat d_\zeta$ cannot generally be taken to be linear in $\zeta$, are fields such as $i(a_f-a_f^\dagger)+\alpha(f)(\hat d_{\beta(f)\zeta}+\hat d_{\beta(f)\zeta}^\dagger)$, which are distinct from the other fields considered in this section even though the vacuum state 1-measurement probability densities are independent of $\beta(f)\zeta$. If we add two displacement function components, as in $i(a_f-a_f^\dagger)+\alpha_1(f)(\hat d_{\beta_1(f)\zeta}+\hat d_{\beta_1(f)\zeta}^\dagger)
+\alpha_2(f)(\hat d_{\beta_2(f)\zeta}+\hat d_{\beta_2(f)\zeta}^\dagger)$ there is a complex modulation of the vacuum state 1-measurement probability density as the proportion of $\beta_1(f)$ to $\beta_2(f)$ changes.
Displacements of the field observable-II {#FieldDeformationII}
========================================
The first deformation of $\hat\phi_f$ that we will discuss in this section is $$\hat\phi_f=i(a_f-a_f^\dagger)(\hat d_\zeta+\hat d_\zeta^\dagger).$$ As in the previous section, this is Hermitian and satisfies microcausality, but the algebra of observables generated by the observable field is finally different, $$[\hat\phi_f,\hat\phi_g]=[(g,f)-(f,g)](\hat d_\zeta+\hat d_\zeta^\dagger)^2,$$ even though the algebra satisfied by the creation and annihilation operators is unchanged. The change in the algebra of observables gives some cause to think that physics associated with this type of construction may be significantly different. $(\hat d_\zeta+\hat d_\zeta^\dagger)^2$ is a central element in the algebra generated by $\hat\phi_f$.
The characteristic function of the vacuum state 1-measurement probability density is $$\begin{aligned}
\chi_P(\lambda|f)&=&\varphi_0(e^{i\lambda\hat\phi_f})\cr
&=&\varphi_0\left(\sum_{j=0}^\infty \frac{(i\lambda)^j i^j (a_f-a_f^\dagger)^j
(\hat d_\zeta+\hat d_\zeta^\dagger)^j}{j!}\right)\cr
\rule[-4ex]{0pt}{9ex}
&=&\varphi_0\left(\sum_{j=0}^\infty \frac{\lambda^{2j}(a_f-a_f^\dagger)^{2j}}{(2j)!}
\frac{(2j)!}{j!^2}\right)\cr
&=&\sum_{j=0}^\infty \frac{(-\lambda^2(f,f))^j}{(2j)!}\frac{(2j)!}{2^j j!}\frac{(2j)!}{j!^2}\cr
&=&{}_1F_1({{\scriptstyle\frac{1}{2}}};1;-2\lambda^2(f,f))=I_0(\lambda^2(f,f))e^{-\lambda^2(f,f)},\end{aligned}$$ where $\varphi_0((a_f-a_f^\dagger)^{2j})=(-(f,f))^j\frac{(2j)!}{2^j j!}$ is a useful identity for the conventional vacuum state. $\chi_P(\lambda|f)$ can be inverse Fourier transformed, using [@GR **7.663**.2 or **7.663**.6], to obtain $$\label{Keqn}
\rho_P(x|f)=\frac{1}{\sqrt{8\pi^3(f,f)}}\exp{\left(-\frac{x^2}{16(f,f)}\right)}K_0\left(\frac{x^2}{16(f,f)}\right).$$ This has variance $2(f,f)$, in contrast to the variance $(f,f)$ for the quantum field $i(a_f-a_f^\dagger)$. $\rho_P(x|f)$ is displayed with variance $2(f,f)=2$ together with the Gaussian for $(f,f)=1$ in figure \[PhiDeformationTwo\].
![The probability density that results from the deformation\
$\hat\phi_f=i(a_f-a_f^\dagger)(\hat d_\zeta+\hat d_\zeta^\dagger)$, with $(f,f)=1$, variance 2 (in red), compared with\
the conventional Gaussian, with $(f,f)$, variance 1 (in blue), and the probability\
density that results from the deformation $\hat\phi_f=i(a_f-a_f^\dagger)(\hat d_\zeta+\hat d_\zeta^\dagger)^2$, with\
$(f,f)=1$, variance 6 (dashed, in red)\[colour on the web\].[]{data-label="PhiDeformationTwo"}](PhiDeformationTwo-resized.eps){width="20em" height="20em"}
The vacuum state probability density $\rho_P(x|f)$ is again independent of $\zeta$; it is infinite at zero, but it is also integrable enough over the real line for all finite moments to exist, which of course we computed explicitly in order to compute $\chi_P(\lambda|f)$.
The probability density $\rho_P(x|f)$ is significantly concentrated both near zero and near $\pm\infty$, relative to the conventional Gaussian probability density. If we compare with a Gaussian that has the same variance, there is a 10 times greater probability of observing a value beyond about 3.66 standard deviations, a 100 times greater probability of observing a value beyond about 4.84 standard deviations, and a 1000 times greater probability of observing a value beyond about 5.76 standard deviations. I suppose $\rho_P(x|f)$ will give a fairly distinctive signature in physics, which future papers will hopefully be able to make evident, and it should be clear fairly quickly whether it can be used to model events in nature.
The characteristic function of the vacuum state $n$-measurement probability density is $$\chi_P(\lambda_1,\lambda_2,...,\lambda_n|f_1,f_2,...,f_n)=
\varphi_0(e^{i\sum_j\lambda_j\hat\phi_{f_j}})=
{}_1F_1({{\scriptstyle\frac{1}{2}}};1;-2\underline{\lambda}^T F\underline{\lambda}),$$ where, as in section \[VacuumDisplacement\], $F$ is the gram matrix $(f_i,f_j)$ and $\underline{\lambda}$ is a vector of the variables $\lambda_i$. For $n=2$, we can inverse Fourier transform this radially symmetric function[^2] using [@GR **7.663**.5], to obtain $$\label{Weqn}
\rho_P(x_1,x_2|f_1,f_2)=
\frac{\exp{\left(-\frac{\XX}{8}\right)}}
{\sqrt{8\pi^3(\XX)\mathrm{det}(F)}},$$ For all $n$, we can confirm, using [@GR **7.672**.2] that the Fourier transform of $$\rho_P(x_1,x_2,...,x_n|f_1,f_2,...,f_n)=
\frac{\exp{\left(-\frac{\XX}{16}\right)}
W_{\frac{n}{4}-\frac{1}{4},\frac{n}{4}-\frac{1}{4}}\left(\frac{\XX}{8}\right)}
{2^{\frac{3n}{4}-\frac{3}{4}}(\XX)^{\frac{n}{4}+\frac{1}{4}}\sqrt{\pi^{n+1}\mathrm{det}(F)}}$$ is ${}_1F_1({{\scriptstyle\frac{1}{2}}};1;-2\underline{\lambda}^T F\underline{\lambda})$, where $W_{a,b}(z)$ is Whittaker’s confluent hypergeometric function. Although these mathematical derivations of probability densities can be derived, and give a distinct insight, the moments, which are essentially what are physically measurable, can be determined more easily from the characteristic functions, or directly from the action of a state on an observable.
We can also compute characteristic functions for higher powers of displacement operators, $\hat\phi_f=i(a_f-a_f^\dagger)(\hat d_\zeta+\hat d_\zeta^\dagger)^k$, $$\begin{aligned}
k=1&\longrightarrow&{}_1F_1({{\scriptstyle\frac{1}{2}}};1;-2\lambda^2(f,f))
=I_0(\lambda^2(f,f))e^{-\lambda^2(f,f)},\\
k=2&\longrightarrow&{}_2F_2({{\scriptstyle\frac{1}{4}}}, {{\scriptstyle\frac{3}{4}}};
{{\scriptstyle\frac{1}{2}}}, 1;-8\lambda^2(f,f)),\\
k=3&\longrightarrow&{}_3F_3({{\scriptstyle\frac{1}{6}}}, {{\scriptstyle\frac{3}{6}}}, {{\scriptstyle\frac{5}{6}}};
{{\scriptstyle\frac{1}{3}}}, {{\scriptstyle\frac{2}{3}}}, 1;-32\lambda^2(f,f)),\\
k=4&\longrightarrow&{}_4F_4({{\scriptstyle\frac{1}{8}}}, {{\scriptstyle\frac{3}{8}}}, {{\scriptstyle\frac{5}{8}}}, {{\scriptstyle\frac{7}{8}}};
{{\scriptstyle\frac{1}{4}}}, {{\scriptstyle\frac{2}{4}}}, {{\scriptstyle\frac{3}{4}}}, 1;-128\lambda^2(f,f)),\\
&\mathit{etc.,}& \end{aligned}$$ which in general have Meijer’s $G$-functions as inverse Fourier transforms [@GR **7.542**.5]. For $k=2$, again using [@GR **7.672**.2], with different substitutions, we can derive the probability density $$\rho_{P2}(x|f)=\frac{1}{\sqrt{64\pi^3(f,f)}}\exp{\left(-\frac{x^2}{64(f,f)}\right)}
K_{\frac{1}{4}}\left(\frac{x^2}{64(f,f)}\right),$$ This has variance $6(f,f)$; it is plotted for $(f,f)=1$ in Figure \[PhiDeformationTwo\]. In general we can multiply $i(a_f-a_f^\dagger)$ by any self-adjoint polynomial in $\hat d_{\beta(f)\zeta}$ and $\hat d_{\beta(f)\zeta}^\dagger$. It will be interesting to discover what range of probability densities this will allow us to construct.
Discussion
==========
This mathematics is essentially quite clear and simple, but it is also rather rich and nontrivial, and there are lots of concrete models. It will be apparent that I do not have proper control of the full range of possibilities. From philosophical points of view that seek a uniquely preferred model and that find the tight constraints of renormalization on acceptable physical models congenial, it will be seen as problematic that there is a plethora of models, but a loosening of constraints accords well with our experience of wide diversity in the natural world, and is no more than a return to the almost unconstrained diversity of classical particle and field models.
It is so far rather unclear how to understand the mathematics as physics, but any interpretation will follow a common (but not universal) quantum field theoretical assumption that we measure probabilities and correlation functions of scalar observables that are indexed by test functions. There are existing ways of discussing condensed matter physics that are fairly amenable to this style of interpretation, but it is likely that we will have to abandon *some* of our existing ways of talking about particles to accommodate this mathematics.
It is also reiterated here, following [@MorganWLQF], that the positive spectrum condition on the energy, which has been so much part of the quantum field theoretical landscape, should be deprecated, because energy (and as well energy density) is unobservable, infinite, and nonlocal. If we think of the random field that is the classical equivalent of a given quantum field, taking $[a_f,a_g^\dagger]=(g,f)+(f,g)$ so that the commutator is real and $[\hat\phi_f,\hat\phi_g]=0$ for all test functions, it is clear that we are discussing an essentially fractal structure, for which differentiation and energy density at a point are undefined. From a proper mathematical perspective, we should consider only finite local observables. We have accepted renormalization formalisms that manage infinities only in lack of a finite alternative, a basis for which this paper and its precursor provide.
The method of section \[FieldDeformationII\] is perhaps more significant mathematically than the methods of sections \[VacuumDisplacement\] and \[FieldDeformationI\], insofar as the quantum field observables of section \[FieldDeformationII\] satisfy modified commutation relations, in common with the methods for constructing nonlinear quantum fields that are presented in [@MorganWLQF]. However, quantum theory somewhat exaggerates the importance of commutation relations between quantum mechanically ideal measurement devices — the trivial commutation relations of classically ideal measurement devices can give a description of experiments that is equally empirically adequate[@MorganVaxjo; @MorganBellRF], and ideal measurement devices between the quantum and the classical can also be used as points of reference[@MorganM].
Physics emphasizes a commitment to observed statistics, which present essentially uncontroversial lists of numbers, but it is far more difficult to describe what we believe we have measured than the statistics and the lists of numbers themselves. It might be said, for example, that “we have measured the momentum of a particle”, and cite a list of times and places where devices triggered, ignoring the delicate questions of (1) whether there is any such thing as “a particle”, (2) whether a particle can be said to have any well-defined properties at all, and (3) whether particles have “momentum” in particular. It makes sense to describe a measurement in such a way, because it forms a significant part of a coordinatization of the measurement that is good enough for the experiment and its results to be reproduced, but an alternative conceptualization can have a radical effect on our understanding.
[9]{}
[^1]: Another possibility, $\hat\phi'_f=a_f+a_f^\dagger+\zeta(f)(\alpha\hat d_\zeta+\alpha^*\hat d_\zeta^\dagger)$, also satisfies microcausality, but is almost trivially seen to be unitarily equivalent to $a_f+a_f^\dagger$, $$e^{{{\frac{1}{2}}}(\alpha\hat d_\zeta-\alpha^*\hat d_\zeta^\dagger)}(a_f+a_f^\dagger)
e^{-{{\frac{1}{2}}}(\alpha\hat d_\zeta-\alpha^*\hat d_\zeta^\dagger)}=
a_f+a_f^\dagger+\zeta(f)(\alpha\hat d_\zeta+\alpha^*\hat d_\zeta^\dagger).$$ This establishes a close enough relationship to the previous section that a longer presentation of this case will not be given here.
[^2]: Recall that the $n$-dimensional inverse Fourier transform of a radially symmetric function $\tilde f(\rho)$ is given by $$\frac{1}{(2\pi)^{\frac{n}{2}}r^{\frac{n}{2}-1}}\int_0^\infty \tilde f(\rho)
\rho^{\frac{n}{2}}J_{\frac{n}{2}-1}(r\rho){{\mathrm{d}}}\rho.$$
|
---
abstract: 'We discuss our targeted search for hypervelocity stars (HVSs), stars traveling with velocities so extreme that dynamical ejection from a massive black hole is their only suggested origin. Our survey, now half complete, has successfully identified a total of four probable HVSs plus a number of other unusual objects. Here we report the most recently discovered two HVSs: SDSS J110557.45+093439.5 and possibly SDSS J113312.12+010824, traveling with Galactic rest-frame velocities at least $+508\pm12$ and $+418\pm10$ km s$^{-1}$, respectively. The other late B-type objects in our survey are consistent with a population of post-main sequence stars or blue stragglers in the Galactic halo, with mean metallicity \[Fe/H\]$_{W_k}=-1.3$ and velocity dispersion $108\pm5$ km s$^{-1}$. Interestingly, the velocity distribution shows a tail of objects with large positive velocities that may be a mix of low-velocity HVSs and high-velocity runaway stars. Our survey also includes a number of DA white dwarfs with unusually red colors, possibly extremely low mass objects. Two of our objects are B supergiants in the Leo A dwarf, providing the first spectroscopic evidence for star formation in this dwarf galaxy within the last $\sim$30 Myr.'
author:
- 'Warren R. Brown, Margaret J. Geller, Scott J. Kenyon, and Michael J. Kurtz'
title: ' Hypervelocity Stars. I. The Spectroscopic Survey'
---
INTRODUCTION
============
Hypervelocity stars (HVSs) travel with velocities so extreme that dynamical ejection from a massive black hole (MBH) is their only suggested origin. First predicted by @hills88, HVSs traveling $\sim$1,000 km s$^{-1}$ are a natural consequence of a MBH in a dense stellar environment like that in the Galactic center. HVSs differ from runaway stars because 1) HVSs are unbound and 2) the classical supernova ejection [@blaauw61] and dynamical ejection [@poveda67] mechanisms that explain runaway stars cannot produce ejection velocities larger than 200 - 300 km s$^{-1}$ [@leonard91; @leonard93; @portegies00; @gualandris04; @dray05]. Depending on the actual velocity distributions of HVSs and runaway stars, some HVSs ejected by the central MBH may overlap with runaway stars in radial velocity.
Following the original prediction of HVSs, @hills91 provided a comprehensive analysis of orbital parameters needed to produce HVSs, and @yu03 expanded the @hills88 analysis to include the case of a binary black hole and to predict HVS production rates. In 2005, Brown and collaborators reported the first discovery of a HVS: a $g'=19.8$ B9 star, $\sim$110 kpc distant in the Galactic halo, traveling with a Galactic rest-frame velocity of at least $+709\pm12$ km s$^{-1}$ (heliocentric radial velocity +853 km s$^{-1}$). Photometric follow-up revealed that the object is a slowly pulsating B main sequence star [@fuentes06]. Only interaction with a MBH can plausibly accelerate a 3 $M_{\sun}$ main sequence B star to such an extreme velocity.
The discovery of the first HVS inspired a wealth of theoretical and observational work. Because HVSs originate from a close encounter with a MBH, HVSs can be used as important tools for understanding the nature and environs of MBHs [@gualandris05; @levin05; @ginsburg06; @holley06; @demarque06]. The trajectories of HVSs also provide unique probes of the shape and orientation of the Galaxy’s dark matter halo [@gnedin05]. Recent discoveries of new HVSs [@edelmann05; @hirsch05; @brown06] are starting to allow observers to place suggestive limits on the stellar mass function of HVSs, the origin of massive stars in the Galactic Center, and the history of stellar interactions with the MBH. Clearly, a larger sample of HVSs will be a rich source for further progress on these issues.
Here we discuss our targeted survey for HVSs and the unusual objects we find in it. To discover HVSs, we have undertaken a radial velocity survey of faint B-type stars, stars with lifetimes consistent with travel times from the Galactic center but which are not a normally expected stellar halo population. This strategy is successful: approximately 1-in-50 of our candidate B stars are HVSs. The first two HVS discoveries from our survey are presented in @brown06. Here we present two further HVS discoveries – one certain HVS and one possible HVS. In addition to HVSs, our survey has uncovered many unusual objects with late B-type colors: post-main sequence stars, young B supergiant stars, DA, DB, and DZ white dwarfs, and one extreme low-metallicity starburst galaxy.
Our paper is organized as follows. In §2 we discuss the target selection and spectroscopic identifications of objects in our survey, now half complete. In §3 we present two new HVS discoveries. In §4 we show that the properties of the other late-B type stars in the sample are consistent with being a Galactic halo population of post-main sequence stars and/or blue stragglers. In §5 we discuss the white dwarfs in the sample, many of which may be unusually low mass DA white dwarfs. In §6 we discuss two young B supergiants in the Leo A dwarf galaxy and one UV bright phase horizontal branch star in the Draco dwarf galaxy. We conclude in §7.
DATA
====
Target Selection
----------------
As @brown06 discuss, HVSs ought to be rare: @yu03 predict there should be $\sim$10$^3$ HVSs in the entire Galaxy. Thus, in any search for HVSs, [*survey volume is important*]{}. Solar neighborhood surveys have not discovered HVSs because, even if they were perfectly complete to a depth of 1 kpc, there is only a $\sim$0.1% chance of finding a HVS in such a small volume. Finding a new HVS among the Galaxy’s $\sim$10$^{11}$ stars also requires selection of targets with a high probability of being HVSs. Our observational strategy is two-fold. Because the density of stars in the Galactic halo drops off as approximately $r^{-3}$, and the density of HVSs drops off as $r^{-2}$ (if they are produced at a constant rate), we target distant stars where we [*maximize the contrast*]{} between the density of HVSs and indigenous stars. Secondly, the stellar halo contains mostly old, late-type stars. Thus we target faint B-type stars, stars with lifetimes consistent with travel times from the Galactic center but which are not a normally expected stellar halo population. O-type stars are more luminous but do not live long enough to reach the halo. A-type stars are also luminous but must be detected against large numbers of evolved blue horizontal branch (BHB) stars in the halo. Based on the @brown05b field BHB luminosity function, we expect only small numbers of hot BHB stars with B-type colors. Our strategy of targeting B-type stars is further supported by observations showing that 90% of the $K<16$ stars in the central $0.5\arcsec$ of the Galactic Center are in fact normal main sequence B stars [@eisenhauer05].
We use Sloan Digital Sky Survey (SDSS) photometry to select candidate B stars by color. Our color selection is illustrated in Fig. \[fig:ugr\], a color-color diagram of stars with B- and A-type colors in the SDSS Fourth Data Release [@adelman06]. @fukugita96 describe the SDSS filter system and the colors of main sequence stars in the SDSS photometric system. We use SDSS point-spread function magnitudes and reject any objects that have bad photometry flags. We compute de-reddened colors using extinction values obtained from @schlegel98. The dotted box in Fig. \[fig:ugr\] indicates the selection region used by @yanny00 to identify BHB candidates. Interestingly, there is a faint group of stars with late B-type colors extending up the stellar sequence towards the ensemble of white dwarfs. We chose our primary candidate B star selection region inside the solid parallelogram defined by: $-0.38<(g'-r')_0<-0.28$ and $2.67(g'-r')_0 + 1.30 <
(u'-g')_0 < 2.67(g'-r')_0 + 2.0$. In addition, we impose $-0.5<(r'-i')_0<0$ to reject objects with non-stellar colors.
{width="3.25in"}
We observe candidate B stars in the magnitude range $17.0<g'_0<19.5$. The bright magnitude limit sets an inner distance boundary $\gtrsim$30 kpc for late B-type stars, a distance beyond that of known runaway B stars [@lynn04; @martin04]. We chose the faint magnitude limit to keep our exposure times below 30 minutes using the 6.5m MMT telescope. In addition, we exclude the region of sky between $b<-l/5 + 50\arcdeg$ and $b>l/5-50\arcdeg$ to avoid excessive contamination from Galactic bulge stars.
There are a total of 430 SDSS DR4 candidate B stars in the primary selection region described above. We have observed 192, or 45% of this total. The average surface number density of targets is 1 per 15 deg$^2$. Thus we have surveyed $\sim$3000 deg$^2$ or 7% of the entire sky. Figure \[fig:sky\] displays the locations of observed candidate B stars in the northern Galactic hemisphere; a handful of stars in the autumn SDSS equatorial stripes are located in the southern Galactic hemisphere and are not displayed.
{width="7.0in"}
In addition, we have observed 55 targets with colors, magnitudes, or positions slightly outside our primary selection region (for example, see Fig.\[fig:ugr2\]). We include the full sample of 247 objects in our discussion below. We note that the region of sky located $40<l<90^{\circ}$ in Fig.\[fig:sky\] lacks HVS discoveries, but this is probably not significant. This region, observed in July 2005, is missing observations of the bluest objects in $(u'-g')_0$ and will be completed in the coming months.
Spectroscopic Observations and Radial Velocities
------------------------------------------------
Observations were obtained with the Blue Channel spectrograph on the 6.5m MMT telescope. Observations were obtained on the nights of 2005 July 10-11, 2005 December 3-5, and 2006 February 22-25. The spectrograph was operated with the 832 line/mm grating in second order, providing wavelength coverage 3650 Å to 4500 Å. Most spectra were obtained with 1.2 Åspectral resolution, however on one night of poor seeing we used a larger slit that provided 1.5 Å spectral resolution for 24 objects. Exposure times ranged from 5 to 30 minutes and were chosen to yield $S/N=15$ in the continuum at 4000 Å. Comparison lamp exposures were after obtained after every exposure.
Radial velocities were measured using the cross-correlation package RVSAO [@kurtz98]. @brown03 describe in detail the cross-correlation templates we use. Errors are measured from the width of the cross-correlation peak, and are added in quadrature with the 9 km s$^{-1}$ systematic uncertainty observed in bright BHB standards. The average uncertainty is $\pm11$ km s$^{-1}$ for the late B-type stars and $\pm40$ km s$^{-1}$ for the DA white dwarfs (with much broader Balmer lines).
Selection Efficiency and Unusual Objects
----------------------------------------
Our candidate B stars include post-main sequence stars and late B blue stragglers, some DA white dwarfs, and a few other unusual objects. We classify the spectral types of the 202 late B stars based on @oconnell73 and @worthey94 line indices as described in @brown03. The spectral types of the stars range from B6 to A1 with an average uncertainty of $\pm1.6$ spectral sub-types. Thus our primary target selection is 84% efficient for selecting stars of late B spectral type. Four of the 202 late B stars, or approximately 1-in-50, are HVSs. In addition, 3 of the late B stars coincide with Local Group dwarf galaxies, which provides special constraints on the nature of those objects.
{width="3.25in"}
Figure \[fig:ugr2\] plots the colors and spectroscopic identifications for the full sample of objects. The solid parallelogram indicates our primary color selection region; the dashed lines show the slightly different color selection regions used on different observing runs. 44 of the objects in Fig. \[fig:ugr2\] are white dwarfs marked by crosses. The white dwarfs are mostly DA white dwarfs, but also include one DB and one DZ white dwarf. Our sample also includes one extreme low-metallicity starburst galaxy, marked by the solid square in Fig. \[fig:ugr2\], that we describe in a separate paper (Kewley et al., in preparation).
HYPERVELOCITY STARS
===================
Our targeted search for HVSs has discovered a total of four probable HVSs. @brown06 report the discovery of the first two HVSs from this survey, and here we report two further HVS discoveries: SDSS J110557.45+093439.5 (hereafter HVS6) and possibly SDSS J113312.12+010824.9 (hereafter HVS7). HVS6 is a faint $g'=19.06\pm0.02$ star with B9 spectral type and travels with a $+606\pm12$ km s$^{-1}$ heliocentric radial velocity. HVS7 is a $g'=17.75\pm0.02$ star with B7 spectral type and travels with a $+531\pm10$ km s$^{-1}$ heliocentric radial velocity. We correct the velocities to the local standard of rest [@hogg05] and remove the 220 km s$^{-1}$ solar reflex motion as follows: $$v_{rf} = v_{helio} + (10\cos{l}\cos{b} + 5.2\sin{l}\cos{b} + 7.2\sin{b}) + 220\sin{l}\cos{b}$$ The minimum Galactic rest frame velocities (indicated $v_{rf}$) of HVS6 and HVS7 are $+508$ and $+418$ km s$^{-1}$, respectively. The minimum Galactic rest-frame velocity of HVS7 is marginally consistent with runaway star mechanisms, but, if it is a main sequence B star, it is unbound to the Galaxy. Thus, for now, we consider HVS7 a HVS. All 7 known HVSs are traveling with large positive radial velocity, consistent with a Galactic center origin.
Figure \[fig:velh\] plots a histogram of Galactic rest-frame velocity for the 202 late B stars in our sample. We calculate the line-of-sight velocity dispersion of the stars using three different methods: 1) fitting a Gaussian to the entire distribution, 2) fitting a Gaussian to just the negative velocity half of the distribution, and 3) simply calculating the dispersion around the mean after clipping the HVSs. All three methods yield equivalent results. Averaging the results of the three methods, our sample has a velocity dispersion of $108\pm5$ km s$^{-1}$ and mean of $-2\pm8$ km s$^{-1}$ consistent with a Galactic halo population.
{width="3.25in"}
HVS6 and HVS7 are 4.7$\sigma$ and 3.9$\sigma$ outliers, respectively, from the velocity distribution. The lower panel of Fig. \[fig:velh\] plots the residuals of the observations from the best-fit Gaussian, normalized by the value of the Gaussian. Stars with velocities below $|v_{rf}|<200$ km s$^{-1}$ show low-significance deviations from a Gaussian distribution. The four HVSs, on the other hand, are 4-6$\sigma$ outliers and are completely off-scale.
In addition to the HVSs, the distribution of velocities in Fig.\[fig:velh\] shows a tail of stars traveling with large positive velocities $v_{rf}>250$ km s$^{-1}$ and no stars traveling with equally large negative velocities. Stars in compact binary systems may produce outliers in the velocity distribution, but such outliers should be distributed symmetrically. Conceivably, the observed asymmetry is the low-velocity tail of HVSs, or it may be the high-velocity tail of runaway stars. Because runaway stars are ejected with low $<300$ km s$^{-1}$ velocities, they follow bound, ballistic trajectories away from and then back onto the disk [e.g. @martin06]. Thus, if the stars in the high velocity tail are runaway stars, they must be very nearby. The exact velocity distribution of runaway stars is, however, unclear. The predictions of @portegies00 are not applicable, for example, because the runaway stars must be low mass, intrinsically faint objects to be located nearby. Moreover, the velocity distribution of HVSs has been calculated for only restrictive sets of circumstances [@hills91; @levin05; @ginsburg06]. Clean predictions of runaway star and HVS velocity distributions are needed to discriminate among the populations in the high velocity tail. Proper motions (as may be measured with the [*Hubble Space Telescope*]{}, [*GAIA*]{}, or the [*Space Interferometry Mission*]{}) will ultimately discriminate between HVSs and runaway stars.
Our low-resolution spectra do not allow determination of exact stellar parameters for HVS6 and HVS7. Stars of late B spectral type are probably post-main sequence stars or main sequence B stars/blue stragglers. We note that the Balmer line widths of HVS6 and HVS7 are too broad to be consistent with those of luminosity class I or II B supergiants. If we assume the HVSs are BHB stars rather than B stars, their blue colors mean they are hot, extreme BHB stars and thus they are intrinsically very faint. The $M_V(BHB)$ relation of @clewley02 yields $M_V(BHB)\simeq+1.6$ and $+1.8$ and heliocentric distance estimates $d_{BHB}\simeq$30 and 15 kpc for HVS6 and HVS7, respectively. In the BHB interpretation, the volume we effectively survey is much smaller than in the B star interpretation. Because the first two HVSs are known B stars [@edelmann05; @fuentes06] and because the B star interpretation implies a production rate probably consistent with @yu03, we assume that HVS6 and HVS7 are B stars for the purpose of discussion. The ultimate discriminant will come from higher resolution, higher signal-to-noise spectroscopy.
We estimate distances for HVS6 and HVS7 by looking at @schaller92 stellar evolution tracks for 3 and 4 $M_{\sun}$ stars with $Z=0.02$. A 3 $M_{\sun}$ star spends 350 Myr on the main sequence with $M_V(3
M_{\sun})\simeq-0.3$. If HVS6 is a 3 $M_{\sun}$ B9 main sequence star, it has a heliocentric distance $d\sim75$ kpc. Using this distance, we now estimate the HVS travel time from the Galactic Center. We make the conservative assumptions that the HVS’s observed velocity is a total space velocity and that its velocity has remained constant. Detailed calculations of HVS trajectories by @gnedin05 show that this simple estimate is reasonably accurate and over-estimates HVS travel times by less than 10% (O.Gnedin, private communication). We estimate the travel time of HVS6 is $\sim$160 Myr, consistent with its 350 Myr main sequence lifetime. By comparison, a 4 $M_{\sun}$ star spends 160 Myr on the main sequence and has $M_V(4 M_{\sun})\simeq-0.9$. If HVS7 is a 4 $M_{\sun}$ B7 main sequence star, it has a heliocentric distance $d\sim55$ kpc and a travel time from the Galactic center of $\sim$120 Myr also consistent with its lifetime. There is a tendency to find HVSs near the end of their lives because the longer they have traveled, the larger the survey volume they populate and the greater the contrast with the indigenous stellar populations.
Our radial velocities provide only a [*lower*]{} limit to the HVSs’ true space velocities. The escape velocity from the Galaxy is approximately 300 km s$^{-1}$ at 50 kpc [@wilkinson99], thus HVS6 is unbound to the Galaxy whether it is a B main sequence star or a BHB star. HVS7, on the other hand, is only unbound if it is a B main sequence star; follow-up spectroscopy is necessary to establish whether it is a “true” HVS.
HVS6 and HVS7 are both present in the USNOB1 [@monet03] catalog but only HVS7 is listed with a proper motion. Averaging the USNOB1 proper motion with that from the GSC2.3 (B. McLean, 2006 private communication), HVS7 has $\mu=10.5\pm9$ mas yr$^{-1}$. If we assume HVS7 is located nearby at $d_{BHB}\simeq$15 kpc consistent with a proper motion detection, then its transverse velocity is $750\pm650$ km s$^{-1}$. Such a velocity would suggest that HVS7 is unbound, but the proper motion measurement is significant at only the 1$\sigma$ level and thus we place little confidence in it.
The new HVSs are not physically associated with any other Local Group galaxy. HVS6 and HVS7 are located at $(l,b)=(243\fdg1,59.\fdg6)$ and $(263\fdg8,57\fdg9)$, respectively (see Fig. \[fig:sky\]). The nearest galaxies to HVS6 are Leo I and Leo II, both $\sim$14$^{\arcdeg}$ away on the sky from HVS6. However, Leo I and Leo II are at distances of $254\pm17$ kpc [@bellazzini04] and $233\pm15$ [@bellazzini05], respectively, many times the estimated distance of HVS6. Thus HVS6 is moving towards Leo I and Leo II at minimum velocities of 330 km s$^{-1}$ and 490 km s$^{-1}$, respectively, and clearly unrelated to those galaxies. The nearest galaxy to HVS7 is the Sextans dwarf 20$^{\arcdeg}$ away on the sky. At a distance of $1320\pm40$ kpc [@dolphin03], Sextans is unrelated to HVS7.
Table \[tab:hvs\] summarizes the properties of all seven known HVSs, four of which were discovered in this survey. The columns include HVS number, Galactic coordinates $(l,b)$, apparent magnitude $g'$, minimum Galactic rest-frame velocity $v_{RF}$ (not a full space velocity), heliocentric distance estimate $d$, travel time estimate from the Galactic Center $t_{GC}$, and catalog identification. We have repeat observations of HVS1, HVS4, and HVS5; their radial velocities are constant within the uncertainties.
HALO STARS
==========
Most objects in our survey are halo stars with late B spectral types, and we now discuss their nature. Stars of late B spectral type are probably main sequence stars / blue stragglers or post-main sequence stars. Unfortunately, main sequence stars and post-main sequence BHB stars share similar effective temperature (color) and surface gravity (spectral line widths), making classification difficult.
Stellar rotation is a useful discriminant between rapidly rotating main sequence B stars [@abt02; @martin04] and slowly rotating BHB stars [@peterson95; @behr03]. However, our low-dispersion spectra do not allow us to measure rotation. Instead, we constrain the nature of the late B type objects by looking at their metallicities and kinematics.
{width="3.25in"}
The strongest indicator of metallicity in our spectra is the 3933 ÅCa[ii]{} K line. The equivalent width of Ca[ii]{} K depends on both temperature and metallicity. To estimate metallicity, we first compute $(\bv)_0$ from the SDSS colors following @clewley05. We then measure the equivalent width of the Ca[ii]{} K line, $W_k$. Finally, we estimate metallicity \[Fe/H\]$_{W_k}$ by interpolating the theoretical curves of @wilhelm99a, assuming $\log{g}=3$ appropriate for a BHB star. We propagate the errors in $(\bv)_0$ and $W_k$ through the @wilhelm99a curves and find that the uncertainty is large, $\pm0.67$ in \[Fe/H\]$_{W_k}$. Moreover, Ca[ii]{} K provides very little leverage on metallicity for the hottest stars $(\bv)_0<-0.05$. For the 105 stars with $(\bv)_0<-0.05$ our metallicity estimates are effectively reduced to a binary measurement: \[Fe/H\]$_{W_k}$$\sim$0 if we see Ca[ii]{} K, and \[Fe/H\]$_{W_k}$$\sim-3$ if Ca[ii]{} K is absent. Note that the @wilhelm99a models restrict our metallicity estimates to $-3 <$ \[Fe/H\]$_{W_k}$ $< 0$.
Figure \[fig:velfeh\] plots metallicities and Galactic rest frame velocities of the late B type objects. We plot hot objects with poor \[Fe/H\]$_{W_k}$ determinations as open symbols. All of the objects are affected at some level by accretion of interstellar material, atomic diffusion in their atmospheres, and observational effects such as interstellar absorption. We do not know the detailed histories of the individual stars, and thus we simply consider the average observed \[Fe/H\]$_{W_k}$ of the sample. Ignoring the objects on the \[Fe/H\]$_{W_k}=0$ and $-3$ boundaries, it is clear that the objects cluster at metal-poor values: the mean metallicity of the sample (excluding objects on the boundaries) is $\overline{{\rm[Fe/H]}_{W_k}}=-1.3$. If instead we assume $\log{g}=4$ appropriate for main sequence/blue straggler stars, additional stars are pushed onto the \[Fe/H\]$_{W_k}=0$ boundary line and the mean metallicity of the sample (excluding objects on the boundaries) increases slightly to $\overline{{\rm[Fe/H]}_{W_k}}=-1.2$. The low mean metallicity of the sample suggests that most objects are not recently-formed main sequence B stars ejected from the disk, but rather the objects are likely post-main sequence stars or old blue stragglers.
The observed $108\pm5$ km s$^{-1}$ velocity dispersion of the late B type objects is also consistent with a Galactic halo population of post-main sequence stars or blue stragglers. Although some have proposed in-situ star formation in the halo [@vanwoerden93; @christodoulou97], there is no evidence for this in modern studies of runaway B stars, including the recent @martin06 study of runaway stars in the Hipparcos catalog. Thus the observed metallicity and velocity distributions suggest that the late B type stars are most likely a Galactic halo population of post-main sequence stars and/or old blue stragglers, and not young runaway B stars ejected from the disk. We hope ultimately to use this sample to provide a useful probe of halo structure.
Table \[tab:bhb\] lists the 202 spectroscopically identified late B type objects, including the 4 HVSs. The columns include RA and Dec coordinates (J2000), $g'$ apparent magnitude, $(u'-g')_0$ and $(g'-r')_0$ color, and our heliocentric velocity $v_{helio}$ and \[Fe/H\]$_{W_k}$ estimate.
WHITE DWARFS
============
44 survey objects are faint white dwarfs, drawn from a largely unexplored region of color-space compared to previous SDSS-based white dwarf spectroscopic surveys [@harris03; @kleinman04; @kilic06]. The objects are almost entirely DA white dwarfs, with colors $-0.4<(u'-g')_0<0.2$ indicating temperatures $10,000 < T_{eff} < 16,000$ K [@kleinman04]. Our color selection region, however, lies at surface gravities $\log{g}<7$ for hydrogen-atmosphere white dwarfs (i.e. to the right of the Bergeron $\log{g}=7$ curve plotted in Fig. 1 of @harris03). Thus the white dwarfs we find are all candidates for objects with unusually low surface gravities and unusually low masses.
The least massive white dwarfs known are $\sim$0.2 $M_{\sun}$ helium-core objects in binary systems containing millisecond pulsars [e.g. @callanan98] or subluminous B (sdB) stars [@heber03; @otoole06]. @liebert04 discuss a 0.18 $M_{\sun}$ helium white dwarf in the SDSS with colors $(u'-g')=0.32$ and $(g'-r')=-0.35$ very similar to our white dwarfs (see Fig. \[fig:ugr2\]). Figure \[fig:wd\] shows the spectra of two white dwarfs in our survey with the most unusually red $(u'-g')$ colors, SDSS J074508.15+182630.0 (top) and SDSS J083303.03+365906.3 (bottom). These objects do not appear to be sdB subdwarfs because their spectra show only very broad hydrogen Balmer lines. It would be very interesting to know whether these white dwarfs are unusually low mass white dwarfs, but detailed modeling is beyond the scope of this paper.
{width="3.25in"}
We search for proper motions in the USNOB1 and GSC2.3 catalogs, and find proper motion measurements for 35 of the 44 white dwarfs, 20 of which are significant at $>3\sigma$ level. The average proper motion of the 20 white dwarfs is 40 mas yr$^{-1}$ with an uncertainty of 7 mas yr$^{-1}$. The late B-type stars, by comparison, have no significant proper motion detections, consistent with their inferred distances. We calculate reduced proper motions for the white dwarfs with proper motion measurements and find values ranging $14<H_{g'}<18$ at $-0.7<(g'-i')_0<-0.5$, which places our objects in the main body of white dwarfs observed by @kleinman04 and @kilic06.
Table \[tab:wd\] lists the 44 spectroscopically identified white dwarfs. The columns include RA and Dec coordinates (J2000), $g'$ apparent magnitude, $(u'-g')_0$ and $(g'-r')_0$ colors, and heliocentric radial velocities $v_{helio}$. We note that broad Balmer lines make for very poor radial velocity determinations. The objects are all DA white dwarfs with two exceptions: SDSS J111133.37+134639.8 is a DB white dwarf and SDSS J151852.49+530121.8 is a DZ white dwarf with strong calcium H and K lines.
UNUSUAL STARS IN DWARF GALAXIES
===============================
B Supergiants in the Leo A Dwarf
--------------------------------
Leo A is an extremely metal-poor, gas-rich Im dwarf galaxy. Stellar population studies show that Leo A contains both very young and very old stellar populations [@tolstoy98; @schulte02; @dolphin02; @vansevicius04]. To date, stellar population studies of Leo A are based entirely on color-magnitude diagrams, all of which reveal a striking “blue plume” of B giants possibly in the galaxy. Here, we discuss the first spectroscopic identifications of two B giants definitely associated with the Leo A dwarf galaxy.
Two stars from our survey, SDSS J095915.12+304410.4 and SDSS J095920.22+304352.7, match Leo A both in position and in velocity. The stars are located 1.2 and 2.0, respectively, from the center of Leo A, well within the 7 $\times$ 4.6 Holmberg diameter of the galaxy [@mateo98]. The stars have heliocentric radial velocity $+20\pm12$ and $+32\pm12$ km s$^{-1}$, respectively, consistent at the 1$\sigma$ level with the velocity of Leo A $+24\pm2$ km s$^{-1}$ measured from 21 cm observations [@young96]. The stars have apparent magnitude $g'=19.90\pm0.03$ and $19.44\pm0.03$, respectively. If the stars are physically associated with Leo A, the galaxy’s distance modulus $(m-M)_0 = 24.51 \pm 0.12$ [@dolphin02] implies that the stars have luminosity $M_V \simeq -4.6$ and $-5.0$, respectively.
Interestingly, the spectra of the two stars in Leo A have unusually narrow Balmer lines for stars in our sample; cross correlation with MK spectral standards indicates that the stars are most likely luminosity class I or II B supergiants. Figure \[fig:b194\] displays a portion of the spectra for SDSS J095915.12+304410.4 ([*upper panel*]{}) and SDSS J095920.22+304352.7 ([*lower panel*]{}), convolved to match the 1.8 Å resolution of MK spectral standards from @gray03. The B9 II ($\gamma$ Lyr) and B9 Ia (HR1035) MK standards are over-plotted as thin lines. It is visually apparent that the observed stars have Balmer line widths between those of the B II and B Ia standards. @garrison84 give luminosities $M_V=-3.1$ for a B9 II star and $M_V=-5.5$ for a B9 Ib star. The luminosities we infer from the distance to Leo A fall between these values, consistent with the spectra. We conclude the two stars are B supergiants in the blue plume of the Leo A dwarf galaxy. Such stars are $\sim$30 Myr old [@schaller92], consistent with star formation age estimates by others from color-magnitude diagrams.
{width="3.25in"}
UV-Bright BHB Star in the Draco Dwarf
-------------------------------------
By chance, another star from our survey is located in the Draco dwarf galaxy. The star, SDSS J172004.07+575110.8, has a spectral type of B9 and an apparent magnitude of $g'=18.44\pm0.02$. The star is also identified as non-variable star \#517 in the classic @baade61 paper. The distance modulus to Draco, $(m-M)_0 = 19.40 \pm 0.15$ [@bonanos04], implies that the star has $M_V \sim -1$, a more difficult luminosity to explain. Unlike the two stars in Leo A, the star in Draco has Balmer line widths inconsistent with B giants. We conclude that the most likely explanation for the star in Draco is that it is a UV-bright, “slow blue phase” horizontal branch star.
The position, velocity, and metallicity of the star in Draco match that of the dwarf galaxy. The star falls within 5 of the center of the Draco, well within the 9 core radius of the galaxy [@irwin95]. The star’s velocity $v_{RF} = -82 \pm 12$ km s$^{-1}$ is consistent at the 1$\sigma$ level with the velocity of Draco $-104 \pm21$ km s$^{-1}$ [@falco99]. Finally, our estimate of the star’s metallicity, \[Fe/H\]$_{W_k}=-1.6\pm0.75$, is consistent with spectroscopic metallicity measurements of Draco’s stellar population that fall into two groups near \[Fe/H\]$=-1.6\pm0.2$ and $-2.3\pm0.2$ [@shetrone01; @lehnert92; @kinman81; @zinn78].
The star in Draco is probably not a main sequence B star, because there is little evidence for young stars in color-magnitude diagrams of Draco [e.g. @bonanos04; @klessen03; @bellazzini02]. It is possible that a B9 main sequence star in Draco is a blue straggler. However, the luminosity of a metal-poor B9 main sequence star is too low to place it at the distance of Draco. A main sequence star with $Z=0.001$ and $T_{eff}=10,500$ K has a mass of 1.7 $M_\sun$ and an absolute magnitude $M_V(B9)=+1.6$ [@schaller92], two-and-a-half magnitudes too faint to be at the distance of Draco.
The star in Draco is also unlikely to be a normal BHB star. The horizontal branch of Draco is well observed and its stars are $20<V<21$ [@klessen03; @bellazzini02]. Moreover, a hot BHB star with spectral type B9 is an intrinsically faint star; the @clewley04 $M_V(BHB)$ relation yields $M_V(BHB)=+1.3$ which is two magnitudes too faint to be at the distance Draco.
Other possibilities, such as a blue-loop Cepheid or a post-AGB star, are also unlikely. Cepheids with masses $>5$ $M_{\sun}$ can travel out of the instability strip on long blue loops [@bono00], but massive stars are unlikely to exist in Draco. Post-AGB stars, stars in the process of blowing off their outer layers to become white dwarfs, can have effective temperatures of $10^4$ K but only for a short time. Although there may be many more AGB stars than BHB stars in Draco, the substantially shorter $10^3 - 10^4$ yr timescale for a post-AGB star to have the correct effective temperature and luminosity (P. Demarque, private communication) suggests that a longer-lived, UV-bright star evolving off of the horizontal branch is a more plausible explanation.
The UV-bright phase is a slow-evolving, helium shell-burning phase that occurs for BHB stars with small hydrogen envelopes. Although the UV-bright phase is more common in metal-rich stars, it occurs in metal-poor stars as well. @yi97 stellar evolution tracks (see their Figure 1) show that metal-poor BHB stars with $\sim0.05$ $M_\sun$ envelopes spend $10^7$ yrs at effective temperatures around 10,000 K and $10^{2.4}$ $L_\sun$. This model provides the exact absolute magnitude $M_V(UV~BHB)=-1$ and spectral type needed to place the star at the distance of Draco, and applies to stars with metallicities ranging from \[Fe/H\]$=-1$ down to $-2.6$. A recent study of BHB stars in Draco identifies $\sim$50 BHB stars in the dwarf galaxy [@klessen03]. If BHB stars spend 150 Myr on the BHB and 10 Myr in the UV-bright phase, then we may expect a few BHB stars in the UV bright phase.
If the star in Draco is a UV-bright BHB star, its spectrum should indicate a low surface gravity. We estimate the surface gravity of the star by measuring the size and steepness of the Balmer jump [@kinman94], and the widths and the shapes of the Balmer lines [@clewley04]. These independent techniques indicate that the star is a low surface gravity star. We conclude that the star in Draco SDSS J172004.1+575111 is most likely UV-bright BHB star.
CONCLUSIONS
===========
In this paper we discuss our targeted survey for HVSs, a spectroscopic survey of stars with late B-type colors that is now half complete. Our survey has discovered a total of four HVSs, or approximately 1-in-50 of our candidates. The first two HVS discoveries are reported in @brown06. Here we report two new HVS discoveries: HVS6 and possibly HVS7, traveling with Galactic rest-frame velocities at least $+508\pm12$ and $+418\pm10$ km s$^{-1}$, respectively. Assuming the HVSs are main sequence B stars, they are at distances $\sim$75 and $\sim$55 kpc, respectively, and have travel times from the Galactic center consistent with their lifetimes.
The remaining late B-type stars have metallicities and kinematics consistent with a Galactic halo population of post main-sequence stars or blue stragglers. However, the line-of-sight velocity distribution shows a tail of objects with large positive velocities. This high velocity tail may be a mix of low-velocity HVSs and high-velocity runaway stars; further theoretical and observational work is needed to understand the nature of the high velocity tail.
Our survey includes many interesting objects besides HVSs. Approximately one-sixth of the objects are DA white dwarfs with unusually red colors, possibly extremely low mass objects.
Two of our objects are luminosity class I or II B supergiants in the Leo A dwarf. Our observations of these B supergiants provide the first spectroscopic evidence for recent $\sim$30 Myr old star formation in Leo A. Another object is an unusual UV bright phase BHB star in the Draco dwarf.
We are continuing our targeted HVS survey of late B-type stars in the SDSS using the MMT telescope. We are also using the Whipple 1.5m telescope to obtain spectroscopy of brighter $15<g'<17$ late B-type objects. Given our current discovery rate, we expect to find perhaps another half dozen HVSs in the coming months. Follow-up high dispersion spectroscopy will provide precise stellar parameters of these stars, and [*Hubble Space Telescope*]{} observations will provide accurate proper motions. Our goal is to discover enough HVSs to allow us to place quantitative constraints on the stellar mass function of HVSs, the origin of massive stars in the Galactic Center, and the history of stellar interactions with the MBH.
We thank R. Zinn and P. DeMarque for helpful conversations, and the referee for a detailed report. We thank M. Alegria, J. McAfee, and A.Milone for their assistance with observations obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. This project makes use of data products from the Sloan Digital Sky Survey, which is managed by the Astrophysical Research Consortium for the Participating Institutions. This work was supported by W. Brown’s Clay Fellowship and the Smithsonian Institution.
[*Facilities:*]{} MMT (Blue Channel Spectrograph)
[78]{} natexlab\#1[\#1]{}
, H. A., [Levato]{}, H., & [Grosso]{}, M. 2002, , 573, 359
, J. K. [et al.]{} 2006, , 162, 38
, W. & [Swope]{}, H. H. 1961, , 66, 300
, B. B. 2003, , 149, 67
, M., [Ferraro]{}, F. R., [Origlia]{}, L., [Pancino]{}, E., [Monaco]{}, L., & [Oliva]{}, E. 2002, , 124, 3222
, M., [Gennari]{}, N., & [Ferraro]{}, F. R. 2005, , 360, 185
, M., [Gennari]{}, N., [Ferraro]{}, F. R., & [Sollima]{}, A. 2004, , 354, 708
, A. 1961, , 15, 265
, A. Z., [Stanek]{}, K. Z., [Szentgyorgyi]{}, A. H., [Sasselov]{}, D. D., & [Bakos]{}, G. [Á]{}. 2004, , 127, 861
, G., [Caputo]{}, F., [Cassisi]{}, S., [Marconi]{}, M., [Piersanti]{}, L., & [Tornamb[è]{}]{}, A. 2000, , 543, 955
, W. R., [Allende Prieto]{}, C., [Beers]{}, T. C., [Wilhelm]{}, R., [Geller]{}, M. J., [Kenyon]{}, S. J., & [Kurtz]{}, M. J. 2003, , 126, 1362
, W. R., [Geller]{}, M. J., [Kenyon]{}, S. J., & [Kurtz]{}, M. J. 2005, , 622, L33
—. 2006, , 640, L35
, W. R., [Geller]{}, M. J., [Kenyon]{}, S. J., [Kurtz]{}, M. J., [Allende Prieto]{}, C., [Beers]{}, T. C., & [Wilhelm]{}, R. 2005, , 130, 1097
, P. J., [Garnavich]{}, P. M., & [Koester]{}, D. 1998, , 298, 207
, D. M., [Tohline]{}, J. E., & [Keenan]{}, F. P. 1997, , 486, 810
, L., [Warren]{}, S. J., [Hewett]{}, P. C., [Norris]{}, J. E., & [Evans]{}, N. W. 2004, , 352, 285
, L., [Warren]{}, S. J., [Hewett]{}, P. C., [Norris]{}, J. E., [Peterson]{}, R. C., & [Evans]{}, N. W. 2002, , 337, 87
, L., [Warren]{}, S. J., [Hewett]{}, P. C., [Norris]{}, J. E., [Wilkinson]{}, M. I., & [Evans]{}, N. W. 2005, , 362, 349
, P. & [Virani]{}, S. 2006, preprint (astro-ph/0603326)
, A. E., [Saha]{}, A., [Claver]{}, J., [Skillman]{}, E. D., [Cole]{}, A. A., [Gallagher]{}, J. S., [Tolstoy]{}, E., [Dohm-Palmer]{}, R. C., & [Mateo]{}, M. 2002, , 123, 3154
, A. E., [Saha]{}, A., [Skillman]{}, E. D., [Dohm-Palmer]{}, R. C., [Tolstoy]{}, E., [Cole]{}, A. A., [Gallagher]{}, J. S., [Hoessel]{}, J. G., & [Mateo]{}, M. 2003, , 125, 1261
, L. M., [Dale]{}, J. E., [Beer]{}, M. E., [Napiwotzki]{}, R., & [King]{}, A. R. 2005, , 364, 59
, H., [Napiwotzki]{}, R., [Heber]{}, U., [Christlieb]{}, N., & [Reimers]{}, D. 2005, , 634, L181
, F. [et al.]{} 2005, , 628, 246
, E. E., [Kurtz]{}, M. J., [Geller]{}, M. J., [Huchra]{}, J. P., [Peters]{}, J., [Berlind]{}, P., [Mink]{}, D. J., [Tokarz]{}, S. P., & [Elwell]{}, B. 1999, , 111, 438
, C. I., [Stanek]{}, K. Z., [Gaudi]{}, B. S., [McLeod]{}, B. A., [Bogdanov]{}, S., [Hartman]{}, J. D., [Hickox]{}, R. C., & [Holman]{}, M. J. 2006, , 636, L37
, M., [Ichikawa]{}, T., [Gunn]{}, J. E., [Doi]{}, M., [Shimasaku]{}, K., & [Schneider]{}, D. P. 1996, , 111, 1748
, R. F., ed. 1984, [The MK process and stellar classification]{} (Toronto: DDO), 275
, I. & [Loeb]{}, A. 2006, preprint (astro-ph/0510574)
, O. Y., [Gould]{}, A., [Miralda-Escud[é]{}]{}, J., & [Zentner]{}, A. R. 2005, , 634, 344
, R. O., [Corbally]{}, C. J., [Garrison]{}, R. F., [McFadden]{}, M. T., & [Robinson]{}, P. E. 2003, , 126, 2048
, A., [Portegies Zwart]{}, S., & [Eggleton]{}, P. P. 2004, , 350, 615
, A., [Zwart]{}, S. P., & [Sipior]{}, M. S. 2005, , 363, 223
, H. C. [et al.]{} 2003, , 126, 1023
, U., [Edelmann]{}, H., [Lisker]{}, T., & [Napiwotzki]{}, R. 2003, , 411, L477
, J. G. 1988, , 331, 687
—. 1991, , 102, 704
, H. A., [Heber]{}, U., [O’Toole]{}, S. J., & [Bresolin]{}, F. 2005, , 444, L61
, D. W., [Blanton]{}, M. R., [Roweis]{}, S. T., & [Johnston]{}, K. V. 2005, , 629, 268
, K., [Sigurdsson]{}, S., [Mihos]{}, C. J., [Feldmeier]{}, J. J., [Ciardullo]{}, R., & [McBride]{}, C. 2006, preprint (astro-ph/0512344)
, M. & [Hatzidimitriou]{}, D. 1995, , 277, 1354
, M. [et al.]{} 2006, , 131, 582
, T. D., [Kraft]{}, R. P., & [Suntzeff]{}, N. B. 1981, in ASSL Vol. 88: Physical Processes in Red Giants, 71–76
, T. D., [Suntzeff]{}, N. B., & [Kraft]{}, R. P. 1994, , 108, 1722
, S. J. [et al.]{} 2004, , 607, 426
, R. S., [Grebel]{}, E. K., & [Harbeck]{}, D. 2003, , 589, 798
, M. J. & [Mink]{}, D. J. 1998, , 110, 934
, M. D., [Bell]{}, R. A., [Hesser]{}, J. E., & [Oke]{}, J. B. 1992, , 395, 466
, P. J. T. 1991, , 101, 562
, P. J. T. 1993, in ASP Conf. Ser. 45, Luminous High-Latitude Stars, ed. D. D. Sasselov, 360
, Y. 2005, preprint (astro-ph/0508193)
, J., [Bergeron]{}, P., [Eisenstein]{}, D., [Harris]{}, H. C., [Kleinman]{}, S. J., [Nitta]{}, A., & [Krzesinski]{}, J. 2004, , 606, L147
, B. B., [Keenan]{}, F. P., [Dufton]{}, P. L., [Saffer]{}, R. A., [Rolleston]{}, W. R. J., & [Smoker]{}, J. V. 2004, , 349, 821
, J. C. 2004, , 128, 2474
—. 2006, preprint (astro-ph/0603238)
, M. L. 1998, , 36, 435
, D. G. [et al.]{} 2003, , 125, 984
, W. O. R. 1973, , 78, 1074
, S. J., [Napiwotzki]{}, R., [Heber]{}, U., [Drechsel]{}, H., [Frandsen]{}, S., [Grundahl]{}, F., & [Bruntt]{}, H. 2006, Baltic Astronomy, 15, 61
, R. C., [Rood]{}, R. T., & [Crocker]{}, D. A. 1995, , 453, 214
, S. F. 2000, , 544, 437
, A., [Ruiz]{}, J., & [Allen]{}, C. 1967, Bol. Obs Tonantzintla Tacubaya, 4, 860
, G., [Schaerer]{}, D., [Meynet]{}, G., & [Maeder]{}, A. 1992, , 96, 269
, D. J., [Finkbeiner]{}, D. P., & [Davis]{}, M. 1998, , 500, 525
, R. E., [Hopp]{}, U., [Drozdovsky]{}, I. O., [Greggio]{}, L., & [Crone]{}, M. M. 2002, , 124, 896
, M. D., [C[ô]{}t[é]{}]{}, P., & [Sargent]{}, W. L. W. 2001, , 548, 592
, E. [et al.]{} 1998, , 116, 1244
, H. 1993, in ASP Conf. Ser. 45: Luminous High-Latitude Stars, ed. D. D. [Sasselov]{}, 11
, V. [et al.]{} 2004, , 611, L93
, R., [Beers]{}, T. C., & [Gray]{}, R. O. 1999, , 117, 2308
, M. I. & [Evans]{}, N. W. 1999, , 310, 645
, G., [Faber]{}, S. M., [Gonzalez]{}, J. J., & [Burstein]{}, D. 1994, , 94, 687
, B. [et al.]{} 2000, , 540, 825
, S., [Demarque]{}, P., & [Kim]{}, Y.-C. 1997, , 482, 677
, L. M. & [Lo]{}, K. Y. 1996, , 462, 203
, Q. & [Tremaine]{}, S. 2003, , 599, 1129
, R. 1978, , 225, 790
[lccccccl]{} HVS1 & 227.3 & 31.3 & 19.8 & +709 & 110 & 160 & SDSS J090745.0+024507$^1$\
HVS2 & 176.0 & 47.1 & 18.8 & +717 & 19 & 32 & US 708$^2$\
HVS3 & 263.0 & -40.9 & 16.2 & +548 & 61 & 100 & HE 0437-5439$^3$\
HVS4 & 194.8 & 42.6 & 18.4 & +563 & 75 & 130 & SDSS J091301.0+305120$^4$\
HVS5 & 146.3 & 38.7 & 17.9 & +643 & 55 & 90 & SDSS J091759.5+672238$^4$\
HVS6 & 243.1 & 59.6 & 19.1 & +508 & 75 & 160 & SDSS J110557.45+093439.5\
HVS7 & 263.8 & 57.9 & 17.7 & +418 & 55 & 120 & SDSS J113312.12+010824.9\
[cccccrr]{} 0:02:33.82 & -9:57:06.8 & $18.578\pm0.021$ & $0.753\pm0.040$ & $-0.328\pm0.040$ & $ -88\pm10$ & $ 0.0\pm0.9$\
0:05:28.14 & -11:00:10.1 & $19.271\pm0.042$ & $1.007\pm0.081$ & $-0.275\pm0.047$ & $-123\pm12$ & $-1.8\pm1.0$\
0:07:52.01 & -9:19:54.3 & $17.440\pm0.017$ & $1.016\pm0.036$ & $-0.276\pm0.039$ & $-119\pm11$ & $-1.6\pm0.6$\
[cccccr]{} 0:28:03.34 & -0:12:13.4 & $18.414\pm0.019$ & $0.517\pm0.032$ & $-0.418\pm0.025$ & $ 97\pm43$\
1:00:44.69 & -0:50:34.1 & $20.111\pm0.062$ & $0.577\pm0.097$ & $-0.300\pm0.068$ & $ 73\pm47$\
1:06:57.83 & -10:08:39.3 & $19.417\pm0.025$ & $0.525\pm0.072$ & $-0.366\pm0.035$ & $ -23\pm47$\
|
---
abstract: 'A decade of X-ray stellar observations with and has led to significant advances in our understanding of the physical processes at work in hot (magnetized) plasmas in stars and their immediate environment, providing new perspectives and challenges, and in turn the need for improved models. The wealth of high-quality stellar spectra has allowed us to investigate, in detail, the characteristics of the X-ray emission across the HR diagram. Progress has been made in addressing issues ranging from classical stellar activity in stars with solar-like dynamos (such as flares, activity cycles, spatial and thermal structuring of the X-ray emitting plasma, evolution of X-ray activity with age), to X-ray generating processes (e.g. accretion, jets, magnetically confined winds) that were poorly understood in the pre-/ era. I will discuss the progress made in the study of high energy stellar physics and its impact in a wider astrophysical context, focusing on the role of spectral diagnostics now accessible.'
author:
- Paola Testa
title: 'X-ray emission processes in stars'
---
n this article I will discuss some recent progress in our understanding of X-ray emission processes in stars, with emphasis towards advances made possible by high-resolution X-ray spectroscopy. This article will necessarily focus on a few selected topics, as tremendous progress has been made in the field in the past decade of and observations, greatly widening our horizons in the study of X-rays from normal stars.
More than half a century of X-ray stellar observations since the first detection of solar X-ray emission [@Friedman51] have revealed the very rich phenomenology of physical processes at work in the outer atmosphere of stars, and their immediate environments. The first systematic observations of X-rays from stars with space observatories revealed that X-ray emission is common in about all types of stars across the Hertzsprung-Russell (HR) diagram though with rather distinct characteristics for different types of stars, pointing to different underlying production mechanisms [@Vaiana81]. Most late-type stars are X-ray sources, often highly variable, with levels of X-ray emission spanning more than four orders of magnitude and saturating at a level of fractional X-ray over bolometric luminosity $L_{\rm X}/L_{\rm bol} \sim 10^{-3}$. The Sun is close to the low activity end of the observed range of X-ray luminosity, with its $L_{\rm X}/L_{\rm bol}$ ranging between $\sim 10^{-7}$ and $\sim 10^{-6}$ during its activity cycle. Massive stars on the other hand typically show low levels of X-ray variability, and $L_{\rm X}/L_{\rm bol} \sim 10^{-7}$, consistent with a scenario where X-rays are produced in shocks due to instabilities in the radiatively driven winds (e.g., [@LucyWhite80; @Owocki88]). Only in a small range of spectral types, from late B to mid-A, are stars observed to be X-ray dark or extremely weak emitters (e.g., [@Czesla07a; @Schroeder07]).
Though the basic characteristics of X-ray stellar emission across the HR diagram had been outlined already by previous X-ray observatories, the high sensitivity and spectral resolution of and have provided novel diagnostics which can probe in detail the physics of hot magnetized plasma. These physical processes are also at work in other very different astrophysical environments albeit on very different energy and temporal scales. High-resolution spectroscopy of stars is producing significant new insights, for instance providing precise temperature and abundance diagnostics, and, for the first time in the X-ray range, diagnostics of density and optical depth.
In the following I will attempt to provide an overview of our current understanding of the X-ray emission mechanisms in massive stars, of the progress in our knowledge of the X-ray activity in solar-like stars, and of selected aspects of the X-ray physics of stars in their early evolution stages in pre-main sequence. In fact, X-ray stellar studies during this past decade have undergone a shift of focus toward the early phases of stellar evolution, and the study of the interplay between circumstellar environment and X-ray activity. E. Feigelson’s article in this same issue addresses the effects of the X-ray emission from the star on its circumstellar environment, on the evolution of the disk, formation of planets, and planetary atmospheres, which are not discussed in this article.
X-ray emission in early-type stars: winds (and magnetic fields) {#sec:hotstars}
===============================================================
Early findings of approximately constant $L_{\rm X}/L_{\rm bol}$ for early-type stars, and the low variability of X-ray emission, were well explained by a model in which X-rays originate in shocks produced by instabilities in the radiatively driven winds of these massive stars (e.g.,[@LucyWhite80; @Owocki88]).
These models yield precise predictions for the shapes and shifts of X-ray emission lines, and models can therefore be tested in detail by deriving information on the line formation radius, overall wind properties, and absorption of overlying cool material. The high spectral resolution of and , and especially the High Energy Transmission Grating Spectrometer (, [@Canizares05]) onboard have revealed a much more complex scenario than the standard model described above. In particular, deviations from the standard model seem to suggest that magnetic fields likely play a significant role in some early-type stars. Magnetic fields have in fact recently been detected in a few massive stars (e.g., [@Donati02]) – most likely fossil fields, because no dynamo mechanism of magnetic field production is predicted to exist for these massive stars since they lack a convective envelope.
High resolution spectra of several massive stars are mostly consistent with the standard wind-shock model, with soft spectra, and blue-shifted, asymmetric and broad ($\sim 1000$ km $s^{-1}$) emission lines: e.g., $\zeta$ Pup [@Cassinelli01], $\zeta$ Ori [@Cohen06]. Other sources, while characterized by the soft emission predicted by wind-shock models, have spectral line profiles that are rather symmetric, unshifted and narrow with respect to model expectations: e.g., $\delta$ Ori [@Miller02], $\sigma$ Ori [@Skinner08]. Furthermore, a few sources have strong hard X-ray emission with many lines narrower than wind-shock model predictions: e.g., $\theta^1$ Ori C [@Gagne05], $\tau$ Sco [@Cohen03]. For this last class of X-ray sources the presence of magnetic fields provides a plausible explanation for the observed deviations from the wind-shock model: the magnetic field can confine the wind which yields hotter plasma and narrower lines, as shown for instance for the case of $\theta^1$ Ori C by Gagn[é]{} et al. through detailed magneto-hydrodynamic simulations which successfully reproduce the observed plasma temperature, $L_{\rm X}$, and rotational modulation [@Gagne05].
An important diagnostic for early-type stars is provided by the He-like triplets (comprising $r$ resonance, $i$ intercombination, and $f$ forbidden lines): the metastable upper level of the $f$ line can be depopulated, populating the upper level of the $i$ transition, through absorption of UV photons. Therefore, the $f/i$ ratio depends on the intensity of the UV field produced by the hot photosphere, i.e. the distance from the photosphere at the location where the given lines form. The $f/i$ ratio is also density sensitive and can be expressed as $R = f/i = R_0 / [1 + \phi/\phi_c + n_e/n_c]$, where $\phi_c$ is a critical value of the UV intensity at the energy coupling the $f$ and $i$ upper levels, and $n_c$ is the density critical value; we note however that densities are generally expected to be below $n_c$. The observed He-like line intensities appear to confirm the wind-shock model when the spatial distribution of the X-ray emitting plasma is properly taken into account [@Leutenegger06]. However there are still unresolved issues. For instance X-ray observations imply opacities that are low and incompatible with the mass loss rates derived otherwise (see e.g., [@Owocki01]).
Cool stars and the solar analogy
================================
The Sun, thanks to its proximity, is at present the only star that can be studied at a very high level of detail, with high spatial and temporal resolution, and it is usually used as a paradigm for the interpretation of the X-ray emission of other late-type stars. However, while the solar analogy certainly seems to apply to some extent to other cool stars, it is not yet well understood how different the underlying processes are in stars with significantly different stellar parameters and X-ray activity levels.
X-ray activity cycles
---------------------
The $\sim 11$ yr cycle of activity is one of the most manifest characteristics of the X-ray emission of the Sun, and yet in other stars it is very difficult to observe. This is because it is intrinsically challenging to carry out regular monitoring of stellar X-ray emission over long enough time scales, and to confidently identify long term cyclic variability from short term variations that are not unusual in cool stars (e.g., flares, rotational modulation). Long term systematic variability similar to the Sun’s cycle has now been observed in three solar-like stars: HD 81809 (G5V, [@Favata08]), 61 Cyg A (K5V, [@Hempelmann06]), $\alpha$ Cen A (G2V, [@Ayres09]). The existence of X-ray cycles in other stars nicely confirms the solar-stellar analogy, and it is also potentially useful in order to better understand the dynamo activity on the Sun, which remains a significant challenge.
The Sun in time
---------------
Studies of large samples of solar-like stars at different evolutionary stages help investigate the evolution of the dynamo processes that are mainly responsible for the X-ray production in these cool stars. In particular, studies of this type carried out with high resolution spectroscopy, while requiring a large investment of time and therefore focusing necessarily on small samples of stars, have nonetheless provided very important insights into the response of the corona to the decline in rotation-powered magnetic field generation and dissipation, and provide details of how X-ray emission on the Sun has evolved over time, as shown for instance by Telleschi et al. [@Telleschi05]. This in turn could be relevant to the evolution of the solar system and the earth’s atmosphere (see Feigelson’s paper in this issue). Within relatively short timescales, during the post T Tauri through early main sequence phase, the efficient mass loss spins down the star significantly. This affects the dynamo process because the stellar rotation rate is one of the most important parameters driving the dynamo. As a consequence, the X-ray activity decreases, with coronal temperature, $L_{\rm X}$, and flare rate all decreasing, as shown in fig. \[fig:Sunlike\] for three solar-like stars spanning ages from $\sim 100$ Myr to $\sim 6$ Gyr.
Element abundances
------------------
The study of element abundances has important implications in the wider astrophysical context and also for stellar physics. For instance, chemical composition is a fundamental ingredient for models of stellar structure since it significantly impacts the opacity of the plasma. Spectroscopic studies of the solar corona have provided a robust body of evidence for element fractionation with respect to the photospheric composition (see e.g., [@Feldman92] and references therein). Furthermore, this fractionation effect appears to be a function of the element First Ionization Potential (FIP), with low FIP elements such as Fe, Si, Mg, found to be enhanced in the corona by a factor of a few, while high FIP elements such as O have coronal abundances close to their photospheric values (e.g., [@Feldman92]). This “FIP effect” has strong implications for the physical processes at work in the solar atmosphere (see e.g., [@Laming04; @Laming09] and references therein). Spectroscopic studies in the extreme ultraviolet have provided the first indication that in other stars as well the chemical composition of coronal plasma is different from that of the underlying photosphere, although with a dependence on FIP that is likely significantly different from that on the Sun (e.g., [@Drake96]). High resolution X-ray spectroscopy with and has for the first time provided robust and detailed information on the chemical composition patterns of hot coronal plasma. Stellar coronae at the high end of the X-ray activity range appear characterized by an [*inverse*]{} FIP effect (IFIP), i.e. with Fe significantly depleted in the corona, compared to the high FIP oxygen (e.g., [@Brinkman01]). Investigation of element abundances in large samples of stars spanning a large range of activity ($L_{\rm X}/L_{\rm bol} \sim 10^{-6}$–$10^{-3}$) find a systematic gradual increase of IFIP effect with activity level (e.g.,[@GAlvarez08]). This trend is shown in fig. \[fig:abund\] for the abundance ratio of low FIP element Mg to high FIP element Ne, derived from spectra for the same sample of stars for which Drake & Testa studied the Ne/O abundance ratio [@DrakeTesta05]. An important caveat to keep in mind is that the stellar photospheric chemical composition is often unknown for the elements of interest, and the [**solar**]{} photospheric composition is instead used as a reference [@Sanz04].
In this context an interesting result is the behavior of Ne/O which remains rather constant over almost the whole observed range of activity [@DrakeTesta05], and, interestingly, this almost constant value is about 2.7 times higher than the adopted solar photospheric value. This might help to shed light on an outstanding puzzle in our understanding of our own Sun. Since Ne cannot be measured in the photosphere – no photospheric Ne lines are present in the solar spectrum – the solar photospheric Ne/O is not constrained. The remarkably constant Ne/O observed in stellar coronae, despite the significantly different properties of these stars, suggests that the observed coronal Ne/O actually reflects the underlying photospheric abundances. If the same value is assumed for the solar photosphere as well, this would help resolve a troubling inconsistency between solar models and data from helioseismology observations [@AntiaBasu05]. It remains unresolved though why the solar coronal Ne/O is found to be systematically lower than in other coronae (e.g., [@Young05]), though this is likely similar to other low activity stars [@Robrade08]. However, Laming [@Laming09] suggests that the low coronal Ne abundance on the Sun might be explained by the same fractionation processes that yield the general FIP effect.
Spatial structuring of X-ray emitting plasma and dynamic events
---------------------------------------------------------------
High spectral resolution in X-rays has made accessible a whole new range of possible diagnostics for the spatial structuring of stellar coronae, for example:
- [**opacity**]{} effects in strong resonance lines yield estimates of path length, and therefore the spatial extent of X-ray emitting structures. Only a handful of sources show scattering effects in their strongest lines, and the derived lengths are very small when compared to the stellar radii, analogous to solar coronal structures [@Testa04b; @Matranga05; @Testa07b].
- [**velocity modulation**]{} derived from line shifts allows us to estimate the spatial distribution of the X-ray emitting plasma at different temperatures, or the contribution of multiple system components to the total observed emission (e.g., [@Brickhouse01; @Chung04; @Ishibashi06; @Huenemoerder06; @Hussain07]). The unprecedented high spectral resolution of is crucial for these studies with a velocity resolution down to $\sim 30$ km s$^{-1}$ (e.g., [@Hoogerwerf04; @Ishibashi06; @Huenemoerder06]).
- [**plasma density**]{}, $n_e$, can be derived from the ratios of He-like triplets ($R = f/i \sim R_0 / [1 + n_e/n_c]$; [@GabrielJordan69]) [^1], therefore providing an estimate of the emitting volumes, since the observed line intensity is proportional to $n_e^2 V$. Several He-like triplet lines lie in the and spectral range covering a wide range of temperatures ($\sim 3-10$ MK from to ), and densities ($\log (n_c[$cm$^{-3}]) \sim 10.5-13.5$ from to ). We note that the unmatched resolving power of is crucial to resolve the numerous blends that affect the Ne and Mg triplets that cover the important $\sim 3-6 \times 10^6$ K range. Studies of plasma densities from He-like triplets in large samples of stars ([@Testa04a] studied , , , and [@Ness04] and ) yield estimates of coronal filling factors which are remarkably small especially for hotter plasma (typically $\ll 1$), but increase with X-ray surface flux [@Testa04a].
- [**flares**]{} can provide clues on the size of the X-ray emitting structures and on the underlying physical processes that produce very dynamic events. The timescale of evolution of the flaring plasma (T, $n_e$) is related to the size of the flaring structure(s), and can be modeled to provide constraints on the loop size (see e.g., [@Reale07] and references therein). Flares we observe in active stars involve much larger amounts of energy than observed on the Sun, with X-ray luminosities reaching values of $10^{32}$ erg s$^{-1}$ and above, i.e. more than two orders of magnitude larger than the most powerful solar flares. It is therefore not obvious that these powerful stellar flares are simply scaled up ($L_{\rm X}$, T, characteristic timescales of evolution) versions of solar flares which we can study and model with a much higher level of detail. Novel diagnostics are provided by high resolution spectra, and time-resolved high resolution spectroscopy of stellar flares is now possible with and , at least for large flares in bright nearby sources. G[ü]{}del et al. [@Guedel02] have studied a large flare observed on Proxima Centauri, observing phenomena analogous to solar flaring events: density enhancement during the flare, supporting the scenario of chromospheric evaporation, and the Neupert effect, i.e. proportionality between soft X-ray emission and the integral of the non-thermal emission (e.g., [@Hudson92]).\
An interesting, and potentially powerful new diagnostic is provided by [**Fe K$\alpha$** ]{}(6.4 keV, 1.94Å) emission, which can be observed in and spectra. On the Sun Fe K$\alpha$ emission has been observed during flares (e.g., [@Parmar84]) and it is interpreted as [**fluorescence**]{} emission following inner shell ionization of [*photospheric*]{} neutral Fe due to hard X-ray coronal emission ($> 7.11$ keV). In this scenario, the efficiency of Fe K$\alpha$ production depends on the geometry, i.e. on the height of the source of hard ionizing continuum, through the dependence on the solid angle subtended and the average depth of formation of Fe K$\alpha$ photons (e.g., [@Bai79; @Drake08]). In cool stars other than the Sun, Fe K$\alpha$ has now been detected in young stars with disks (see next section) where the fluorescent emission is thought to come from the cold disk material, and in only two, supposedly diskless, sources during large flares: the G1 yellow giant HR 9024 [@Testa08a], and the RS CVn system II Peg [@Osten07; @Ercolano08]. For HR 9024 the observations can be matched in detail with a hydrodynamic model of a flaring loop yielding an estimate for the loop height $h \sim 0.3 R_{\star}$ [@Testa07a], and an [*effective height*]{} for the fluorescence production of $\sim 0.1 R_{\star}$ ($R_{\star}$ being the stellar radius). These values compare well with the value derived from the analysis of the measured Fe K$\alpha$ emission, $h \lesssim 0.3 R_{\star}$.
Young stars: powerful coronae, accretion, jets, magnetic fields and winds
=========================================================================
X-ray emission from young stars is presently one of the hot topics in X-ray astrophysics. Stellar X-rays are thought to significantly affect the dynamics, heating and chemistry of protoplanetary disks, influencing their evolution (see article by E.Feigelson in this same issue). Also, irradiation of close-in planets increases their mass loss rates possibly to the extent of complete evaporation of their atmospheres (e.g., [@Penz08]).
Young stars are typically characterized by strong and variable X-ray emission (e.g., [@Preibisch05]), and many recent and studies have been investigating whether their coronae might just be powered up versions of their evolved main sequence counterparts, or whether other processes might be at work in these early evolutionary stages. For example, the observations have addressed the issue of accretion-related X-ray emission processes in accreting (classical) T Tauri stars (CTTS), on which material from a circumstellar disk is channeled onto the central star by its magnetic field.
CTTS have observed X-ray luminosities that are systematically smaller by about a factor 2 than non accreting TTS (WTTS), (e.g., [@Preibisch05]). It is not yet clear however if accretion might suppress or obscure coronal X-rays, or instead, whether higher X-ray emission levels might increase photoevaporation of the accreting material, modulating the accretion rate [@Drake09].
Accretion related X-ray production
----------------------------------
High resolution spectroscopy has proved crucial for probing the physics of X-ray emission processes in young stars. The first high resolution X-ray spectrum of an accreting TTS, TW Hya, has revealed obvious peculiarities [@Kastner02] with respect to the coronal spectra of main sequence cool stars:
- [**very soft emission**]{}: the X-ray spectrum of TW Hya is characterized by a temperature of only few MK ($\sim 3$ MK) whereas coronae with comparable X-ray luminosities ($L_{\rm X} \sim 10^{30}$ erg s$^{-1}$) typically have strong emission at temperatures $\gtrsim$ 10 MK.
- [**high $n_e$**]{}: the strong cool He-like triplets of Ne and O have line ratios that imply very high densities ($n_e \gtrsim 10^{12}$ cm$^{-3}$), whereas in non-accreting sources typical densities are about two orders of magnitude lower.
- [**abundance anomalies**]{}: the X-ray spectrum of TW Hya is characterized by very low metal abundances, while Ne is extremely high [@Stelzer04; @Drake05] when compared to other stellar coronae.
These peculiar properties strongly suggest that the X-ray emission of TW Hya is originating from shocked accreting plasma. Indeed, the observed X-ray spectra of some of these sources have been successfully modeled as accretion shocks [@Gunther07; @Sacco08]. High resolution spectra subsequently obtained for other CTTS have confirmed unusually high $n_e$ from the lines [@Schmitt05; @Gunther06; @Argiroffi07; @Robrade07], indicating that in these stars at least some of the observed X-rays are most likely produced through accretion-related mechanisms. We note that TW Hya is the CTTS for which the cool X-ray emission produced in the accretion shocks is the most prominent with respect to the coronal emission, while all other CTTS for which high resolution spectra have been obtained have a much stronger coronal component. For these latter sources we are able to probe accretion related X-rays only thanks to the high spectral resolution which allows us to separate the two components. Recent studies of optical depth effect in strong resonance lines in CTTS provide confirmation of the high densities derived from the He-like diagnostics [@Argiroffi09]. Another diagnostic of accretion related X-ray production mechanisms is offered by the / ratio which, in accreting TTS, is much larger than in non-accreting TTS or main sequence stars [@Guedel07] (see also fig.\[fig:HAe\]). Herbig AeBe stars, young intermediate mass analogs of TTS, appear to share the same properties [@Robrade07].
Flaring activity and coronal geometry
-------------------------------------
X-ray emission of young stars is characterized by very high levels of X-ray variability pointing to very intense flaring activity in the young coronae of TTS. This is beautifully demonstrated by the Orion Ultradeep Project (COUP) of almost uninterrupted (spanning about 13 days) observations of the Orion Nebula Cluster star forming region[^2]. Hydrodynamic modeling of some of the largest flares of TTS imply, for some of these sources, very large sizes for the flaring structures ($L \gtrsim 10 R_{\star}$). This may provide evidence of a star-disk connection [@Favata05]. However, follow-up studies of these flares indicate that the largest structures seem to be associated with non-accreting sources, consistent with the idea that in accreting sources, the inner disk, reaching close to the star, might truncate the otherwise very large coronal structures [@Getman08]. In a few of these sources, with strong hard X-ray spectra, Fe K$\alpha$ emission has been observed (see e.g., [@Tsujimoto05] for a survey of Orion stars). The Fe K$\alpha$ emission is generally interpreted as fluorescence from the circumstellar disk, however in a few cases the observed equivalent widths are extremely high and apparently incompatible with fluorescence models (see e.g., [@Czesla07; @Giardino07]). This apparent discrepancy could either be due to partial obscuration of the X-ray emission of the flare [@Drake08] or could instead point to different physical processes at work, for instance impact excitation [@Emslie86].
Herbig Ae stars
---------------
In their pre-main sequence phase, intermediate mass stars appear to be moderate X-ray sources (e.g., [@Stelzer09] and references therein). Their X-ray emission characteristics are overall similar to the lower mass TTS (hot, variable), possibly implying that the same X-ray emission processes are at work in the two classes of stars, or that the emission is due to unseen TTS companions. However, a handful of Herbig Ae stars show unusually soft X-ray emission: e.g., AB Aur [@Telleschi07], HD 163296 [@Swartz05; @Gunther09]. High resolution spectra have been obtained for these stars, together with HD 104237 [@Testa08]. One similarity with the high resolution spectra of CTTS, appears to be the presence of a soft excess (/), compared to coronal sources, as shown by [@Gunther09] (see figure \[fig:HAe\]). However their He-like triplets are generally compatible with low density, at odds with the accreting TTS (with maybe the exception of HD 104237, possibly indicating higher $n_e$). AB Aur and HD 104237 have X-ray emission that seems to be modulated on timescales comparable with the rotation period of the A-type star therefore rendering the hypothesis that X-ray emission originates from low-mass companions less plausible.
Conclusions
===========
The past decade of stellar observations has led to exciting progress in our understanding of the X-ray emission processes in stars, also shifting in the process the perspective of stellar studies which are now much more focused on star and planet formation. In particular high resolution X-ray spectroscopy, available for the first time with and , is now playing a crucial role in constraining and developing models of X-ray emission, e.g., for early-type stars, late-type stellar coronae, and in the case of young stars, by providing a unique means for probing accretion related X-ray emission processes, as well as the opportunity to examine the effects of X-rays on the circumstellar environment.
Progress and some open issues
-----------------------------
X-ray emission processes in early-type stars now present a much more complex scenario, in which magnetic fields also likely play a key role. Some puzzling results found for several massive stars concern the hard, variable X-ray spectra with relatively narrow lines, which cannot be explained by existing models.
Spectroscopic studies of large samples of stars have provided robust findings on chemical fractionation in X-ray emitting plasma, which now require improved models to understand the physical processes yielding the observed abundance anomalies.
A satisfactory understanding of activity cycles is lacking even for our own Sun, and recent discoveries of X-ray cycles on other stars can provide further constrains for dynamo models.
We are now taking the first steps in studying flares with temporally resolved high resolution spectra, and this will greatly help constrain our models and really test whether the physics of these dynamic events, in the extreme conditions seen in some cases (e.g., $T \gtrsim 10^8$ K), are still the same as for solar flares. At present, the effective areas are often insufficient to obtain good S/N at high spectral resolution on the typical timescales of the plasma evolution during these very dynamic events. The International X-ray Observatory (IXO), in the planning stages for a launch about a decade from now, will make a large number of stars accessible for this kind of study.
In young stars, a very wide range of phenomena are observed to occur, and while this young field has already offered real breakthroughs there is still a long way to go to understand the details of accretion, jets, extremely large X-ray emitting structures, the influence of X-rays on disks and planets, and the interplay between accretion and X-ray activity.
This work has greatly benefited greatly from discussions with several people, and, in particular, I would like to warmly thank Jeremy Drake and Manuel G[ü]{}del. I also would like to thank Hans Moritz G[ü]{}nther for permission to use original figure material. This work has been supported by NASA grant GO7-8016C.
Friedman H., Lichtman S.W. & Byram E.T. (1951) Photon counter measurements of solar X-rays and Extreme Ultraviolet light. Ph. Rv. 83:1025-1030. Vaiana, G.S. et al. (1981) Results from an extensive Einstein stellar survey. ApJ 245:163-182. Lucy L.B. & White R.L. (1980) X-ray emission from the winds of hot stars. ApJ 241:300-305. Owocki S.P., Castor J.I. & Rybicki G.B. (1988) Time-dependent models of radiatively driven stellar winds. I - Nonlinear evolution of instabilities for a pure absorption model. ApJ 335:914-930. Czesla S., & Schmitt J.H.H.M. (2007) Are magnetic hot stars intrinsic X-ray sources? A&A 465:493-499. Schr[" o]{}der C. & Schmitt J.H.M.M. (2007) X-ray emission from A-type stars. A&A 475:677-684. Canizares C.R. et al. (2005) The Chandra High-Energy Transmission Grating: Design, Fabrication, Ground Calibration, and 5 Years in Flight. PASP 117:1144-1171. Donati J.F. et al. (2002) The magnetic field and wind confinement of [$\theta^{1}$]{} Orionis C. MNRAS 333:55-70. Cassinelli J.P., Miller N.A., Waldron W.L., MacFarlane J.J. & Cohen D.H. (2001) Chandra Detection of Doppler-shifted X-Ray Line Profiles from the Wind of [$\zeta$]{} Puppis (O4 F). ApJL 554:55-58. Cohen D.H. et al. (2006) Wind signatures in the X-ray emission-line profiles of the late-O supergiant [$\zeta$]{} Orionis. MNRAS 368:1905-1916. Miller N.A., Cassinelli J.P., Waldron W.L., MacFarlane J.J. & Cohen D.H. (2002) New challenges for wind shock models: the Chandra spectrum of the hot star [$\delta$]{} Orionis. ApJ 577:951-960. Skinner S.L et al. (2008) High-resolution Chandra X-ray imaging and spectroscopy of the [$\sigma$]{} Orionis cluster. ApJ 683:796-812. Gagn[é]{} M. et al. (2005) Chandra HETGS multiphase spectroscopy of the young magnetic O star [$\theta^1$]{} Orionis C. ApJ 628:986-1005. Cohen D.H. et al. (2003) High-Resolution Chandra Spectroscopy of [$\tau$]{} Scorpii: A Narrow-Line X-Ray Spectrum from a Hot Star. ApJ 586:495-505. Leutenegger M.A., Paerels F.B.S., Kahn S.M. & Cohen D.H. (2006) Measurements and analysis of Helium-like triplet ratios in the X-ray spectra of O-type stars. ApJ 650:1096-1110. Owocki S.P. & Cohen D.H. (2001) X-ray line profiles from parameterized emission within an accelerating stellar wind. ApJ 559:1108-1116. Favata F., Micela G., Orlando S., Schmitt J.H.M.M. & Sciortino S. (2008) The X-ray cycle in the solar-type star HD 81809. XMM-Newton observations and implications for the coronal structure. A&A 490:1121-1126. Hempelmann A. et al. (2006) Coronal activity cycles in 61 Cygni. A&A 460:261-267. Ayres T.R. (2009) The Cycles of [$\alpha$]{} Centauri. ApJ 696:1931-1949. Telleschi A. et al. (2005) Coronal evolution of the Sun in time: high-resolution X-ray spectroscopy of solar analogs with different ages. ApJ 622:653-679. Feldman U. (1992) Elemental abundances in the upper solar atmosphere. Phys. Scr. 46:3:202-220. Laming J.M. (2004) A unified picture of the first ionization potential and inverse first ionization potential effects. ApJ 614:1063-1072. Laming J.M. (2009) Non-Wkb models of the first ionization potential effect: implications for solar coronal heating and the coronal Helium and Neon abundances. ApJ 695:954-969. Drake J.J., Laming J.M. & Widing K.G. (1996) The FIP effect and abundance anomalies in late-type stellar coronae. IAU Colloq. 152: Astrophysics in the Extreme Ultraviolet, Ed. S. Bowyer & R.F. Malina, 97. Brinkman A.C. et al. (2001) First light measurements with the XMM-Newton reflection grating spectrometers: Evidence for an inverse first ionisation potential effect and anomalous Ne abundance in the Coronae of HR 1099. ApJL 365:324-328. Garc[í]{}a-Alvarez D., Drake J.J., Kashyap V.L., Lin L. & Ball B. (2008) Coronae of young fast rotators. ApJ 679:1509-1521. Drake J.J. & Testa P. (2005) The ‘solar model problem’ solved by the abundance of neon in nearby stars. Nature 436:525-528. Sanz-Forcada J., Favata F. & Micela G. (2004) Coronal versus photospheric abundances of stars with different activity levels. A&A 416:281-290. Antia H.M. & Basu S. (2005) The discrepancy between solar abundances and helioseismology. ApJL 620:129-132. Young, P.R. (2005) The Ne/O abundance ratio in the quiet Sun. A&A 444:L45-L48. Robrade J., Schmitt J.H.M.M. & Favata F. (2008) Neon and oxygen in low activity stars: towards a coronal unification with the Sun. A&A 486:995-1002. Testa P., Drake J.J., Peres G. & DeLuca E.E. (2004) Detection of X-ray resonance scattering in active stellar coronae. ApJL 609:L79-L82. Matranga M., Mathioudakis M., Kay H.R.M. & Keenan F.P. (2005) Flare X-ray observations of AB Doradus: evidence of stellar coronal opacity. ApJL 621:L125-L128. Testa P., Drake J.J., Peres G. & Huenemoerder D.P. (2007) On X-ray optical depth in the coronae of active stars. ApJ 665:1349-1360. Brickhouse N.S., Dupree A.K & Young P.R. (2001) X-Ray Doppler Imaging of 44i Bootis with Chandra. ApJL 562:75-78. Chung S.M., Drake J.J., Kashyap V.L., Lin L. & Ratzlaff P.W. (2004) Doppler Shifts and Broadening and the Structure of the X-Ray Emission from Algol. ApJ 606:1184-1195. Ishibashi K., Dewey D., Huenemoerder D.P. & Testa, P. (2006) Chandra/HETGS observations of the Capella system: the primary as a dominating X-ray source. ApJL 644:L1171-L1120. Huenemoerder D.P., Testa P. & Buzasi D.L. (2006) X-ray spectroscopy of the Contact binary VW Cephei. ApJ 650:1119-1132. Hussain, G.A.J. et al. (2007) The coronal structure of AB Doradus determined from contemporaneous Doppler imaging and X-ray spectroscopy. MNRAS 377:1488-1502. Hoogerwerf R., Brickhouse N.S. & Mauche C.W. (2004) The radial velocity and mass of the white dwarf of EX Hydrae measured with Chandra. ApJ 610:411-415. Gabriel A.H. & Jordan C. (1969) Interpretation of solar helium-like ion line intensities. MNRAS 145:241. Testa P., Drake J.J. & Peres G. (2004) The density of coronal plasma in active stellar coronae. ApJ 617:508-530. Ness J.-U., G[ü]{}del M., Schmitt J.H.M.M., Audard M. & Telleschi A. (2004) On the sizes of stellar X-ray coronae. A&A 427:667-683. Reale F. (2007) Diagnostics of stellar flares from X-ray observations: from the decay to the rise phase. A&A 417:271-279. G[" u]{}del M., Audard M., Skinner S.L. & Horvath M.I. (2002) X-ray evidence for flare density variations and continual chromospheric evaporation in Proxima Centauri. ApJL 580:L73-L76. Hudson H.S., Acton L.W., Hirayama T. & Uchida Y. (1992) White-light flares observed by YOHKOH. PASJ 44:L77-L81. Parmar A.N. et al. (1984) SMM observations of K-[$\alpha$]{} radiation from fluorescence of photospheric iron by solar flare X-rays. ApJ 279:866-874. Bai T. (1979) Iron K-[$\alpha$]{} fluorescence in solar flares - A probe of the photospheric iron abundance. Sol. Phys. 62:113-121. Drake J.J., Ercolano B. & Swartz D.A. (2008) X-Ray-fluorescent Fe K[$\alpha$]{} lines from stellar photospheres. ApJ 678:385-393. Testa P. et al. (2008) Geometry diagnostics of a stellar flare from fluorescent X-rays. ApJL 675:L97-L100. Osten R.A. et al. (2007) Nonthermal hard X-ray emission and iron K[$\alpha$]{} emission from a superflare on II Pegasi. ApJ 654:1052-1067. Ercolano B., Drake J.J., Reale F., Testa P. & Miller J.M. (2008) Fe K[$\alpha$]{} and hydrodynamic loop model diagnostics for a large flare on II Pegasi. ApJ 688:1315-1319. Testa P., Reale F., Garcia-Alvarez D., & Huenemoerder D.P. (2007) Detailed diagnostics of an X-ray flare in the single giant HR 9024. ApJ 663:1232-1243. Penz T., Micela G. & Lammer H. (2008) Influence of evolving stellar X-ray luminosity distribution on exoplanetary mass loss. A&A 477:309-314. Preibisch T. et al. (2005) The origin of T Tauri X-ray emission: new insights from the Chandra Orion Ultradeep Project. ApJS 160:401-422. Drake J.J., Ercolano B., Flaccomio E., & Micela G. (2009) X-ray photoevaporation-starved T Tauri accretion. ApJL 699:35-38. Kastner J.H., Huenemoerder D.P., Schulz N.S., Canizares C.R. & Weintraub D.A. (2002) Evidence for accretion: high-resolution X-ray spectroscopy of the classical T Tauri star TW Hydrae. ApJ 567:434-440. Stelzer B. & Schmitt J.H.M.M. (2004) X-ray emission from a metal depleted accretion shock onto the classical T Tauri star TW Hya. A&A 418:687-697. Drake J.J., Testa P. & Hartmann L. (2005) X-Ray diagnostics of grain depletion in matter accreting onto T Tauri stars. ApJL 627:149-152. G[ü]{}nther H.M., Schmitt J.H.M.M., Robrade J. & Liefke C. (2007) X-ray emission from classical T Tauri stars: accretion shocks and coronae?. A&A 466:1111-1121. Sacco G.G. et al. (2008) X-ray emission from dense plasma in classical T Tauri stars: hydrodynamic modeling of the accretion shock. A&A 491:L17-L20. Schmitt J.H.M.M., Robrade J., Ness J.U., Favata F. & Stelzer B. (2005) X-rays from accretion shocks in T Tauri stars: The case of BP Tau. A&A 432:L35-L38. G[ü]{}nther H.M., Liefke C., Schmitt J.H.M.M., Robrade J. & Ness J.U. (2006) X-ray accretion signatures in the close CTTS binary V4046 Sagittarii. A&A 459:L29-L32. Argiroffi C., Maggio A. & Peres G. (2007) X-ray emission from MP Muscae: an old classical T Tauri star. A&A 465:5-8. Robrade J. & Schmitt J.H.M.M. (2007) X-rays from RU Lupi: accretion and winds in classical T Tauri stars. A&A 473:229-238. Argiroffi C. et al. (2009) X-ray optical depth diagnostics of T Tauri accretion shocks. A&A 507:939-948. G[" u]{}del M. & Telleschi A. (2007) The X-ray soft excess in classical T Tauri stars. A&A 474:L25-L28. Favata F. et al. (2005) Bright X-Ray flares in Orion young stars from COUP: evidence for star-disk magnetic fields?. ApJS 160:469-502. Getman K.V. et al. (2008) X-ray flares in Orion young stars. II. Flares, magnetospheres, and protoplanetary disks. ApJ 688:437-455. Tsujimoto M. et al. (2005) Iron fluorescent line emission from young stellar objects in the Orion Nebula. ApJS 160:503-510. Czesla S., & Schmitt J.H.H.M. (2007) The nature of the fluorescent iron line in V 1486 Orionis. A&A 470:L13-L16. Giardino G. et al. (2007) Results from Droxo. I. The variability of fluorescent Fe 6.4 keV emission in the young star Elias 29. High-energy electrons in the star’s accretion tubes?. A&A 475:891-900. Emslie A.G., Phillips K.J.H. & Dennis B.R. (1986) The excitation of the iron K-[$\alpha$]{} feature in solar flares. Sol. Phys. 103:89-102. Stelzer B., Robrade J., Schmitt J.H.M.M. & Bouvier J. (2009) New X-ray detections of Herbig stars. A&A 493:1109-1119. Telleschi A. et al. (2007) The first high-resolution X-ray spectrum of a Herbig star: AB Aurigae. A&A 468:541-556. Swartz D.A. et al. (2005) The Herbig Ae star HD 163296 in X-rays. ApJ 628:811-816. G[ü]{}nther H.M. & Schmitt J.H.M.M. (2009) The enigmatic X-rays from the Herbig star HD 163296: Jet, accretion, or corona?. A&A 494:1041-1051. Testa P., Huenemoerder D.P., Schulz N.S. & Ishibashi K. (2008) X-Ray emission from young stellar objects in the [$\epsilon$]{} Chamaeleontis group: the Herbig Ae star HD 104237 and associated low-mass stars. ApJ 687:579-597.
{width="16cm"}
![ Abundance ratio of high FIP Ne to low FIP Mg for a sample of stars covering a wide range of activity. The abundance ratio is derived through a ratio of combination of H-like and He-like resonance lines, which is optimized to make the ratio largely temperature insensitive, as in [@DrakeTesta05]. The sample of spectra is the same analyzed by Drake & Testa [@DrakeTesta05]. []{data-label="fig:abund"}](plot_nemg_col.ps){width="10cm"}
![ The ratio of / vs. oxygen luminosity for a large sample of main sequence and pre-main sequence stars shows the soft excess in high-resolution spectra of CTTS and HAe stars with respect to main sequence and non accreting stars. The figure is a modified version of fig.7 of G[ü]{}nther et al. ([@Gunther09]), where the data point for HD 104237 (using measured fluxes from [@Testa08]) has been added. []{data-label="fig:HAe"}](fig_softexc_guenther_n.ps){width="8cm"}
[^1]: For cool stars the UV field is typically too weak to affect the He-like lines (which it does for hot stars as mentioned above) and therefore the $f/i$ ratio is mainly sensitive to the plasma density, above a critical density value which depends on the specific triplet (see [@GabrielJordan69]).
[^2]: Movies of this dataset are available at http://www.astro.psu.edu/coup/.
|
---
abstract: |
The ability to automatically detect certain types of cells or cellular subunits in microscopy images is of significant interest to a wide range of biomedical research and clinical practices. Cell detection methods have evolved from employing hand-crafted features to deep learning-based techniques. The essential idea of these methods is that their cell classifiers or detectors are trained in the pixel space, where the locations of target cells are labeled. In this paper, we seek a different route and propose a convolutional neural network (CNN)-based cell detection method that uses encoding of the output pixel space. For the cell detection problem, the output space is the sparsely labeled pixel locations indicating cell centers. We employ random projections to encode the output space to a compressed vector of fixed dimension. Then, CNN regresses this compressed vector from the input pixels. Furthermore, it is possible to stably recover sparse cell locations on the output pixel space from the predicted compressed vector using $L_1$-norm optimization. In the past, output space encoding using compressed sensing (CS) has been used in conjunction with linear and non-linear predictors. To the best of our knowledge, this is the first successful use of CNN with CS-based output space encoding. We made substantial experiments on several benchmark datasets, where the proposed CNN + CS framework (referred to as CNNCS) achieved the highest or at least top-3 performance in terms of F1-score, compared with other state-of-the-art methods.
**Keywords**: Cell Detection, Convolutional Neural Network, Compressed Sensing.
author:
- Yao Xue
- Nilanjan Ray
bibliography:
- 'refs.bib'
title: Cell Detection in Microscopy Images with Deep Convolutional Neural Network and Compressed Sensing
---
Introduction
============
Automatic cell detection is to find whether there are certain types of cells present in an input image (e.g. microscopy images) and to localize them in the image. It is of significant interest to a wide range of medical imaging tasks and clinical applications. An example is breast cancer, where the tumor proliferation speed (tumor growth) is an important biomarker indicative of breast cancer patients’ prognosis. In practical scenario, the most common method is routinely performed by pathologists, who examine histological slides under a microscope based on their empirical assessments, which could be really accurate in several cases, but generally is slow and prone to fatigue induced errors.
Cell detection and localization constitute several challenges that deserves our attention. First, target cells are surrounded by clutters represented by complex histological structures like capillaries, adipocytes, collagen etc. In many cases, the size of the target cell is small, and consequently, it can be difficult to distinguish from the aforementioned clutter. Second, the target cells can appear very sparsely (only in tens), moderately densely (in tens of hundreds) or highly densely (in thousands) in a typical 2000-by-2000 pixel high resolution microscopy image as shown in Fig. \[fig:example1\]. Additionally, significant variations in the appearance among the targets can also be seen. These challenges render the cell detection/localization/counting problems far from being solved at the moment, in spite of significant recent progresses in computer vision research.
![Left picture shows a microscopy image with two target cells annotated by yellow crosses on their centers. Right top pictures give details about the two target cells whose nuclei are in mitotic phase. Right bottom pictures provide more examples of mitotic figures.[]{data-label="fig:example1"}](45.jpg){width="13cm"}
In recent years, object detection has been significantly advanced following the big success by deep learning. However, cell detection or localization task is not simply a sub-task of a general object detection, which typically deals with extended objects, such as humans and vehicles that occupy a significant portion of the field of view in the image. Extended object detection and localization have witnessed much progress in the computer vision community. For example, Region-based Convolutional Neural Networks (R-CNN) [@rcnn] and its variants [@fast-rcnn], [@faster-rcnn], Fully Convolutional Networks (FCN) [@FCN] with recent optimization [@redmon2016yolo9000] have become the state-of-the-art algorithms for the extended object detection problem. These solutions cannot be easily translated to cell detection, since assumptions and challenges are different for the latter. For example, for an extended object, localization is considered successful if a detection bounding box is 50% overlapping with the actual bounding box. For cell detection, tolerance is typically on a much tighter side in order for the localization to be meaningful.
**Conventional cell detection approaches**
In the last few decades, different cell recognition methods had been proposed [@Meijering12]. Traditional computer vision based cell detection systems adopt classical image processing techniques, such as intensity thresholding, feature detection, morphological filtering, region accumulation, and deformable model fitting. For example, Laplacian-of-Gaussian (LoG) [@LoG] operator was a popular choice for blob detection; Gabor filter or LBP feature [@LBP] offers many interesting texture properties and had been attempted for a cell detection task [@LoG-cellSeg].
Conventional cell detection approaches follow a “hand-crafted feature representation”+“classifier” framework. First, detection system extracts one (or multiple) kind of features as the representation of input images. Image processing techniques offer a range of feature extraction algorithms for selection. After that, machine learning based classifiers work on the feature vectors to identify or recognize regions containing target cells. “Hand-crafted feature representation”+“classifier” approaches suffer from the following limitations:
\(1) It is a non-trivial and difficult task for humans to select suitable features. In many cases, it requires significant prior knowledge about the target cells and background.
\(2) Most hand-crafted features contain many parameters that are crucial for the overall performance. Consequently, users need to perform a lot of trial-and-error experiments to tune these parameters.
\(3) Usually, one particular feature is not versatile enough. The feature may often be tightly coupled with a particular type of target cell and may not work well when presented with a different type of target cell.
\(4) The performance of a hand-crafted feature-based classifier soon reaches an accuracy plateau, even when trained with plenty of training data.
**Deep learning based cell detection approaches**
In comparison to the conventioal cell detection methods, deep neural networks recently has been applied to a variety of computer vision problems, and has achieved better performance on several benchmark vision datasets [@ImageNet], [@rcnn], [@FCN]. The most compelling advantage of deep learning is that it has evolved from fixed feature design strategies towards automated learning of problem-specific features directly from training data [@LeCun-nature]. By providing massive amount of training images and problem-specific labels, users do not have to go into the elaborate procedure for the extraction of features. Instead, deep neural network (DNN) is subsequently optimized using a mini-batch gradient descent method over the training data, so that the DNN allows autonomic learning of implicit relationships within the data. For example, shallow layers of DNN focus on learning low-level features (such as edges, lines, dots), while deep layers of DNN form more abstract high-level semantic representations (such as probability maps, or object class labels).
With the advent of deep learning in the computer vision community, it is no wonder that the state-of-the-art methods in cell detection are based on deep neural networks. The essential idea behind all these methods is that detectors are trained as a classifier in the image pixel space, either as a pixel lableing [@Alpher21] or as a region proposal network [@CasNN]. Thus, these methods predict the $\left\lbrace x,y\right\rbrace $-coordinates of cells directly on a 2-D image. Because, target cell locations are sparse in an image, the classifiers in these methods face the class imbalance issue. Moreover, target cells are often only subtly different from other cells. Thus, these methods tend produce significant amount of false positives.
**Compressed sensing-based output encoding**
In this work, deviating from past approaches, we introduce output space encoding in the cell detection and localization problem. Our observation is that the output space of cell detection is quite sparse: an automated system only needs to label a small fraction of the total pixels as cell centroid locations. To provide an example, if there are $5000$ cells present in an image of size $2000$-by-$2000$ pixels, this fraction is $5000/(2000*2000) = 0.00125$, signifying that even a dense cell image is still quite sparse in the pixel space.
Based on the observation about sparse cell locations within a microscopy image, we are motivated to apply compressed sensing (CS) [@Alpher16] techniques in the cell detection task. First, a fixed length, compressed vector is formed by randomly projecting the cell locations from the sparse pixel space. Next, a deep CNN is trained to predict the encoded, compressed vector directly from the input pixels (i.e., microscopy image). Then, $L_1$ norm optimization is utilized to recover sparse cell locations. We refer to our proposed cell detection framework as CNNCS (convolutional neural network + compressed sensing).
Output space encoding/representation/transformation sometimes yields more accurate predictions in machine learning [@ECOC], [@RAkEL]. In the past, CS-based encoding was used in conjunction with linear and non-linear predictions [@CS], [@Bayesian-CS], [@output-space-thesis]. We believe, the proposed CNNCS is the first such attempt to solve cell detection and localization that achieved competitive results on benchmark datasets. There are several advantages of using CS-based output encoding for cell detection and localization. First, the compressed output vector is much shorter in length than the original sparse pixel space. So, the memory requirement would be typically smaller and consequently, there would be less risk of overfitting. Next, there are plenty of opportunities to apply ensemble averages to improve generalization. Furthermore, CS-theory dictates that pairwise distances are approximately maintained in the compressed space [@Alpher16], [@Alpher17]. Thus, even after output space encoding, the machine learner still targets the original output space in an equivalent distance norm. From earlier research, we also point out a generalization error bound for such systems. Our contribution is summarized below:
First, this is the first attempt to combine deep learning with CS-based output encoding to solve cell detection and localization problem.
Second, we try to overcome the aforementioned class imbalance issue by converting a classification problem into a regression problem, where sparse cell locations are distributed by a random projection into a fixed length vector as a target for the regression.
Third, we introduce redundancies in the CS-based output encoding that are exploited by CNN to boost generalization accuracy in cell detection and localization. This redundancies also help to reduce false detections.
Fourth, we demonstrate that the proposed CNNCS framework achieves competitive results compared to the state-of-the-art methods on several benchmark datasets and challenging cell detection contests.
Background and Related Work {#bg}
===========================
General Object Detection
------------------------
Prior to deep learning, general object detection pipeline consisted of feature extraction followed by classifiers or detectors. Detection had traditionally been addressed using the handcrafted features such as SIFT [@sift], HOG [@HOG], LBP [@LBP], etc. At that time, progress in object detection greatly depended on the invention of more discriminative hand-crafted features. Following the big success of Convolutional Neural Network (CNN) in image classification task [@ImageNet], deep learning model have been widely adopted in the computer vision community. For example, Region-based Convolutional Neural Networks (R-CNN) [@rcnn] and its variants [@fast-rcnn], [@faster-rcnn], Fully Convolutional Networks (FCN) [@FCN] with recent optimization [@redmon2016yolo9000] have become the state-of-the-art algorithms for the extended object detection problems.
Once again, cell detection or localization task is not simply a sub-task of a general object detection, those state-of-the-art solutions for general object detection are able to provide useful clues to right direction, but cannot be easily translated to cell detection.
Cell Detection and Localization
-------------------------------
The state-of-the-art methods in detection and localization today include deep learning techniques for cell detection and localization. Recently a deep Fully Convolutional Network (FCN) [@FCN] was proposed for the image segmentation problem and had shown remarkable performance. Soon after the FCN is proposed, [@Alpher25] presented a FCN-based framework for cell counting, where their FCN is responsible for predicting a spatial density map of target cells, and the number of cells can be estimated by an integration over the learned density map. Slightly similar to [@Alpher25], a cascaded network [@CasNN] has been proposed for cell detection. [@CasNN] uses a FCN for candidate region selection, and then a CNN for further discrimination between target cells and background.
In [@DNN-IDSIA], a mitosis detection method has been proposed by CNN-based prediction followed by ad-hoc post processing. As the winner of ICPR 2012 mitosis detection competition, [@DNN-IDSIA] used deep max-pooling convolutional neural networks to detect mitosis in breast histology images. The networks were trained to classify each pixel in the images, using a patch centered on the pixel as context. Then post processing was applied to the network output. In [@AggNet16], expectation maximization has been utilized within deep learning framework in an end-to-end fashion for mitosis detection. This work presents a new concept for learning from crowds that handle data aggregation directly as part of the learning process of the convolutional neural network (CNN) via additional crowd-sourcing layer. It is the first piece of work where deep learning has been applied to generate a ground-truth labeling from non-expert crowd annotation in a biomedical context.
Compressed Sensing
------------------
During the past decade, compressed sensing or compressive sensing (CS) [@Alpher16] has emerged as a new framework for signal acquisition and reconstruction, and has received growing attention, mainly motivated by the rich theoretical and experimental results as shown in many reports [@Alpher17], [@Alpher18], [@Alpher16], and so on. As we know, the Nyquist-Shannon sampling theorem states that a certain minimum sampling rate is required in order to reconstruct a band-limited signal. However, CS enables a potentially large reduction in the sampling and computation costs for sensing/reconstructing signals that are sparse or have a sparse representation under some linear transformation (e.g. Fourier transform).
Under the premise of CS, an unknown signal of interest is observed (sensed) through a limited number of linear observations. Many works [@Alpher17], [@Alpher18], [@Alpher16] have proven that it is possible to obtain a stable reconstruction of the unknown signal from these observations, under the general assumptions that the signal is sparse (or can be represented sparsely with respect to a linear basis) and matrix discoherence. The signal recovery techniques typically rely on a convex optimization with a penalty expressed by $\emph{L}_1$ norm, for example orthogonal matching pursuit (OMP) [@OMP] and dual augmented Lagrangian (DAL) method [@DAL].
Proposed Method
===============
System Overview
---------------
The proposed detection framework consists of three major components: (1) cell location encoding phase using random projection, (2) a CNN based regression model to capture the relationship between a cell microscopy image and the encoded signal $y$, and (3) decoding phase for recovery and detection. The flow chart of the whole framework is shown in Fig. \[baseline-system-overview\].
![The system overview of the proposed CNNCS framework for cell detection and localization.[]{data-label="baseline-system-overview"}](system.jpg){width="13cm"}
During training, the ground truth location of cells are indicated by a pixel-wise binary annotation map $B$. We propose two cell location encoding schemes, which convert cell location from the pixel space representation $B$ to a compressed signal representation $y$. Then, training pairs, each consisting of a cell microscopy image and the compressed signal $y$, train a CNN to work as a multi-label regression model. We employ the Euclidean loss function during training, because it is often more suitable for a regression task. Image rotations may be performed on the training sets for the purpose of data augmentation as well as making the system more robust to rotations.
During testing, the trained network is responsible for outputting an estimated signal $\hat{y}$ for each test image. After that, a decoding scheme is designed to estimate the ground truth cell location by performing $L_1$ minimization recovery on the estimated signal $\hat{y}$, with the known sensing matrix.
Cell Location Encoding and Decoding Scheme
------------------------------------------
### Encoding Schemes
In the CNNCS framework, we employ two types of random projection-based encodings as described below.
**Scheme-1: Encoding by Reshaping**
For the cell detection problem, cells are often annotated by pixel-level labels. The most common way is to attach a dot or cross at the center of every cell, instead of a bounding box around the cell. So, let us suppose there is a pixel-wise binary annotation map $B$ of size $w$-by-$h$, which indicates the location of cells by labeling 1 at the pixels of cell centroids, otherwise labeling 0 at background pixels. To vectorize the annotation map $B$, the most intuitive scheme is to concatenate every row of $B$ into a binary vector $f$ of length $wh$. Thus, a positive element in $B$ with $\{x,y\}$ coordinates will be encoded to the $[x+h(y-1)]$-th position in $f$. $f$ is also a $k$-sparse signal, so, there are at most $k$ non-zero entries in $f$. Here, we refer this intuitive encoding scheme as “Scheme-1: Encoding by Reshaping”.
After the vector $f$ is generated, we apply a random projection. CS theory guarantees that $f$ could be fully represented by linear observations $y$: $$y=\Phi f,$$ provided the sensing matrix $\Phi$ satisfies a restricted isometry property (RIP) condition [@Alpher17], [@Alpher18]. In many cases, $\Phi$ is typically a $M \times N$ ($M \ll N=hw$) random Gaussian matrix. Here, the number of observations $M$ is much smaller than $N$, and obeys: $M \geq C_M k log(N)$, where $C_M$ is a small constant greater than one.
**Scheme-2: Encoding by Signed Distances**
For the encoding scheme-1, the space complexity of the interim result $f$ is $\mathcal{O}(wh)$. For example, to encode the location of cells in a 260-by-260 pixel image, scheme-1 will produce $f$ as a 67,600-length vector; so that in the subsequent CS process, a huge sensing matrix in size of $M$-by-67600 is required in order to match the dimension of $f$, which will make the system quite slow, even unacceptable for larger images. To further optimize the encoding scheme, we propose a second scheme, where the coordinates of every cell centroid are projected onto multiple observation axes. We refer the second encoding scheme as “Scheme-2: Encoding by Projection.”
To encode location of cells, we create a set of observation axes $OA=\left\lbrace oa_{l} \right\rbrace, l=1,2,\dots, L$, where $L$ indicates the total number of observation axes used. The observation axes are uniformly-distributed around an image (See Fig. \[encoding\], left-most picture) For the $l$-th observation axis $oa_{l}$, the location of cells is encoded into a $R$-length ($R=\sqrt{w^{2}+h^{2}}$) sparse signal, referred as $f_{l}$ (See Fig. \[encoding\], third picture). We calculate the perpendicular signed distances ($f_{l}$) from cells to $oa_{l}$. Thus, $f_{l}$ contains signed distances, which not only measure the distance, but also describe on which side of $oa_{l}$ cells are located. After that, the encoding of cell locations under $oa_{l}$ is $y_{l}$, which is obtained by the following random projection: $$y_{l}=\Phi f_{l},$$ We repeat the above process for all the $L$ observation axes and obtain each $y_{l}$. After concatenating all the $y_{l}$, $l=1,2,\dots, L$, the final encoding result $y$ is available, which is the joint representation of cells location. The whole encoding process is illustrated by Fig. \[encoding\].
{width="17cm"}
For encoding scheme-2, the size of the sensing matrix $\Phi$ is $M$-by-$\sqrt{w^{2}+h^{2}}$. In comparison, encoding scheme-1 requires a much larger sensing matrix of size $M$-by-$wh$. The first advantage of encoding scheme-2 is that it dramatically reduces the size of the sensing matrix, which is quite helpful for the recovery process, especially when the size of images is large. Secondly, the encoding result $y$ carries **redundant** information about cell locations. In the subsequent decoding phase, averaging over the redundant information makes the final detection more reliable. More details can be found in experiments section. A final point is that in case more than one cell locations are projected to the same bin in a particular observation axis, such a conflict will not occur for the same set of cells at other observation axes.
### Decoding Scheme
Accurate recovery of $f$ can be obtained from the encoded signal $y$ by solving the following $L_1$ norm convex optimization problem: $$\hat{f}=\arg\min_f \| f \|_1 \qquad \text{subject to}\qquad y=\Phi f$$ After $\hat{f}$ is recovered, every true cell is localized $L$ times, i.e. with $L$ candidate positions predicted. The redundancy information allows us to estimate more accurate detection of a true cell.
The first two images of Fig. \[decoding\] from left present examples of the true location signal $f$ and decoded location signal $\hat{f}$, respectively. The noisy signed distances of $\hat{f}$ are typically very close to each observation axis. That is why we create observation axes outside of the image space, so that these noisy distances can be easily distinguished from true candidate distances. This separation is done by mean shift clustering, which also groups true detections into localized groups of detections. Two such groups (clusters) are shown in Fig. \[decoding\], where the signed distances formed circular patterns of points (in green) around ground truth detections (in yellow). Averaging over these green points belonging to a cluster provides us a predicted location (in red) as shown in Fig. \[decoding\].
![Cell Location Decoding Scheme. From left to right: true location signal $f$, decoded location signal $\hat{f}$ and detection results. Yellow crosses indicate the ground-truth location of cells, green crosses are the candidates points, red crosses represent the final detected points.[]{data-label="decoding"}](decoding.jpg){width="13cm"}
Signal Prediction by Convolutional Neural Network
-------------------------------------------------
{width="17cm"}
We utilize a CNN to build a regression model between a cell microscopy image and its cell location representation: compressed signal $y$. We employ two kinds of CNN architectures. One of them is AlexNet [@ImageNet], which consists of 5 convolution layers + 3 fully connected layers; the other is the deep residual network (ResNet) [@ResNet] where we use its 152-layer model. In both the architectures, the loss function is defined as the Euclidean loss. The dimension of output layer of AlexNet and ResNet has been modified to the length of compressed signal $y$. We train the AlexNet model from scratch, in comparison, we perform fine-tuning on the weights in fully-connected layer of the ResNet.
To prepare the training data, we generate a large number of square patches from training images. Along with each training patch, there is a signal (i.e. the encoding result: $y$), which indicates the location of target cells present in the patch. After that, patch rotation is performed on the collected training patches for data augmentation and making the system rotation invariant.
The trained CNN not only predicts the signal from its output layer, the feature maps learned from its Conv layers also provide rich information for recognition. Fig. \[heatmap\] visualizes the learned feature maps, which represents the probabilistic score or activation maps of target cell regions (indicated by green boxes in the left image) during training process. It can be observed that higher scores are fired on the target regions of score masks, while most of the non-target regions have been suppressed more and more with training process going on.
To further optimize our CNN model, we apply Multi-Task Learning (MTL) [@Caruana1997]. During training a CNN, two kinds of labels are provided. The first kind is the encoded vector: $y$, which carries the pixel-level location information of cells. The other kind is a scalar: cell count ($c$), which indicates the total number of cells in a training image patch. We concatenate the two kinds of labels into the final training label by $label = \left\lbrace y,\lambda c\right\rbrace$, where $\lambda$ is a hyper parameter. Then, Euclidean loss is applied on the fusion label. Thus, supervision information for both cell detection and cell counting can be jointly used to optimize the parameters of our CNN model.
Theoretical Justification
-------------------------
**Equivalent Targets for Optimization**
We first show that from the optimization standpoint, compressed vector is a good proxy for the original, sparse output space. This result directly follows from the CS theory. As mentioned before, $f$ indicates the cell location represented in pixel space, and $y$ is the cell location represented in compressed signal space. They follow the relationship: $y=\Phi f$, where $\Phi$ is the sensing matrix. Let us assume that $f_p$ and $f_g$ are respectively the prediction and ground-truth vectors in the pixel space. Similarly, we have $y_p$ and $y_g$ as their compressed counterparts, respectively.
**Claim:** $\left \| y_{g}-y_{p} \right \|$ and $\left \| f_{g}-f_{p} \right \|$ are approximately equivalent targets for optimization.
**Proof:** According to the CS theory, a sensing matrix $\Phi \in \mathbb{R}^{m\times d}$ should satisfy the $\left( k,\delta \right)-restricted\ isometry\ property \left(\left(k,\delta \right)-RIP\right)$, which states that for all $k-$sparse $f\in\mathbb{R}^{d}$, $\delta\in\left( 0,1\right)$, the following holds [@Alpher17], [@Alpher18], [@Alpher16]: $$\left(1-\delta \right)\|f\|\leq\|\Phi f\|\leq\left(1
+\delta \right)\|f\|.$$
Note that if the sensing matrix $\Phi$ satisfies $(2k,\delta)$-RIP, then (4) also holds good. Now replace $f$ with $\left(f_{g}-f_{p}\right)$ and note that $\left(f_{g}-f_{p}\right) $ is $2k$-sparse. Thus, $$\left(1-\delta \right)\left \| f_{g}-f_{p} \right \|\leq \left \|y_{g}-y_{p} \right \|\leq \left(1+\delta \right)\left \| f_{g}-f_{p} \right \|.$$ From the right hand side inequality, we note that if $\left \| f_{g}-f_{p} \right \|$ is small, then $\left \| y_{g}-y_{p} \right \|$ would be small too. In the same way, if $\left \| y_{g}-y_{p} \right \|$ is large, then the inequality implies that $\left \| f_{g}-f_{p} \right \|$ would be large too. Similarly, from the left hand side inequality, we note that if $\left \| f_{g}-f_{p} \right \|$ is large then $\left \| y_{g}-y_{p} \right \|$ will be large, and if $\left \| y_{g}-y_{p} \right \|$ is small then $\left \| f_{g}-f_{p} \right \|$ will small too. These relationships prove the claim that from the optimization perspective $\left \| y_{g}-y_{p} \right \|$ and $\left \| f_{g}-f_{p} \right \|$ are approximately equivalent.\
**A Bound on Generalization Prediction Error**\
In this section we mention a powerful result from [@CS]. Let $h$ be the predicted compressed vector by the CNN, $f$ be the ground truth sparse vector, $\hat{f}$ be the reconstructed sparse vector from prediction, and $\Phi$ be the sensing matrix. Then the generalization error bound provided in [@CS] is as follows: $$\|\hat{f}-f \|^{2}_{2}\leq C_{1}\cdot\|h-\Phi f \|^{2}_{2}+C_{2}\cdot sperr(\hat{f},f),$$ where $C_1$ and $C_2$ are two small constants and $sperr$ measures how well the reconstruction algorithm has worked [@CS]. This result demonstrates that expected error in the original space is bound by the expected errors of the predictor and that of the reconstruction algorithm. Thus, it makes sense to apply a very good machine learner such as deep CNN that can minimize the first term in the right hand side of (6). On the other hand, DAL provides one of the best $L_1$ recovery algorithms to minimize the second term in the right side of (6).
Experiments
===========
Datasets and Evaluation Criteria
--------------------------------
We utilize seven cell datasets, on which CNNCS and other comparison methods are evaluated. The 1st dataset [@Nuclei-data] involves 100 H$\&$E stained histology images of colorectal adenocarcinomas. The 2nd dataset [@bacterial-data] consists of 200 highly realistic synthetic emulations of fluorescence microscopic images of bacterial cells. The 3rd dataset [@our-ECCV] comprises of 55 high resolution microscopic images of breast cancers double stained in red (cytokeratin – epithelial marker) and brown (nuclear – proliferative marker). The 4th dataset is the ICPR 2012 mitosis detection contest dataset [@ICPR-2012] including 50 high-resolution (2084-by-2084) RGB microscope slides of Mitosis. The 5th dataset [@ICPR-2014] is the ICPR 2014 grand contest of mitosis detection, which is a follow-up and an extension of the ICPR 2012 contest on detection of mitosis. Compared with the contest in 2012, the ICPR 2014 contest is much more challenging, which contains way more images for training and testing. The 6th dataset is the AMIDA-2013 mitosis detection dataset [@AMIDA-2013], which contains 676 breast cancer histology images belonging to 23 patients. The 7th dataset is the AMIDA-2016 mitosis detection dataset [@AMIDA-2016], which is an extension of the AMIDA 2013 contest on detection of mitosis. It contains 587 breast cancer histology images belonging to 73 patients for training, and 34 breast cancer histology images for testing with no ground truth available. For each dataset, the annotation that represents the location of cell centroids is shown in Fig.\[dataset\], details of datasets are summarized in Table. \[data-table\].
[\*[22]{}[c]{}]{} Cell Dataset & Size & Ntr/Nte & AC\
Nuclei [@Nuclei-data]& 500$\times$500 & 50/50 & 310.22\
Bacterial [@bacterial-data] & 256$\times$256 & 100/100 & 171.47\
Ki67 Cell [@our-ECCV] & 1920$\times$2560 & 45/10 & 2045.85\
ICPR 2012 [@ICPR-2012] & 2084$\times$2084 & 35/15 & 5.31\
ICPR 2014 [@ICPR-2014] & 1539$\times$1376 & 1136/496 & 4.41\
AMIDA 2013 [@AMIDA-2013] & 2000$\times$2000 & 447/229 & 3.54\
AMIDA 2016 [@AMIDA-2016] & 2000$\times$2000 & 587/34 & 2.13\
![Dataset examples and their annotation.[]{data-label="dataset"}](dataset-h.jpg){width="13cm"}
For evaluation, we adopt the criteria of the ICPR 2012 mitosis detection contest [@ICPR-2012], which is also adopted in several other cell detection contests. A detection would be counted as true positive ($TP$) if the distance between the predicted centroid and ground truth cell centroid is less than $\rho$. Otherwise, a detection is considered as false positives ($FP$). The missed ground truth cells are counted as false negatives ($FN$). In our experiments, $\rho$ is set to be the radius of the smallest cell in the dataset. Thus, only centroids that are detected to lie inside cells are considered correct. The results are reported in terms of Precision: $P=TP/(TP+FP)$ and Recall: $R=TP/(TP+FN)$ and $F_1$-score: $F_1=2PR/(P+R)$ in the following sections.
Experiments with Encoding Scheme-1
----------------------------------
To evaluate, we carry out performance comparison experiment between CNNCS and three state-of-the-art cell detection methods (“FCN-based” [@Alpher25], “Le.detect” [@Alpher21], “CasNN” [@CasNN]). In this experiment, the scheme-1: encoding by reshaping is applied in CNNCS.
For the four methods to provide different values of Precision-Recall as shown in Fig. \[per\], we tune hyper parameters of every method. With scheme-1, CNNCS has a threshold $T$ to apply on the recovered sparse signal $\hat{f}$ before re-shaping it to a binary image $B$. $T$ is used to perform cell vs. non-cell binary classification and can be treated as a hyper parameter during training. In “FCN-based” [@Alpher25], there is also a threshold applied to the local probability-maximum candidate points to make final decision about cell or non-cell. Similarly, in the first step of “Le.detect” [@Alpher21], researchers use a MSER-detector (a stability threshold involved here) to produce a number of candidate regions, on which their learning procedure determines which of these candidates regions correspond to cells. In the first experiment, we analyze the three methods using Precision-Recall curves by varying their own thresholds.
Fig. \[per\] presents Precision-Recall curves on three cell datasets. All the four methods give reliable detection performances in the range of recall=\[0.1-0.4\]. After about recall=0.6, the precision of “FCN-based” [@Alpher25] drops much faster. This can be attributed to the fact that “FCN-based” [@Alpher25] works by finding local maximum points on a cell density map. However, the local maximum operation fails in several scenarios, for example when two cell density peaks are close to each other, or large peak may covers neighboring small peaks. Consequently, to obtain the same level of recall, “FCN-based” [@Alpher25] provides many false detections.
![Precision and recall curves of four methods on three datasets.[]{data-label="per"}](F1-Bacterial.jpg "fig:"){width="8cm"} ![Precision and recall curves of four methods on three datasets.[]{data-label="per"}](F1-Ki67.jpg "fig:"){width="8cm"} ![Precision and recall curves of four methods on three datasets.[]{data-label="per"}](F1-Nuclei.jpg "fig:"){width="8cm"}
Furthermore, it also can be observed that CNNCS has an improvement over “Le.detect” [@Alpher21] (red line clearly outperforms black line under varying recall values). This can be largely explained by the fact that traditional methods (no matter if [@Alpher21] or [@Alpher25] is used) always try to predict the coordinates of cells directly on a 2-D image. The coordinates are sensitive to system prediction bias or error, considering the nature of cell detection that cells are small and quite dense in most cases. It is not surprising that “Le.detect” [@Alpher21] will miss some cells and/or detect other cells in wrong locations. In comparison, CNNCS transfers the cell detection task from pixel space to compressed signal space, where the location information of cells is no longer represented by $\left\lbrace x,y\right\rbrace $-coordinates. Instead, CNNCS performs cell detection by regression and recovery on a fixed length compressed signal. Compared to $\left\lbrace x,y\right\rbrace$-coordinates representation, the compressed signal is more robust to system prediction errors. For example, as shown in the right top corner of Fig. \[heatmap\], even though there are differences between the ground-truth compressed signal and predicted compressed signal, the whole system can still give reliable detection performance as shown in Fig. \[per\].
To get a better idea of the CNNCS method, we visualize a set of cell images with their detected cells and ground-truth cells in Fig. \[baseline-example\]. It can be observed that CNNCS is able to accurately detect most cells under a variety of conditions.
![Detection results. Ground-truth: red, Prediction: blue. Left part shows the ground truth signal and the predicted sparse signal that carries the location information of cells; right part shows the ground-truth and detected cells.[]{data-label="baseline-example"}](baseline-example.jpg){width="13cm"}
Experiments with Encoding Scheme-2
----------------------------------
### Experiment on ICPR 2012 mitosis detection dataset
To evaluate the performance of encoding scheme 2, we carry out the second group of performance comparison experiments. In the first experiment, we apply the proposed method on the ICPR 2012 mitosis detection contest dataset, which consists of 35 training images and 15 testing images. For the training process, we extracted image sub-samples (260-by-260) with no overlap between each other from the 35 training images. After that every 90$^\circ$ image rotation is performed on each sub-sample for data augmentation, resulting in a total of 8,960 training dataset. In addition, we perform random search to tune the three hyper parameters in scheme-2: (1) the number of rows in sensing matrix: $M$, (2) the number of observation lines: $L$ and (3) the importance ($\lambda$) of cell count during MTL. After that, the best performance is achieved when $M=112, L=27, \lambda=0.20$. Furthermore, we trained five CNN models to reduce the performance variance introduced by a single model and to improve the robustness of the whole system. Recently, deep residual network (ResNet) introduces residual connections into deep convolutional networks and has yielded state-of-the-art performance in the 2015 ILSVRC challenge [@ResNet]. This raises the question of whether there is any benefit in introducing and exploiting more recent CNN architectures into the cell detection task. Thus, in the experiment, we have explored the performance of CNNCS with different neural network architectures (AlexNet and ResNet). Finally, CNNCS gets the **highest** F1-score among all the comparison methods, details are summarized in Table \[ICPR-2012-table\].
[\*[22]{}[c]{}]{} Method & Precision & Recall & F$_1$-score\
UTRECHT & 0.511 & 0.680 & 0.584\
NEC [@NEC] & 0.747 & 0.590 & 0.659\
IPAL [@IPAL] & 0.698 & 0.740 & 0.718\
DNN [@DNN-IDSIA] &0.886&0.700&0.782\
RCasNN [@CasNN] &0.720&0.713&0.716\
CasNN-single [@CasNN] &0.738&0.753&0.745\
CasNN-average [@CasNN] &0.804&0.772&0.788\
\[11pt/3pt\]
CNNCS-AlexNet &0.860&0.788&0.823\
CNNCS-ResNet &0.867&0.801&0.833\
CNNCS-ResNet-MTL &0.872&0.805&[**0.837**]{}\
Compared to the state-of-the-art method: CasNN-average [@CasNN], CNNCS with ResNet and MTL achieved a better performance with $F_1$-score 0.837. It can be observed from Table \[ICPR-2012-table\] that the precision of our method outperforms the previous best precision by 0.06-0.07, and recall also has recorded about 0.02 improvement. This phenomenon can be attributed to the detection principle of our method, where every ground-truth cell is localized with multiple candidate points guaranteed to be around the true location, then the average coordinates of these candidates is computed as the final detection. As a result, localization closer to the true cell becomes more reliable compared to other methods, thus leading to a higher precision. In addition, an improvement of $F_1$-score from 0.833 to 0.837 achieved by MTL demonstrates that the knowledge jointly learned from cell detection and cell counting provides further benefits at negligible additional computations.
### Experiment on ICPR 2014 mitosis detection dataset
In the second experiment, we evaluated CNNCS on the ICPR 2014 contest of mitosis detection dataset (also called MITOS-ATYPIA-14), which is a follow-up and an extension of the ICPR 2012 contest on detection of mitosis. Compared with the contest in 2012, the ICPR 2014 contest is much more challenging, which contains more images for training and testing. It provides 1632 breast cancer histology images, 1136 images for training, 496 images for testing. Each image is in the size of 1539$\times$1376. We divide the training images into training set (910 images) and validation set (226 images). We perform random search on the validation set to optimize the hyper parameters. The best performance on MITOS-ATYPIA-14 dataset is achieved when $M=103, L=30, \lambda=0.24$. On the test dataset, we have achieved the **highest** F1-score among all the participated teams. The F1-score of all the participated teams are shown in Table \[ICPR-2014-table\]. As we see, the CNNCS method shows significant improvement compared to the results of other teams in all the histology slice groups. On an average, CNNCS has almost doubled the F1-score of teams at the second place.
[\*[22]{}[c]{}]{} Slice group & CUHK & MINES & YILDIZ & STRAS & CNNCS\
A06 &0.119&0.317&0.370&0.160&[**0.783**]{}\
A08 &0.333&0.171&0.172&0.024&[**0.463**]{}\
A09 &0.593&0.473&0.280&0.072&[**0.660**]{}\
A19 &0.368&0.137&0.107&0.011&[**0.615**]{}\
Average &0.356&0.235&0.167&0.024&[**0.633**]{}\
### Experiment on AMIDA 2013 mitosis detection dataset
The third experiment was performed on the AMIDA-2013 mitosis detection dataset, which contains 676 breast cancer histology images, belonging to 23 patients. Suspicious breast tissue is annotated by at least two expert pathologists, to label the center of each cancer cell. We train the proposed CNNCS method using 377 images, validate on 70 training images and test it on the testing set of AMIDA-2013 challenge that has 229 images from the last 8 patients. We employ ResNet as the network architecture with data balancing and MTL in the training set. Similar to previous experiments, we perform random search on the validation set to optimize the hyper parameters. The best performance on AMIDA-2013 dataset is achieved when $M=118, L=25, \lambda=0.32$. Finally among all the 17 participated teams, we achieve the **third highest** F1-score=0.471, which is quite close to the second place, and has a significant improvement over the fourth place method [@AggNet16]. For details, Table \[AMIDA-2013-table\] summarizes the comparison between CNNCS and other methods.
[\*[22]{}[c]{}]{} Method & Precision & Recall & F$_1$-score\
IDSIA [@DNN-IDSIA] &0.610&0.612&0.611\
DTU &0.427&0.555&0.483\
AggNet [@AggNet16] &0.441&0.424&0.433\
CUHK &0.690&0.310&0.427\
SURREY &0.357&0.332&0.344\
ISIK &0.306&0.351&0.327\
PANASONIC &0.336&0.310&0.322\
CCIPD/MINDLAB &0.353&0.291&0.319\
WARWICK &0.171&0.552&0.261\
POLYTECH/UCLAN &0.186&0.263&0.218\
MINES &0.139&0.490&0.217\
SHEFFIELD/SURREY &0.119&0.107&0.113\
SEOUL &0.032&0.630&0.061\
UNI-JENA &0.007&0.077&0.013\
NIH &0.002&0.049&0.003\
\[11pt/3pt\]
CNNCS &0.3588&0.5529&0.4352\
### Experiment on AMIDA 2016 mitosis detection dataset
In the fourth experiment, we participated in the AMIDA-2016 mitosis detection challenge (also called TUPAC16), which is a follow-up and an extension of the AMIDA-2013 contest on detection of mitosis. Its training dataset has 587 breast cancer histology images in size of 2000$\times$2000, belonging to 73 patients. Its test dataset contains 34 breast cancer histology images in the same size without publicly available ground truth labels.
[\*[22]{}[c]{}]{} Team & F$_1$-score\
Lunit Inc. &0.652\
IBM Research Zurich and Brazil &0.648\
Contextvision (SLDESUTO-BOX) &0.616\
The Chinese University of Hong Kong &0.601\
Microsoft Research Asia &0.596\
Radboud UMC &0.541\
University of Heidelberg &0.481\
University of South Florida &0.440\
Pakistan Institute of Engineering and Applied Sciences &0.424\
University of Warwick &0.396\
Shiraz University of Technology &0.330\
Inha University &0.251\
\[11pt/3pt\]
CNNCS (on validation set) &0.634\
We train the proposed CNNCS method using randomly chosen 470 training images and validate on the remaining 117 training images. Additionally, we apply the following ensemble averaging technique to further increase precision and recall values. Originally, we have partitioned every test image into about 100 non-overlapping patches. Instead of starting the partitioning from the top-left corner, now we set the starting point of the first patch from {offset, offset}. The offset values are set as 0, 20, 40,..., 160, and 180 (i.e. every 20 pixel) resulting in a total of 10 different settings. Under every offset setting, CNNCS method is run on all the generated patches and provides detection results. Then, we merge detection results from all the offset settings. The merging decision rule is that if there are 6 or more detections within a radius of 9 pixels, then we accept average of these locations as our final detected cell center. Other implementation settings are similar to the settings in the experiment of AMIDA-2013. Finally, we achieved F1-score=0.634 on the validation set (becuase of the lack of publicly available test set), which is the **third highest** in all the 15 participated teams. Table \[AMIDA-2016-table\] provides more details of the contest results. Furthermore, Fig.\[200-results\] provides twelve examples of our detection results in the AMIDA-2016 grand challenge of mitosis detection.
![Results on AMIDA-2016 dataset. Yellow cross indicates the ground-truth position of target cells. Green cross indicates cell position predicted by an observation axis. Red cross indicates the final detected cell position, which is the average of all green crosses.[]{data-label="200-results"}](200-results.jpg){width="13cm"}
Conclusion
==========
This is the first attempt demonstrating that deep convolutional neural network can work in conjunction with compressed sensing-based output encoding schemes toward solving a significant medical image processing task: cell detection and localization from microscopy images. In this work, we made substantial experiments on several mainstream datasets and challenging cell detection contests, where the proposed CNN + CS framework (referred to as CNNCS) achieved very competitive (the highest or at least top-3 in terms of F1-score) results compared to the state-of-the-art methods in cell detection task. In addition, the CNNCS framework has the potential to be trained in End-to-End manner, which is our near future plan and could further boost performance.
|
---
bibliography:
- 'ff.bib'
title: |
**Optimal strategies in the Fighting Fantasy gaming system: influencing stochastic dynamics by gambling with limited resource**\
Iain G. Johnston${}^{1,2}$\
${}^1$ Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Bergen, Bergen, Norway\
${}^2$ Alan Turing Institute, London, UK
---
Abstract {#abstract .unnumbered}
========
Fighting Fantasy is a popular recreational fantasy gaming system worldwide. Combat in this system progresses through a stochastic game involving a series of rounds, each of which may be won or lost. Each round, a limited resource (‘[<span style="font-variant:small-caps;">luck</span>]{}’) may be spent on a gamble to amplify the benefit from a win or mitigate the deficit from a loss. However, the success of this gamble depends on the amount of remaining resource, and if the gamble is unsuccessful, benefits are reduced and deficits increased. Players thus dynamically choose to expend resource to attempt to influence the stochastic dynamics of the game, with diminishing probability of positive return. The identification of the optimal strategy for victory is a Markov decision problem that has not yet been solved. Here, we combine stochastic analysis and simulation with dynamic programming to characterise the dynamical behaviour of the system in the absence and presence of gambling policy. We derive a simple expression for the victory probability without [<span style="font-variant:small-caps;">luck</span>]{}-based strategy. We use a backward induction approach to solve the Bellman equation for the system and identify the optimal strategy for any given state during the game. The optimal control strategies can dramatically enhance success probabilities, but take detailed forms; we use stochastic simulation to approximate these optimal strategies with simple heuristics that can be practically employed. Our findings provide a roadmap to improving success in the games that millions of people play worldwide, and inform a class of resource allocation problems with diminishing returns in stochastic games.\
**Keywords:** stochastic game; Markov decision problem; stochastic simulation; dynamic programming; resource allocation; stochastic optimal control; Bellman equation
Introduction {#introduction .unnumbered}
============
Fantasy adventure gaming is a popular recreational activity around the world. In addition to perhaps the best-known ‘Dungeons and Dragons’ system [@gygax1974dungeons], a wide range of adventure gamebooks exist, where a single player takes part in an interactive fiction story [@costikyan2007games]. Here, the reader makes choices that influence the progress through the book, the encounters that occur, and the outcomes of combats. In many cases, die rolls are used to provide stochastic influence over the outcomes of events in these games, particularly combat dynamics. These combat dynamics affect the game outcome, and thus the experience of millions of players worldwide, yet have rarely been studied in detail.
Here we focus on the stochastic dynamics of a particular, highly popular franchise of gamebooks, the Fighting Fantasy (FF) series [@green2014you]. This series, spanning over 50 adventure gamebooks (exemplified by the famous first book in the series, ‘The Warlock of Firetop Mountain’ [@jackson2002warlock]), has sold over $20$ million copies and given rise to a range of associated computer games, board games, and apps, and is currently experiencing a dramatic resurgence in popularity [@bbc; @osterberg2008rise]. Outside of recreation, these adventures are used as pedagogical tools in the huge industry of game design [@zagal2015fighting] and in teaching English as a foreign language [@philips1994role].
In FF, a player is assigned statistics ([<span style="font-variant:small-caps;">skill</span>]{} and [<span style="font-variant:small-caps;">stamina</span>]{}), dictating combat proficiency and endurance respectively. Opponents are also characterised by these combat statistics. Combat proceeds iteratively through a series of ‘attack rounds’. In a given round, according to die rolls, the player may draw, win or lose, respectively. These outcomes respectively have no effect, damage the opponent, and damage the player. The player then has the option of using a limited resource ([<span style="font-variant:small-caps;">luck</span>]{}) to apply control to the outcome of the round. This decision can be made dynamically, allowing the player to choose a policy based on the current state of the system. However, each use of [<span style="font-variant:small-caps;">luck</span>]{} is a gamble [@maitra2012discrete; @dubins1965inequalities], where the probability of success depends on the current level of the resource. If this gamble is successful, the player experiences a positive outcome (damage to the opponent is amplified; damage to the player is weakened). If the gamble is unsuccessful, the player experiences a negative outcome (damage to the opponent is weakened, damage to the player is amplified). The optimal strategy for applying this control in a given state has yet to be found.
This is a stochastic game [@shapley1953stochastic; @adlakha2015equilibria] played by one or two players (the opponent usually has no agency to decide strategies) on a discrete state space with a finite horizon. The game is Markovian: in the absence of special rules, the statistics of the player and opponent uniquely determine a system state, and this state combined with a choice of policy uniquely determine the transition probabilities to the next state. The problem of determining the optimal strategy is then a Markov decision problem (MDP) [@bellman1957markovian; @kallenberg2003finite]. In an MDP, a decision-maker must choose a particular strategy for any given state of a system, which evolves according to Markovian dynamics. In FF combat, the decision is always binary: given a state, whether or not to use the dimishing resource of [<span style="font-variant:small-caps;">luck</span>]{} to attempt to influence the outcome of a given round.
The study of stochastic games and puzzles is long established in operational research [@smith2007dynamic; @bellman1965application] and has led to several valuable and transferrable insights [@smith2007dynamic; @little1963algorithm]. Markov analysis, dynamic programming, and simulation have been recently used to explore strategies and outcomes in a variety of games, sports, and TV challenges [@lee2012comparison; @smith2007dynamic; @johnston2016endless; @perea2007dynamic; @percy2015strategy; @clarke2003dynamic]. Specific analyses of popular one-player recreational games with a stochastic element including Solitaire [@rabb1988probabilistic; @kuykendall1999analyzing], Flip [@trick2001building], Farmer Klaus and the Mouse [@campbell2002farmer], Tetris [@kostreva2003multiple], and The Weakest Link [@thomas2003best]. These approaches typically aim to identify the optimal strategy for a given state, and, in win/lose games, the overall probability of victory over all possible instances of the game [@smith2007dynamic]. In stochastic dynamic games, counterintuitive optimal strategies can be revealed through mathematical analysis, not least because ‘risking points is not the same as risking the probability of winning’ [@neller2004optimal].
The FF system has some conceptual similarities with the well-studied recreational game Pig, and other so-called ‘jeopardy race games’ [@neller2004optimal; @smith2007dynamic], where die rolls are used to build a score then a decision is made, based on the current state of the system, whether to gamble further or not. Neller & Presser have used a value iteration approach to identify optimal strategies in Pig and surveyed other similar games [@neller2004optimal]. In FF combat, however, the player has potential agency both over their effect on the opponent and the opponent’s effect on them. Further, resource allocation in FF is a dynamic choice and also a gamble [@maitra2012discrete; @dubins1965inequalities], the success probability of which diminishes as more resource is allocated. The probability of a negative outcome, as opposed to a positive one, therefore increases as more resource is used, providing an important ‘diminishing returns’ consideration in policy decision [@deckro2003modeling]. In an applied context this could correspond to engaging in, for example, espionage and counterespionage [@solan2004games], with increasing probability of negative outcomes with more engagement in these covert activites.
The optimal policy for allocating resource to improve a final success probability has been well studied in the context of research and development (R&D) management [@heidenberger1999research; @baye2003strategic; @canbolat2012stochastic; @gerchak1999allocating]. While policies in this field are often described as ‘static’, where an initial ‘up-front’ decision is made and not updated over time, dynamic policy choices allowing updated decisions to be made based on the state of the system (including the progress of competitors) have also been examined [@blanning1981variable; @hopp1987sequential; @posner1990optimal]. Rent-seeking ‘contest’ models [@clark1998contest] also describe properties of the victory probability as a function of an initial outlay from players. The ‘winner takes all’ R&D model of Canbolat *et al.*, where the first player to complete development receives all the available payoff, and players allocate resource towards this goal [@canbolat2012stochastic], bears some similarity to the outcomes of the FF system. The model of Canbolat *et al.* did not allow dynamic allocation based on the current system state, but did allow a fixed cost to be spread over a time horizon, and computed Nash equilibria in a variety of cases under this model.
A connected branch of the literature considers how to allocate scarce resource to achieve an optimal defensive outcome [@golany2015allocating; @valenzuela2015multiresolution], a pertinent question both for human [@golany2009nature] and animal [@clark1992inducible] societies. Both optimisation and Nash equilibrium approaches are used in these contexts to identify solutions to the resource allocation problem under different structures [@golany2015allocating; @valenzuela2015multiresolution]. The FF system has such a defensive component, but the same resource can also be employed offensively, and as above takes the unusual form of a gamble with a diminishing success probability.
We will follow the philosophy of these optimisation approaches to identify the optimal strategy for allocating resource to maximise victory probability from a given state in FF. We first describe the system and provide solutions for victory probability in the case of no gambling and gambling according to a simple policy. Next, to account for the influence of different gambling policies on the stochastic dynamics of the game, we use a backwards-induction approach to solve the Bellman equation for the system and optimise victory probability from any given state. We then identify simple heuristic rules that approximate these optimised strategies and can be used in practise.
Game dynamics {#game-dynamics .unnumbered}
-------------
Within an FF game, the player has nonnegative integer statistics called [<span style="font-variant:small-caps;">skill</span>]{}, [<span style="font-variant:small-caps;">stamina</span>]{}, and [<span style="font-variant:small-caps;">luck</span>]{}. [<span style="font-variant:small-caps;">skill</span>]{} and [<span style="font-variant:small-caps;">luck</span>]{} are typically $\leq 12$; [<span style="font-variant:small-caps;">stamina</span>]{} is typically $\leq 24$, although these bounds are not required by our analysis. In a given combat, the opponent will also have [<span style="font-variant:small-caps;">skill</span>]{} and [<span style="font-variant:small-caps;">stamina</span>]{} statistics. We label the [<span style="font-variant:small-caps;">skill</span>]{}, [<span style="font-variant:small-caps;">stamina</span>]{}, and [<span style="font-variant:small-caps;">luck</span>]{} of the player (the ‘hero’) as $k_h, s_h,$ and $l$ respectively, and the opponent’s [<span style="font-variant:small-caps;">skill</span>]{} and [<span style="font-variant:small-caps;">stamina</span>]{} as $k_o$ and $s_o$. Broadly, combat in the FF system involves a series of rounds, where differences in [<span style="font-variant:small-caps;">skill</span>]{} between combatants influences how much [<span style="font-variant:small-caps;">stamina</span>]{} is lost in each round; when one combatant’s [<span style="font-variant:small-caps;">stamina</span>]{} reaches zero or below, the combat is over and that combatant has lost. The player may choose to use [<span style="font-variant:small-caps;">luck</span>]{} in any given round to influence the outcome of that round. More specifically, combat proceeds through Algorithm 1.
**Algorithm 1. FF combat system.**
1. Roll two dice and add $k_h$; this is the player’s attack strength $A_h$.
2. Roll two dice and add $k_o$; this is the opponent’s attack strength $A_o$.
3. If $A_h = A_o$, this attack round is a draw. Go to 6.
4. If $A_h > A_o$, the player has won this attack round. *Make decision* whether to use [<span style="font-variant:small-caps;">luck</span>]{}.
1. If *yes*, roll two dice to obtain $r$. If $r \leq l$, set $s_o = s_o - 4$. If $r > l$, set $s_o = s_o - 1$. For either outcome, set $l = l - 1$. Go to 6.
2. If *no*, set $s_o = s_o - 2$. Go to 6.
5. If $A_h < A_o$, the opponent has won this attack round. *Make decision* whether to use [<span style="font-variant:small-caps;">luck</span>]{}.
1. If *yes*, roll two dice to obtain $r$. If $r \leq l$, set $s_h = s_h - 1$. If $r > l$, set $s_h = s_h - 3$. For either outcome, set $l = l - 1$. Go to 6.
2. If *no*, set $s_h = s_h - 2$. Go to 6.
6. If $s_h > 0$ and $s_o \leq 0$, the player has won; if $s_o > 0$ and $s_h \leq 0$, the opponent has won. Otherwise go to 1.
It will readily be seen that these dynamics doubly bias battles in favour of the player. First, the opponent has no opportunity to use [<span style="font-variant:small-caps;">luck</span>]{} to their benefit. Second, when used offensively (to support an attack round that the player has won), the potential benefit (an additional 2 damage; Step 4a in Algorithm 1) outweighs the potential detriment (a reduction of 1 damage; Step 5a in Algorithm 1). When used defensively, this second bias is absent.
We will retain the [<span style="font-variant:small-caps;">skill</span>]{}, [<span style="font-variant:small-caps;">stamina</span>]{}, [<span style="font-variant:small-caps;">luck</span>]{} terminology throughout this analysis. However, the concepts here can readily be generalised. A player’s [<span style="font-variant:small-caps;">skill</span>]{} can be interpreted as their propensity to be successful in any given competitive round. Their [<span style="font-variant:small-caps;">stamina</span>]{} can be interpreted as the amount of competitive losses they can suffer before failure. [<span style="font-variant:small-caps;">luck</span>]{} is the resource that a player can dynamically allocate to gamble on influencing the outcome of competitive rounds, that diminishes with use (so that the success probability of the gamble also diminishes). For example, within the (counter)espionage analogy above, [<span style="font-variant:small-caps;">skill</span>]{} could be perceived as a company’s level of information security, [<span style="font-variant:small-caps;">stamina</span>]{} its potential to lose sensitive information before failure, and [<span style="font-variant:small-caps;">luck</span>]{} the resource that it can invest in (counter)espionage.
Analysis {#analysis .unnumbered}
========
In basic combat dynamics, [<span style="font-variant:small-caps;">skill</span>]{} does not change throughout a battle. The probabilities $p_w, p_d, p_l$ of winning, drawing, an losing an attack round depend only on [<span style="font-variant:small-caps;">skill</span>]{}, and remain constant throughout the battle. We therefore consider $P^t(s_h, s_o, l)$: the probability, at round $t$, of being in a state where the player has [<span style="font-variant:small-caps;">stamina</span>]{} $s_h$ and [<span style="font-variant:small-caps;">luck</span>]{} $l$, and the opponent has [<span style="font-variant:small-caps;">stamina</span>]{} $s_o$. A discrete-time master equation can readily be written down describing the dynamics above:
$$\begin{aligned}
& P^{t+1}(s_h, s_o, l) = \nonumber \\
& \,\, p_w (1 - \lambda_1(s_h, s_o+2, l)) P^{t}(s_h, s_o+2, l) + p_w \lambda_1(s_h, s_o+4, l+1) q(l+1) P^{t}(s_h, s_o+4, l+1) \nonumber \\
& \,\,+ p_w \lambda_1(s_h, s_o+1, l+1) (1-q(l+1)) P^{t}(s_h, s_o+1, l+1) + p_l (1 - \lambda_0(s_h+2, s_o, l)) P^{t}(s_h+2, s_o, l) \nonumber \\
& \,\,+ p_l \lambda_0(s_h+3, s_o, l+1) (1-q(l+1)) P^{t}(s_h+3, s_o, l+1) + p_l \lambda_0(s_h+1, s_o, l+1) q(l+1) P^{t}(s_h+1, s_o, l+1) \nonumber \\
& \,\,- (1-p_d) P^{t}(s_h, s_o, l). \label{eqn1} \end{aligned}$$
Here, $\lambda_i(s_h, s_o, l)$ reflects the decision whether to use [<span style="font-variant:small-caps;">luck</span>]{} after a given attack round outcome $i$ (player loss $i = 0$ or player win $i = 1$), given the current state of the battle. This decision is free for the player to choose. $q(l)$ and $(1-q(l))$ are respectively the probabilities of a successful and an unsuccessful test against [<span style="font-variant:small-caps;">luck</span>]{} $l$. We proceed by first considering the case $\lambda_0 = \lambda_1 = 0$ (no use of [<span style="font-variant:small-caps;">luck</span>]{}). We then consider a simplified case of constant [<span style="font-variant:small-caps;">luck</span>]{}, before using a dynamic programming approach to address the full general dynamics.
We are concerned with the *victory probability* $v_p$ with which the player is eventually victorious, corresponding to a state where $s_h \geq 1$ and $s_o \leq 0$. We thus consider the ‘getting to a set’ outcome class of this stochastic game [@maitra2012discrete], corresponding to a ‘winner takes all’ race [@canbolat2012stochastic]. We start with the probability of winning, drawing, or losing a given round. Let $k_h$ be the player’s [<span style="font-variant:small-caps;">skill</span>]{} and $k_o$ be the opponent’s [<span style="font-variant:small-caps;">skill</span>]{}. Let $D_1$, ..., $D_4$ be random variables drawn from a discrete uniform distribution on $[1, 6]$, and $X = D_1 + D_2 - D_3 - D_4$. Let $\Delta_k = k_h - k_o$. Then, the probabilities of win, draw, and loss events correspond respectively to $p_w = P(X + \Delta_k > 0), p_d = P(X + \Delta_k = 0), p_l = P(X + \Delta_k < 0)$.
$X$ is a discrete distribution on $[-10, 10]$. The point density $P(X = i)$ is proportional to the $(i+10)$th sextinomial coefficient, defined via the generating function $\left( \sum_{j = 0}^5 x^j \right)^n$, for $n = 5$ (specific values $\frac{1}{1296} \{1,\allowbreak 4,\allowbreak 10,\allowbreak 20,\allowbreak 35,\allowbreak 56,\allowbreak 80,\allowbreak 104,\allowbreak 125,\allowbreak 140,\allowbreak 146,\allowbreak 140,\allowbreak 125,\allowbreak 104,\allowbreak 80,\allowbreak 56,\allowbreak 35,\allowbreak 20,\allowbreak 10,\allowbreak 4,\allowbreak 1\}$; OEIS sequence A063260 [@oeis]). Hence
$$\begin{aligned}
p_w(\Delta_k) & = \sum_{j = -\Delta_k+1}^{10} P (X = j) \\
p_d(\Delta_k) & = P (X = -\Delta_k) \\
p_l(\Delta_k) & = \sum_{j = -10}^{-\Delta_k-1} P(X = j)\end{aligned}$$
Dynamics without luck-based control {#dynamics-without-luck-based-control .unnumbered}
-----------------------------------
We first consider the straightforward case where the player employs no strategy, never electing to use [<span style="font-variant:small-caps;">luck</span>]{}. We can then ignore $l$ and consider steps through the $(s_h, s_o)$ [<span style="font-variant:small-caps;">stamina</span>]{} space, which form a discrete-time Markov chain. Eqn. \[eqn1\] then becomes
$$\begin{aligned}
P^{t+1}(s_h, s_o, l) = p_w P^t(s_h, s_o+2, l) + p_l P^t(s_h+2, s_o, l) - (1-p_d) P^t(s_h, s_o, l). \label{eqn2}\end{aligned}$$
We can consider a combinatorial approach based on ‘game histories’ describing steps moving through this space [@maitra2012discrete]. Here a game history is a string from the alphabet $\{W, D, L\}$, with the character at a given position $i$ corresponding to respectively to a win, draw, loss in round $i$. We aim to enumerate the number of possible game histories that correspond to a given outcome, and assign each a probability.
We write $w, d, l$ for the character counts of $W, D, L$ in a given game history. A victorious game must always end in $W$. Consider the string describing a game history omitting this final $W$. First leaving out $D$s, we have $(w-1)$ $W$s and $l$ $L$s that can be arranged in any order. We therefore have $n(w,l) = \binom{w - 1 + l}{l}$ possible strings, each of length $w-1+l$. For completeness, we can then place any number $d$ of $D$s within these strings, obtaining
$$n(w,l,d) = \binom{w - 1 + l}{l} \binom{w - 1 + l + d}{d}.$$
Write $\sigma_h = \lfloor s_h/2 \rfloor$ and $\sigma_o = \lfloor s_o/2 \rfloor$, describing the number of rounds each character can lose before dying. Then, for a player victory, $w = \sigma_o$ and $l \leq \sigma_h - 1$. $d$ can take any nonnegative integer value. The appearance of each character in a game string is accompanied by a multiplicative factor of the corresponding probability, so we obtain
$$v_p = p_w^{\sigma_o} \sum_{l=0}^{\sigma_h-1} \sum_{d = 0}^{\infty} p_l^l p_d^d \binom{w - 1 + l}{l} \binom{w - 1 + l + d}{d},$$
where the probability associated with the final $W$ character has now also been included. After some algebra, and writing $\rho_w = p_w / (1-p_d)$ and $\rho_l = p_l / (1-p_d)$, this expression becomes
$$v_p = p_w^{\sigma_o} (p_l + p_w)^{-\sigma_o} \Bigg( \rho_w^{-\sigma_o} - p_l^{\sigma_h} (p_l + p_w)^{-\sigma_h} \times \binom{\sigma_h + \sigma_o - 1}{\sigma_h} \, {}_2 F_1 \left( 1, \sigma_h + \sigma_o, \sigma_h + 1, \rho_l \right) \Bigg)
\label{basiceqn}$$
Eqn. \[basiceqn\] is compared with the result of stochastic simulation in Fig. \[basiccompare\], and shown for various [<span style="font-variant:small-caps;">skill</span>]{} differences $\Delta_k$ and $(s_h, s_o)$ initial conditions. Intuitively, more favourable $\Delta_k > 0$ increase $v_p$ and less favourable $\Delta_k < 0$ decrease $v_p$ for any given state, and discrepancies between starting $s_h$ and $s_o$ also influence eventual $v_p$. A pronounced $s {\ (\mathrm{mod}\ 2)}$ structure is observed, as in the absence of [<span style="font-variant:small-caps;">luck</span>]{}, $s = 2n$ is functionally equivalent to $s = 2n-1$ for integer $n$. For lower initial [<span style="font-variant:small-caps;">stamina</span>]{}s, $v_p$ distributions become more sharply peaked with $s$ values, as fewer events are required for an eventual outcome.
![**Victory probability in the absence of [<span style="font-variant:small-caps;">luck</span>]{}-based strategy.** (i) Comparison of predicted victory probability $v_p$ from Eqn. \[basiceqn\] with stochastic simulation. (ii) $v_p$ behaviour as [<span style="font-variant:small-caps;">skill</span>]{} difference $\Delta_k$ changes.[]{data-label="basiccompare"}](figs1.png){width="8cm"}
Analytic dynamics with simplified [<span style="font-variant:small-caps;">luck</span>]{}-based control {#analytic-dynamics-with-simplified-luck-based-control .unnumbered}
------------------------------------------------------------------------------------------------------
To increase the probability of victory beyond the basic case in Eqn. \[basiceqn\], the player can elect to use [<span style="font-variant:small-caps;">luck</span>]{} in any given round. We will first demonstrate that the above history-counting analysis can obtain analytic results when applied to a simplified situation when [<span style="font-variant:small-caps;">luck</span>]{} is not depleted by use, so that the only limit on its employment is its initial level [@blanning1981variable]. We then employ a dynamic programming scheme to analyse the more involved general case.
First consider the case where [<span style="font-variant:small-caps;">luck</span>]{} is only used offensively (step 4a in Algorithm 1), and is used in every successful attack round. For now, ignore losses and draws. Then every game history consists of $A$s and $B$s, where $A$ is a successful offensive use of [<span style="font-variant:small-caps;">luck</span>]{} and $B$ is an unsuccessful offensive use of [<span style="font-variant:small-caps;">luck</span>]{}.
Consider the game histories that lead to the opponent losing exactly $n$ [<span style="font-variant:small-caps;">stamina</span>]{} points *before* the final victorious round. There are $\lfloor n/4 \rfloor + 1$ string lengths that can achieve this, which are $L = n - 3k$, where $k$ runs from $0$ to $\lfloor n/3 \rfloor$. The strings with a given $k$ involve $n-4k$ failures and $k$ successes.
If we make the simplifying assumption that [<span style="font-variant:small-caps;">luck</span>]{} is not depleted with use, every outcome of a [<span style="font-variant:small-caps;">luck</span>]{} test has the same success probability $q(l) = q$. Then the problem is simplified to finding the number of ways of arranging $k$ $A$s and $(n-4k)$ $B$s for each possible string:
$$N(k) = \frac{(n-3k)!}{k! (n-4k)!}$$
Now, for every string with a given $k$, with corresponding string length $n-3k$, we can place $l$ $L$s and $d$ $D$s as before, giving
$$N(k; n, l, d) = \frac{(n-3k)!}{k! (n-4k)!} \binom{n-3k + l}{l} \binom{n-3k + l + d}{d}.$$
The complete history involves a final victorious round. For now we will write the probability of this event as $p_f$, then the probability associated with this set of histories is
$$\begin{aligned}
P(k; n, l, d, p_f) = p_f \frac{(n-3k)!}{k! (n-4k)!} q^k (1-q)^{n-4k} p_l^l \binom{n-3k + l}{l} p_d^d \binom{n-3k + l + d}{d} \label{stringcount}\end{aligned}$$
and, as $P(k; n, p_f) = \sum_{l=0}^{\sigma_{h}-1} \sum_{d=0}^{\infty} P(k; n, l, d, p_f)$,
$$\begin{aligned}
P(k; n, p_f) = & p_f \frac{1}{k!(n-4k)!} (1-p_d)^{3k-n-2-\sigma_{h}} (p_w-p_w q)^{n-4k} (p_w q)^k \rho^{-n-1} (n-3k)! \nonumber \\
& \times \bigg[ (1-p_d)^{\sigma_h+1} \rho^{3k} + p_l^{\sigma_h} (p_l+p_d-1) \rho^n \binom{n-3k+\sigma_h}{\sigma_h} {}_2F_1(1, 1-3k+n+\sigma_h, 1+\sigma_h, p_l/(1-p_d)) \bigg]\end{aligned}$$
where $\rho = (p_d+p_l-1)/(p_d-1)$. Hence
$$P_n(n, p_f) = \sum_{k=0}^{n/3} P(k; n, p_f)$$
Now consider the different forms that the final victorious round can take. The opponent’s [<span style="font-variant:small-caps;">stamina</span>]{} can be reduced to 4 followed by an $A$, 3 followed by $A$, 2 followed by $A$, or 1 followed by $A$ or $B$. If we write $P(m; X)$ for the probability of reducing the opponent’s [<span style="font-variant:small-caps;">stamina</span>]{} to $m$ then finishing with event $X$,
$$v_p = P(4; A)+P(3;A)+P(2;A)+P(1;A)+P(1;B)$$
hence
$$\begin{aligned}
v_p = P_n(s_o-4, q)+P_n(s_o-3, q)+P_n(s_o-2, q) +P_n(s_o-1, q)+P_n(s_o-1, (1-q)). \label{dumbluckeqn}\end{aligned}$$
Fig. \[dumbluck\](i) compares Eqn. \[dumbluckeqn\] and stochastic simulation, and shows that use of [<span style="font-variant:small-caps;">luck</span>]{} can dramatically increase victory probability in a range of circumstances. Similar expressions can be derived for the defensive case, where [<span style="font-variant:small-caps;">luck</span>]{} is solely used when a round is lost, and with some relaxations on the structure of the sums involved the case where [<span style="font-variant:small-caps;">luck</span>]{} is not used in every round can also be considered.
Clearly, for constant [<span style="font-variant:small-caps;">luck</span>]{}, the optimal strategy is to always use [<span style="font-variant:small-caps;">luck</span>]{} when the expected outcome is positive, and never when it is negative. The expected outcomes for offensive and defensive strategies can readily be computed. For offensive use, the expected damage dealt when using [<span style="font-variant:small-caps;">luck</span>]{} is
$$\langle d \rangle = 1 \times (1-q) + 4 \times q \label{leqn1}$$
In the absence of [<span style="font-variant:small-caps;">luck</span>]{}, $d = 2$ damage is dealt, so the expected outcome using [<span style="font-variant:small-caps;">luck</span>]{} is beneficial if $\langle d \rangle > 2$. We therefore obtain $q > \frac{1}{3}$ as the criterion for employing [<span style="font-variant:small-caps;">luck</span>]{}. For comparison, the probability of scoring 6 or under on two dice is $q(6) = 0.417$ and the probability of scoring 5 or under on two dice is $q(5) = 0.278$.
For defensive use, the expected damage received when using [<span style="font-variant:small-caps;">luck</span>]{} in an attack round is
$$\langle d \rangle = 1 \times q + 3 \times (1-q) \label{leqn2}$$
In the absence of [<span style="font-variant:small-caps;">luck</span>]{}, $d = 2$ damage is taken, and we thus now obtain $q > \frac{1}{2}$ as our criterion. For comparison, the probability of scoring 7 or under on two dice is $q(7) = 0.583$.
The counting-based analyses above assume that [<span style="font-variant:small-caps;">luck</span>]{} stays constant. The dynamics of the game actually lead to [<span style="font-variant:small-caps;">luck</span>]{} diminishing every time is it is used. Instead of the outcome of a [<span style="font-variant:small-caps;">luck</span>]{} test being a constant $q$, it now becomes a function of how many tests have occurred previously.
To explore this situation, consider the above case where [<span style="font-variant:small-caps;">luck</span>]{} is always used offensively and never defensively. Describe a given string of [<span style="font-variant:small-caps;">luck</span>]{} outcomes as an ordered set $\mathcal{V}$ of indices labelling where in a string successes occur. For example, the string of length $L = 5$ with $\mathcal{V} = {2, 4}$ would be $BABAB$. Let $\mathcal{V}' = \mathcal{V}^c \setminus \{1, ..., L\}$. Then
$$P(\mathcal{V}) = \prod_{i \in \mathcal{V}} q(i) \prod_{i \in \mathcal{V}'} (1-q(i))$$
where $q(i)$ is the probability of success on the $i$th [<span style="font-variant:small-caps;">luck</span>]{} test.
As above, for a given $k$, $L = n-3k$, and we let $\mathcal{S}(k)$ be the set of ordered sets with $k$ different elements between 1 and $n-3k$. Then the probability of a given string of $A$s and $B$s arising is
$$P(n) = \sum_{k=0}^{n/3} \sum_{\mathcal{V} \in \mathcal{S}(k)} \prod_{i \in \mathcal{V}} q(i) \prod_{i \in \mathcal{V}'} (1-q(i))$$
We can no longer use the simple counting argument in Eqn. \[stringcount\] to compute the probabilities of each history, because each probability now depends on the specific structure of the history. It will be possible to enumerate these histories exhaustively but the analysis rapidly expands beyond the point of useful interpretation, so we turn to dynamic programming to investigate the system’s behaviour.
![**Victory probability with constant [<span style="font-variant:small-caps;">luck</span>]{} employed offensively.** (i) Eqn. \[dumbluckeqn\] compared to stochastic simulation. (ii) Constant luck outcomes for intermediate and low [<span style="font-variant:small-caps;">luck</span>]{}, and for high [<span style="font-variant:small-caps;">luck</span>]{} mitigating low $\Delta_k$ values.[]{data-label="dumbluck"}](figs2.png){width="8cm"}
Stochastic optimal control with dynamic programming {#stochastic-optimal-control-with-dynamic-programming .unnumbered}
---------------------------------------------------
In a game with a given $\Delta_k$, we characterise every state of the system with a tuple $\mathcal{S} = \{s_h, s_o, l, O\}$ where $O$ is the outcome (win or loss) of the current attack round. The question is, given a state, should the player elect to use [<span style="font-variant:small-caps;">luck</span>]{} or not?
A common approach to identify the optimal strategy for a Markov decision problem in a discrete state space is to use the Bellman equation [@bellman1957markovian; @kirk2012optimal], which in our case is simply
$$v_p(\mathcal{S}) = \max_a \left( \sum_{\mathcal{S}'} P_a (\mathcal{S}, \mathcal{S}') v_p (\mathcal{S}') \right).$$
Here, $a$ is a strategy dictating what action to take in state $\mathcal{S}$, $P_a(\mathcal{S}, \mathcal{S}')$ is the probability under strategy $a$ of the transition from state $\mathcal{S}$ to state $\mathcal{S}'$, and $v_p(\mathcal{S})$ is the probability-to-victory of state $\mathcal{S}$. The joint problem is to compute the optimal $v_p$, and the strategy $a$ that maximises it, for all states. To do so, we employ a dynamic programming approach of backward induction [@bellman1957markovian], starting from states where $v_p$ is known and computing backwards through potential precursor states.
![**Dynamic programming scheme.** The discrete state space of the game is represented by horizontal ($s_h$) and vertical ($s_o$) axes, layers ($l$) and a binary outcome at each position ($O$). (i) First, $v_p = 0$ is assigned to states corresponding to loss ($s_h \leq 0$, blue) and $v_p = 1$ is assigned to states corresponding to victory ($s_o \leq 0$, red). Then, iteratively, all states where all outcomes have an assigned $v_p$ are considered. (ii) Considering outcomes after a loss in the current state (here, ${1, 1, l, 0}$, highlighted red). The probability-to-victory for a ‘no’ strategy corresponds to the solid line; the probability-to-victory for a ‘yes’ strategy consists of contributions from both dashed lines (both outcomes from a [<span style="font-variant:small-caps;">luck</span>]{} test). (iii) Same process for a win outcome from the current state. (iv) Now the previous node has characterised probabilities-to-victory for both loss and win outcomes, a next iteration of nodes can be considered. Now, the probability-to-victory from a ‘yes’ strategy has a term corresponding to the potential outcomes from the previously characterised node.[]{data-label="dyncartoon"}](dynscheme-dec2.png){width="12cm"}
The dynamic programming approach first assigns a probability-to-victory $v_p$ for termination states (Fig. \[dyncartoon\](i)). For all ‘defeat’ states with $s_h \leq 0, s_o > 0$, we set $v_p = 0$; for all ‘victory’ states with $s_o \leq 0, s_h > 0$, set $v_p = 1$ (states where both $s_h \leq 0$ and $s_o \leq 0$ are inaccessible). We then iteratively consider all states in the system where $v_p$ is fully determined for a loss outcome from the current round (Fig. \[dyncartoon\](ii)), and similarly for a win outcome (Fig. \[dyncartoon\](iii)).
For states involving a loss outcome, we compute two probability propagators. The first corresponds to the strategy where the player elects to use [<span style="font-variant:small-caps;">luck</span>]{}, and is of magnitude
$$\begin{aligned}
p_y = &\sum_{\mathcal{S}'} P_y (\mathcal{S}, \mathcal{S}') v_p(\mathcal{S}') \nonumber \\
= & \, q(l) p_l v_p(\{s_h-1, s_o, l-1, 0\}) + (1-q(l)) p_l v_p(\{s_h-3, s_o, l-1, 0\}) \nonumber \\
& + \, q(l) p_w v_p(\{s_h-1, s_o, l-1, 1\}) + (1-q(l)) p_w v_p(\{s_h-3, s_o, l-1, 1\})\end{aligned}$$
the second corresponds to the strategy where the player does not use [<span style="font-variant:small-caps;">luck</span>]{}, and is
$$\begin{aligned}
p_n = & \, \sum_{\mathcal{S}'} P_n (\mathcal{S}, \mathcal{S}') v_p(\mathcal{S}') \nonumber \\
= & \,p_l v_p(\{s_h-2, s_o, l, 0\}) + p_w v_p(\{s_h-2, s_o, l, 1\}).\end{aligned}$$
When considering the probability-to-victory for a given state, we hence consider both the next $(s_h, s_o, l)$ combination that a given event will lead to, and also both possible outcomes (win or loss) from this state. For a given state, if $p_y > p_n$, we record the optimal strategy as using [<span style="font-variant:small-caps;">luck</span>]{} and record $v_p = p_y$; otherwise we record the optimal strategy as not to use [<span style="font-variant:small-caps;">luck</span>]{} and record $v_p = p_n$. In practise we replace $p_y > p_n$ with the condition $p_y > (1 + 10^{-10}) p_n$ to avoid numerical artefacts, thus requiring that the use of [<span style="font-variant:small-caps;">luck</span>]{} has a *relative* advantage to $v_p$ above $10^{-10}$.
We do the same for states involving a win outcome from this round, where the two probability propagators are now
$$\begin{aligned}
p_y = & \, q(l) p_l v_p(\{s_h, s_o-4, l-1, 0\}) + (1-q(l)) p_l v_p(\{s_h, s_o-1, l-1, 0\}) \nonumber \\
& +\, q(l) p_w v_p(\{s_h, s_o-4, l-1, 1\}) + (1-q(l)) p_w v_p(\{s_h, s_o-1, l-1, 1\});\end{aligned}$$
and
$$\begin{aligned}
p_n = & \,p_l v_p(\{s_h, s_o-2, l, 0\}) + p_w v_p(\{s_h, s_o-2, l, 1\}).\end{aligned}$$
Each new pair of states for which the optimal $v_p$ is calculated opens up the opportunity to compute $v_p$ for new pairs of states (Fig. \[dyncartoon\](iv)). Eventually a $v_p$ and optimal strategy is computed for each outcome, providing a full ‘roadmap’ of the optimal decision to make under any circumstance. This full map is shown in Supplementary Fig. \[suppfig\], with a subset of states shown in Fig. \[fullluckdyn-analytic\].
{width="17cm"}
{width="17cm"}
We find that a high [<span style="font-variant:small-caps;">luck</span>]{} score and judicious use of [<span style="font-variant:small-caps;">luck</span>]{} can dramatically enhance victory probability against some opponents. As an extreme example, with a [<span style="font-variant:small-caps;">skill</span>]{} detriment of $\Delta_k = -9$, $s_h = 2, s_o = 23, l = 12$, use of [<span style="font-variant:small-caps;">luck</span>]{} increased victory probability by a factor of $10^{18}$, albeit to a mere $2.3 \times 10^{-19}$. On more reasonable scales, with a [<span style="font-variant:small-caps;">skill</span>]{} detriment of $\Delta_k = -4$, $s_h = 22, s_o = 19, l = 12$, use of [<span style="font-variant:small-caps;">luck</span>]{} increased victory probability 1159-fold, from $9.8 \times 10^{-6}$ to 0.011 (the highest fold increase with the final probability greater than 0.01). With a [<span style="font-variant:small-caps;">skill</span>]{} detriment of $\Delta_k = -2$, $s_h = 22, s_o = 21, l = 12$, use of [<span style="font-variant:small-caps;">luck</span>]{} increased victory probability 21-fold, from 0.010 to 0.22 (the highest fold increase with the initial probability greater than 0.01). Using encounters that appear in the FF universe [@outofthepit], for a player with maximum statistics $k_h = 12, s_h = 24, l = 12$, optimal use of [<span style="font-variant:small-caps;">luck</span>]{} makes victory against an adult White Dragon ($k_o = 15, s_o = 22$) merely quite unlikely ($v_p = 0.046$) rather than implausible ($v_p = 4.4 \times 10^{-4}$), and victory against a Hell Demon ($k_o = 14, s_o = 12$) fairly straightforward ($v_p = 0.78$) rather than unlikely ($v_p = 0.28$).
### Structure of optimal policy space {#structure-of-optimal-policy-space .unnumbered}
There is substantial similarity in optimal policy choice between several regions of state space. For large [<span style="font-variant:small-caps;">skill</span>]{} deficiencies $\Delta_k \leq -6$ (low probability of victory) the distribution of optimal strategies in [<span style="font-variant:small-caps;">stamina</span>]{} space is the same for a given $l$ for all $\Delta_k$. For higher $l$, this similarity continues to higher $\Delta_k$; for $l = 12$, only 5 points in [<span style="font-variant:small-caps;">stamina</span>]{} space have different optimal strategies for $\Delta_k = -9$ and $\Delta_k = -4$. At more reasonable victory probabilities, a moderate transition is apparent between $l = 6$ and $l = 5$, where the number of points in [<span style="font-variant:small-caps;">stamina</span>]{} space where the optimal strategy involves using [<span style="font-variant:small-caps;">luck</span>]{} decreases noticeably. This reflects the lower expected advantage for $l=5$ (Eqns. \[leqn1\]-\[leqn2\]).
The interplay of several general strategies is observed in the optimal structures. First note that $\lceil s / 4 \rceil$ gives the number of hits required for defeat if a hit takes 4 [<span style="font-variant:small-caps;">stamina</span>]{} points (a successful offensive [<span style="font-variant:small-caps;">luck</span>]{} test) and $\lceil s / 2 \rceil$ gives the number of hits required for defeat in the absence of strategy. These scales partition [<span style="font-variant:small-caps;">stamina</span>]{} space by the number of rounds required for a given outcome and hence dictate several of the ‘banded’ structures observable in strategy structure. For example, at $s_h = 2$, it is very advantageous for the player to attempt to mitigate the effect of losing another round. Almost all circumstances display a band of defensive optimal strategy at $s_h = 2$.
At $s_o = 3$, a successful offensive [<span style="font-variant:small-caps;">luck</span>]{} test is very advantageous (immediate victory). An unsuccessful offensive test, leading to $s_o = 2$, is not disadvantageous to the same extent: we still need exactly one successful attack round without [<span style="font-variant:small-caps;">luck</span>]{}, as we would if we had not used [<span style="font-variant:small-caps;">luck</span>]{} and achieved $s_o = 1$ instead. A strip of strategy 3 (or strategy 1) at $s_o = 3$ is thus the next most robustly observed feature, disappearing only when victory probability is already overwhelmingly high. Many other structural features result from a tradeoff between conserving [<span style="font-variant:small-caps;">luck</span>]{} and increasing the probability of encountering this advantageous region. An illustration of the broad layout of optimal strategies is shown in Fig. \[fullluckdyn-illus\]; a more fine-grained analysis is provided in the Appendix.
Simulated dynamics with heuristic [<span style="font-variant:small-caps;">luck</span>]{} strategies {#simulated-dynamics-with-heuristic-luck-strategies .unnumbered}
---------------------------------------------------------------------------------------------------
While the dynamic programming approach above gives the optimal strategy for any circumstance, the detailed information involved does not lend itself to easy memorisation. As in Smith’s discussion of solitaire, ‘the curse of dimensionality applies for the state description, and most wise players use heuristics’ [@smith2007dynamic]. We therefore consider, in addition to the semi-quantitative summary in Fig. \[fullluckdyn-illus\], coarse-grained quantitative ‘strategies’ that, rather than specifying an action for each branch of the possible tree of circumstances, use a heuristic rule that is applied in all circumstances.
Among the simplest strategies are ‘open-loop’ rules, where the current state of the player’s and the opponent’s [<span style="font-variant:small-caps;">stamina</span>]{} does not inform the choice of whether to use [<span style="font-variant:small-caps;">luck</span>]{}. We may also consider ‘feedback’ rules, where the current state of [<span style="font-variant:small-caps;">stamina</span>]{}s informs this choice. We parameterise [<span style="font-variant:small-caps;">luck</span>]{} strategies by a threshold $\tau$; [<span style="font-variant:small-caps;">luck</span>]{} will only be used if the player’s current [<span style="font-variant:small-caps;">luck</span>]{} score is greater than or equal to $\tau$.
We consider the following [<span style="font-variant:small-caps;">luck</span>]{} strategies:
1. Never use [<span style="font-variant:small-caps;">luck</span>]{} ($\lambda_0 = \lambda_1 = 0$ as above);
2. Use [<span style="font-variant:small-caps;">luck</span>]{} in every non-draw round (i.e. to both ameliorate damage and enhance hits, $\lambda_0 = \lambda_1 = \Theta(l > \tau)$);
3. Use [<span style="font-variant:small-caps;">luck</span>]{} defensively (i.e. only to ameliorate damage, $\lambda_1 = 0, \lambda_0 = \Theta(l > \tau)$);
4. Use [<span style="font-variant:small-caps;">luck</span>]{} offensively (i.e. only to enhance hits, $\lambda_0 = 0, \lambda_1 = \Theta(l > \tau)$);
5. Use [<span style="font-variant:small-caps;">luck</span>]{} defensively if $l > \tau$ and $s_h < 6$;
6. Use [<span style="font-variant:small-caps;">luck</span>]{} offensively if $l > \tau$ and $s_o > 6$;
7. Use [<span style="font-variant:small-caps;">luck</span>]{} if $l > \tau$ and $s_h < 6$;
8. Use [<span style="font-variant:small-caps;">luck</span>]{} if $l > \tau$ and $s_h < 4$;
9. Use [<span style="font-variant:small-caps;">luck</span>]{} if $l > \tau$ and $s_h < s_o$.
Strategy 0 is the trivial absence of [<span style="font-variant:small-caps;">luck</span>]{} use; strategies 1-3 are ‘open-loop’ and 4-8 are ‘feedback’. Strategies 2 and 4 are defensive; strategies 3 and 5 are offensive; strategies 1, 6, 7, 8 are both offensive and defensive.
The analytic structure determined above is largely visible in the simulation results (Supplementary Fig. \[fullluckdyn-sim\]). The main differences arise from the construction that the same strategy is retained throughout combat in the simulation study. For low $\Delta_k$, high $l$, the favouring of strategy 3 except at low $s_h$ (strategy 1) and $s_o < 2$ (strategy 2) is clear. At low $s_h$, some feedback strategies gain traction: for example, strategy 8, which is identical to strategy 1 when $l$ is high and $s_h < s_o$, but avoids depleting [<span style="font-variant:small-caps;">luck</span>]{} if ever $s_h \geq s_o$, is more conservative than always using strategy 1. Similarly, strategy 5 provides a feedback alternative to strategy 3, using [<span style="font-variant:small-caps;">luck</span>]{} only at $s_o > 6$, which is favoured at intermediate $l$ and high $s_o$ where it is beneficial to preserve [<span style="font-variant:small-caps;">luck</span>]{}. These feedback-controlled options mimic the state-specific application of strategy found in the optimal cases in Fig. \[fullluckdyn-analytic\].
Decreasing $l$ introduces the sparsification of the strategy 3 regions – an effect which, as with the analysis, is more pronounced at higher $\Delta_k$. In simulations, at higher $\Delta_k$, the increasingly small relative increases in victory probability mean that several regions do not display substantial advantages to using strategy. However, at moderate to positive $\Delta_k$, the increased takeover of strategy 2 is observed for high $s_o$, low $s_h$. Strategy 4, a more conservative feedback version of strategy 2, also experiences support in these regions, competing with strategy 2 particularly in bands where $s_h {\ (\mathrm{mod}\ 2)} = 0$, as described in the analytic case. The above banded structures of strategies 2 and 3, and the ubiquitous strategy 2 (or equivalent) at $s_h = 2$, are also observed for intermediate $l$.
Under these simplified rulesets, what is the optimal threshold $\tau$ above which we should use [<span style="font-variant:small-caps;">luck</span>]{}? Straightforward application of Eqns. \[leqn1\]-\[leqn2\] are supported by simulation results (Supplementary Fig. \[taudist\]), where the probability that a given $\tau$ appears in the highest $v_p$ strategy peaked at $\tau = 6$ for strategy 1 and $\tau = 5$ for strategy 3.
A heuristic approach harnessing these insights, at the broad level of Fig. \[fullluckdyn-illus\], might be as follows. Employ strategy 3 (offensive [<span style="font-variant:small-caps;">luck</span>]{} use), unless: (i) initial [<span style="font-variant:small-caps;">luck</span>]{} is high or $s_o$ is low, in which case strategy 1 (unconditional [<span style="font-variant:small-caps;">luck</span>]{} use); (ii) $k_h > k_o$ and (i) is not true, in which case strategy 2 (defensive [<span style="font-variant:small-caps;">luck</span>]{} use); or (iii) $k_o > k_h$ and $s_o \leq 2$ and (i) is not true, in which case strategy 2 (defensive [<span style="font-variant:small-caps;">luck</span>]{} use). At the next level of detail, feedback control in the form of strategies 4-8 may be beneficial in some cases, notably when [<span style="font-variant:small-caps;">luck</span>]{} depletion is likely to become an issue; regions where this is likely to be the case can be read off from Supplementary Fig. \[fullluckdyn-sim\]. Finally, the optimal strategy for any given state can be read off from Supplementary Fig. \[suppfig\].
Discussion {#discussion .unnumbered}
==========
We have examined the probability of victory in an iterated, probabilistic game that plays a central role in a well-known and widespread interactive fiction series. The game can be played with or without ‘strategy’, here manifest by the consumption of a limited resource to probabilistically influence the outcome of each round.
Several interesting features of the FF combat system make it potentially noteworthy with respect to similar systems [@canbolat2012stochastic; @neller2004optimal; @smith2007dynamic]. The allocation of resource is dynamic and depends on system state [@blanning1981variable; @hopp1987sequential; @posner1990optimal]. The use of resource can both increase the probability of a positive outcome for the player, or a negative outcome for the opponent. Use of this resource does not guarantee a positive outcome: its use is a gamble [@maitra2012discrete; @dubins1965inequalities] that may negatively affect the player. The probability of this negative effect increases as more resource is used, providing an important consideration in the decision of whether to invoke this policy in the face of ‘diminishing returns’ [@deckro2003modeling].
The fact that gambles are dynamically used to influence iterated stochastic dynamics complicates the analysis of the game. To employ dynamic programming, state space was labelled both by the current statistics of the players and by the outcome of the current stochastic round. The policy decision of whether to gamble thus depends both on current statistics and the outcome of a round.
This analysis reveals several structural properties of the system that are not specific to the FF context of this study. The case of a limited resource being gambled against to both amplify successes and mitigate failures (with the probability of a positive outcome diminishing as resource is used) bears some resemblance to, for example, a conceptual picture of espionage and counterespionage [@solan2004games]. For example, resource could be allocated to covert operations with some probability of amplifying thefts and mitigating losses of information, with increased probability of negative outcomes due to discovery as these covert activites are employed more. Our analysis then reveals the strategies to be employed in different scenarios of information security ([<span style="font-variant:small-caps;">skill</span>]{}) and robustness to information loss ([<span style="font-variant:small-caps;">stamina</span>]{}).
In the absence of strategy, we find a closed-form expression for victory probability that takes an intuitive form. When strategy is included, dramatic increases in victory probability are found. The strong advantages provided by successful use of resource towards the ‘endgame’, where a successful gamble will produce instant victory or avoid instant defeat, shapes the structure of the optimal policy landscape. When little resource is available, complex structures emerge in the optimal landscape that depend on the tradeoff between using resource in the current state or ‘saving’ it in case of a more beneficial state later (‘risking points is not the same as risking the probability of winning’ [@neller2004optimal]). When default victory is unlikely, using resource to reinforce rare success probabilities is a favoured strategy; when default victory is likely, using resource to mitigate rare loss probabilities is favoured. The specific optimal policy in a given state is solved and can be reasonably approximated by more heuristic strategies [@smith2007dynamic]. Interestingly, there is little performance loss when these heuristics are ‘open-loop’, in the sense that policy choice only depends on a round’s outcome and coarsely on the amount of current resource. ‘Feedback’ strategies additionally based on the statistics of the two players did not provide a substantial advantage as long as endgame dynamics were covered.
Numerous FF gamebooks embellish the basic combat system, where weapons, armour, and other circumstances led to different [<span style="font-variant:small-caps;">stamina</span>]{} costs or different rules. In the notoriously hard ‘Crypt of the Sorcerer’ [@cryptbook], if the final opponent (with $k_o = 12$) wins two successive rounds the player is instantly killed, altering the Markovian nature of the combat system (and substantially decreasing victory probability). A simulation study [@crypt], while not employing an optimised [<span style="font-variant:small-caps;">luck</span>]{} strategy, estimated a 95% probability of this instant death occurring (and a 0.11% probability of overall victory through the book). Further expansion of this analysis will generalise the potential rulesets, and hence allow the identification of optimal strategies in more situations.
Another route for expansion involves optimising victory probability while preserving some statistics, for example enforcing that $s_h > s^*$ or $l > l^*$ at victory, so that some resource is retained for the rest of the adventure after this combat. Such constraints could readily be incorporated through an initial reallocating $v_p$ over system states in the dynamic programming approach (Fig. \[dyncartoon\](i)), or by expanding the definition of the score being optimised to include some measure of desired retention in addition to $v_p$.
In an era of artificial intelligence approaches providing effective but essentially uninterpretable strategies for complex games [@campbell2002deep; @lee2016human; @gibney2016google], more targetted analyses still have the potential to inform more deeply about the mechanisms involved in these strategies. Further, mechanistic understanding makes successful strategies readily available and simple to implement in the absence of computational resource. We hope that this increased interpretability and accessibility both contribute to the demonstration of the general power of these approaches, and help improve the experience of some of the millions of FF players worldwide.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful to Steve Jackson, Ian Livingstone, and the many other FF authors for creating this system and the associated worlds. The author also thanks Daniel Gibbs and Ellen Røyrvik for productive discussions.
Appendix {#appendix .unnumbered}
========
Detailed structure of optimal policy space {#detailed-structure-of-optimal-policy-space .unnumbered}
------------------------------------------
**Lower $\Delta_k$.** In this regime, for $s_h = 2$, there is a band of strategy 2, reflecting the fact that [<span style="font-variant:small-caps;">luck</span>]{} is best spent attempting to avoid the fatality of a lost attack round. For certain $s_{o}$ regions, strategy 3 is optimal. For example, when $l$ is low, strategy 3 appears when $s_o = 3$ or $4$. Here, the probability of winning an attack round is very low, so even a small chance of a successful [<span style="font-variant:small-caps;">luck</span>]{} test amplifying the outcome to a killing blow is worth taking. These bands of strategy 3 propagate as $l$ increases, occupying strips between $s_o = 3$ and $s_o = 4n$. One $s_{opp}$ value is omitted from each band, at $s_o = 2 + 4n$. It will be noticed that these values are those where, if [<span style="font-variant:small-caps;">luck</span>]{} is not employed, a successful round ($s_o - 2$) will decrease $\lceil s_o / 4 \rceil$, whereas if [<span style="font-variant:small-caps;">luck</span>]{} is employed and is not successful ($s_o - 1$), that next band will not be reached. When [<span style="font-variant:small-caps;">luck</span>]{} is a limited resource it does not pay to ‘spend’ it in these cases.
Where strategy 2 and 3 bands overlap, intuitively, strategy 1 appears, using [<span style="font-variant:small-caps;">luck</span>]{} for either outcome. As $l$ increases further ($l \geq 8$) other regions of [<span style="font-variant:small-caps;">stamina</span>]{} space display a preference for strategy 2, reflecting the fact that substantial [<span style="font-variant:small-caps;">luck</span>]{} is now available relative to the likely number of remaining attack rounds, so that is pays to invest [<span style="font-variant:small-caps;">luck</span>]{} defensively as well as offensively. As $l$ increases further still, the expanding regions of strategy 2 overlap more with existing strategy 3 regions, leading to an expansion of strategy 1 in low $s_h$ regions.
Column-wise, beginning at $\Delta_k = -6$, some horizontal bands in the the strategy 3 region sparsify into $s_h {\ (\mathrm{mod}\ 2)} = 1$ bands. This banding reflects the ubiquitous presence of strategy 2 when $s_h = 2$, to potentially defer the final fatal loss and allow one more attack round. If $s_h {\ (\mathrm{mod}\ 2)} = 1$, this region will not be reached unless the player uses a (non-optimal) defensive strategy elsewhere, so [<span style="font-variant:small-caps;">luck</span>]{} can be freely invested in offensive strategy. If $s_h {\ (\mathrm{mod}\ 2)} = 0$, the battle has some probability (high if $\Delta_k$ is low) of encountering the $s_h = 2$ state, so there is an advantage to preserving [<span style="font-variant:small-caps;">luck</span>]{} for this circumstance.
**Higher $\Delta_k$.** Patterns of strategies 2-3 remains largely unchanged as $\Delta_k$ increases, until the regions of strategy 3 at lower $l$ start to become sparsified, breaking up both row-wise and column-wise. Row-wise breaks, as above, occur at $s_o = 2+4n$, so that low [<span style="font-variant:small-caps;">luck</span>]{} is not used if the outcome will not cross a $\lceil s_o / 4 \rceil$ band. Eventually these breaks are joined by others, so that for e.g. $\Delta_k = -4$, $l = 4$ strategy 3 only appears in bands of $s_o = 3 + 4n$. Here, a successful [<span style="font-variant:small-caps;">luck</span>]{} test shifts the system to the next $\lceil s_o / 4 \rceil$ band, while an unsuccessful [<span style="font-variant:small-caps;">luck</span>]{} test means that the next band can be reached with a successful attack round without using [<span style="font-variant:small-caps;">luck</span>]{}. Column-wise, strategy 2 expands in $s_h {\ (\mathrm{mod}\ 2)} = 0$ bands at higher $\Delta_k$, and the ongoing sparsification of strategy 3 regions for low $l$ as $\Delta_k$ increases.
For higher $l$, as $\Delta_k$ increases, the sparsification of strategy 3 regions occurs in tandem with an expansion of strategy 2 regions. Strategy 2 becomes particularly dominant in high $s_o$, low $s_h$ regions. Here, the player expects to win most attack rounds, and the emphasis shifts to minimising losses from the rare attack rounds that the opponent wins, so that the player has time to win without their [<span style="font-variant:small-caps;">stamina</span>]{} running out. A banded structure emerges (for example, $\Delta_k = 2, l=12$) where for $s_o {\ (\mathrm{mod}\ 4)} \leq 1$ strategy 2 is favoured and for $s_o {\ (\mathrm{mod}\ 4)} \geq 2$ strategies 1 and 3 are favoured. This structure emerges because of the relatively strong advantage of a successful [<span style="font-variant:small-caps;">luck</span>]{} test at $s_o = 3$ and $s_o = 4$, leading to immediate victory, and the propagating advantage of successful [<span style="font-variant:small-caps;">luck</span>]{} tests that lead to this region (wherever $s_o {\ (\mathrm{mod}\ 4)} \geq 2$). In the absence of this advantage ($s_o {\ (\mathrm{mod}\ 4)} \leq q$), the best strategy is to minimise damage from rare lost rounds until the advantageous bands are obtained by victories without depleting [<span style="font-variant:small-caps;">luck</span>]{}.
At higher yet $\Delta_k$, victory is very likely, and the majority of [<span style="font-variant:small-caps;">stamina</span>]{} space for high $l$ is dominated by the defensive strategy 2 – minimising the impact of the rare attack rounds where the opponent wins. At intermediate $l$, strategies 2 and 3 appear in banded formation, where defensive [<span style="font-variant:small-caps;">luck</span>]{} use is favoured when $s_h$ is even, to increase the number of steps needed to reach $s_h = 0$.
{width="16cm"}
{width="17cm"}
![**Probability of [<span style="font-variant:small-caps;">luck</span>]{} threshold $\tau$ appearing in an optimal strategy.** Histogram shows $\tau$ values appearing in heuristic approaches to optimise $v_p$ in stochastic simulation, featuring strategies 1-3 from the text.[]{data-label="taudist"}](plotthresh.png){width="8cm"}
|
---
author:
- |
Yuting Ma\
`yma@stat.columbia.edu`\
Department of Statistics\
Columbia University\
New York, NY 10027
- |
Tian Zheng\
`tzheng@stat.columbia.edu`\
Department of Statistics\
Columbia University\
New York, NY 10027
bibliography:
- 'sDist\_SADM\_firstRevision.bib'
title: 'Boosted Sparse Non-linear Distance Metric Learning'
---
**Abstract**
This paper proposes a boosting-based solution addressing metric learning problems for high-dimensional data. Distance measures have been used as natural measures of (dis)similarity and served as the foundation of various learning methods. The efficiency of distance-based learning methods heavily depends on the chosen distance metric. With increasing dimensionality and complexity of data, however, traditional metric learning methods suffer from poor scalability and the limitation due to linearity as the true signals are usually embedded within a low-dimensional nonlinear subspace. In this paper, we propose a nonlinear sparse metric learning algorithm via boosting. We restructure a global optimization problem into a forward stage-wise learning of weak learners based on a rank-one decomposition of the weight matrix in the Mahalanobis distance metric. A gradient boosting algorithm is devised to obtain a sparse rank-one update of the weight matrix at each step. Nonlinear features are learned by a hierarchical expansion of interactions incorporated within the boosting algorithm. Meanwhile, an early stopping rule is imposed to control the overall complexity of the learned metric. As a result, our approach guarantees three desirable properties of the final metric: positive semi-definiteness, low rank and element-wise sparsity. Numerical experiments show that our learning model compares favorably with the state-of-the-art methods in the current literature of metric learning.
<span style="font-variant:small-caps;">Keywords</span>: [Boosting, Sparsity, Supervised learning]{}
INTRODUCTION
============
Beyond its physical interpretation, distance can be generalized to quantify the notion of similarity, which puts it at the heart of many learning methods, including the $k$-Nearest Neighbors ($k$NN) method, the $k$-means clustering method and the kernel regressions. The conventional Euclidean distance treats all dimensions equally. With the growing complexity of modern datasets, however, Euclidean distance is no longer efficient in capturing the intrinsic similarity among individuals given a large number of heterogeneous input variables. This increasing scale of data also poses a curse of dimensionality such that, with limited sample size, the unit density of data points is largely diluted, rendering high variance and high computational cost for Euclidean-distance-based learning methods. On the other hand, it is often assumed that the true informative structure with respect to the learning task is embedded within an intrinsic low-dimensional manifold [@johnson1984extensions], on which model-free distance-based methods, such as $k$NN, are capable of taking advantage of the inherent structure. It is therefore desirable to construct a generalized measure of distance in a low-dimensional nonlinear feature space for improving the performance of classical distance-based learning methods when applied to complex and high dimensional data.
We first consider the Mahalanobis distance as a generalization of the Euclidean distance. Let $\{\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n\}$ be a set of points in a feature space $\mathcal{X} \subseteq \mathbb{R}^p$. The Mahalanobis distance metric parameterized by a weight matrix $W$ between any two points $\mathbf{x}_i$ and $\mathbf{x}_j$ is given by: $$\label{eq: Mahalanobis_dist}
d_W(\mathbf{x}_i, \mathbf{x}_j) = \sqrt{(\mathbf{x}_i - \mathbf{x}_j)^T W (\mathbf{x}_i - \mathbf{x}_j)},$$ where $W \in \mathbb{R}^{p \times p}$ is symmetric positive semi-definite (PSD), denoted as $W \succeq 0$. The Mahalanobis distance can also be interpreted as the Euclidean distance between the points linearly transformed by $L$: $$\label{eq: MahalanobisDist2}
d_W(\mathbf{x}_i, \mathbf{x}_j) = ||L(\mathbf{x}_i - \mathbf{x}_j)||_2,$$ where $LL^T = W$ can be found by the Cholesky Decomposition. From a general supervised learning perspective, a “good” Mahalanobis distance metric for an outcome $y$ at $\mathbf{x}$ is supposed to draw samples with similar $y$ values closer in distance based on $\mathbf{x}$, referred to as the *similarity objective*, and to pull dissimilar samples further away, referred to as the *dissimilarity objective*, in the projected space.
There has been considerable research on the data-driven learning of a proper weight matrix $W$ for the Mahalanobis distance metric in the field of *distance metric learning*. Both accuracy and efficiency of distance-based learning methods can significantly benefit from using the Mahalanobis distance with a proper $W$ [@compsurvey]. A detailed comparison with related methods is presented in Section \[related\_work\]. While existing algorithms for metric learning have been shown perform well across various learning tasks, each is not sufficient in dealing with some basic requirements collectively. First, a desired metric should be flexible in adapting local variations as well as capturing nonlinearity in the data. Second, in high-dimensional settings, it is preferred to have a sparse and low-rank weight matrix $W$ for better generalization with noisy inputs and for increasing interpretability of the fitting model. Finally, the algorithm should be efficient in preserving all properties of a distance metric and be scalable with both sample size and the number of input variables.
In this paper, we propose a novel method for a local sparse metric in a nonlinear feature subspace for binary classification, which is referred to as *sDist*. Our approach constructs the weight matrix $W$ through a gradient boosting algorithm that produces a sparse and low-rank weight matrix in a stage-wise manner. Nonlinear features are adaptively constructed within the boosting algorithm using a hierarchical expansion of interactions. The main and novel contribution of our approach is that we mathematically convert a global optimization problem into a sequence of simple local optimization via boosting, while efficiently guaranteeing the symmetry and the positive semi-definiteness of $W$ without resorting to the computationally intensive semi-definite programming. Instead of directly penalizing on the sparsity of $W$, *sDist* imposes a sparsity regularization at each step of the boosting algorithm that builds a rank-one decomposition of $W$. The rank of the learned weight matrix is further controlled by the sparse boosting method proposed in [@sparseboosting]. Hence, three important attributes of a desirable sparse distance metric are automatically guaranteed in the resulting weight matrix: positive semi-definiteness, low rank and element-wise sparisty. Moreover, our proposed algorithm is capable of learning a sparse metric on nonlinear feature space, which leads to a flexible yet highly interpretable solution. Feature selection might be carried out as a spontaneous by-product of our algorithm that provides insights of variable importance not only marginally but also jointly in higher orders.
Our paper is organized as follows. In Section 2 we briefly illustrate the motivation for our method using a toy example. Section \[boosting\_metric\] dissects the global optimization for linear sparse metric learning into a stage-wise learning via gradient boosting algorithm. Section \[section\_nonlinear\] extends the framework proposed in Section \[boosting\_metric\] to the nonlinear sparse metric learning by hierarchical expansion of interactions. We summarize some related works in Section \[related\_work\]. Section \[practical\_remark\] provides some practical remarks on implementing the proposed method in practice. Results from numerical experiments are presented in Section \[numeric\_exp\]. Finally, Section \[conclusion\] concludes this paper by summarizing our main contributions and sketching several directions of future research.
AN ILLUSTRATIVE EXAMPLE
=======================
![An illustrative example of the XOR binary classification problem. *Left*: Training dataset is consisted of sample points from two classes that are distributed on four clusters aligned at the crossing diagonals on a three-dimensional plane. 200 data points are generated from four bivariate Gaussian distributions and are projected into the designated three-dimensional plane as illustrated in the figure. *Right*: The transformed subspace learned by the $sDist$ algorithm. Two horizontal dimensions are the first two input variables selected by $sDist$, that is, $x_1$ and $x_2$ in this case. The vertical dimension $z$ is the first principal component of the transformed subspace defined as $ L \phi(\mathbf{x})$, $LL^T=W$, which displays the overall shape of the surface on which new distances are computed. The colors on the grid indicates the true class probability of each class on the log scale. The yellow color indicates high probability in the class generative probability distribution and the red color indicates low probability. Since it is difficult to visualize two overlapping probability distributions in one plane, we use the same color scale for both classes and just focus on the magnitude of class generative probability distributions in each area.[]{data-label="fig: toy"}](demo_surface.pdf){width="\textwidth"}
Before introducing the details of the *sDist* algorithm, we offer here a toy example in Figure \[fig: toy\] to illustrate the problem of interest. The *left* panel of Figure \[fig: toy\] demonstrates the binary classification problem XOR (Exclusive OR) in a 3-dimensional space, which is commonly used as a classical setting for nonlinear classification in the literature. In the original space, sample points cannot be linearly separated. In this setting, sample points with the same class label are distributed in two clusters positioned diagonally from each other. In the original space, sample points cannot be linearly separated. It is also observed that the vertical dimension $x_3$ is redundant, as it provides no additional information regarding the class membership aside from $x_1$ and $x_2$. Hence, it is expected that there exists a nonlinear subspace on which points on the opposite diagonals of the tilted surface are closer to each other. Moreover, the subspace should be constructed solely based on a minimum set of variables that are informative about the class membership. The *right* panel of Figure \[fig: toy\] is the transformed subspace learned by the proposed *sDist* algorithm, which is only based on the informative variables $x_1$ and $x_2$. In particular, the curved shape of the resulted surface ensures that sample points with the same class label are drawn closer and those with opposite label are pulled further apart.
BOOSTED LINEAR SPARSE METRIC LEARNING {#boosting_metric}
=====================================
In this section, we first discuss the case of learning a linear sparse metric. Extension to nonlinear metric is discussed in Section 4. Assume that we are given a dataset $\mathcal{S} = \{\mathbf{x}_i, y_i\}$, $i=1, \dots, N$, $\mathbf{x}_i \in \mathcal{X} \subseteq \mathbb{R}^p$, where $\mathcal{X}$ is the input feature space and $p$ is the number of dimensions of the input vector[^1]. The class label $y_i \in \{-1, 1\}$. Consider an ideal scenario where there exists a metric parametrized by $W$ such that, in the $W$-transformed space, classes are separable. Then a point should, on average, be closer to the points from the same class than to the ones from the other class in its local neighborhood. Under this proposition, we propose a simple but intuitive discriminant function at $\mathbf{x}_i$ between classes characterized by $W$: $$\label{eq: def_f}
f_{W, k}(\mathbf{x}_i) = d_{W,k}^-(\mathbf{x}_i) - d_{W,k}^+(\mathbf{x}_i)$$ with $$\begin{aligned}
d_{W,k}^-(\mathbf{x}_i) &= \frac{1}{k} \sum_{j \in S^-_k(\mathbf{x}_i)} (\mathbf{x}_i - \mathbf{x}_j)^T W (\mathbf{x}_i - \mathbf{x}_j) \\
d_{W,k}^+(\mathbf{x}_i) &= \frac{1}{k} \sum_{j \in S^+_k(\mathbf{x}_i)} (\mathbf{x}_i - \mathbf{x}_j)^T W (\mathbf{x}_i - \mathbf{x}_j)
\end{aligned}$$ where $S^+_k(\mathbf{x}_i)$ and $S^-_k(\mathbf{x}_i)$ are the set of $k$ nearest neighbors of $\mathbf{x}_i$ with the same class labels and with the opposite class labels as $y_i$, respectively. Without any prior information, the local neighborhoods are first identified using the Euclidean distance [^2]. When the domain knowledge of local similarity relationships are available, local neighborhoods can be constructed with better precision. The predicted class label is obtained by $\hat{y} = 1$ if $\hat{f}_W(x) >0$ and $\hat{y} = -1$ otherwise. For simplicity, we drop $k$ in the notations $f_{W, k}(\cdot)$, $d_{W,k}^-$, and $d_{W,k}^+$ as $k$ is fixed throughout the algorithm.
The base classifier in serves as a continuous surrogate function of the $k$NN classifier, which is differentiable with respect to the weight matrix $W$. Instead of using the counts of the negative and the positive sample points in local neighborhoods, we adopt the continuous value of distances between two class to indicate the local affinity to the negative and the positive classes. Detailed comparison of the performance of the proposed classifier with the $k$NN classifier at different values of $k$ can be found in the Appendix \[App:AppendixA\]. It is shown that $f_W$ in achieves lower test error with small values of $k$ that is commonly used in the neighborhood-based methods. Furthermore, as we will show in the following, the differentiability of $f_W$ enables smooth optimization on $W$ which facilitates a faster and more stable learning algorithm.
Alternatively, $f_W(\mathbf{x}_i)$ can be represented as an inner product between the weight matrix $W$ and the data information matrix $D$, defined below, which contains all information of training sample point $\mathbf{x}_i$ for classification: $$\begin{aligned}
\hat{f}_W(\mathbf{x}_i) = \langle D_{i}, W \rangle,\end{aligned}$$ where $$D_{i} = \frac{1}{k} \left[ \sum_{j \in S^-_k(\mathbf{x}_i)} (\mathbf{x}_i - \mathbf{x}_j) (\mathbf{x}_i - \mathbf{x}_j) ^T - \sum_{j \in S^+_k(\mathbf{x}_i)} (\mathbf{x}_i - \mathbf{x}_j) (\mathbf{x}_i - \mathbf{x}_j) ^T \right]$$ and $\langle \cdot, \cdot \rangle$ stands for the inner product for vectorized matrices. Since the matrics $D_i$’s can be pre-calcuated without the intervention of $W$, this alternative formulation of $\hat{f}_W(\mathbf{x}_i)$ suggests a computationally efficient optimization of $W$ while keeping $D_i$’s fixed.
For learning $W$, we evaluate the performance of the classifier $f_W(\mathbf{x}_i)$ using the exponential loss, which is commonly used as a smooth objective function in binary classification: $$\begin{aligned}
\label{eq: loss_1}
L(\mathbf{y}, f_W) & = \sum\limits_{i=1}^{N} L(y_i, f_W(\mathbf{x}_i) ) = \sum\limits_{i=1}^{N} \exp(-y_i \langle D_i, W \rangle )\end{aligned}$$ Our learning task is then translated to derive a weight matrix $W$ on the original feature space that minimizes the loss function in . The optimization of this objective function, however, is generally intractable for high dimensional data. Our proposed method, *sDist*, seeks solution in minimizing objective function via optimizing adaptable sub-problems such that a feasible solution can be achieved. In short, the building block of *sDist* are: a gradient boosting algorithm which learns a rank-one update of the weight matrix $W$ at each step; a sparsity regularization on each rank-one update to enforce the element-wise sparsity and while preserving the positive semi-definiteness simultaneously, and a sparse boosting criterion that controls the total number of boosting steps to achieve overall sparsity and low rank of the resulting weight matrix.
Metric Learning via Boosting
----------------------------
In the distance metric learning literature, much effort has been put forward to learn the weight matrix $W$ by solving a single optimization problem globally, as in [@DMLXing] and [@LMNN]. However, the optimization turns out to be either computationally intractable or susceptible to local optima with noisy high-dimensional inputs.
Boosting [@freund1995boosting] offers a stagewise alternative to a single complex optimization problem. The motivation for boosting is that one can use a sequence of small improvements to derive a better global solution. Under the classification setting, boosting combines the outputs of many *weak learners* trained sequentially to produce a final aggregated classifier. Here, a weak learner is a classifier that is constructed to be only modestly better than a random guess. Subsequent weak learners are trained with more weights on previously misclassified cases, which reduces dependence among the trained learners and produces a final learner that is both stable and accurate. Such an ensemble of weak learners has been proven to be more powerful than a single complex classifier and has better generalization performance [@ESL]. In [@PSDBoost] and [@AdaBoostDML], a boosting algorithm has been implemented for learning a full distance metric, which has motivated the proposed algorithm in this paper. Their important theorem on trace-one semi-definite matrices is central to the theoretical basis of our approach.
Adopting a boosting scheme, *sDist* is proposed to learn a weight matrix $W$ in a stepwise fashion to avoid over-fitting to the training data in one optimization process. To construct the gradient boosting algorithm, we first decompose the learning problem into a sequence of weak learners. It is shown in [@PSDBoost] that for any symmetric positive semi-definite matrix $W \in \mathbb{R}^{p \times p}$ with trace one, it can be decomposed into a linear convex span of symmetric positive semi-definite rank-one matrices: $$W = \sum\limits_{m=1}^{M} w_m Z_m, \quad \mbox{rank}(Z_m)=1 \mbox{ and } tr(Z_m) = 1,$$ where $w_m \geq 0$, $m=1, \dots, M$, and $\sum\limits_{i=1}^{M} w_m= 1$. We define the vector of weights $\mathbf{w} = (w_1, w_2, \dots, w_M)$. The parameter $M \in \mathbb{Z}^+$ is the number of boosting iterations. Since any symmetric rank-one matrix can be written as an outer product of a vector to itself. We further decompose $W$ as $$\label{eq: decomp_psd2}
W = \sum\limits_{m=1}^{M} w_m \xi_m \otimes \xi_m, \quad ||\xi_m||_2 = 1 \mbox{ for all } m=1, 2, \dots, M.$$
Based on the decomposition in , we propose a gradient boosting algorithm that, within each step $m$, learns a rank-one matrix $Z_m = \xi_m \otimes \xi_m$ and its non-negative weight $w_m$. Each learned $Z_m$ can be considered as a small transformation of the feature space in terms of scaling and rotation. We use the following base learner in the gradient boosting algorithm: $$\label{eq: def_WL}
g_m(\mathbf{x}_i) = \langle D_i, Z_m \rangle.$$
In consecutive boosting steps, the target discriminant function is constructed as a stage-wise additive expansion. At the $m^{th}$ step, the aggregated discriminant function is updated by adding the base learner $g_m(\cdot)$ with weight $w_m$ to the existing classifier with weight matrix $\hat{W}_{m-1}$ that is learned from the previous $m-1$ steps: $$\begin{aligned}
f_{W_m}(\mathbf{x}_i) & = f_{W_{m-1}}(\mathbf{x}_i) + w_m g_m(\mathbf{x}_i) \\
& = \langle D_i, \sum\limits_{j=1}^{m-1} w_j Z_j \rangle + w_m \langle D_i, Z_m \rangle \\
& = \langle D_i, \hat{W}_{m-1} + w_m Z_m \rangle = \langle D_i, \hat{W}_m \rangle
\end{aligned}$$ where the resulting composite $\hat{W}_m$ is shown to be a weighted sum of $Z_m$’s learned from all previous steps. Therefore, the rank-one matrices obtained at each boosting step are assembled to construct the desired weight matrix, reversing the decomposition in . In this process, the required symmetry and positive semi-definiteness of weight matrix are automatically preserved without imposing any constraint. Moreover, the number of total boosting steps $M$ caps the rank of the final weight matrix. Thus, we can achieve an optimal reduced rank distance metric by using an appropriate $M$, which is discussed in Section 3.3.
In the gradient boosting algorithm, the learning goal is to attain the minimum of the loss function in . It is achieved by adapting a steepest-descent minimization in the functional space of $f_W$ in , which is characterized by the weight matrix $W$. The optimization problem in each boosting step is divided into two sub-steps, for $m = 1, \dots, M$:
- **Finding the rank-one matrix $Z_{m}$ given the previous aggregation $\hat{W}_{m-1}$**. The residuals from the previous $m-1$ steps are: $$\label{eq: GBA_residual}
r_i^{(m)} = \left[- \frac{\partial L(y_i, f)}{\partial f } \right]_{f = f_{\hat{W}_{m-1}}} = y_i \exp( -y_i f_{\hat{W}_{m-1}}(\mathbf{x}_i))$$ for $ i= 1, \dots, n$. The subsequent rank-one matrix $Z_m$ is obtained by minimizing the loss function on the current residuals for a new weak learner $g(\cdot)$ in , that is, $$\label{eq: GBA_2}
\begin{aligned}
Z_m &= \underset{Z \in \mathbb{R}^{p \times p},\ \mbox{rank}(Z) = 1}{\operatorname{arg \min}} \ \sum\limits_{i=1}^{n} L(r_i^{(m)}, g(\mathbf{x}_i)) = \underset{Z}{\operatorname{arg \min}} \sum\limits_{i=1}^{n} \exp( -r_i^{(m)} \langle D_i, Z \rangle ).
\end{aligned}$$ Since $$r_i^{(m)} g_m(\mathbf{x}_i) = r_i^{(m)} \langle D_i, Z_m \rangle = r_i^{(m)} \langle D_i, \xi_m \otimes \xi_m \rangle = \xi_m^T (r_i^{(m)} D_i) \xi_m,$$
the objective of is equivalent to identifying $$\label{eq: GBA_22}
\begin{aligned}
\xi_m & = \underset{\xi \in \mathbb{R}^{p},\ || \xi ||_2=1}{\operatorname{arg \min}} \ \sum\limits_{i=1}^{ n} \exp (- \xi^T r_i^{(m)} D_i \xi),
\end{aligned}$$ and rank-one update of weight matrix is calculated as $Z_m = \xi_m \otimes \xi_m$.
However, is non-convex and suffers from local minima and instability. Instead of pursuing the direct optimization on the objective function in , we resort to an approximation of it by the first order Taylor expansion, which is commonly used in optimizing non-convex exponential objective functions. It allow us to take advantage of the exponential loss in the binary classification task as well as avoid the expensive computational cost of considering a higher order of expansion. This approximation results in a simpler convex minimization problem : $$\label{eq: GBA_pca}
\xi_m = \underset{\xi \in \mathbb{R}^{p}, \ || \xi ||_2 =1}{\operatorname{arg \min}} -\xi^T A_m \xi$$ where $A_m = \sum\limits_{i=1}^{n} r_i^{(m)} D_i $. It is worthnoting that solving is equivalent to computing the the eigenvector associated with the largest eigenvalue of $A_m$ via eigen-decomposition.
- **Finding the positive weight $w_m$ given $Z_{m}$**: The optimal weight in the $m^{th}$ step minimizes given the learned $Z_m$ from the previous step. With $g_m(\mathbf{x}_i) = \langle D_i, Z_m \rangle$: $$\label{eq: GBA_w}
\begin{aligned}
\tilde{w}_m & = \underset{w \geq 0}{\operatorname{arg \min}} \sum\limits_{i=1}^{n} L(y_i, f_{\hat{W}_{m-1} + w Z_m}(\mathbf{x}_i)).
\end{aligned}$$ $\tilde{w}_m$ in is obtained by solving $$\frac{\partial L}{\partial \omega} = - \sum\limits_{i=1}^{n} r_i^{(m)} g_m(\mathbf{x}_i) \exp(- w y_i g_m(\mathbf{x}_i)) = 0$$ with simple algorithms such as the bisection algorithm [@boyd2004convex]. The vector of weights $\mathbf{w}$ is obtained by normalizing $\mathbf{w} = \frac{\tilde{\mathbf{w}}}{||\tilde{\mathbf{w}}||_2}$.
At last, the weight matrix $W_m$ is updated by $$\label{eq: agg_W}
\hat{W}_m = \hat{W}_{m-1} + w_m Z_m$$ The full algorithm is summarized in Algorithm \[alg: algo1\] in Section 4.
Sparse Learning and Feature Selection
-------------------------------------
In the current literature of sparse distance metric learning, a penalty of sparsity is usually imposed on the columns of the weight matrix $W$ or $L$, which is inefficient in achieving both element-wise sparsity and low rank in the resulting $W$. For instance, Sparse Metric Learning via Linear Programming (SMLlp) [@SMLlp] is able to obtain a low-rank $W$ but the resulting $W$ is dense, rendering it not applicable to high-dimensional datasets and being lack of feature interpretability. Other methods, such as Sparse Metric Learning via Smooth Optimization (SMLsm) [@SMLsm], cannot preserve the positive semidefiniteness of $W$ while imposing constraints for element-wise sparsity and reduced rank. These methods often rely on the computationally intensive projection to the positive-semidefinite cone to preserve the positive semi-definiteness of $W$ in their optimization steps. With the rank-one decomposition of $W$, we achieve element-wise sparsity and low rank of the resulting weight matrix simultaneously by regularizing both $\xi$ at each boosting step and the total number of boosting steps $M$.
First, we enforce the element-wise sparsity by penalizing on the $l_1$ norm of $\xi$. This measure not only renders a sparse linear transformation of the input space but also select a small subset of features relevant to the class difference as output at each step. The optimization in is replaced by a penalized minimization problem: $$\label{eq: Sparse_1}
\xi_m = \underset{\xi \in \mathbb{R}^{p}, \ || \xi ||_2 =1}{\operatorname{arg \min}} \ -\xi^T A_m \xi + \lambda_{\xi} \sum\limits_{j=1}^{p} |\xi_j|$$ where $\lambda_{\xi} > 0$ is the regularizing parameter on $\xi$.
As pointed out in Section 3.1, can be solved as a eigen-decomposition problem. The optimization problem in , appended with a single sparsity constraint on the eigenvector associated with the largest eigenvalue, is shown in [@tpower] as a sparse eigenvalue problem. We adopt a simple yet effective solution of the truncated iterative power method introduced in [@tpower] for obtaining the largest sparse eigenvectors with at most $\kappa$ nonzero entries. Power methods provide a scalable solution for obtaining the largest eigenvalue and the corresponding eigenvector of high-dimensional matrices without using the computationally intensive matrix decomposition. The truncated power iteration applies the hard-thresholding shrinkage method on the largest eigenvector of $A_m$, which is summarized in Algorithm \[alg: power\_method\] in Appendix \[App:AppendixB\].
Using parameter $\kappa$ in the sparse eigenvalue problem spares the effort of tuning the regularizing parameter $\lambda_{\xi}$ indefinitely to achieve the desirable level of sparsity. Under the context of *sDist*, $\kappa$ indeed controls the level of *module effect* among input variables, namely, the joint effect of selected variables on the class membership. Inputs that are marginally insignificant can have substantial influence when joined with others. The very nature of the truncated iterative power method enables us to identify informative variables in groups within each step. These variables are very likely to constitute influential interaction terms that explain the underlying structure of decision boundary which are hard to discern marginally. This characteristic is deliberately utilized in the construction of nonlinear feature mapping adaptively, which is discussed in detail in Section \[section\_nonlinear\]. In practice, the value of $\kappa$ can be chosen based on domain knowledge, depending on the order of potential interactions among variables in the application. Otherwise, we use cross-validation to select the ratio between $\kappa$ and the number of features $p$, denoted as $\rho$, at each boosting step as it is often assumed that the number of significant features is relatively proportional to the total number of features in real applications.
Sparse Boosting
---------------
The number of boosting steps $M$, or equivalently the number of rank-one matrices, bounds the overall sparsity and the rank of resulted weight matrix. Without controlling over $M$ from infinitely large, the resulted metric may fail to capture the low-dimensional informative representation of the input variable space. Fitting with infinitely many weak learners without regularization will produce an over-complicated model that causes over-fitting and poor generalization performance. Hence, in addition to sparsity control over $\xi$, we incorporate an automatic selection of the number of weak learners $M$ into the boosting algorithm by formulating it as an optimization problem. This optimization imposes a further regularization on the weight matrix $W$ to enforce a low-rank structure. Therefore, the resulting $W$ is ensured to have reduced rank if the true signal lies in a low dimensional subspace as well as guaranteeing the overall element-wise sparsity.
To introduce the sparse boosting for choosing an $M$, we first rewrite the aggregated discriminant function at the $m^{th}$ step as a hat operator $\Upsilon_m$, mapping the original feature space to the reduced and transformed space, i.e., $\Upsilon_m: X \rightarrow \tilde{X}_m$, in which $\tilde{X}_m$ is the transformed space by $\hat{L}_m$, $\hat{L}_m \hat{L}_m^T = \hat{W}_m$. Therefore, we have $$f_{\hat{W}_m} (x) = \langle D, \hat{W}_m \rangle = f (\Upsilon_m(X)).$$ Here $\Upsilon_m$ is uniquely defined by the positive semi-definiteness of $\hat{W}_m$. Hence, we define the complexity measure of the boosting process at the $m^{th}$ step by the generalized definition of degrees of freedom in [@green1994nonparametric]: $$\label{eq: def_C}
C_m = tr(\Upsilon_m)= tr(\hat{L}_m).$$ With the complexity measure in , we adopt the sparse boosting strategy introduced in [@sparseboosting]. First, let the process carry on for a large number, $M$, of iterations; then the optimal stopping time $\hat{m}$ is the minimizer of the stopping criterion $$\label{eq: opt_m}
\hat{m} = \underset{1 \leq m \leq M}{\operatorname{arg \min}} \ \left\{ \sum\limits_{i=1}^{N} L(y_i, f_{\hat{W}_m}(\mathbf{x}_i)) \right\}+ \lambda_C C_m$$ where $\lambda_C > 0$ is the regularizing parameter for the overall complexity of $W$.
This objective is rather intuitive: $\xi_m$’s are learned as sparse vectors and thus $Z_m = \xi_m \otimes \xi_m$ has nonzero entries mostly on the diagonal at variables selected in $\xi_m$. Therefore, $tr(\hat{L_m})$ is a good approximation of the number of selected variables, which explicitly indicates the level of complexity of the transformed space at step $m$.
BOOSTED NONLINEAR SPARSE METRIC LEARNING {#section_nonlinear}
========================================
The classifier defined in works well only when the signal of class membership is inherited within a linear transformation of the original feature space, which is rarely the case in practice. In this section, we introduce nonlinearity in metric learning by learning a weight matrix $W$ on a nonlinear feature mapping of the input variable space $\phi(\mathbf{x}): \mathcal{R}^p \rightarrow \mathcal{R}^{\tilde{p}}$, where $\tilde{p} \geq p$. The nonlinear discriminant function is defined as $$\label{eq: def_f_exp}
f_W^{\phi} (\mathbf{x}_i) = \langle D_i^{\phi}, W_m \rangle$$ where $$\begin{aligned}
\label{eq: D_phi}
D_i^{\phi} & = \frac{1}{k} \sum_{j \in S_k^-(\mathbf{x}_i)} [\phi^{(m)}(\mathbf{x}_i) - \phi^{(m)}(\mathbf{x}_j)][\phi^{(m)}(\mathbf{x}_i) - \phi^{(m)}(\mathbf{x}_j)]^T\\
& - \frac{1}{k} \sum_{j \in S_k^+(\mathbf{x}_i)} [\phi^{(m)}(\mathbf{x}_i) - \phi^{(m)}(\mathbf{x}_j)][\phi^{(m)}(\mathbf{x}_i) - \phi^{(m)}(\mathbf{x}_j)]^T \nonumber\end{aligned}$$ Learning a “good” feature mapping in the infinite-dimensional nonlinear feature space is infeasible. In [@LMCA], Torresani and Lee resort to the “kernel” trick and construct the Mahalanobis distance metric on the basis expansion of kernel functions in Reproducing Kernel Hilbert Space. Taking a different route, Kedem *et al* [@NonlinearDML] abort the reliance on the Mahalanobis distance metric and learn a distance metric on the non-linear basis functions constructed by regression trees. Although these methods provide easy-to-use “black box” algorithms that offers extensive flexibility in modeling a nonlinear manifold, they are sensitive to the choices of model parameters and are subject to the risk of overfitting. The superfluous set of basis functions also hinders the interpretability of the resulting metric model with respect to the relevant factors of class separation.
In this paper, we restrict the feature mapping $\phi(\mathbf{x})$ to the space of polynomial functions of the original input variables $x_1, \dots, x_p$. The construction of nonlinear features is tightly incorporated within the boosted metric learning algorithm introduced in Section \[boosting\_metric\]. Accordingly, a proper metric is learned in concert with the building of essential nonlinear mappings suggested in the data.
We initialized $\phi(\mathbf{x}) = (x_1, x_2, \dots, x_p)^T$ as the identity mapping at step $0$. In the following steps, based on the optimal sparse vector $\xi$ learned from the regularized optimization problem , we expand the feature space by only including interaction terms and polynomial terms among the nonzero entries of $\xi$, that is, the selected features. Such strategy allows the boosting algorithm to benefit from the flexibility introduced by the polynomials without running into overwhelming computational burden and storage need. In comparison, the full polynomial expansion results in formidable increase in dimensionality of the information matrices $D^{\phi}_i$’s to as much as $(2^p)^2$.
The polynomial feature mapping also permits selection of significant nonlinear features. Kernel methods are often preferred in nonlinear classification problems due to its flexible infinite-dimensional basis functions. However, for the purpose of achieving sparse weight matrix, each basis function need to be evaluated for making the selection toward a sparse solution. Hence, using kernel methods in such a case is computationally infeasible due to its infinite dimensionality of basis functions. By adaptively expanding polynomial features, optimizing on the expanded feature space is able to identify not only significant input variables but also informative interaction terms and polynomial terms.
Before we layout the details of the adaptive feature expansion algorithm, we define the following notions: Let $\mathcal{C}_m = \{\tilde{x}_1, \dots, \tilde{x}_{p_m}\}$ be the set of candidate variables at step $m$, where $\tilde{x}$ represents the candidate feature, and $\tilde{p}_m$ is the cardinality of the set $\mathcal{C}_m$, that is, the number of features at step $m$. The set $\mathcal{C}_m$ includes the entire set of original variables as well as the appended interaction terms. Denote $\mathcal{S}_m$ as the cumulative set of the unique variables selected up to step $m$, and $\mathcal{A}_m$ be the set of variables being newly selected in step $m$. Then,
- : Set $\mathcal{C}_0 = \{\tilde{x}_1=x_1, \dots, \tilde{x}_p=x_p \}$, the set of the original variables.
- : Select $\mathcal{A}_1 \subset \mathcal{C}_0$ by the regularized optimization in with prespecifed $| \mathcal{A}_1 | =\kappa$. $$\mbox{Set} \quad \mathcal{S}_1 = \mathcal{A}_1; \quad \mathcal{C}_1 = \mathcal{C}_0 \cup ( \mathcal{S}_1 \otimes \mathcal{A}_1 )$$ where the operator “$\otimes$" is defined as $$\mathcal{S}_1 \otimes \mathcal{A}_1 = \{\tilde{x}_i \tilde{x}_j: \tilde{x}_i \in \mathcal{S}_1, \tilde{x}_j \in \mathcal{A}_1 \}$$
- , $m = 2, \dots, M$: Select $\mathcal{A}_m \subset \mathcal{C}_{m-1}$. Then $$\label{eq: def_CC}
\mathcal{S}_m = \mathcal{S}_{m-1} \cup \mathcal{A}_m, \quad \mathcal{C}_m = \mathcal{C}_{m-1} \cup (\mathcal{S}_m \otimes \mathcal{A}_m )$$
Then $\phi(\mathbf{x})$ at the $m^{th}$ step of the algorithm is defined as $\phi^{(m)}(\mathbf{x}) \triangleq X_{\mathcal{C}_{m-1}}$, the vector[^3] whose components are elements in $\mathcal{C}_{m-1}$
It is worthnoting that, in updating $D_i^{\phi^{(m)}}$, there is no need to compute the entire matrix, the cost of which is on the order of $np_{m}^3$. Instead, taking advantage of the existing $D_i^{\phi^{(m-1)}}$, it is only required to add $\delta_m \triangleq (p_m - p_{m-1})$ rows of pairwise products between the newly added terms and currently selected ones and to make the resulting matrix symmetric. The extra computational cost is reduced to $O(n \delta_m^3)$ and $\delta_m \ll p_m $ when $p$ is large. Therefore, the method of expanding the feature space in the step-wise manner is tractable even with large $p$. Since we only increase the dimension of feature space by a degree less than $\frac{1}{2}(\delta_m\kappa + \kappa)$ at each step with $M$ controlled by the sparse boosting, the proposed hierarchical expansion is computationally feasible even with high-dimensional input data.
We integrate the adaptive feature expansion for nonlinear metric learning into the boosted sparse metric learning algorithm in Section \[boosting\_metric\]. The final algorithm is summarized in Algorithm \[alg: algo1\]. The details of how to choose the value of parameters $\kappa$, $\lambda_C$ and $M$ are elaborated in Section \[practical\_remark\]
Input Parameters: $\kappa$, $M$, and $\lambda_C$ 1) Initialization: $\hat{W}_0 = I_{p \times p}$; $\mathcal{C}_0 = \{\tilde{x}_1=x_1, \dots, \tilde{x}_p=x_p \}$; residuals $r_i^{(0)} = y_i$, $i=1, 2, \dots, n$. 2) For $m=1$ to $M$: (a) Define the nonlinear feature mapping $\phi^{(m)}(\mathbf{x}) = X_{\mathcal{C}_{m-1}}$; Update $D_i^{\phi^{(m)}}$ according to (b) $A_m = \sum\limits_{i=1}^{n} r_i^{(m)} D_i^{\phi^{(m)}}$. (c) Get $\xi_m$ from the regularized minimization problem: $$\begin{aligned}
\xi_m = \underset{\xi \in \mathbb{R}^{p_m}, || \xi ||_2 =1}{\operatorname{arg \min}} -\xi^T A_m \xi + \lambda_{\xi} \sum\limits_{j=1}^{p_m} |\xi_j|,\end{aligned}$$ by the truncated iterative power method (Algorithm \[alg: power\_method\] in Appendix \[App:AppendixB\]) with corresponding $\kappa$. (d) Based on the sparse solution of $\xi_m$,update $\mathcal{A}_m, \mathcal{S}_m$ and $\mathcal{C}_m$. $g_m(\mathbf{x}_i) = \xi_m^T D_i^{\phi^{(m)}} \xi_m$ for $i=1, 2, \dots, n$. (e) Get $w_m$ from by the bisection algorithm. (f) Compute residuals $r_i^{(m)}$ based on : $$r_i^{(m+1)} = r_i^{(m)} \exp(-y_i \omega_m g_m(\mathbf{x}_i)), \mbox{for } i=1, \dots, n.$$ (g) Update the weight matrix: $$\hat{W}_m = \mathcal{I}_m^T \hat{W}_{m-1} \mathcal{I}_m + w_m \xi_m \xi_m^{T}$$ where $\mathcal{I}_m= (I_{p_{m-1} \times p_{m-1}}, \mathbf{0}_{p_{m-1} \times p_{m} - p_{m-1}})$, where $I_{p \times p}$ is the $p$ by $p$ identity matrix and $\mathbf{0}_{p \times q}$ is the zero matrix of dimension $p$ by $q$. 3) Determine the optimal stopping time by solving $$\hat{m} = \underset{1 \leq m \leq M}{\operatorname{arg \min}}\sum\limits_{i=1}^{N} L(y_i, \hat{W}_m) + \lambda_C C_m.$$ Then set the output $\hat{W} = \hat{W}_{\hat{m}}$.
\[alg: algo1\]
RELATED WORK {#related_work}
============
There is an extensive literature devoted on the problem of learning a proper $W$ for the Mahalanobis distance. In this paper, we focus on the problem of supervised metric learning for classification in which class labels are given in the training sample. In the following, we categorize related methods in the literature into four groups: 1) global metric learning, 2) local metric learning, 3) sparse metric learning, and 4) nonlinear metric learning.
Global metric learning aims to learn a $W$ that addresses the similarity and dissimilarity objectives at all sample points. Probability Global Distance Metric (PGDM) learning [@DMLXing] is an early representative method of this group. In PGDM, the class label ($y$) is converted into pairwise constraints on the metric values between pairs of data points in the feature ($\mathbf{x}$) space: equivalence (similarity) constraints that similar pairs (in $y$) should be close (in $\mathbf{x}$) by the learned metric; and in-equivalence (dissimilarity) constraints that dissimilar ones (in $y$) should be far away (in $\mathbf{x}$). The distance metric is then derived to minimize the sum of squared distances between data points with the equivalence constraints, while maintaining a lower bound for the ones with the in-equivalence constraints. The global optimum for this convex optimization problem is derived using Semi-Definite Programming (SDP). However, the standard SDP by the interior point method requires $O(p^4)$ storage and has a worst-case computational complexity of approximately $O(p^{6.5})$, rendering it computationally prohibitive for large $p$. Flexible Metric Nearest Neighbor (FMNN) [@friedman1994flexible] is another method of this group, which, instead, adapts a probability framework for learning a distance metric with global optimality. It assumes a logistic regression model in estimating the probability for pairs of observations being similar or dissimilar based on the learned metric, yet suffering poor scalability as well.
The second group of methods, local metric learning methods, learn $W$ by pursuing similarity objective within the local neighborhoods of observations and a large margin at the boundaries between different classes. For examples, see the Neighborhood Component Analysis (NCA) [@NCA] and the Large Margin Nearest Neighbor (LMNN) [@LMNN]. NCA learns a distance metric by stochastically maximizing the probability of correct class-assignment in the space transformed by $L$. The probability is estimated locally by the Leave-One-Out (LOO) kernel density estimation with a distance-based kernel. LMNN, on the other hand, learns $W$ deterministically by maximizing the margin at class boundary in local neighborhoods. Adapting the idea of PGDM while focusing on local structures, it penalizes on small margins in distance from the query point to its similar neighbors using a hinge loss. It has been shown in [@LMNN] that LMNN delivers the state-of-the-art performance among most distance metric learning algorithms. Despite its good performance, LMNN and its extensions suffers from high computational cost due to their reliance on SDP similar to PGDM. Therefore, they always require data pre-processing for dimension reduction, using *ad-hoc* tools, such as the Principal Component Analysis (PCA), when applied to high-dimensional data. A survey paper [@compsurvey] provides a more thorough treatment on learning a linear and dense distance metric, especially from the aspect of optimization.
When the dimension of data increases, learning a full distance metric becomes extremely computationally expensive and may easily run into overfitting with noisy inputs. It is expected that a sparse distance matrix would produce a better generalization performance than its dense counterparts and afford a much faster and efficient distance calculation. Sparse metric learning is motivated by the demand of learning appropriate distance measures in high-dimensional space and can also lead to supervised dimension reduction. In the sparse metric learning literature, sparsity regularization can be introduced in three different ways: on the rank of $W$ for learning a low-rank $W$, (e.g., [@LMCA], [@msNCA], [@SMLlp], [@semiSML]), on the elements of $W$ for learning an element-wise sparse $W$ [@logDetSML], and the combination of the two [@SMLsm]. All these current strategies suffer from various limitations and computational challenges. First, a low-rank $W$ is not necessarily sparse. Methods such as [@SMLlp] impose penalty on the trace norm of $W$ as the proxy of the non-convex non-differentiable rank function, which usually involves heavy computation and approximation in maintaining both the status of low rank and the positive semi-definiteness of $W$. Searching for an element-wise sparse solution as in [@logDetSML] places the $l_1$ penalty on the off-diagonal elements of $W$. Again, the PSD of the resulting sparse $W$ is hard to maintain in a computationally efficient way. Based on the framework of LMNN, Ying *et al.* [@SMLsm] combine the first two strategies and penalize on the $l_{(2,1)}$ norm[^4] of $W$ to regularize the number of non-zero columns in $W$. Huang *et al.* [@GSML] proposed a general framework of sparse metric learning. It adapts several well recognized sparse metric learning methods with a common form of sparsity regularization $tr(SW)$, where $S$ varies among methods serving different purposes. As a limitation of the regularization, it is hard to impose further constraint on $S$ to guarantee PSD in the learned metric.
As suggested in , the Mahalanobis distance metric implies a linear transformation of the original feature space. This linearity inherently limits the applicability of distance metric learning in discovering the potentially nonlinear decision boundaries. It is also common that some variables are relevant to the learning task only through interactions with others. As a result, linear metric learning is at the risk of ignoring useful information carried by the features beyond the marginal distributional differences between classes. Nonlinear metric learning identifies a Mahalanobis distance metric on a nonlinear mappings of the input variables, introducing nonlinearity via well-designed basis functions on which the distances are computed. Large Margin Component Analysis (LMCA )[@LMCA] maps the input variables onto a high-dimensional feature space $\mathcal{F}$ by a nonlinear map $\phi: \mathcal{X} \rightarrow \mathcal{F}$, which is restricted to the eigen-functions of a Reproducing Kernel Hilbert Space (RKHS) [@aronszajn1950theory]. Then the learning objective is carried out using the “kernel trick” without explicitly compute the inner product. LMCA involves optimizing over a non-convex objective function and is slow in convergence. Such heavy computation limits its scalability to relatively large datasets. Kedem *et al.* [@NonlinearDML] introduce two methods for nonlinear metric learning, both of which derived from extending LMNN. $\chi^2$-LMNN uses a nonlinear $\chi^2$-distances for learning a distance metric for histogram data. The other method, GB-LMNN, exploits the gradient boosting algorithm that learns regression trees as the nonlinear basis functions. GB-LMNN relies on the Euclidean distance in the nonlinearly expanded features space without an explicit weight matrix $W$. This limits the interpretability of its results. Current methods in nonlinear metric learning are mostly based on black-box algorithms which are prone to overfit and have limited interpretability of variables.
PRACTICAL REMARKS {#practical_remark}
=================
When implementing Algorithm \[alg: algo1\] in practice, the performance of the *sDist* algorithm can be further improved in terms of both accuracy and computational efficiency by a few practical techniques, including local neighborhood updates, shrinkage, bagging and feature sub-sampling. We numerically evaluate the effect of the following parameters on a synthetic dataset in Section \[numeric\_exp\].
As stated in Section \[boosting\_metric\], the base classifier $f_{W,k}(x_i)$ in is constructed based on local neighborhoods. Without additional domain knowledge about the local similarity structure, we search for local neighbors of each sample point using the Euclidean distance. While the actual neighbors found in the truly informative feature subspace may not be well approximated by the neighbors found in the Euclidean space of all features, the learned distance metrics in the process of the boosting algorithm can be used to construct a better approximation of the true local neighborhoods. The revised local neighborhoods are helpful in preventing the learned metric from overfitting to the neighborhoods found in the Euclidean distance and thus reducing overfitting to the training samples. In practice, we update local neighborhoods using the learned metric at a number of steps in the booting algorithm. The frequency of the local neighborhood updates is determined by the trade-off between the predictive accuracy and the computational cost for re-computing distances between pairs of sample points. The actual value of updating frequency varies in real data applications and can be tuned by cross-validation.
In addition to the sparse boosting in which the number of boosting steps is controlled, we can further regularize the learning process by imposing a shrinkage on the rank-one update at each boosting step. The contribution of $Z_m$ is scaled by a factor $0 < \nu \leq 1$ when it is added to the current weight matrix $W_{m-1}$. That is, step 2g in Algorithm \[alg: algo1\] is replaced by $$\begin{aligned}
\hat{W}_m = \mathcal{I}_m^T \hat{W}_{m-1} \mathcal{I}_m + \nu w_m \xi_m \xi_m^T.\end{aligned}$$ The parameter $\nu$ can be regarded as controlling the learning rate of the boosting procedure. Such a shrinkage helps in circumventing the case that individual rank-one updates of the weight matrix fit too closely to the training samples. It has been empirically shown that smaller values of $\nu$ favor better generalization performance and require correspondingly larger values of $M$ [@friedman2001greedy]. In practice, we use cross-validation to determine the value of $\nu$.
***B**ootstrap **Ag**gregat**ing*** (bagging) has been demonstrated to improve the performance of a noisy classifier by averaging over weakly correlated classifiers [@ESL]. Correlations between classifiers are diminished by random subsampling. In the gradient boosting algorithm of $sDist$, we use the same technique of randomly sampling a fraction $\eta$[^5], $0 < \eta \leq 1$, of the training observations to build each weak learner for learning the rank-one update. This idea has been well exploited in [@friedman2002stochastic] with tree classifiers, and it is shown that both accuracy and execution speed of the gradient boosting can be substantially improved by incorporating randomization into the procedure. The value of $\eta$ is usually taken to be 0.5 or smaller if the sample size is large, which is tuned by cross-validation in our numerical experiments. In particular to our algorithm, bagging substantially reduces the training set size for individual rank-one updates so that $D_i$ can be computed on the fly more quickly without being pre-calculated, avoiding the need of computational memory. As a result, in applications with large sample sizes, bagging not only benefits the test error but also improves computational efficiency.
In high-dimensional applications, it is likely that the input variables are correlated, which translates to high variance in the estimation. As *sDist* can be viewed as learning an ensemble of nonlinear classifiers, high correlation among features can deteriorate the performance of the aggregated classifier. To resolve it, we employ the same strategy as in random forests [@breiman2001random] of random subsampling on features to reduce the correlation among weak learners without greatly increasing the variance. At each boosting step $m$, we randomly select a subset of features of size $\tilde{p}_m$ from the candidate set $C_m$, where $\kappa < \tilde{p}_m \leq p_m$, on which $D_i$’s is constructed with dimension $\tilde{p}_m \times \tilde{p}_m$. The optimization in is then executed on a much smaller scale and select $\kappa$ significant features from the random subset. As with bagging, feature subsampling enables fast computation of $D_i$’s without pre-calculation. We use $\tilde{p}_m = \sqrt{p_m}$ at the $m^{th}$ boosting step, which is suggested in [@breiman2001random]. Although feature subsampling will reduce the chance of selecting the significant features at each boosting step, it shall be emphasized that bagging on training samples and feature subsampling should be accompanied by shrinkage and thus more boosting steps correspondingly. It is shown in [@ESL] that subsampling without shrinkage leads to poor performance in test samples. With sufficient number of boosting steps, the algorithm manages to identify many informative features without including a dominant number of irrelevant ones. While the actual value of $M$ depends on the applications, in general we suggest a large value of $M$ in order to cover most of the informative features in the random subsampling. Since the computational complexity of the proposed algorithm is linearly scalable in the number of boosting steps $M$ while quadratic in the feature dimension $p$, feature subsampling is more computationally efficient even with large $M$. Hence, in high-dimensional setting, reducing the dimension of feature set to $\sqrt{p}$ makes the algorithm substantially faster. Moreover, via feature subsampling, the resulting weight matrix has much less complexity measure defined in as compared to the one without feature subsampling at each boosting step. As the sparse boosting approach optimizes over a tradeoff between prediction accuracy and the complexity of the weight matrix, the resulting $W$ would still be sufficiently sparse. Therefore, the feature subsampling with large number of boosting steps does not contradict with the goal of searching for sparse solutions.
However, there is no rule of thumb for choosing the value of $M$ in advance. Since each application has different underlying structure of its significant feature subspace as well as involving with different level of noise, the actual value of $M$ varies case by case. In general, we suggest a large number of $M$, from 500 to 2000, that is proportional the number of features $p$. When feature subsampling is applied, $M$ should be increase in an order of $\sqrt{p}$. Since the sparse boosting process is implemented, overfitting is effectively controlled even with large $M$ and thus it is recommended to start with considerably large value of $M$. Otherwise, we use cross-validation to evaluate different choices of $M$’s.
NUMERICAL EXPERIMENTS {#numeric_exp}
=====================
In this section, we present both simulation studies and real-data applications to evaluate the performance of the proposed $sDist$ algorithm. The algorithm is implemented with the following specifications. We use 5-fold cross-validations to determine the degree of sparsity for each rank-one update $\rho$, choosing from candidate values $\{0.05, 0.1, 0.2\}$. The same cross-validation is also applied to the tune overall complexity regularizing parameter $\lambda_C \in \{0.001, 0.01, 0.1, 1, 10\} $. In order to control the computation cost and to ensure interpretability of the selected variables and polynomial features, we impose an upper limit on the maximum order of polynomial of the expanded features. That is, when the polynomial has an order greater than a cap value, we stop adding it to the candidate feature set. For our experiments, the cap order is set to be 4. Namely, we expect to see maximally four-way interactions. The total number of boosting steps $M$ is set to be 2000 for all simulation experiments. While by sparse boosting, the actual numbers of weak learners used vary from case to case. Throughout the numerical experiments, the reported test errors are estimated using the $k$-Nearest Neighbor classifier with $k=3$ under the tuned parameter configuration.
The performance of $sDist$ is compared with several other distance metric learning methods, with the $k$-Nearest Neighbor ($k$NN) representing the baseline method with no metric learning, Probability Global Distance Metric (PGDM)[@DMLXing], Large Margin Nearest Neighbor (LMNN) [@LMNN], Sparse Metric Learning via Linear Programming (SMLlp) [@SMLlp], and Sparse Metric Learning via Smooth Optimization (SMLsm) [@SMLsm]. PGDM [^6] [@DMLXing] is a global distance metric learning method that solves the optimization problem: $$\begin{aligned}
\underset{W \succeq 0}{\operatorname{\min}} \quad & \sum_{y_i = y_j} (\mathbf{x}_i - \mathbf{x}_j)^T W (\mathbf{x}_i - \mathbf{x}_j) \\
\mbox{s.t.} \quad & \sum_{y_i \neq y_l} (\mathbf{x}_i - \mathbf{x}_l)^T W (\mathbf{x}_i - \mathbf{x}_l) \geq 1.\end{aligned}$$ LMNN [^7] learns the weight matrix $W$ by maximizing the margin between classes in local neighborhoods with a semi-definite programming. That is, $W$ is obtained by solving: $$\begin{aligned}
\underset{W \succeq 0, \ \xi_{ijl} \geq 0}{\operatorname{\min}} \quad & (1-\mu) \sum\limits_{i=1}^{n} \sum\limits_{j \in \mathcal{S}^+_k(\mathbf{x}_i)} (\mathbf{x}_i - \mathbf{x}_j)^T W (\mathbf{x}_i - \mathbf{x}_j) + \mu \sum\limits_{i=1}^{n} \sum\limits_{j \in \mathcal{S}^+_k(\mathbf{x}_i)} \sum\limits_{l \in \tilde{\mathcal{S}}^-(\mathbf{x}_i)} \xi_{ijl},\\
\mbox{s.t.} \quad & (\mathbf{x}_i - \mathbf{x}_l)^T W(\mathbf{x}_i - \mathbf{x}_l) - (\mathbf{x}_i - \mathbf{x}_j)^T W (\mathbf{x}_i - \mathbf{x}_j) \geq 1 - \xi_{ijl},\end{aligned}$$ where $\xi_{ijl}$’s are slack variables and $\tilde{\mathcal{S}}^-(\mathbf{x}_i) \triangleq \{ l | y_l \neq y_i \mbox{ and } d_I(\mathbf{x}_i, \mathbf{x}_l) \leq \underset{j \in \mathcal{S}^+_k(\mathbf{x}_i)}{\operatorname{\max}} d_I(\mathbf{x}_i, \mathbf{x}_j) \}$. In the experiments, we use $\mu = 0.5$ as suggested in [@LMNN]. SMLlp aims at learning a low rank weight matrix $W$ by optimizing over the linear projection $L \in \mathbb{R}^{p \times D}$ with $D \leq p$ in : $$\begin{aligned}
\underset{L \in \mathbb{R}^{p \times D}, \ \xi_{ijl} \geq 0}{\operatorname{\min}} \quad & \sum\limits_{(i, j, l) \in \mathcal{T}} \xi_{ijl} + \mu \sum\limits_{r=1}^{p} \sum\limits_{s=1}^{D} |L_{rs}|,\\
\mbox{s.t.} \quad & \| L\mathbf{x}_i - L\mathbf{x}_j \|^2_2 \leq \| L\mathbf{x}_i - L\mathbf{x}_l \|^2_2 + \xi_{ijl}, \quad \forall \ (i,j,l) \in \mathcal{T},
\end{aligned}$$ where $\mathcal{T} \in \{(i,j,l) \ | \ j = \mathcal{S}^+_1(\mathbf{x}_i), \ l = \mathcal{S}^-_1(\mathbf{x}_i) \}$. In a similar manner, SMLsm[^8] learns a low-rank weight matrix $W$ by employing a $l_{(2,1)}$ norm on the weight matrix $W$ to enforce column-wise sparsity. It is cast into the minimization problem: $$\begin{aligned}
\underset{U \in \mathcal{O}^p}{\operatorname{\min}} \ \underset{W \succeq 0, \ \xi_{ijl} \geq 0}{\operatorname{\min}} \quad & \sum\limits_{(i, j, l) \in \mathcal{T}} \xi_{ijl} + \mu \sum\limits_{r=1}^{p} \left( \sum\limits_{s=1}^{D} W_{rs}^2 \right)^{\frac{1}{2}},\\
\mbox{s.t.} \quad & 1+ (\mathbf{x}_i - \mathbf{x}_j)^T U^T W U (\mathbf{x}_i - \mathbf{x}_j) \leq (\mathbf{x}_i - \mathbf{x}_l)^T U^T W U (\mathbf{x}_i - \mathbf{x}_l) + \xi_{ijl}, \quad \forall \ (i,j,l) \in \mathcal{T},
\end{aligned}$$ where $\mathcal{O}^p$ is the set of $p-$dimensional orthonormal matrices.
The effectiveness of distance metric learning in high-dimensional datasets heavily depends on the computational complexity of the learning method. PGDM deploys a semi-definite programming in the optimization for $W$ which is in the order of $O(p^2 + p^3 + n^2p^2)$ for each gradient update. LMNN requires a computation complexity of $O(p^4)$ for optimization. SMLsm converges in $O(p^3/\epsilon)$, where $\epsilon$ is the stopping criterion for convergence. In comparison, $sDist$ runs with a computational complexity of approximately $O(M[(\kappa p + p)\kappa \log p + np^2])$ where $M$ is the number of boosting iterations and $\kappa$ is the number of nonzero entries in rank-one updates. In practice, $sDist$ can be significantly accelerated by applying the modifications in the Section 5, in which $p$ is substituted by $\tilde{p}$ and $n$ is substituted by $\eta n$.
We construct two simulations settings that are commonly used as classical examples for nonlinear classification problems in the literature, the “double ring” case and the “XOR” case. In Figure 2, the left most column of the figures indicates the contour plots of high class probability for generating sample points in a 3-dimensional surface, whereas the nput variable space is expanded to a much greater space of $p=50$, where irrelevant input variables represent pure noises. Figure \[fig: sim\_fig\] (*top row*) shows a simulation study in which sample points with opposite class labels interwine in a double rings, and Figure \[fig: sim\_fig\] (*bottom row*) borrows the illustrative example of “XOR” classification in Section 2. The columns 2-4 in Figure \[fig: sim\_fig\] illustrate the transformed subspaces learned by $sDist$ algorithm at selected iterations. Since the optimal number of iterations is not static and due to the space limit, we show only the first iteration, the last iteration determined by sparse boosting, and the middle iteration, which is rounded half of the optimal number of iterations. It is clearly shown in Figure \[fig: sim\_fig\] that the surfaces transformed by the learned distance metric correctly capture the structures of the generative distributions. In the “double ring” example (*top row*), the learned surface sinks in the center of the plane while the rim bends upward so that sample points in the “outer ring” are drawn closer in the transformed surface. The particular shape owes to the quadratic polynomial of the two informative variables chosen in constructing $W$, shown as the parabola in cross-sectional grid lines. In the “XOR” example (*bottom row*), the diagonal corners are curved toward the same directions. The interaction between the two informative variables is selected in additional to original input variable, which is essential in describing this particular crossing nonlinear decision boundary. *sDist* also proves highly computationally efficient, achieving approximate optimality within a few iterations.
![Transformed subspaces corresponding to metrics learned for nonlinear binary classification problems. The first column shows the simulations setups. *Upper*: Sample points are drawn from a “double rings” distribution. Shown are the contour plot of the generative class probability on a 3-dimensional surface. *Lower*: Sample points are drawn from the classical XOR scenario. Columns 2-4 demonstrate how the metric learning algorithm transofrm the feature space at selected iterations. The vertical dimensions is computed as the first principle components of the transformed feature space $L \phi(\mathbf{x})$, where $LL^T=W$. Note that the solid lines in the contour plots only show the geodesic lines of high probabilities in the class generation probability distributions. The generated class labels are not separable.[]{data-label="fig: sim_fig"}](experiment_demo.pdf){width="\textwidth"}
We also compare the performance of $sDist$ with other metric learning methods under different values of dimensions $p$ and sample sizes $N$ to demonstrate its scalability and its strength in obtaining essentially sparse solution in high-dimensional datasets. In this case, we generate the sample points from the “double ring ” example and the “XOR” example with the numbers of informative variables being 10% of the total dimensions, ranging from 100 to 5000. The results of these two cases are shown in Table \[tab: sim\_DR\] and Table \[tab: sim\_XOR\] respectively. It is noted that $sDist$ achieves relatively low test errors as compared to the competing methods, especially in high dimensional settings. *sDist* is also proved to be scalable to datasets with large sample sizes and with high-dimensional inputs.
[l|rrr|rrr|rrr]{}\
& & &\
& & & & & & & & &\
$k$NN & 0.310 & 0.40 & 0.488 & 0.311 & 0.426 & 0.478 & 0.308 & 0.475 & 0.489\
PGDM & 0.320 & 0.355 & 0.389 & 0.312 & 0.356 & 0.377 & 0.337 & 0.340 & 0.412\
LMNN & 0.230 & 0.280 & 0.290 & 0.245 & 0.291 & 0.289 & 0.246 & 0.303 & 0.315\
SMLsm & 0.222 & 0.289 & 0.250 & **0.169** & 0.200 & 0.249 & 0.199 & 0.276 & 0.330\
sDist & **0.143** & **0.189** & **0.192** & 0.177 & **0.183** & **0.191** & **0.168** & **0.179** & **0.202**\
Bayes Rate & 0.130 & 0.150 & 0.160 & 0.154 & 0.156 & 0.144 & 0.160 & 0.154 & 0.156\
[l|rrr|rrr|rrr]{}\
**** & & &\
& & & & & & & & &\
$k$NN & 0.355 & 0.410 & 0.491 & 0.420 & 0.446 & 0.499 & 0.397 & 0.500 & 0.500\
PGDM & 0.221 & 0.355 & 0.383 & 0.289 & 0.356 & 0.360 & 0.354 & 0.350 & 0.403\
LMNN & **0.145** & 0.280 & 0.274 & 0.188 & 0.213 & 0.239 & 0.198 & 0.231 & 0.299\
SMLsm & 0.207 & 0.307 & 0.333 & 0.277 & 0.291 & 0.337 & 0.242 & 0.378 & 0.420\
sDist & 0.157 & **0.199** & **0.192** & **0.169** & **0.183** & **0.225** & **0.193** & **0.187** & **0.221**\
Bayes Rate & 0.130 & 0.160 & 0.160 & 0.133 & 0.177 & 0.181 & 0.155 & 0.144 & 0.138\
The performance of *sDist* is also evaluated on three public datasets, presented in Table \[tab: real\_data\_err\]. For each dataset, we randomly split the original data into a 70% training set and a 30% testing set, and repeat for 20 times. Parameter values are tuned by cross-validation similarly as the simulation studies. The reported test errors in Table \[tab: real\_data\_err\] are the averages over 20 random splits on the datasets. The reported running times are the average CPU times for one execution[^9]. We also obtain the average percentage of features selected by various sparse metric learning methods in Figure \[fig: num\_var\].
[p[2cm]{} || r R[2cm]{} | r R[2cm]{} | r R[2cm]{} ]{} **Data Statistics** & & &\
\# Inputs & & &\
\# Instances & & &\
& Test Error & Running Time (sec) & Test Error & Running Time (sec) & Test Error & Running Time (sec)\
$k$NN & 0.13& 0.01 &0.14 & 2.07 & 0.46 & 9.05\
PGDM &0.07& 37.80 &0.09& 960.47 & 0.31 & 2527.82\
LMNN &0.06& 20.06 & 0.08 & 1960.94 & 0.39 & 1323.64\
SMLsm &0.09 & 173.19 & 0.09 & 1293.97 & 0.41 & 2993.97\
*sDist* &**0.05** & 27.49 & **0.07** & 473.07 & **0.09** & 689.64\
![The average percentages of variables (features) selected in the final metrics learned by different algorithms as well as the average running times. The percentage of *sDist* is calculated as the ratio of the total number of selected features over $p_{m*}$, where $p_{m*}$ is the dimension of the candidate set defined in at the optimal stopping iteration $m*$ selected by the sparse boosting method in Section 3.3.[]{data-label="fig: num_var"}](var_time.pdf){width="\textwidth"}
We first compare various distance metric learning methods on the Iononsphere dataset [@UCI] [^10]. This radar dataset represents a typical small dataset. It contains mixed data types, which poses a challenge to most of the distance-based classifiers. From Table \[tab: real\_data\_err\], we see that *sDist* and other metric learning methods significantly reduce the test errors by learning a nonlinear transformation of the input space, as compared to the ordinary $k$NN classifier. *sDist* particularly achieves the best performance by screening out a large proportion of noises. The marginal features selected by different methods are compared in Figure \[fig: num\_var\]. Features selected by *sDist* are mostly interactions within a single group of variables, suggesting an interesting underlying structure of the data for better interpretation.
SECOM [@UCI] [^11] contains measurements from sensors for monitoring the function of a modern semi-conductor manufacturing process. This dataset is a representative real-data application in which not all input variables are equally valuable. The measured signals from the sensors contain irrelevant information and high noise which mask the true information from being discovered. Under such scenario, accurate feature selection methods are proven to be effective in reducing test error significantly as well as identifying the most relevant signals [@UCI]. As shown in Table \[tab: real\_data\_err\], *sDist* [^12] demonstrates dominant performance over the other three methods with an improvement about 33% over the original $k$NN using the Euclidean distance. As compared to SMLsm, another sparse metric learning method, *sDist* shows much better scalability with a large number of input variables in terms of CPU time.
MADELON is an artificial dataset used in the NIPS 2003 challenge on feature selection [^13] [@UCI] [@guyon2006feature][@guyon2007competitive]. It contains sample points with binary labels that are clustered on the vertices of a five dimensional hypercube, in which these five dimensions constitute 5 informative variables. Fifteen linear combinations of these five variables were added to form a set of 20 (redundant) informative variables while the other 480 variables have no predictive power on class label. In Table \[tab: real\_data\_err\], *sDist* shows excellent performance compared to the other competing methods in terms of both predictive accuracy and computational efficiency. The test error achieved by *sDist* also outperforms states-of-the-art methods beyond the distance metric learning literature on the Madelon dataset [@kursa2010feature] [@suarez2014genetic] [@turki2014weighted]. *sDist* also attains the sparsest solution as shown in Figure \[fig: num\_var\], with 15.2% of features selected in the final weight matrix. Its outstanding performance indicates the importance of learning the low-dimensional manifold in high-dimensional data, particularly for the cases with low signal-to-noise ratio.
We also experimented different configurations of the tuning parameters introduced in the algorithm and the practical remarks on the Madelon dataset, including the frequency of local neighborhood updates, bagging fraction $\eta$, and the degree of sparsity for rank-one updates $\rho$. The performances in terms of both training error and validation error are shown in Figure \[fig: madelon\_tune\] for both the $k$NN classifier and the base classifier $f_W$ defined in . Particularly, it is evident that updating neighborhood more frequently seems to reduce validation error. The gain in performance diminishes as the frequency increases beyond a certain level. In practice, we suggest updating the local neighborhood every 50 steps as a tradeoff between the accuracy and the computational cost. In this example, the best performances of both classifiers are achieved at the bagging fraction 0.3 or 0.5 when the degree of sparsity $\rho$ is small. While as $\rho$ is large, the errors monotonically decrease as the bagging fraction increases. In practice, we suggest a bagging fraction 0.5 for moderate-size datasets and 0.3 for large datasets. When the true informative subspace is of relatively low-dimensional, as in the case of the Madelon dataset, both training errors and validation errors are reduced with small values of $\rho$. Sparse rank-one updates benefit the most from the boosting algorithm for progressive learning and prevent overfitting at each single step, while in other cases, the optimal value of $\rho$ depends on the underlying sparsity structure.
![Sensitivity analysis of different configurations of the tuning parameters in the *sDist* algorithm: frequency of local neighborhood updates, the bagging fraction $\eta$, and the degree of sparsity $\rho$ using the Madelon dataset. Training errors and testing errors are reported for both $k$NN classifier and the base classifier $f_W$ in based on 20 randomly partitioned 5-fold cross-validations.[]{data-label="fig: madelon_tune"}](madelon_tune_order1.pdf){width="\textwidth"}
CONCLUSIONS {#conclusion}
===========
In this paper, we propose an adaptive learning algorithm for finding a sparse Mahalanobis distance metric on a nonlinear feature space via a gradient boosting algorithm for binary classification. We especially introduced sparsity control that results in automatic feature selection. The *sDist* framework can be further extended in several directions. First, our framework can be generalized to multiclass problems. The base discriminant function in can be extended for a multiclass response variable in a similar fashion as in [@zhu2009multi] for multiclass AdaBoost. More specifically, the class label $c_i$ is recoded as a $K$-dimensional vector $\mathbf{y}_i = \{y_{i,1}, y_{i,2}, \dots, y_{i,K}\}^T$ with $K$ being the number of classes. Here $y_{ij} = 1$ if $c_i = j$ and $-\frac{1}{K-1}$ otherwise. Then a natural generalization of loss function in is given by: $$L(\mathbf{y}, f_W^{\phi}) = \sum\limits_{i=1}^{n} \exp\left( -\frac{1}{K} \mathbf{y}_i^T f_W^{\phi}(\mathbf{x}_i) \right).$$ The other way is to redefine the local positive/negative neighborhood as the local similar/dissimilar neighborhood as in [@LMNN], where the[ *similar*]{} points refer to sample points with the same class label and the [*dissimilar*]{} ones are those with different class labels. However, a rigorous discussion on the extension to muticlass problems requires comprehensive analysis. It is not entirely straightforward in how to exactly address multiclass labels in metric learning, or whether the learning goal is to determine a common metric for all classes or to construct different metrics between pairs of classes. Due to the limited scope of this paper, we will leave these questions in future studies.
Furthermore, in the proposed $sDist$ algorithm, we approached the fitting of nonlinear decision boundary through interaction expansion and local neighborhoods. It has been noted that distance measures have close connections with kernel functions, which is commonly used for nonlinear learning methods in the literature. Integrating the nonlinear distance metric learning with kernel methods will lead to more flexible and powerful classifiers.
Comparison between the Classifier $f_W$ in and the $k$NN Classifier {#App:AppendixA}
===================================================================
The classifier in can be considered as a continuous surrogate function of the $k$-Nearest Neighbor classifier, which is differentiable with respect to $W$. In Figure \[fig: compare\_knn\_fw\], we show the performance of the kNN classifier and $f_W$ in at different values of $k$ on the real dataset Ionosphere. It suggests that, with small $k$ ($k \leq 11)$ which is normally used in neighborhood-based method, $f_W$ consistently outperforms $k$NN classifier with aligned pattern in terms of the average test errors based on 20 randomly partitioned cross-validations.
![Comparison between $f_W$ in and the $k$NN classifier in terms of the average test errors based on 20 randomly partitioned 5-fold cross-validations using the real dataset Ionosphere.[]{data-label="fig: compare_knn_fw"}](compare_knn_fw.pdf){width="\textwidth"}
Truncated Power Method {#App:AppendixB}
======================
At each boosting step, we solve the constrained optimization problem in using the truncated power method as given in Algorithm 2.
Input: $A_m \in \mathbb{R}^{p_m \times p_m}$, $\kappa \in \{1, 2, \dots, p_m\}$, and the regularizing parameter $\lambda_0 > 0$ 1) Initialization: $A_0 = A_m$, $\xi_0 = \frac{\mathbf{1}_{p_m}}{\sqrt{p_m}}$ 2) Iteration: For $t = 1, 2, \dots$, repeat until convergence (a) Update $\lambda_t = 10\lambda_0$ until $A_t = A_{t-1} + \lambda_t I_p$ becomes positive semi-definite. (b) Compute $\hat{\xi}_t = \frac{A_t \xi_{t-1}}{||A_t \xi_{t-1}||}$. (c) Let $F_t = \mbox{supp}(\hat{\xi}_t, \kappa)$ be the indices of $\hat{\xi}_t$ with the largest $\kappa$ absolute values. Compute $\tilde{\xi}_t = \mbox{Truncate}(\hat{\xi}_t, F_t)$. (d) Normalize $\xi_t = \frac{\tilde{\xi}_t}{||\tilde{\xi}_t||}$. Output: $\xi_m = \xi_t$
\[alg: power\_method\]
It is worth noting that $A_m$ in each step of gradient boosting is not guaranteed to be positive semi-definite. Thus, to ensure that the objective function to be non-decreasing, we add a positive diagonal matrix $\tilde{\lambda} I_p$ to the matrix $A$ for $\tilde{\lambda}$ large enough such that $\tilde{A} = A + \tilde{\lambda}I_p$ is positive semi-definite and symmetric. Such change only adds a constant term to the objective function, which produces a different sequence of iterations, and there is a clear tradeoff. If $\tilde{\lambda}$ dominates $A$, the objective function becomes approximately a squared norm, and the algorithm tends to terminate in only a few iterations. In the limiting case of $\tilde{\lambda} \rightarrow \infty$, the method will not move away from the initial iterate. To handle this issue, we adapt a stochastic method that gradually increase $\tilde{\lambda}$ during the iterations and we do so only when the monotonicity is violated, as shown in the step 1 of Algorithm \[alg: power\_method\]. This truncated power method allows fast computation of the largest $\kappa$-sparse eigenvalue. For s high-dimensional but sparse matrix $A_m$, it also supports sparse matrix computation, which decreases the complexity from $O(p^3)$ to $O(\kappa p T)$, where $T$ is the number of iterations.
[^1]: For simplicity, we only consider datasets with numerical features in this paper, on which distances are naturally defined.
[^2]: In Section \[practical\_remark\], we introduce a practical solution that updates local neighborhoods regularly as the boosting algorithm proceeds.
[^3]: Here $X = [\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n]^T$. When $\mathcal{C}$ is a set of variable or interactions of variables, $X_{\mathcal{C}}$ represents the columns of $X$ (or products of columns of $X$) listed in $\mathcal{C}$.
[^4]: The $l_{(2,1)}$ norm of $W$ is given by: $||W||_{(2,1)} = \sum\limits_{h=1}^{p} (\sum\limits_{k=1}^{p} W_{hk}^2 )^{\frac{1}{2}}$ [@SMLsm]
[^5]: The parameter $\eta$ is referred as the “bagging fraction” in the following.
[^6]: Source of Matlab codes: <http://www.cs.cmu.edu/%7Eepxing/papers/Old_papers/code_Metric_online.tar.gz>
[^7]: Source of Matlab codes: [ http://www.cse.wustl.edu/\~kilian/code/code.html]( http://www.cse.wustl.edu/~kilian/code/code.html)
[^8]: Source of Matlab codes: <http://www.albany.edu/~yy298919/software.html>
[^9]: Running time of *sDist* for datasets ionosphere, SECOM, Madelon are based on $M=100$, 500, and 500 respectively with the configurations that achieve the best predictive performance. The $sDist$ algorithm is implemented on R (version $3.1.3$) on x86\_64 Redhat Linux GNU system. Other competing algorithms are implemented on Matlab ($R2014a$) on the same operating system.
[^10]: Available at
[^11]: The data is available at <https://archive.ics.uci.edu/ml/datasets/SECOM>. The original data is trimmed by taking out variables with constant values and variables with more than 10% of missing values so that the dimension is reduced from 591 to 414. Observations with missing value after the trimming on variables are discarded in this experiment, which reduces the sample size to 1436.
[^12]: Due to the heterogeneity in the input variables, we standardized the input variable matrix before implementing the *sDist* algorithm. In the nonlinear expansions, selected interaction terms are also scaled before being added to the candidate set $\mathcal{C}$.
[^13]: The data is available at <https://archive.ics.uci.edu/ml/datasets/Madelon>. We use both the train data and the validation data. The 5-fold cross-validation is performed on the combined dataset.
|
---
abstract: 'Relativistic jets are the most energetic manifestation of the active galactic nucleus (AGN) phenomenon. AGN jets are observed from the radio through gamma-rays and carry copious amounts of matter and energy from the sub-parsec central regions out to the kiloparsec and often megaparsec scale galaxy and cluster environs. While most spatially resolved jets are seen in the radio, an increasing number have been discovered to emit in the optical/near-IR and/or X-ray bands. Here we discuss a spectacular example of this class, the 3C 111 jet, housed in one of the nearest, double-lobed FR II radio galaxies known. We discuss new, deep [*Chandra*]{} and [*HST*]{} observations that reveal both near-IR and X-ray emission from several components of the 3C 111 jet, as well as both the northern and southern hotspots. Important differences are seen between the morphologies in the radio, X-ray, and near-IR bands. The long (over 100 kpc on each side), straight nature of this jet makes it an excellent prototype for future, deep observations, as it is one of the longest such features seen in the radio, near-IR/optical and X-ray bands. Several independent lines of evidence, including the X-ray and broadband spectral shape as well as the implied velocity of the approaching hotspot, lead us to strongly disfavor the EC/CMB model and instead favor a two-component synchrotron model to explain the observed X-ray emission for several jet components. Future observations with [*NuSTAR*]{}, [*HST*]{}, and [*Chandra*]{} will allow us to further constrain the emission mechanisms.'
author:
- 'Devon Clautice, Eric S. Perlman, Markos Georganopoulos, Matthew L. Lister, Francesco Tombesi, Mihai Cara, Herman L. Marshall, Brandon Hogan, Demos Kazanas'
title: 'The Spectacular Radio-Near-IR-X-ray Jet of 3C 111: X-ray Emission Mechanism and Jet Kinematics'
---
Introduction
============
One of the milestone discoveries of [*Chandra*]{} was the X-ray emission from nearly 100 quasar and radio galaxy jets, as well as their hotspots[[^1]]{}. The latter are high brightness regions where the jets collide with the intergalactic medium. In the radio and optical, the emission from these sites is synchrotron in nature. This guarantees the presence of X-ray emission, via the Synchrotron Self Compton (SSC) process. The discrepancy between the observed X-ray fluxes and the predictions of SSC models is often glaring [e.g., @schwartz00; @wilson01; @samb04; @marshall05], with the X-rays commonly being orders of magnitude brighter than the SSC prediction if equipartition magnetic fields are assumed. [@tavecchio00] and [@celotti01] proposed to explain this excess X-ray emission as external Compton (EC) scattering of cosmic microwave background (CMB) photons by the jet’s relativistic electrons. This requires jets with bulk Lorentz factor $\Gamma \sim 10$ that are oriented close to the line of sight for nearly their entire length. Alternatively [@dermer02], the X-rays may be synchrotron emission from high energy electrons suffering Compton losses in the Klein-Nishina regime. These particles are often required to be in a separate high-energy population [@hardcastle04; @hardcastle06]. In this case, the X-ray and optical emission require [*in situ*]{} particle acceleration, as the radiating particles have lifetimes of a few to hundreds of years, much shorter than the particle’s time to travel down the jet. Those emissions would then provide an excellent probe of the physics in jet regions where particle acceleration is happening. A third model [upstream Compton, @georg03], proposes a decelerating jet, with electrons in the faster, upstream flow scattering photons produced in the slow downstream flow, thus contributing to the X-ray emission.
Discriminating between these models relies on several diagnostics, including component spectral energy distributions (SEDs) and differences between radio and X-ray jet morphology [@Jester06; @Kharb12]). We have proposed two diagnostics that can rule out the EC/CMB model. The first of these [@Krawczynski:11; @Poutanen:93] relies on the fact that except for scatterings from low-energy particles ($\gamma \sim 1$) inverse-Comptonized CMB radiation should be unpolarized, reflecting the unpolarized nature of the seed photon population. This diagnostic was first used by [@cara13] to almost completely rule out the EC/CMB model for one quasar jet, PKS 1136-135. Another diagnostic [@georg06] relies on the fact that the observed synchrotron emission at IR and lower energies must also be Comptonized, resulting in a minimum level of GeV gamma-ray emission. This has ruled out the EC/CMB model for the jets of 3C 273 and PKS 0637-752 [@meyer14; @meyer15]). Finally, in a few FR IIs (e.g., Pictor A, [@hardcastle15; @gentry15; @marshall10]) that are viewed at larger angles, the broadband SED even suggests synchrotron emission without requiring a separate, high-energy electron population.
{width="10.7cm"} {width="6.7cm"}
{width="17.5cm"}
With all of these different possibilities, one of the most basic needs for investigating models of both jet emission and physics is to find ideal testing grounds. Only a very few prototype jets, that are bright in several bands at low redshifts, and minimally bent, are known. Here we discuss a new, prototype jet. 3C 111 is a powerful FR II radio galaxy [@FR:74] at $z=0.0485$ [@HB91]. Our [*HST*]{} images (§§2-3) show that the host galaxy is a bright giant elliptical with somewhat distorted outer isophotes, and several prominent companions within 50 kpc. On parsec scales, VLBI observations show component speeds as high as 8$c$ in the approaching, northern jet [@lister13]. Shallow, 10 ks [*Chandra X-ray Observatory*]{} survey observations by [@hogan10] revealed X-ray emission from three knots in the northern jet (which we call K30, K61 and K97) and the northern hotspot (NHS). The jet is extremely long (nearly 4 arcminutes) and its host galaxy resides in a rich optical environment. Here we discuss the results of new, deep observations with both [*Chandra*]{} and the [*Hubble Space Telescope*]{} (HST). These observations not only confirm the results of [@hogan10] but also reveal near-IR and X-ray emission from several components in the 3C 111 jet, as well as the southern (receding) hotspot.
This paper is laid out as follows: In Section 2, we describe our observations and data reduction methods. Section 3 shows the results and discusses the broadband spectrum of the jet components. We close in Section 4 by stating our conclusions. Throughout this paper we assume $\Omega_m=0.27$, $\Omega_{\Lambda}=0.73$, $\Omega_r=0$ and $H_0=71\mbox{ km}\mbox{
s}^{-1}\mbox{ Mpc}^{-1}$.
Observations and Data Reductions
================================
[*Chandra*]{} Observations
--------------------------
[*Chandra*]{} has observed 3C 111 three times with the ACIS-S. In 2008, a shallow, 10 ks survey observation (dataset 701719) was taken [@hogan10], which discovered the X-ray emission from the jet. On 10-11 January 2013, we obtained much deeper observations (dataset 702798), for a total on-source time of 127 ks. These observations were gathered using alternating exposure mode, with interleaved frame times of 1.5s ($\times 4$) and 0.3s ($\times 1$) during each cycle. This was done in order to enable us to minimize the effect of pileup in the region of the quasar nucleus, while at the same time keeping the majority of the time optimized for detection of fainter sources in a broader field. It reduced efficiency by 15%, giving us a total exposure time of 92 ks (1.5s frame time only), but allowed us to discriminate inner jet knots from emission due to the AGN in the innermost 10 arcseconds, where pileup is a factor, by using the 0.3s frame time data (17 ks exposure time). These observations were augmented on 4-5 November 2014 by ACIS/HETG observations (150 ks, PI F. Tombesi, dataset 703007).
All observations were reduced in CIAO version 4.8.0, using CALDB v. 4.7.0, with standard screening criteria and calibration files provided by the [*Chandra*]{} X-Ray Center. Pixel randomization was removed, and only events in grades 0, 2 Ð- 4, and 6 were retained. We also checked for flaring background events. No significant flaring events were found, so that we did not have to filter by time. We subsampled the native [*Chandra*]{} resolution by 4, leading to a pixel scale of 0.123 arcsec/pixel. Datasets 702798 and 703007 were combined to obtain the images discussed in this paper. We chose not to incorporate the much shallower dataset 701719 into that analysis because of its poor statistics. To show the extended structure, we smoothed the X-ray image adaptively using the CIAO task [*csmooth*]{}, smoothing only below a minimum significance of 4. [[^2]]{}
[*HST*]{} Observations
----------------------
{width="8.6cm"} {width="8.6cm"} {width="8.6cm"} {width="8.6cm"}
[*HST*]{} observed 3C 111 on 30 January 2013 for three orbits, using the Wide-field Camera 3 (WFC3). Images were gathered both in the UVIS channel using the F850LP filter (1 orbit) and in the IR channel using the F160W filter (2 orbits). Because of the size of the 3C 111 jet-hotspots system, we restricted [*HST*]{}’s orientation so that the jet fell along a chip diagonal in both observations. Unfortunately for ease of scheduling we had to leave a 10-degree allowance on the allowed position angle (PA), and the PA that was used placed emission from the NHS at the edge of the field of the UVIS/F850LP observation. To compensate for this, we located archival observations obtained on 26 February 1996 with the Wide-Field and Planetary Camera 2 (WFPC 2) with the F791W filter (PI Meisenheimer). These latter observations include only the northern hotspot and some of the northern jet. Because of the small field of view, two pointings were necessary in the IR channel, while one pointing was deemed adequate in the UVIS. In addition, we used a standard, 2-position dither pattern at each location in each band. This, combined with the multiple readouts, was more than adequate to remove bad pixels and cosmic rays in the IR/F160W observation, but in the UVIS/ F850LP observation it was not adequate, and there were a significant number of pixels that had cosmic ray strikes in both images. In addition to the above there exist two shorter observations obtained with NICMOS and WFC2 (PI Sparks). Table 1 gives details of all [*HST*]{} observations.
All [*HST*]{} images were re-calibrated in PYRAF using the most up-to-date reference files (i.e., flat field, distortion correction table, etc.) obtained from the STScI Calibration Database system. We corrected for charge transfer efficiency (CTE) effects in the UVIS/F850LP data using the recipe of [@And2012] and in the WFPC2 data using the recipe of [@Dolphin:00] and [@Riess:00]. In the UVIS/F850LP data we also pre-processed the data using L.A. Cosmic[^3] [@vanDok:01] prior to drizzling. This significantly decreased the number of cosmic rays affecting the final image. We used the `Astrodrizzle` task [@Gonzaga:12] from the `STSCI_PYTHON` package to drizzle-combine the images for each of the two filter combinations. Besides combining the images, `Astrodrizzle` distortion-corrects the images, performs image flat-fielding, cosmic-ray rejection, image alignment, and other tasks. Prior to any analysis, the HST data had to be galaxy-subtracted. This was done using the tasks `ellipse` and `bmodel`. The model fitting was done iteratively, excluding nearby stars and galaxies (note that 3C 111 lies in a fairly dense cluster of galaxies). Local background regions were used to determine the blank sky noise emission for each source aperture. Sigma clipping was used to eliminate any pixel values that deviated beyond 3 sigma from the median. Photon noise was estimated by multiplying the weight map created by `Astrodrizzle` with our science image (in counts/second) to obtain the number of counts in each sky- subtracted source region. Read noise was taken from the header values in each image; dark current was estimated from the dark reference file indicated in the header.
Aperture photometry was done on the images using the apertures shown in Figure 1. Aperture correction was done following the recommendations of the WFC3 Data Handbook [@rajan11], while for the WFPC2 dataset it was done following [@holtzmann95]. Conversion to flux units was performed by multiplying image data in electrons/s by the corresponding and values for all images. 3C 111 is at a low galactic latitude ($b_{II}=8.8^\circ$), relatively near the Taurus molecular cloud (the nearest large star-forming region in our Galaxy). Ungerer et al. (1985), in their detailed optical and radio study, pointed out that the region of the cloud in front of 3C 111 is not the densest part (see Figure 3 of Ungerer et al. 1985). This result is also supported by the results of the XMM-Newton Extended Survey of the Taurus Molecular Cloud project (Güdel et al. 2007). Due to the presence of this molecular cloud, galactic extinction is unfortunately high, with $A_V=4.5$ mag assuming a standard $R_V=3.1$ [@SF11; @SFD98]. We note that [@meisenheimer97] used a much lower value for the extinction to 3C 111, stemming from the earlier survey of [@burheil82] (see also §3).
Results
=======
The 3C 111 jet can be seen across the electromagnetic spectrum, from the radio through the X-rays. In Figure 1, we show our deep [*Chandra*]{} imaging of 3C 111, along with archival VLA imaging [@leahy97]. X-ray emission is evident in at least 8 jet regions, plus the northern and southern hotspot. This emission is also seen in the near-IR, as shown in Figure 2, which shows close-ups of three jet regions in the F160W image, respectively the northern hotspot, inner jet and southern hotspots. The near-IR image shows emission from most, but not all X-ray emitting jet regions. In the F850LP and F791W images, the only jet or jet-related emission that can be seen comes from the northern hotspot. This is likely a result of the high Galactic extinction towards 3C 111. Most of the panels in Figures 1 and 2 show one image as greyscale and another as contours, allowing us to compare the morphology in different bands. To aid in this comparison, we named the northern jet features using the distance in arcseconds from the nucleus. Thus, as an example, knot K14 has its flux maximum 14 from the nucleus.
Jet Morphology
--------------
There is significant evidence of differences between the radio, near-IR and X-ray morphology, as seen in Figures 1 and 2, as well as in Figure 3, which shows the profile of relative flux (each normalized to 1 at an arbitrary point) along the jet in the [*Chandra*]{}, F160W, and VLA images. We note that there are strong differences between the radio, near-IR and X-ray fluxes. The near-IR and X-ray morphology are discussed in detail here. The radio morphology will be discussed in more detail in a future paper, where we also discuss follow-on deep JVLA observations. In the next sub-section, we will discuss the spectral energy distribution of the jet features, including the X-ray and optical spectral indices for the knots where it was possible to extract such information. In registering the three data sets to a common frame of reference, we assumed the VLA map to be the fiducial, adhering to the usual IAU standard. The [*HST*]{} images were registered to this frame by hand, as the Guide Star Catalog alignment always has errors of near arcsecond level, assuming that the optical and radio AGN core positions were identical. To register the [*Chandra*]{} data to this frame, we followed the CIAO thread “Correcting Absolute Astrometry", using CIAO task [*wavdetect*]{} to match sources in the 2MASS catalog in the [*Chandra*]{} images. This yielded a final offset of about 0.2 from the radio. We also merged the data from datasets 702798 and 703007 using [*reproject\_obs*]{} in CIAO. Following this, the $1\sigma $ errors in the positions from the [*HST*]{} image are $<0.02\arcsec$, while those in the X-ray image are $\pm 0.16\arcsec$ relative to the radio frame of reference according to [@Rots11], although to be conservative for this purpose we used 0.3.
![Relative flux as a function of distance from the nucleus for the approaching jet of 3C 111, as seen in the radio ([*VLA*]{} image, blue trace), near-IR ([*HST/WFC3 IR/F160W*]{} image, green trace) and X-rays ([*Chandra*]{} image, red trace). Each of the three traces was extracted from a slice $1.476\arcsec$ wide, along a vector extending from the nucleus of the galaxy through the NHS. Major knot regions are labeled above the traces. See §2 for details on the alignment of the three images, and see §3 for discussion.](3C111_plot_120_smoothed.png){width="9.5cm"}
Components within $\sim$ 20 of the nucleus have flux profiles in the X-rays that are mixed with that of the unresolved nuclear source due to the [*Chandra*]{} PSF (see Fig. 3), and are somewhat piled up in the long frame time, undispersed [*Chandra*]{} image, and within 10-15 the knots also lie within the galaxy seen in the optical/near-IR image. However, despite this, we can make a few remarks about how the radio, optical and X-ray morphologies compare, using the short frame time data from the interleaved dataset (702798) as well as the Order 0 HETG image (dataset 703007). Knot K9’s X-ray morphology (Figure 1) has a strong peak towards its upstream end that is not seen in the radio. Unfortunately, however, it lies too close to the diffraction spike in the F160W image to fully characterize in the near-IR. Knot K14 appears to peak slightly closer to the nucleus in the radio image than in either the near-IR or X- rays. X-ray emission is clearly seen downstream of that component extending nearly continuously to knot K30. That emission is not seen in either the near-IR or the radio (the near-IR emission that is present is more likely due to subtle galaxy features in the same region, as shown in the middle panel of Figure 2). That X-ray emission includes a knot seen only in the X-rays, knot K22.
Knot K30, seen in all three bands, has an X-ray flux peak that is located significantly upstream of either the radio or near-IR one (Fig 4), with the near-IR peak located closer to the nucleus than the radio one. The X-ray flux from K30 also declines much more quickly with increasing distance from the nucleus than in either the near-IR or radio, which show similar decline rates (Figure 3). The K40-K45 region is also complex. The radio flux of K40 displays two peaks, with the near-IR peak associated with the one closer to the nucleus. The X-ray emission, however, does not peak until 42 from the nucleus, where there is an apparent radio minimum. The radio emission picks up again between 43-45, while through that region and extending out to nearly 50, the X-ray emission appears roughly continuous and there is no significant X-ray knot at 51 from the nucleus as there is in the radio. Moving further out, there is a flux maximum at about 55 from the nucleus in the radio image that is not seen in the X-ray or F160W images. Knot K61, which represents an apparent ’kink’ in the jet, is seen in both the radio and X-rays. Its X-ray morphology has a ’corkscrew’ like appearance that is not prominent in the radio, where only its downstream half can be seen. In the optical/near-IR, K61 unfortunately lies very near a bright foreground star and so while there is possible emission in the near-IR it lies too close to the star or its diffraction spikes to have confidence in its detection and/or measure a flux.
![Close-up views of the K30 (top) and NHS (bottom) regions of the [*Chandra*]{} image, showing the location of the flux maxima in the radio (blue), near-IR (green), and X-ray (red) bands. The sizes of the error bars on each position are shown. Radio contours overlaid in cyan.](morphology_K30_2.png "fig:"){width="9.cm"} ![Close-up views of the K30 (top) and NHS (bottom) regions of the [*Chandra*]{} image, showing the location of the flux maxima in the radio (blue), near-IR (green), and X-ray (red) bands. The sizes of the error bars on each position are shown. Radio contours overlaid in cyan.](morphology_NHS_2.png "fig:"){width="8.5cm"}
Four regions are seen within the extended lobes. Within the northern lobe we see knots K97 and K107, as well as the flux maximum of the NHS itself. While these three hotspots have outwardly similar morphologies in the X-ray, near-IR and radio, close examination reveals important differences. In particular, it is only in the radio that one appreciates the extent of the northern lobe, which extends for over 30 in a ’plume’ shape that includes both K97 and K107, In the X-ray and near-IR, we see only the three hotspots (with K97 barely detected), plus extensions to the NHS in two directions, the first being back upstream pointing at K107, and the second one pointing off to the southwest parallel to the flux contours defining the lobe’s southern edge. The latter could indicate material outflowing from the hotspot, similar to what has been postulated for the 3C 273 jet by [@Roser96], while the general shape of the jet in that region indicates that the jet does bend as it enters the lobe. A close look at the NHS itself also reveals that its flux maximum is not located at the same position in the radio, near-IR/optical and X-ray, with the X-ray maximum seen upstream of the maxima seen in the near-IR/optical and radio (which are aligned with each other). This misalignment, which is suggestive but not firm at the 2.5-3 $\sigma$ significance level, is shown in Figures 1 and 2, and quantified in Fig. 4. In addition, we also see for the first time X-ray and near-IR emission from the SHS. The radio and near-IR emission from the SHS flux are well aligned (Fig. 2), while there simply are not enough photons detected in the [*Chandra*]{} image to firm up the comparison between its X-ray and optical flux maximum position, as only $32 \pm 8$ counts are seen from the SHS and the X-ray emission is significantly extended.
Jet Spectral Energy Distribution
--------------------------------
We have extracted fluxes and spectral energy distributions (SEDs) for all jet and hotspot regions. The sky regions used are shown in Figure 1 as green ellipses. The results are given in Table 2. Where a component is not detected in a given band, we give a 2$\sigma$ upper limit. The optical and near-IR fluxes were extracted from the galaxy-subtracted images and are corrected for extinction using the published value of $A_V$. In addition, for the near-IR and optical fluxes we also subtracted the average flux from a radial ring at the same distance from the nucleus, in order to minimize galaxy subtraction residuals. This was necessary because of the rather disturbed morphology of the host galaxy as well as the presence of several bright, nearby companion galaxies as well as bright stars. To convert the optical and near-IR count rates into fluxes we used the header information from SYNPHOT. By default, these assume a flat spectrum ($\alpha \approx 0, F_\nu \propto \nu^{-\alpha}$); however, the errors from this assumption are typically $<5\%$ in these wide bands. The fluxes in a given band are assumed to be centered at the band’s pivot wavelength.
The X-ray spectra of the three brightest regions in the 3C 111 jet (knots K30 and K61, and the NHS) were extracted using [*specextract*]{}. The extraction regions used are shown on Figure 1. Background spectra were obtained using annular regions at the same radii as the components itself, and excluding the readout streak. We used unweighted ARFs and RMFs and corrected for the PSF. Spectral fitting was done in [*Sherpa*]{} using [*XSpec*]{} models xsphabs and xspowerlaw.
Determining the correct column density of absorption for 3C 111 is complicated, as it is known that the source shows an additional absorbing column in excess of the Galactic value of $N_H=3.0 \times 10^{21} {\rm cm^{-2}}$ (e.g., [@1998MNRAS.299..410R; @2011MNRAS.418.2367B; @2013MNRAS.434.2707T]). For this analysis, we have used Galactic absorption with a column density of $N_H = 8.6 \pm 0.02 \times 10^{21} {\rm ~cm^{-2}}$. This was determined from the [*Chandra*]{} HETG data set, the full analysis of which will be discussed in a future paper (Tombesi et al., in prep.). The other [*Chandra*]{} data sets of 3C 111 suffer from pileup in the region of the quasar nucleus, making it impossible to determine accurately the value of $N_H$ from them $-$ [*e.g.,*]{} using our 0.3s frame time data to fit the absorption gives a value of $5.04 \times 10^{21} {\rm ~cm^{-2}}$. An $N_H$ of $8.6 \times 10^{21} {\rm ~cm^{-2}}$ is consistent with previous expectations (see also [@2013MNRAS.434.2707T]).
We used the CSTAT statistic in Sherpa as well as the Simplex (aka Nelder-Mead) fitting optimization method because of their robustness in low-signal cases. These fits were all checked using the Monte-Carlo method, and the results matched those of Simplex. The CSTAT statistic in [*Sherpa*]{} is equivalent to the Cash statistic but allows for easier checking of the goodness-of-fit. We checked the goodness-of-fit in two ways: first, by looking at the reduced statistic; and second, by running a simulation of the model and using [*plot\_cdf*]{} to check that the cumulative distribution function had a median at about 0.5. The fitting was done for 0.5-7 keV on unbinned data. The flux was determined from the [*calc\_energy\_flux*]{} function, over a range of 0.5 to 7 keV. Simulations were also used to determine the error in the flux value. Errors in flux are given at 68% confidence, while the error in photon index and normalization are given at 90% confidence intervals. This yielded the X-ray spectral indices given in Table 3. As can be seen, all three regions have X-ray spectral indices between $\alpha=0.76$ to $\alpha=1.01$. The X-ray fluxes from other jet regions were corrected for scattered light from the AGN itself using annular regions at the same radius as the component itself. The X-ray count rates for all jet regions were converted to flux assuming Galactic absorption. For the three regions where X-ray spectral fitting was possible, we used the power-law fits given in Table 3. For all other regions, we used a power-law index of $\alpha=0.87$, equal to the average of the three regions fit.
{width="8.5cm"} {width="8.5cm"}\
{width="8.5cm"} {width="8.5cm"}
We show the resulting SEDs for all the components in Figure 5. For regions K30, K61 and the NHS, we use the fitted X-ray flux and spectral index. Fig. 5 also includes ground-based K and R-band fluxes for the NHS that were previously published in [@meisenheimer97], corrected with updated values for the Galactic extinction (squares in the lower right panel, see discussion in §2.2), as well as a 1.3 mm flux from IRAM [@meisenheimer89]. The 1.3 mm IRAM point lies very close to the power law ($\alpha_R=0.85$) extrapolated from the 8.4 GHz observation of [@leahy97]. The apparent discrepancy between our F160W flux (Table 2, circles in Fig. 5) and the extrapolation of the K-to-R band spectral index from [@meisenheimer97] merits further discussion. We chose a slightly larger aperture than [@meisenheimer97], to include faint extended flux not seen by those authors, as shown in Figure 6. This is only 3% of the total, and both after and before this, our F791W flux is within 1$\sigma$ of the [@meisenheimer97] K to R band extrapolation. While it is possible that our flux in F160W is incorrect, we consider this unlikely given our careful choice of a source-free background region (Figure 6) and the well-established nature of the HST flux scale. Alternately, the K-band flux measurement of [@meisenheimer97] was affected by either poor background subtraction or poor flux calibration. We favor this explanation, as due to the crowded field (Figures 3, 6) it is likely that the background region in a ground-based image, like that of [@meisenheimer97], would include flux from one or more neighboring objects, thus causing an apparent underestimate of the source flux. We were unable to confirm this with the authors of [@meisenheimer97], however).
{width="8.6cm"}
As can be seen, most of the jet components have diverse SED shapes that naively can be fit by synchrotron emission from a single electron population. For example, knots K45 and K97 appear to be fit reasonably well by single power laws extending up to X-ray energies, and most other knots have X-ray flux that is significantly below the extrapolation of the radio-near-IR power law. However, we do not favor this simple interpretation, as in the NHS the fitted X-ray spectral slope is much harder than the extrapolation of the radio to optical synchrotron component, while in knot K61 the X-ray flux is a factor of about 4 higher than a simple extrapolation of the radio to near-IR power law. Thus a second emission component is necessary to fit the SED of these jet knots and possibly others. In broad terms, such a spectral shape has been seen before in other quasar jets [e.g., PKS 0637-752, knots WK7.8 and WK8.7, @mehta09], and requires either a contribution from another, inverse-Compton mechanism (the so-called EC/CMB mechanism), or alternately a second, entirely distinct high-energy electron population to account for the X-ray emission. Here, however, the fitted X-ray spectra combined with the fact that the X-ray emission of knot K30 and the NHS has a maximum at a different location than the near-IR or optical emission, makes the two-component synchrotron interpretation much more likely. Additionally, a Doppler factor of $\delta \gtrsim 45$ is required to explain the observed properties of the NHS flux if EC/CMB is the dominant emission mechanism at work (see §4.2).
Physical Considerations
=======================
The jet of 3C 111 is unique for several reasons. Chief among these are the fact that both the approaching and receding hotspots can be seen in all bands, and its extreme length, with X-ray and near-IR components seen in the jet for more than 100 arcseconds. The data we present here can be used to place a variety of constraints on both the kinematics of the jet as well as the X-ray emission mechanism. In §4.1, we use the detection of both the approaching and receding hotspots, as well as VLBA observations, to comment on the kinematics of the jet, while in §4.2 we discuss efforts to model the X-ray spectrum and broadband spectral energy distribution of the brightest knots to constrain their emission mechanism in the X-rays.
Jet Deceleration
----------------
The flux ratio between the northern and southern hotspots can be used to determine the permitted values for $\beta$ and $\theta$ by using $$\frac{S_{1}}{S_{2}} = \left( \frac{1 + \beta ~{\rm cos} \theta}{1 - \beta ~{\rm cos}\theta}\right)^{2+\alpha}$$
(e.g., [@2012rjag.book.....B]). We do this individually for the radio, near-IR, and X-ray bands. Here, $\theta$ is the angle to the line of sight, $\beta = v/c$, and $\alpha$ is the spectral index for each band (0.85 for radio, 1.50 for near-IR, and 0.83 for X-ray; see Figure 5, lower right panel and discussion in §3.2). The jet/counterjet hotspot flux ratio differs significantly between bands: $3.34 \pm .01$ in the radio, $9.03 \pm 0.36$ in the near-IR, and $5.34 \pm 1.61$ in the X-ray. This equation makes the assumptions that the jet and counterjet are exactly identical and 180$^{\circ}$ apart. [@jorstad2005] used VLBA observations and determined the most likely viewing angle to be $18.1 \pm 5.0$ degrees on the parsec scale. We found the permitted range of $\beta$ and $\theta$ for the VLBA scale by using their value for the transverse $\beta_T$ apparent to solve
$$\beta_T = \frac{\beta ~{\rm sin}\theta}{1 - \beta ~{\rm cos} \theta}.$$
Figure \[beta\_plot\] shows the $\beta$ vs $\theta$ plot for the parsec-scale VLBA results as well as the $\sim$100 kiloparsec-scale hotspots using our data. We see a clear deceleration from $\beta \sim$ 0.96 at the parsec scale to $\beta \sim$ 0.2-0.4 at the hotspot, with the velocity of the radio-emitting plasma significantly slower than that of the X-ray- and near-IR-emitting plasma. This is consistent with the two-component synchrotron model due to the fact that the radio- and X-ray-emitting electron populations appear to be moving at significantly different velocities, however it may require that the near-IR-emitting electrons do not occupy the entire jet cross-section, as in the simplest version of this scenario the near-IR and radio emission come from the same spectral component. Given the relatively modest beaming we find, it is interesting that no jet components are seen in the counterjet between the nucleus and SHS. Additional [*HST*]{} and [*Chandra*]{} observations are required to better constrain the near-IR spectral index and elaborate on these issues. [@2015JKAS...48..299O] more recently used VLBI observations to constrain the viewing angle of 3C 111 on mas scales to $\theta \lesssim 20$ degrees and the intrinsic velocity to $\beta \gtrsim 0.98$, in agreement with the findings of [@jorstad2005]. Given the large assumptions and the probable complex structure and dynamics of the hotspot regions, this analysis serves to place an upper limit on the amount of beaming in the jet. The analysis is inconsistent with a highly-beamed jet, as we would expect the jet/counterjet hotspot flux ratio to be larger if beaming were higher.
The spectral index used for the radio is based on the assumption that the slope is constant up to the near-IR. We plan to improve on this value in a future paper where we analyze JVLA observations (C, X, and Ku bands) of 3C 111. A harder spectral index for the radio would increase the likely value for $\beta$, however the offset would not be large enough to bring it into agreement with the near-IR, where the $\Delta \beta \sim 0.1$. This uncertainty does not affect the small $\Delta \beta$ between the X-ray and near-IR, though the near-IR spectral slope could change a small amount with additional [*HST*]{} bands to fit the slope.
While the viewing angle has a rather large uncertainty, the $\beta$ value is much more constrained. The relative difference in $\beta$ between bands is preserved no matter the viewing angle, adding to the evidence that there are two electron populations moving at significantly different speeds.
The jet to counterjet length ratio is in relatively good agreement with the radio jet to counterjet flux ratio. The approaching jet is $\sim$ 121 arcsec in length and the counterjet is $\sim 74$ arcsec in length, giving a length ratio of 0.61. For a jet moving at a constant speed $\beta$ and angle $\theta$, we expect the ratio of the lengths to be equal to $(1 - \beta cos \theta) / (1 + \beta cos \theta)$. This matches well with our observed value for $\theta=18.1^\circ$, giving a value of $\beta=0.254$ (Fig. 7), although this depends on how the approaching and receding jets decelerate (e.g., [@ryle1967]) and whether there is bending in either jet.
![Plot of $\beta = v/c$ vs viewing angle for the VLBA scale (solid black line) and kpc-scale radio (blue), near-IR (green), and X-ray (red). 1$\sigma$ uncertainties shown as shaded regions. The dotted lines indicate the VLBA-scale likely viewing angle of $18.1 \pm 5.0$ degrees.[]{data-label="beta_plot"}](beta_plot_revised2.pdf){width="8.6cm"}
Modeling of the Spectral Energy Distribution
--------------------------------------------
The spectral indices we have obtained for K30, K61, and the NHS are all such that they must lie on either the low-energy tail or near the turnover of the second emission component. Synchrotron and EC/CMB models predict differing slopes for the emission from the very lowest energy electrons, namely $\alpha = -1/3$ for synchrotron and $\alpha = -1$ for EC/CMB (e.g., [@dermer2009; @stawarz2008]). If the observed spectral index at any part of the low-energy tail were to become significantly harder than $-1/3$, then that would rule out synchrotron as the dominant emission mechanism.
Figure \[alpha\_plot\] shows the spectral indices for various overlapping energy ranges. All three regions are in good agreement with constant spectral slopes across the entire 0.5-7.0 keV band.
![X-ray spectral indices for various overlapping energy ranges with error bars for 68% and 95% confidence intervals. Energy ranges: 0.5-2, 0.5-3, 0.5-3.5, 1-4, 1.5-4.5, 2-5, 2.5-5.5, 3-7 keV. The labeled dashed lines indicate the predicted spectral indices for the low-energy tail of the synchrotron and EC/CMB models.[]{data-label="alpha_plot"}](alpha_plot1_revised.pdf "fig:"){width="8.6cm"} ![X-ray spectral indices for various overlapping energy ranges with error bars for 68% and 95% confidence intervals. Energy ranges: 0.5-2, 0.5-3, 0.5-3.5, 1-4, 1.5-4.5, 2-5, 2.5-5.5, 3-7 keV. The labeled dashed lines indicate the predicted spectral indices for the low-energy tail of the synchrotron and EC/CMB models.[]{data-label="alpha_plot"}](alpha_plot2_revised.pdf "fig:"){width="8.6cm"} ![X-ray spectral indices for various overlapping energy ranges with error bars for 68% and 95% confidence intervals. Energy ranges: 0.5-2, 0.5-3, 0.5-3.5, 1-4, 1.5-4.5, 2-5, 2.5-5.5, 3-7 keV. The labeled dashed lines indicate the predicted spectral indices for the low-energy tail of the synchrotron and EC/CMB models.[]{data-label="alpha_plot"}](alpha_plot3_revised.pdf "fig:"){width="8.6cm"}
Using the parsec-scale viewing angle of $18.1$ degrees and the associated values for $\beta$ from Figure \[beta\_plot\], we can make approximations for the values of $\Gamma$ and $\delta = \lbrack \Gamma (1 - \beta ~{\rm cos} \theta) \rbrack^{-1}$ in order to model the SED for the synchrotron and EC/CMB cases for the NHS.
Figure \[SED\_model\] shows several attempts at modeling the SED of K61 and the NHS with varying parameters for the synchrotron model using the Compton Sphere suite[[^4]]{}. In the case of K61, our near-IR and X-ray data serve to constrain the low-energy tail of the second emission component. However, because the near-IR spectrum for K61 is not available, we are not able to determine whether the detected flux is dominated by the first or second emission components $-$ the spectrum could be either falling in the near-IR as the first synchrotron component dies off, or it could be rising as the second emission component ramps up. Future [*HST*]{} observations would allow us to constrain which emission component is responsible for the detected near-IR flux. We have plotted two example models for the second emission component showing these possibilities using a magnetic field strength ranging over $B = (1-3.2) \times 10^{-5}$ G, with $\gamma_{max} = (3.6-10) \times 10^{9}$, and $\gamma_{min} = (1.3-3.6) \times 10^7$, with a comoving luminosity of $2.15 \times 10^{42}\ {\rm erg\ s^{-1}}$. The magnetic field strength $B$ and fitted $\gamma_{max}$ values translate to a radiative lifetime of $\sim$ 100 years, which is difficult to explain without distributed [*in situ*]{} acceleration $-$ this requirement can be relaxed by using a lower value of $B$.
Varying several of the input parameters can have a large effect on the shape of the curve above 7 keV for K61 and especially in the case of the NHS. The bottom of Figure \[SED\_model\] shows several representative models for the SED of the NHS near the [*NuSTAR*]{} energy band. Unlike K61, the low-energy tail of the second emission component of the NHS is not constrained by the radio or near-IR data. The models shown here vary wildly in emission above 7 keV, where the magnetic field strengths ranges over $B = (0.2-1) \times 10^{-4}$ G, with $\gamma_{max} = (1.9-100) \times 10^{8}$, and $\gamma_{min} = (5.2-27) \times 10^3$, with a comoving luminosity of $1 \times 10^{43}\ {\rm erg\ s^{-1}}$. Future observations using [*NuSTAR*]{} would allow us to constrain the SED up to $\sim$ 80 keV.
If the X-ray emission is due only to EC/CMB, then an estimate of the magnetic field strength can be made using
$$\frac{S_{sync}}{S_{IC}} = \frac{(2 \times 10^4 T)^{(3-p)/2} {B_{\mu G}^{(1+p)/2}}}{8 \pi \rho}$$
[@felten1966], where $\rho = \Gamma^2 \rho_0 (1 + z)^4$ is the apparent energy density of the CMB at redshift z, $\rho_0 = 4.19 \times 10^{-13}$ erg cm$^{-3}$ is the local CMB energy density, the apparent temperature of the CMB is $\delta T$, and the temperature of the CMB is $T = 2.728(1 + z) K$. This calculation gives a value of $B \approx 7.9 \times 10^{-5}$ G. While this is comparable to that quoted for other jets where the EC/CMB model is used to model their X-ray emission, in this case a comoving luminosity of $\sim 10^{51} {\rm erg\ s^{-1}} $ is required to fit the model to our X-ray data. We feel this is unrealistic, as it would violate the Eddington limit by many orders of magnitude. For that reason, we have not shown it in any figure.
Additionally, assuming an equipartition magnetic field, a Doppler factor of $\delta \sim 45$ is required for EC/CMB to explain the observed X-ray/radio NHS flux even for the case of $\theta = 0$ degrees using standard formulae [@2002ApJ...565..244H]. The required beaming is highly unlikely given the observed properties of the 3C 111 jet, [*e.g.*]{} the observed brightness of the SHS and the lack of obvious blazar properties.
We do not have many data points with which to constrain the model of the low-energy synchrotron component, especially in K30 and K61. We expect to be able model its SED well in a follow-up paper using JVLA observations of the jet. As well, additional [*HST*]{} and [*Chandra*]{} observations would help to better constrain the near-IR to optical and X-ray spectral indices of the components, and perhaps also constrain the X-ray emission mechanism of additional components.
![Representative models for the SED of the high-energy synchrotron component along with our binned X-ray data for K61 ([*Top*]{}) and the NHS ([*Bottom*]{}). Dashed vertical lines represent the boundaries of the energy range that [*NuSTAR*]{} is capable of observing.[]{data-label="SED_model"}](SEDmodel_K61.pdf "fig:"){width="8.6cm"} ![Representative models for the SED of the high-energy synchrotron component along with our binned X-ray data for K61 ([*Top*]{}) and the NHS ([*Bottom*]{}). Dashed vertical lines represent the boundaries of the energy range that [*NuSTAR*]{} is capable of observing.[]{data-label="SED_model"}](SEDmodel_NHS.pdf "fig:"){width="8.6cm"}
Conclusions
===========
We have presented new [*Chandra*]{} and [*HST*]{} observations of 3C 111 that reveal that its jet has eight X-ray and near-IR/optical emitting components, which extend for 121 arcsec (355 kpc deprojected length) from its AGN nucleus in the approaching jet, and also reveal the hotspot emission on the counterjet side. The 3C 111 jet is remarkable for several reasons. While some other jets are comparably long, no other known jet boasts the same combination of length, number of visible components and low redshift that 3C 111 does. For example, the jet of Pic A [@marshall10; @gentry15; @hardcastle15], which is similarly straight, longer in angular extent (almost 4’), and is about 30% nearer, has only three components that have been detected in the near-IR, while the jet of 3C 273 [@Jester06], which extends for a somewhat greater distance from its host galaxy and is somewhat brighter, is nearly $4\times$ as far at a redshift $z=0.158$.
The analysis discussed in this paper strongly disfavors the EC/CMB model as the dominant X-ray emission mechanism in several of the components of 3C 111’s jet. The hotspot flux ratio for each of the bands we have shows the jet to have decelerated to, at most, $\beta \sim 0.4$. This, combined with a relatively high viewing angle of $\theta \sim 18.1^\circ$ based on VLBA observations, demands a power requirement many orders of magnitude above the Eddington limit for EC/CMB to be the dominant X-ray emission mechanism of the jet.
We instead favor a two-component synchrotron model. Morphological comparison between radio, near-IR, and X-ray bands for K30 and the NHS show the X-ray flux maxima to be significantly upstream of the maxima in the radio, suggesting the presence of two separate electron populations with distinct energy distributions in these regions. This evidence is compounded by the analysis of the jet/counterjet hotspot flux ratio for each band, which shows the near-IR- and X-ray-emitting electrons to be moving at a significantly faster velocity than that of the radio-emitting electron population.
We have made efforts to model the spectral energy distribution of the high-energy synchrotron emission and determine how future observations using [*NuSTAR*]{} can be used to constrain the emission mechanism. Future [*HST*]{} and [*Chandra*]{} observations will allow us to put further constraints on the spectral energy distribution models for the jet components we have analyzed and test the emission mechanism of additional jet components.
These results are based on observations made by the [*Chandra*]{} X-ray Observatory (datasets 702798 and 703007) and [*Hubble Space Telescope*]{} (program 13114), as well as the Very Large Array ([*VLA*]{}, program AB534). EP, DC and FT acknowledge support for this work by the National Aeronautics and Space Administration (NASA) through Chandra awards G03-14113A (EP, DC) and G04-15103A (FT) issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astronomical Observatory for and on behalf of the National Aeronautics and Space Administration under contract NAS8-03060. EP and DC also acknowledge support from HST grant GO-13114.01, which was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research made use of Astropy, a community-developed core Python package for Astronomy [@astropy], hosted at http://www.astropy.org. This research also made use of APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com.
Anderson, J., MacKenty, J., Baggett, S., & Noeske, K., 2012, http://www.stsci.edu/hst/wfc3/ins\_performance/CTE/
Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, , 558, A33
Ballo, L., Braito, V., Reeves, J. N., Sambruna, R. M., & Tombesi, F. 2011, , 418, 2367
Boettcher, M., 2012, Chapter 2, Relativistic Jets from Active Galactic Nuclei, Edited by M. Boettcher, D.E. Harris, ahd H. Krawczynski, 425 pages. Berlin: Wiley, 2012
Burstein, D., & Heiles, C., 1982, AJ, 87, 1165
Cara, M., Perlman, E. S., Uchiyama, Y., et al., 2013, ApJ, 773, 186
Celotti, A., Ghisellini, G., & Chiaberge, M., 2001, , 321, L1
Dermer, C., D. & Atoyan, A. M., 2002, ApJ, 586, L81
Dermer, C. D., & Atoyan, A. 2004, ApJ, 611, L9
Dermer, C. D., Finke, J. D., Krug, H., Böttcher, M. 2009, , 692, 32-46
Dolphin, A., 2000, PASP, 112, 1397
Fanaroff, B. L., & Riley, J. M. 1974, , 167, 31
Felten, J. E., & Morrison, P. 1966, , 146, 686
Gentry, E. S., Marshall, H. L., Hardcastle, M. J., et al., 2015, ApJ, 808, 92
Georganopoulos, M., Perlman, E. S., Kazanas, D., McEnery, J., 2006, ApJ, 653, L5
Georganopoulos, M., & Kazanas, D., 2004, ApJ, 604, L81
Georganopoulos, M. & Kazanas, D., 2003, ApJ, 589, L5
Gonzaga, S., Hack, W., Fruchter, A., & Mack, J., 2012, “The DrizzlePac Handbook, Version 1.0" (Baltimore: STScI)
Güdel, M., Briggs, K. R., Arzner, K., et al., 2007, A& A, 468, 353
Hardcastle, M. J., Harris, D. E., Worrall, D. M., et al., 2004, ApJ, 612, 729
Hardcastle, M. J., Lenc, E., Birkinshaw, M., et al., 2016, , 455, 3526
Hardcastle, M. J., 2006, MNRAS, 366, 1465
Harris, D. E., & Krawczynski, H. 2002, , 565, 244
Hewitt, A., & Burbidge, G., 1991, ApJS, 75, 297
Hogan, B., Lister, M. L., Kharb, P., Marshall, H. L., Cooper, N. J., 2011, ApJ, 730, 92
Holtzman, J. A., Hester, J. J., Casertano, S., et al., 1995, PASP, 107, 156
Jester, S., Harris, D. E., Marshall, H. L., Meisenheimer, K., 2006, ApJ, 648, 900
Jorstad, S. G., Marscher, A. P., Lister, M. L., et al. 2005, , 130, 1418
Kharb, P., O’Dea, C. P., Tilak, A., et al., 2012, AJ, 748, 81
Krawczynski, H., 2012, ApJ, 744, 30
Leahy, J. P., Black, A. R. S., Dennett-Thorpe, J., et al., 1997, MNRAS, 291, 20
Lister, M. L., Cohen, M. H., Homan, D. C., et al., 2009, AJ, 138, 1874
Lister, M. L., Aller, M. F., Aller, H. D., et al., 2013, AJ, 146, 120
Marshall, H. L. Schwartz, D. A., Lovell, J. E. J., et al., 2005, ApJS, 156, 13
Marshall, H. L., Hardcastle, M. J., Birkinshaw, M., et al., 2010, ApJ, 714, L213
Mehta, K. T., Georganopoulos, M., Perlman, E. S., Padgett, C. A., Chartas, G., 2009, ApJ, 690, 1706
Meyer, E. T., & Georganopoulos, M., 2014, ApJ, 780, L27
Meyer, E. T., Georganopoulos, M., Sparks, W. B., Godfrey, L., Lovell, J. E. J., Perlman, E. S., 2015, ApJ, 805, 154
Meisenheimer, K., Röser, H.-J., Hiltner, P. R., Yates, M. G., Longair, M. S., Chini, R., Perley, R. A., 1989, A& A, 219, 63
Meisenheimer, K., Yates, M. G., & Röser, H.-J., 1997, A& A, 325, 57
Oh, J., Trippe, S., Kang, S., et al. 2015, Journal of Korean Astronomical Society, 48, 299
Poutanen, J., & Vilhu, O. 1993, , 275, 337
Rajan, A., 2011, WFC3 Data Handbook, STScI
Reynolds, C. S., Iwasawa, K., Crawford, C. S., & Fabian, A. C. 1998, , 299, 410
Riess, A., 2000, WFPC2 ISR 00-04
Röser, H.–J., Conway, R. G., & Meisenheimer, K., 1996, A& A, 314, 414
Rots, A. H., Budavari, T., 2011, ApJS, 192, 8
Ryle, M., Sir, & Longair, M. S. 1967, , 136, 123
Sambruna, R. M., Gambill, J. K., Maraschi, L., 2004, ApJ, 608, 698
Schwartz, D. A., Marshall, H. L., Lovell, J. E. J. et al., 2000, ApJ, 540, L69
Schlafly, E. F., & Finkbeiner, D. P., 2011, ApJ, 737, 103
Schlegel, D. J., Finkbeiner, D. P., & Davis, M., 1998, ApJ, 500, 525
Stawarz, [Ł]{}., & Petrosian, V. 2008, , 681, 1725
Tavecchio, F., Maraschi, L., Sambruna, R. M., Urry, C. M. , 2000, ApJ, 544, L23
Tombesi, F., Reeves, J. N., Reynolds, C. S., Garc[í]{}a, J., & Lohfink, A. 2013, , 434, 2707
Ungerer, V., Nguyen-Quang-Rieu, Mauron, N., Brillet, J., 1985, A& A, 146, 23
van Dokkum, P. G., 2001, PASP, 113, 1420
Wilson, A. S., Young, A. J., & Shopbell, P. L., 2001, ApJ, 547, 740
[^1]: see e.g., https://hea-www.harvard.edu/XJET/
[^2]: This smoothed image was not used for scientific measurements, but is useful for illustrative purposes.
[^3]: see http://www.astro.yale.edu/dokkum/lacosmic/
[^4]: Found at http://astro.umbc.edu/compton
|
---
abstract: 'In this notes we shall describe the relation of a certain class of simple random curves arising in 2D statistical mechanics models in the scaling limit, which can be described dynamically by Stochastic L[œ]{}wner Evolutions (SLE), and the equivalent Renormalisation Group (RG) theoretic interpretation in Conformal Field Theory, as a fixed point of the RG flow. Further, we shall recall the relation of this random curves with String Theory, and how one can derive a general measure on such random paths, by using weighted regularised determinants, which come from sections of twisted line bundles. Importantly, the null vector at level two in the Verma module for the highest-weight representation of the Virasoro algebra corresponds to a generalised Doob-Getoor $h$-transform.'
author:
- |
Roland Friedrich\
\
\
\
title: '**A Renormalisation Group approach to Stochastic L[œ]{}wner Evolutions and the Doob $h$-transform**'
---
Introduction
============
This text is a somewhat extended version of a talk I gave at the workshop “RENORMALIZATIONÓ at the Max-Planck-Institute for Mathematics in Bonn, in December 2006.
Its aim is to explain to a non-specialist audience, how Stochastic L[œ]{}wner Evolutions (SLE) are connected with the Renormalisation Group (RG), but also what the underlying global geometric picture is. In doing so, I shall restrict myself to the case of the unit disc, the original model, and leave out the general theory for arbitrary Riemann surfaces, as developed in [@F; @FK; @K; @KS]. During the process of writing, I felt free to expand on results which I obtained several years ago.
There has been two major directions to construct measures on random loops or intervalls, namely the one pursued by P. Malliavin [@M] and the other one by O. Schramm, who initiated SLE [@Sch], but also G. Lawler, W. Werner [@LSW], (LSW), and M. Aizenman [@AB]. Both was taking place roughly around 1999. The (technical) foundations of SLE were laid down by Schramm and S. Rohde in [@RSch], and subsequently applied by LSW to open probabilistic problems.
The parts of the theory underlying SLE which we are going to present here, aims at giving a unified treatment which is capable to incorporate both approaches mentioned at the beginning, but also to explain all the general results physically, as proposed e.g., in [@LSW]. The main tool to achieve this are determinant line bundles, i.e. domain-wise regularised determinants, and the so-called “Virasoro Uniformisation" (VU), which had been co-invented by M. Kontsevich, quite some time ago, and which generalises parts of the work of Kirillov, Yurev and Neretin. Also, (VU) is at the heart of the fundamental approach to CFT on Riemann surfaces, by Tsuchiya, Ueno and Yamada [@TUY].
Finally, from a mathematical perspective the main point is that the null vector at level two in the Verma module of the highest-weight representation of the Virasoro algebra corresponds to the Doob-Getoor $h$-transform. To consider harmonic sections with respect to the generator of the Diffusion process on the determinant line bundle is required if one wants to have martingales. In turn martingales are dictated by the physical set-up, namely to derive from models of Statistical Mechanics in a dynamical fashion measures on simple paths.
A detailed and comprehensive companion text to the material presented here, is [@F1].
Different point of views
========================
Let us start with the following example, as depicted in Figure \[SAW\]. There we have a hexagonal lattice approximating a closed disc. The yellow coloured hexagons compose the interior in the discretisation of the disc and the white ones, the boundary, indicated by the circle.
![A simple curve on a hexagonal lattice connecting two boundary points. Topological set-up.[]{data-label="SAW"}](SAW.pdf)
Further we have drawn a simple polygonal path connecting two boundary points and strictly laying in the interior of the disc.
From a discrete, combinatorial and enumerative point of view, we can ask several natural questions. E.g. how many discrete simple path are connecting the same two boundary points, thereby always running inside the circle or what is the (Euclidean) length of the longest path. It is important to note, that here boundary and interior are a priori just topological notions, merely giving the roughest binary framework to start with.
If we would keep the circle fixed but replace our lattice by another of a finer grid size, we still can ask the same questions but some of them become increasingly unnatural. Also, the combinatorial complexity of the problem would start to explode, so that instead of keeping track of all possibilities, we would be forced to resort to statistical methods. But what does that exactly mean and what are the relevant quantities we should look for.
Well, by now many people and groups invented their questions and methods, as the above problem could arise e.g. in Polymer Physics.
A particular challenge would be to construct probability measures on such discrete paths and to understand how all this scales as the mesh size becomes smaller and smaller. Also, what should the continuum object be and what are the natural measures to start with, as the same set can support different ones.
Again, depending on the context, e.g. dynamical generation of paths or “static sets", several classes of measures have been considered and proposed.
![A simple curve on a hexagonal lattice connecting two boundary points. Boundary condition set-up.[]{data-label="SAW_Boundary"}](SAW_Boundary.pdf)
As we shall discuss in the next section, Statistical Mechanics and in particular Lattice Models, provide a rich source of such curves in form of interfaces. But, there are some fine points one has to be aware of, as the simple topological set-up will be altered by the notion of boundary conditions, as depicted in Figure \[SAW\_Boundary\], and further explained latter. However, boundary conditions intrinsically reveal that we are taking up a macroscopic stance on the problem, which shifts our focus on what to consider as being natural and interesting.
Motivation from Statistical Mechanics
=====================================
Besides percolation, as originally, the two-dimensional Ising model has by now become a widely adapted way in the literature to motivate SLE and to connect it with statistical mechanics respectively CFT, [@FK; @F; @K].
So let us start with a simply connected and bounded domain $D$ in the plane $\mathbb{R}\cong\mathbb{C}$ and a triangular lattice (TL) of mesh size $\delta>0$, “just" covering the closed set $\overline{D}$ . We shall assume that the boundary of the domain is sufficiently smooth and the mesh size enough fine to avoid cumbersome complications.
A configuration of spins is a function $\sigma$ on the set of vertices $V(TL)$ with values $\pm 1$. We may now pass to the dual hexagonal lattice, and colour the hexagons enclosing a vertex with value $-1$ black and those with value $+1$ white. In the percolation setting this would correspond to site (or vertex) percolation on a triangular grid.
A sure way to get an interface (domain wall), as illustrated in Figure \[Hexagon\], is to take two marked points $A$ and $B$ on the boundary $\partial D$ and then to fix the value of the spin sitting in the centre of exactly one hexagon black, if that one intersects the boundary segment from $B$ to $A$ and similarly white, for that boundary part which belongs to the segment from $A$ to $B$, assuming a counter-clockwise orientation.
![The interface associated with the two-dimensional Ising model.[]{data-label="Hexagon"}](Percolation2.pdf)
The curve which arises is a Jordan curve (no branching) connecting vertices such that on the left we always have spin $-1$ and to the right $+1$.
In the case of our model with this particular choice of boundary conditions, we have an object that persists on all finite scales and should therefore be a good candidate to proceed to an object which can be regarded as a “natural" scaling limit, i.e. a macroscopic observable.
However as a look at Figure \[Hexagon\] reveals, once the particular (red) interface is fixed, we could change some of the values of the spins corresponding to boundary parts, without altering the shape of the curve itself, which implies that the set of all configurations having that particular interface, is much bigger than just those compatible with the fixed “domain wall" boundary conditions.
The situation at the discrete level is summarised schematically in Figure \[configurations\]. The (yellow-blue) square represents the space of all possible configurations, which is fibered over the space of all possible boundary values, i.e. the state space, and corresponds to the case of “free boundary values".
![The configuration space as a fibered space over the state space, principal fibration, or the non-local fibration over curves.[]{data-label="configurations"}](Configurations.pdf)
The state space can be subdivided into two disjoint non-empty sets, which are labelled as “curves" respectively “no curves" and which contain those boundary conditions which may produce a curve $\gamma$ connecting the two marked points $A$ and $B$, respectively not. It is important to note that the fibre over a state from “curves" may contain configurations which do not include a domain wall running from $A$ to $B$. This is indicated by the (blue) triangle $\{\sigma\}_{\gamma}$ over the set “curves".
There is one special state, labelled $AB$, which corresponds to the boundary conditions described as “domain wall boundary conditions", (cf. Fig. \[Hexagon\]). Now, the fibre over the state $AB$ is composed of configurations that do always contain a Jordan arc running from $A$ to $B$ and therefore we may call it the “special fibre".
We have yet another (partial) fibration. Namely the space of configurations with a domain wall (blue triangle) is fibered over the set of particular curves. In our Figure \[configurations\], this is depicted by the vertical line labelled “particular curves" with the projection map $\pi$. It is important to note, that say for a curve $\gamma_0$, the intersection of the fibres $\pi^{-1}(\gamma_0)$ and $\text{pr}^{-1}(AB)$ is not just one point (configuration), but indeed does contain several.
Gibbs measures with “proper" boundary conditions
------------------------------------------------
The Gibbs measure on configurations naturally induces a probability measure on the random curves we just considered. However, the boundary conditions we choose, play an essential role.
So, in the thermodynamical equilibrium at the absolute temperature $T>0$ the Gibbs measure is described by $$d\mu_G(\sigma):=\frac{1}{Z} e^{-\beta E[\sigma]},\qquad \beta:=\frac{1}{T}\quad\mbox{(inverse temperature)}~,$$ where the functional $E[\sigma]$, denotes the energy of a configuration and the numerical pre-factor $Z>0$ serves as normalisation, such that the sum of all elementary events ads up to 1. It is defined as $$Z:=\sum_{\{\sigma\}} e^{-\beta E[\sigma]}~,$$ with the sum extending over all configurations $\{\sigma\}$ and it is called the partition function. The measure for a fixed domain $D$ depends on two parameters, namely the temperature but also the underlying mesh size, i.e. $\mu_G=\mu_{G(T,\delta)}$.
If we choose fixed $AB$-boundary conditions for the calculation of the partition function, then we are restricting ourselves to the special fibre over $AB$, and take its Boltzmann-weighted volume $Z_{{\sigma}_{AB}}$ as normalisation factor. For free boundary conditions we would restrict the summation to the set of all configurations which contain a domain wall, i.e. to the set $\{\sigma\}_{\gamma}$ (blue triangle) with $Z_{{\sigma}_{\gamma}}$ as the corresponding partition function.
The natural equivalence relation on the set of configurations $\{\sigma\}_{\gamma}$ is given by declaring two realisations $\sigma_1$ and $\sigma_2$ as being equivalent if they include the same chordal domain wall, connecting the two marked boundary points, cf. Fig. \[Hexagon\]. Again, we can restrict the equivalence relation to the fibre $\text{pr}^{-1}(AB)$ only. In any case, the situation yields the fibration with base space the particular curves and the equivalent configuration corresponding to the fibres, respectively to their intersection with $\text{pr}^{-1}(AB)$.
The probability measure on the quotient space, with simple events the particular curves $\gamma$, is the image measure, either with respect to $Z_{{\sigma}_{AB}}$ or $Z_{{\sigma}_{\gamma}}$, i.e. $\pi_*\mu_{G_{AB}}$ resp. $\pi_*\mu_{G_{\sigma_{\gamma}}}$.
Although the geometric set of simple random curves is the same (topological set-up), by taking different boundary conditions we arrive at having two different measures on the same set.
Now, and this is important to note, the proofs (analytic or numeric for the Ising model) [@Sm; @GC] concerning the scaling limit of the above measures, i.e. the approach of the thermodynamic limit along the critical temperature $T_c$, deal with the family of measures on the special fibre $\text{pr}^{-1}(AB)$.
Further, as we shall discuss in the next section, on general grounds it was / is expected that the scaling limit is a measure, supported on simple paths, which is conformally invariant.
Let us close with the following remarks. The procedure just described, is not particular to the Ising model. We can choose other models, e.g. $Q$-states Potts model (i.e. different Boltzmann-weights) as well as other lattices (not necessarily hexagonal) with appropriate boundary conditions (e.g. wired and free). For lattice models there are other “good" probabilistic events. In the Ising model slightly below the critical temperature $T_c$ we observe the sea of nested Jordan curves of domain boundaries.
![Central charge zero random field of loops on a torus, which has a dual description as a percolation model. (Result from unpublished work on Random Fields and SLE, 2000/2001)[]{data-label="cd"}](labyrinthe.jpg)
Again, by passing to $T_c$ and simultaneously rescaling we obtain a dense collection of closed, non-intersecting loops, carrying a scale invariant probability distribution which would be also derived from the series of Gibbs measures as we explained in the case of chordal lines. (cf. [@KS])
Measures as Domain Functionals, the ALPS-correspondence and SLE
===============================================================
So-far our discussion centred around a fixed domain and the measures associated with it, supported on simple curves and connecting two previously chosen distinct boundary points. Now, we could ask, how the measure would change if we keep the domain, but vary the points, or even more generally, also vary the underlying domains. So, what we have in fact is a domain functional $$\label{dom_funct}
D_{A,B}\mapsto\mu_{D_{A,B}}$$ where $D$ denotes the domain, and $A,B\in\partial D$, two distinguished (and ordered) boundary points.
In two dimensions, there exists a very strong statement, the Riemann mapping theorem, which tells that any two simply connected domains are conformally related, and if the domains are bounded by Jordan curves, then this mapping can be extended as a homeomorphism to the whole boundary, according to CarathŽodory’s theorem.
Now it was conjectured [@Ai; @LPS], that these measures should be conformally invariant in the scaling limit, which can be expressed as the following commutative diagram, $$\begin{CD}
D_{A,B}@>f>> D'_{A',B'}\\
@V F_{\text{ALPS}} VV @V F_{\text{ALPS}}VV\\
\mu_{D_{A,B}} @>f_*>> \mu_{D'_{A',B'}}
\end{CD}$$ with the morphism $f$ being a conformal equivalence which respects the ordered marked points, i.e. $f(A)=A'$ resp. $f(B)=B'$, and as objects Jordan domains with two marked and ordered points, $\text{\texttt{JDom}}_{\bullet,\bullet}$ . As we shall see latter, this category can be enlarged to include also slit domains. $f_*$ denotes the induced measurable mapping, i.e. $F_{\text{ALPS}}(f)=:f_*$, where $F_{\text{ALPS}}$ stands for the “Aizenman–Langlands–Pouilot–Saint-Aubin" correspondence (functor).
Let us note that because of conformal invariance and because the set of morphisms between any two objects in $\text{\texttt{JDom}}_{\bullet,\bullet}$ has more than one object, in fact a real continuum, it is enough to study just one reference model, e.g. the upper half-plane $\mathbb{H}$ with $0$ and $\infty$ as the marked points.
The category $\text{\texttt{JDom}}_{\bullet,\bullet}$ has several equivalent parametrisations. Namely, if we consider first the set $\text{\texttt{J}}^{\infty}$ of smooth Jordan curves, then one has the double quotient [@AMT] $$\text{SU}(1,1){\setminus}\Diff_+(S^1) / \text{SU}(1, 1)$$ as the base space, and as fibre model of the total space, the torus (minus the diagonal), i.e., $S^1\times S^1\setminus\{\text{diagonal}\}$. Sections then correspond to domains with two distinct marked boundary points.
As these measures are translational invariant, it is enough to look at $\text{\texttt{J}}^{\infty}_0$, i.e., all smooth Jordan curves surrounding the origin. The uniformising application with domain the unit disc $\D$, is unique if we require it to preserve the origin $0$, and to have either a strictly positive derivative at $0$ or to map $1$ onto one of the marked points.
For symmetry reasons we take the first normalisation. Then the manifold $\cal M$ of all such univalent maps corresponds to a contractible subset in (cf. [@F]) $${\mathrm{Aut}}({\OO})=\{a_1z+a_2 z^2+a_3 z^3\dots,~ 0<a_1\}\hookrightarrow\C^{\N^{*}},$$ which itself is a contractible space. By the Bieberbach-DeBranges theorem the coefficients of elements in $\cal M$ satisfy $a_n\leq n\cdot a_1$. Similarly, we have
$$\begin{CD}
\{\text{conformally invariant measures}\}_{\kappa>0}@>>>{\cal M}\times (S^1\times S^1\setminus\{\text{diagonal}\})@>\pi>>{\cal M}\hookrightarrow{\mathrm{Aut}}({\OO})~.
\end{CD}$$ Then the trivial bundle, with fibre (almost) a torus, parametrise, up to one real positive constant $\kappa$, conformally invariant probability measures on simple paths which connect two distinct boundary points, as we shall see.
It was O. Schramm’s original insight [@Sch], to use L[œ]{}wner’s slit mapping to describe these random traces dynamically and to classify then all possible conformally invariant measures on them, i.e. to show the existence of $\kappa$.
Physically, the existence of such a parameter can be understood as labelling models from Statistical Mechanics that contribute a measure, as previously explained. However, it is a deep and fundamental fact of the two-dimensional conformally invariant realm, that one parameter is enough, as we shall see in Section \[RG\_sec\].
Schramm-L[œ]{}wner Evolutions
-----------------------------
One of the basic observations in deriving the “driving function" in the dynamical approach to random Jordan arcs is, besides a symmetry argument, continuity and the previously introduced notion of conformal invariance, the following Markovian type property (stated for the category $\text{\texttt{JDom}}_{\bullet,\bullet}$):
For a domain $D$ with non-degenerate boundary, let ${\cal{W}}(D_{A,B})$ be the set of Jordan arcs in $D$ with endpoints $A$ and $B$. Denote by $\{\mu_{D_{A,B}}\}$ a family of probability measures on Jordan arcs in the complex plane such that $$\mu_{D_{A,B}}({\cal W}(D_{A,B})=1~.$$ Then the Markovian-type property says for $\gamma$ a random Jordan arc, that if $\gamma'$ is a sub-arc of $\gamma$ which has $A$ as one endpoint and whose other endpoint we denote by $A'$, then the conditional distribution of $\gamma$ given $\gamma'$ is $$\label{E:markov}
\mu_{D_{A,B}|_{\gamma'}}=\mu_{(D\backslash\gamma')_{A',B}}~.$$
So, the only compatible driving function which satisfies the above requirements has to be proportional to standard one-dimensional Brownian motion, which leads to the following facts [@Sch; @RSch].
The chordal SLE$_\kappa$ curve $\gamma$ in the upper half-plane $\HH$ describes the growth of simple random curves emerging from the origin and aiming at infinity, as follows:
\[loewner\_eq\] For $z\in\HH$, $t\geq0$ define $g_t(z)$ by $g_0(z)=z$ and $$\frac{\partial g_t(z)}{\partial t} = \frac{2}{ g_t(z) - W_t}
\label{lowner}.$$
The maps $g_t$ are normalised such that $g_t (z) = z + o(1) $ when $z \to \infty$ and $W_t:= \sqrt{\kappa}\,B_t$ where $B_t(\omega)$ is the standard one-dimensional Brownian motion, starting at 0 and with variance $\kappa>0$. Given the initial point $g_0(z)=z$, the ordinary differential equation (\[lowner\]) is well defined until a random time $\tau_z$ when the right-hand side in (\[lowner\]) has a pole. There are two sets of points that are of interest, namely the preimage of infinity $\tau^{-1}(\infty)$ and its complement. For those in the complement we define: $$\label{Khull}
K_{t}:=\overline{\{z\in\HH: \tau(z)<t\}}$$ The family $(K_t)_{t\geq0}$, called hulls, is an increasing family of compact sets in $\overline{\HH}$ where $g_t$ is the uniformising map from $\HH\setminus K_t$ onto $\HH$. Further there exists a continuous process $(\gamma_t)_{t\geq0}$ with values in $\overline{\HH}$ such that $\HH\setminus K_t$ is the unbounded connected component of $\HH\setminus\gamma[0,t]$ with probability one. This process is the trace of the SLE${}_{\kappa}$ and it can be recovered from $g_t$, and therefore from $W_t$, by $$\gamma_t = \lim_{z\rightarrow W_t, z\in\HH} g_t^{-1}(z)~.$$ The constant $\kappa$ characterises the nature of the resulting curves. For $0<\kappa\leq 4$, SLE${}_{\kappa}$ traces over simple curves, for $4<\kappa<8$ self-touching curves (curves with double points, but without crossing its past) and, finally, if $8\leq\kappa$ the trace becomes space filling.
Now, for another simply connected domain $D$ with two boundary points $A,B\in\partial D$ the chordal $SLE_{\kappa}$ in $D$ from $A$ to $B$ is defined as $$K_t(D_{A,B}):=h^{-1}(K_t(\HH,0,\infty))$$ where $K_t(\HH,0,\infty)$ is the hull as in (\[Khull\]) and $h$ is the conformal map from $D$ onto $\HH$ with $h(A)=0$ and $h(B)=\infty$.
Before we end this section, let us rewrite (\[lowner\]) in It[ô]{} form, by setting $f_t(z):=g_t(z)-W_t$, which now satisfies the stochastic differential equation $$\label{Loewner-Ito}
df_t(z)=\frac{2}{f_t(z)} dt-dW_t~.$$ For a non-singular boundary point $x\in\R$, we can read off the generator $A$ for the It[ô]{}-diffusion $X_t:=f_t(x)$ as $$A=2\frac{1}{x}\frac{d}{dx}-\frac{\kappa}{2}\frac{d^2}{dx^2}~.$$ Defining the first order differential operators $$\label{Witt}
\ell_n:=-x^{n+1}\frac{d}{dx}\qquad n\in\Z~,$$ we obtain $$A=\frac{\kappa}{2}\ell^2_{-2}-2\ell_{-1}~.$$ Let us note, that the differential operators (\[Witt\]) form a representation of the Witt algebra [@FW1; @FW2]. We shall come back to this matters latter, where we shall see, how the $A$-harmonic functions correspond to null-vectors in a Verma module of the Virasoro algebra.
Renormalisation Group flow and conformally invariant measures {#RG_sec}
=============================================================
Before we proceed, we shall recall Zamolodchikov’s $c$-Theorem for two-dimensional field theories [@Z].
Let us be given a (Euclidean) Field Theory with action functional $S[{\bf g},a]$ depending on an (infinite) set of dimensionless parameters, ${\bf g}=(g_1, g_2,\dots)$, the “coupling constants", and an (ultraviolet) “cut-off" $a$ such that the action is obtained as an integral of local densities, i.e. $S=\int \sigma({\bf g}, a, x) dx$.
The fundamental assumption is the existence of a one-parameter group of motions $R_t$ in the space $Q$ of coupling constants ${\bf g}$, $R_t:Q\rightarrow Q$, with the property that a field theory described by an action $S[R_t{\bf g}, e^t a]$ is equivalent to the original theory with the action $S[{\bf g}, a]$ modulo correlations. This means that all correlation functions calculated in the two theories are the same at scales $x\gg e^ta$ and $t>0$. The components of the vector fields which generate the renormalisation group (RG) flow are called “$\beta$-functions", i.e. $$\label{Z1}
\frac{dg_i}{dt}=\beta_i({\bf g})~.$$
Then the following properties hold true for the RG:
1. There exists a positive function $c(g)\geq 0$ which decreases monotonically, i.e., $$\label{Z2}
\frac{d}{dt}c=\beta_i(g)\frac{\partial}{\partial g_i}c(g)\leq 0$$ with equality only obtained at the fixed points of the RG-flow, i.e., at $g=g_*$, $(\beta_i(g_*)=0)$.
2. The “critical" fixed points are stationary for $c(g)$, i.e. $\beta_i(g)=0\Rightarrow \partial c/\partial g_i=0$. Further, at the critical fixed points the corresponding $2D$ field theory is a Conformal Field Theory, with generators $L_n$, $n\in\Z$, of the infinite symmetry algebra, the Virasoro algebra, satisfying the commutation relations $$\label{Z3}
[L_n, L_m]=(n-m) L_{n+m}+\frac{\tilde{c}}{12}(n^3-n)\,\delta_{n+m, 0}~,$$ with $\tilde{c}$ the “central charge" and which is a function of the fixed points, i.e., $\tilde{c}=\tilde{c}(g_*)$.
3. The value of $c(g)$ at the fixed point $g_*$ coincides with the corresponding central charge in (\[Z3\]), i.e., $c(g_*)=\tilde{c}(g_*)$.
![The $\mbox{SLE}_{\kappa}$ claim: $\mbox{SLE}_{\kappa}$ is the unstable fixed point of the renormalisation group flow.[]{data-label="SLE_RG"}](scaling_limit)
Now, in Figure \[SLE\_RG\], we have summarised the process of approaching the scaling limit of the two-parameter family of Gibbs measures $d\mu_G(N,T)$, $N\sim1/\delta^2$, for a given Boltzmann weight, i.e., for a particular lattice model. Then, as just discussed, the corresponding fixed point in the RG flow is described by a CFT, characterised by its central charge $c$. Additionally the limiting measure should be conformally invariant and supported on random simple chordal paths, i.e., by SLE for some $\kappa$.
Now, the “Main Identity" which functionally relates Schramm’s diffusion constant $\kappa$ of the SLE process with CFT, has been derived by various people very early [@BB; @FW2; @K]. The identity reads $$\label{kappa=c}
c=\frac{(\kappa-6)(3\kappa-8)}{2\kappa}~.$$
Therefore, we know from the $c$-Theorem how to relate the RG flow and SLE.
The link we have just derived here is of considerable mathematical interest, as it connects dynamical systems, representation theory and stochastic analysis very deeply. It is certainly worth, to be pursued further.
The String model for the Wilson loop and SLE
============================================
The Functional Integral approach to Schramm-L[œ]{}wner Evolutions was introduced in [@F; @FK; @K] and further extended in [@BF_corr] to Liouville Theory.
The physical problem for the Wilson loop in (Classical) String Theory is to sum over all two-dimensional real surfaces having a closed Jordan curve $C$ in $\R^n$ as boundary.
Historically, this sum $Z$, the partition function, has been defined and calculated (approximately) according to the classical “Dual String" methods, by applying the Nambu-Goto action, by Eguchi, Durhuus, Olesen, Nielsen, Petersen, Brink, Di Vecchia, Howe, Deser, Zumino, LŸscher, Symanzik and Weisz for “dual strings" however it was Polyakov’s approach and his specific action, centred around the Dirichlet integral, which made the problem much more approachable. His action was then subsequently extended to the same problem, i.e, to non-closed strings, by others like Friedan, but decisively by O. Alvarez [@A]. In any case, the underlying physical object is a Functional Integral, with the action being the specific part, and the mathematical problem left over, to interpret it “rigorously". For mathematical details of String Theory, see, e.g. [@AJPS].
The “partition function" is obtained from the well-known functional integral $$\label{partition_integral}
Z:=\int_{\{g\}}\int_{\{X\}} e^{-S[g, X]}\, [Dg][DX]~,$$ which is properly computed over all embeddings $\{X\}$ and all Riemannian metrics $\{g\}$.
The action $S=S[g, X]$, for surfaces $\Sigma$ with boundary $\partial \Sigma$, contains besides the Dirichlet Energy $D[g,X]$ of the embedding, also other terms, to make the theory renormalisable, and in its simplest version reads in local co-ordinates, $$\begin{aligned}
\label{PA_action}
S[g,X] & := & D[g,X]+\text{metric terms} \\
& = & \frac{1}{2}\int_{\Sigma} g^{ij}\frac{\partial X^{\mu}}{\partial x^i}\frac{\partial X^{\mu}}{\partial y}\sqrt{\det g}\, dx dy+\frac{1}{2\pi}\int_{\Sigma} K_g\sqrt{\det g}\, dx dy+\frac{1}{2\pi}\int_{\partial \Sigma} k_g |dx| \end{aligned}$$ where $K_g$ is the Gauss curvature of the metric $g$ and $k_g$ the geodesic curvature of $\partial M$ according to $g$.
Subsequently we shall not discuss the metric contributions further as they basically enter via the Gauss-Bonnet theorem. Also, we shall only treat regular metrics, although singularities have a profound effect [@BF_corr; @KS].
Now, in Conformal Field Theory, the marginals, i.e, the integral over all embeddings for a fixed metric is the important quantity, as we recall. This will also reveal the analogy with our earlier discussion in the statistical mechanics approach.
Spaces of mappings and the $H^{1/2}$ space on the circle
--------------------------------------------------------
The Sobolev space $H^{1/2}(S^1,\R)/\R$ of $L^2(S^1)$ real functions with mean-value zero on the circle can be identified with the sequence space $${\ell}_2^{1/2}=\{~u\equiv(u_0, u_1, u_1,\dots)~|~u_i\in\C~\text{and}~ \{\sqrt{n}\,u_n\}~\text{is square summable}~\}~.$$ Part of the importance of this Hilbert space comes from the facts that one can interpret its vectors as boundary values of real harmonic functions on the unit disc, $\D$, with finite Dirichlet energy but also as it characterises the subset of quasi-symmetric (qs.) homeomorphisms of the set of all homeomorphisms of $S^1$, [@NS]. The Poisson integral representation gives then a harmonic extension of the space $H^{1/2}$, which is also an isometric isomorphism of Hilbert spaces. For the disc the extension can explicitly be written down in terms of the formula of Douglas.
Let us consider a compact real two-dimensional smooth surface $\Sigma$ with non-empty and non-degenerate boundary $\partial \Sigma$, homeomorphic to the unit circle. Let us also fix a metric $g$, the background metric, on $\Sigma$ and let us consider a map $$h:\Sigma\rightarrow\R^n~,$$ mapping $\partial \Sigma$ diffeomorphically and orientation preserving onto the contour $C$.
For $h$ harmonic with respect to the metric $g$ we would have for the Laplacian $\Delta_g$, $$\label{J2.4.1}
\Delta_{g} h\equiv 0.$$ The fact which permits to progress further in the endeavour of defining the path integral is first that any embedding $X:\Sigma\rightarrow\R^n$ compatible with the boundary conditions, i.e., $X(\partial\Sigma)=C$ can be decomposed as $$\label{J2.4.2}
X=h_X+X_0$$ where $h_X$ is the unique harmonic map with $h_X|_{\partial\Sigma}\equiv X|_{\partial\Sigma}$ and $X_0|_{\partial\Sigma}=0$. This yields the affine space $$h_X+H^{1/2}_0(\Sigma, \R^n)$$ with $H^{1/2}_0(\Sigma, \R^n)$ denoting the Sobolev space of all maps from $\Sigma$ to $\R^n$ with vanishing boundary values.
An appropriate Hilbert space of states would be ${\cal H}:=H^{1/2}(\partial\Sigma, C)$.
![The harmonic section $\tilde{h}$ (harmonic extension) of the space of all embeddings of the boundary $\partial\Sigma$ of the surface. The boundary values compose the space of states.[]{data-label="h-section"}](Harmonic_Section.pdf)
Let $f\in {\cal H}$ and let us denote by $\tilde{f}$ its harmonic extension, i.e. the unique harmonic function with boundary value $f$. The situation is schematised in Figure \[h-section\].
In the next few lines we shall deal only with one component of the field. The integral over the fibre $\pi^{-1}(f)$, (cf. Fig. \[h-section\]), $$\label{ }
\Psi[f]:=\int_{\pi^{-1}(f)} e^{-S[g,X]}\, [DX]$$ is the continuum version of the partition sum with a specific choice of boundary conditions. Then the following calculation, with the previous notational conventions, yields for the Dirichlet energy ($*$ Hodge star), $$\begin{aligned}
2\cdot S[X] & = & \int_{\Sigma} dX\wedge *dX =\int_{\Sigma}d(\tilde{f}+X_0)\wedge * d(\tilde{f}+X_0)\\
& = & \int_{\Sigma} d\tilde{f}\wedge* d\tilde{f}+\int_{\Sigma} dX_0\wedge * dX_0+2\int_{\Sigma} dX_0\wedge d\tilde{f}\end{aligned}$$ Since $\Delta_g \tilde{f}=d*d\tilde{f}=0$ by definition for the harmonic function $\tilde{f}$, we obtain $$\label{ }
S[\tilde{f}+X_0]=S[X_0]+S[\tilde{f}]=S[X_0]+\frac{1}{2}\int_{\partial \Sigma}f*d\tilde{f}~.$$ Therefore the partition function factorises into $$\int_{\pi^{-1}(f)} e^{-S[g,X]}\, [DX]= e^{-\frac{1}{2}\int_{\partial \Sigma}f*d\tilde{f}}\cdot \int_{\{X_0\}} e^{-S[X_0]}\, [DX_0]~,$$ which gives (for one component) $$\label{SurfaceState}
\Psi[f]=\left[\frac{\det(\Delta_g)}{\operatorname{Area}(\Sigma,g)}\right]^{-1/2} \,\cdot\,{{\mathrm e}}^{-\frac{1}{2}\int_{\partial \Sigma} f* d\tilde{f}}~,$$ and where it is understood that the determinants are regularised. Note that we are dealing with bordered surfaces, and therefore we do not have zero modes.
To obtain the full marginal $Z[g]$, we have to integrate over all possible boundary values. The appropriate measure $\mu$, has been considered from a Gaussian point of view by G. Segal and I. Frenkel, and for “Unitarising Measures for the Virasoro algebra", by P. Malliavin, H. Airault and A. Thalmaier [@M; @AMT]; (cf. the support of measures in [@AMT]). Technically $$\label{SFM}
\int_{\tilde{h}({\cal H})} e^{-D[g,\tilde{h}]}\,[D\tilde{h}]:=\int_{\varphi\in\operatorname{Hom}_{qs}(S^1)} e^{-\operatorname{E}[\tilde{\varphi},g_0]} d\mu_{\text{SFM}}(\varphi)=:j[{\bf t},C]$$ where we have denoted the measure on $\text{Hom}_{qs}(S^1)$ by $d\mu_{\text{SFM}}$, $E[\tilde{\varphi},g_0]$ stands for the Dirichlet integral of the harmonic extension induced by $\varphi$, which also requires now $\Sigma$ to have a complex structure, and finally ${\bf t}$ denotes the conformal class of the metric $g_0$ compatible with the structure on $\Sigma$, i.e. a point in TeichmŸller space (cf. [@AJPS]).
The expression (\[SFM\]) is independent of the choices made, with the crucial exception that it depends on the conformal class of the metric.
So, for every fixed metric $g$, the marginal integral in (\[partition\_integral\]) over embeddings equals $$\label{Det_zeta}
Z[g]=\int_{\operatorname{Emb}(\Sigma)} e^{-S[X,g]} [DX]:=j[{\bf t},C]\cdot\left[\frac{\det(\Delta_g)}{\operatorname{Area}(\Sigma,g)}\right]^{-n/2}$$
If we restrict to “planar" surfaces with the Wilson loop $C$ as boundary, i.e. to embeddings of the unit disc into $\R^n$, then the dependence on the conformal class is simple, as all complex structures on the disc are equivalent. It gives also (part) of the relevant partition function for SLE.
However, the regularisation procedure introduces the so-called conformal anomaly, which renders the expressions covariant. Also, for singular metrics interesting effects show up which correspond to exotic versions of the SLE-process [@BF_corr], e.g. SLE$(\kappa,\rho)$ [@LSW], or measure the degree of non-commutativity of SLE, cf. [@FK; @LSW].
Now, in CFT the important objects are the marginals $Z[g]$, and they are naturally grouped together as a determinant line bundle, as we shall discuss next.
Determinant line bundles and flat connections
---------------------------------------------
In String Theory but also in Conformal Field Theory, the partition function is considered to be a section of a determinant line bundle. Here we shall briefly recall how one can derive a measure on random paths, by using regularised determinants [@FK; @F; @K; @KS].
Since to every Jordan domain we can associate the determinant of the Laplacian (with respect to the Euclidean metric and Dirichlet boundary conditions), i.e., $\det(\Delta_D):=\det(\Delta_{g_{\operatorname{Eucl.}}})$, we get a trivial bundle over ${\cal M}$, where $f\in{\cal M}$ denotes the uniformising map from the unit disc $\D$ onto the domain $D$, containing the origin. $$\begin{CD}
\det(\Delta_{f(\D)})\\
@V \pi VV \\
{\cal M}
\end{CD}$$ The uniformising map provides us also with a natural connection which allows us to compare the regularised determinants at different points. It has its origin in Polyakov’s string theory [@P] and was then subsequently extended in the works [@A; @OPS; @HZ]. Let us consider the space ${\cal F}$ of all flat metrics on $\D$ which are conformal to the Euclidean metric, obtained by pull-back. Namely, for $D\in\text{\texttt{JDom}}$ let $f:\D\rightarrow D$ be a conformal equivalence, and define $$\phi:=\log|f'|~.$$ This gives a correspondence of harmonic functions on $\D$ with the category $\text{\texttt{JDom}}$ and by Weyl rescaling with ${\cal F}$ via $$ds=|f'||dz|=e^{\phi}|dz|~.$$ To fix the $\operatorname{SU}(1,1)$-freedom, which gives classes of isometric metrics, we divide the state space $H^{1/2}(S^1)$ by the Mšbius group of the disc. Henceforth we shall work with the equivalence classes, so, e.g. $0$ corresponds to the orbit of the Euclidean metric under $\operatorname{SU}(1,1)$.
The connection reads [@OPS]: $$\label{PA_rel}
\det(\Delta_D)=e^{-\frac{1}{6\pi}\oint_{S^1}(\frac{1}{2}\phi{*}d\phi
+\phi|dz|)}\cdot\det(\Delta_{\D})$$ Next we would like to show a group property which is essential, as it translates latter into the Markov property, on which conformally invariant measures on paths hinge. Let us consider the sequence of conformal maps between domains $\D, D, G$: $$\begin{CD}
\D@>f>> D@>g>>G~.
\end{CD}$$ Then the relation of $\det(\Delta_G)$ and $\det(\Delta_\D)$ is obtained via $
\frac{d}{dz}g(f(z))=g'(f(z))\cdot f'(z),
$ and $$\log|g'(f(z))\cdot f'(z)|=\underbrace{\log|g'(f(z))|}_{=:\psi(z)}+\underbrace{\log|f'(z)|}_{=:\phi(z)}~.$$ Further by using the property of harmonic functions, i.e., $$\oint_{S^1}\frac{1}{2}(\phi\partial_n\psi+\psi\partial_n\phi)=0~,$$ gives $$\det(\Delta_G)=
\underbrace{e^{-\frac{1}{6\pi}\oint_{S^1}(\frac{1}{2}\psi(f(z)){*}d\psi(f(z))
+\psi(f(z))|dz|)}}_{I.}\cdot\underbrace{e^{\frac{1}{6\pi}\oint_{S^1}(\frac{1}{2}\phi{*}d\phi+\phi|dz|)}\cdot\det(\Delta_{\D})}_{II.}$$ where $$\begin{aligned}
I. & = & \oint_{\partial D}(\frac{1}{2}\tilde{\psi}{*}d\tilde{\psi}~
+\tilde{\psi}|dw|)\qquad\text{with~ $\tilde{\psi}(w):=\log|g'(w)|$}~, \\
II. & = & \det(\Delta_D) \end{aligned}$$ This also shows, that we can consider the determinant for the unit disc as the origin in an infinite affine space.
Let us note, that one should be careful with some of the signs in the literature (cf. [@A; @OPS; @HZ]). Then according to expression (\[Det\_zeta\]), for the partition function with two-dimensional target space, we have to take the inverse on both sides in (\[PA\_rel\]), i.e., $$\label{PFunction_rel}
Z_D=e^{\frac{1}{6\pi}\oint_{S^1}(\frac{1}{2}\phi{*}d\phi
+\phi|dz|)}\cdot Z_{\D}~.$$ The mapping $f$ which maps the unit disc $\mathbb{D}$ onto the slit unit disc $\mathbb{D}\setminus{[1-\sqrt{t},1]}$, can be given by: $$f(z)=\frac{\sqrt{1+t}-\sqrt{t+\left(\frac{z-1}{z+1}\right)^2}}{\sqrt{1+t}+\sqrt{t+\left(\frac{z-1}{z+1}\right)^2}}~.$$ But for such singular disturbances of the boundary the variation formula (\[PA\_rel\]) “breaks down", as the Weyl rescaled metric acquires singularities on the boundary. In order to compute nevertheless the variation of the partition function, one has to resort to a regularisation procedure, which involves the Schwarzian derivative (cf. [@FK]).
However, as the expression (\[SurfaceState\]) showed, we have also to specify boundary conditions, i.e. a state in which our system should be.
So, if we translate the “domain-wall" boundary conditions, we get another contribution, which contains the information of the points where the boundary conditions change. In the continuum this corresponds to jump discontinuities along the boundary for the Dirichlet problem. We shall treat here the unit disc with jumps of width $2\lambda$ at $1$ and $-1$. The general case is then obtained by composition with a conformal map.
The harmonic function compatible with these boundary conditons is $$\label{harmonic_jump}
u(r e^{it})=\Re\left[\frac{2\lambda}{\pi i}\left(\log(r e^{it}-1)-\log(r e^{it}+1)\right)\right]$$ where we have assumed counter-clockwise orientation of the circle. However, the resulting contribution according (\[SurfaceState\]) has to be regularised in order to have a finite quantity, $Z_{\lambda}$, depending on the points of discontinuity and the height of the jump(s). But, this also breaks the conformal invariance, such that the quantity varies covariantly over the space of domains, i.e., $Z_{\lambda}=Z_{\lambda}(\text{domain})$ .
![Harmonic function which satisfies the jump boundary conditions with $\lambda=1$ on the arc from $\pi/4$ to $\pi$ and $-1$ on the rest, for the Dirichlet problem. In the centre the unit disc is displayed.[]{data-label="SLE_RG"}](Jump_Sol.pdf)
Nevertheless, this reveals the global geometry of our partition functions in case of the Bosonic free field, i.e. for $c=1$: $$\begin{CD}
\label{correlators}
\pi^*(\det(\Delta_D)^{-1})\otimes Z_{\lambda}@>>> {\cal M}\times (S^1\times S^1\setminus\{\text{diagonal}\})\\
@. @V\pi VV\\
\det(\Delta_D)^{-1}@>>>{\cal M}\hookrightarrow{\aut}({\OO})
\end{CD}$$
However, what we see is that in the continuum compared with the lattice, there are infinitely many possibilities from which we have to choose the proper ones. The possible choices compose the spectrum and are dictated by the scaling dimensions $h$ of the “boundary fields", which can be obtained from highest-weight representations of the Virasoro algebra [@C; @BPZ]. They relate to $\lambda$ as $h=\text{const.}\cdot \lambda^2$. In the next section we shall derive this purely from probability theory.
To summarise, there are a number of variants of SLE, consisting of a Riemannian bordered surface $X$ (oriented, otherwise general topology) with marked points $x_1,\dots, x_n$ on the boundary and $y_1,\dots, y_m$ in the bulk. Analytic coordinates (or merely $1$-jets, i.e. dependence only on the first derivative) at the marked points are given. Then the partition function $Z$ is a positive function of such configurations. It has a tensor dependence on analytic co-ordinates (i.e. it transforms as $\prod_i(dz_i)^{h_i}\prod_j |dw_j|^{2h_j}$, $z_i$ local co-ordinate at $x_i$, $w_j$ local coordinate at $y_j$ ), and depends on the metric as $\det(\Delta_D)^{-c}$, where $c$ is the central charge, a real constant. Further, it should be positive and equal to the renormalised partition function in the lattice approximation.
These partition functions are null vectors of canonical Virasoro representations, and they correspond to correlators in Conformal Field Theory [@FW1; @FW2; @F; @FK; @K; @KS; @BB]. The (simplest) version for the disc with two marked points is now stated.
Let a configuration $(D, A, B)$, consisting of a simply connected domain $D$, with metric $g$ smooth up to the boundary and two marked boundary points $A, B$, with analytic local co-ordinates, be given. The partition function $Z_{\operatorname{SLE}_{\kappa}}$ of chordal $\operatorname{SLE}_{\kappa}$ is: $$Z_{\operatorname{SLE}_{\kappa}} = |{\det}_D|^{\otimes c}\otimes |T^*_A\partial D|^{\otimes h}\otimes |T^*_B\partial D|^{\otimes h}=\det(\Delta_D)^{-c}\cdot Z_h\equiv\langle\psi(A)\psi(B)\rangle~,$$ where $$\begin{aligned}
\label{c-charge}
\nonumber
c &=& 1-\frac{3}{2}\cdot\frac{(\kappa-4)^2}{\kappa}=-\frac{(\kappa-6)(3\kappa-8)}{2\kappa}~\text{is the central charge, and}\\
h &=& h(\kappa) =\frac{6-\kappa}{2\kappa}~\text{the highest-weight.}\end{aligned}$$ $\langle~\rangle$ denotes the unnormalised correlator of the boundary condition changing operators, corresponding to the boundary field $\psi$, of weight $h$.
Let us comment on the above Theorem. The fundamental and central relation is (\[c-charge\]), which ensures that the partition function really behaves as a section of a bundle. Note also the different meanings of $\det$.
Let us discuss the case $\kappa=3$, briefly. According to formula (\[c-charge\]) we get $c=1/2$ and $h=h_{1;2}=1/2$ which would correspond to the Ising model. The weight of the tensor is a half-order differential and the two-point function can be expresses by means of the Szeg[ő]{} kernel for spin $\frac{1}{2}$, i.e., for fermions. This little, but important, example should be seen as a general paradigm which tells us that a correlator with multiple insertions can be derived and expressed in terms of the Green’s function, as a Vandermonde determinant, as follows from Wick’s theorem.
Doob’s $h$-transform, Virasoro null-vectors and SLE
===================================================
This is a key section as it gives a unified treatment of the directions initiated by Schramm, Lawler and Werner, and it further connects with the one started by Malliavin. Our theory relies on the “Virasoro Uniformisation" (VU), which in case of the disc, corresponds to the theory developed by A.A. Kirillov, D. Yur’ev [@KY] and Y. Neretin. In addition they took also the underlying infinite KŠhler geometry into account. We shall now build on [@F].
Let us begin by briefly explaining the main idea. In Figure \[RW\], we have illustrated the path space which is identical for all models from statistical mechanics, producing simple curves, e.g. domain walls. However, every such model contributes its own measure on the path space.
If we parametrise it, we get from the measure in the statistical mechanics model different families of probability densities, whose evolution we have again indicated. As our earlier discussion showed, it is the central charge, resp. $\kappa$ which labels different measures.
Also, to every point corresponds a partition function, which is composed of the regularised determinant and the information on the boundary conditions. The measures on the path space and the transformation properties of this partition functions under the stochastic evolution, i.e., the lifted process, have to form a conformally invariant pair. In particular, the conformal image of any trajectory has to be again a martingale, up to reparametrisation. This requirement gives us the necessary link between the parameters involved, i.e., $\kappa, c, h$, but also explains the appearance of degenerate highest-weight representations of the Virasoro algebra.
![Different models with different Gibbs measures produce different probability densities on the same path space in ${\cal M}\subset\aut({\OO})$. The time evolution of the $\kappa$-densities is governed by a second-order differential operator in Hšrmander form, given in terms of the explicit Lie fields ${\cal L}_n$.[]{data-label="RW"}](RW.pdf)
### Path space and Weil-Petersson metric
Recall the space ${\cal M}$. An element of it can be written as $c_1z+c_2 z^2+c_3 z^3+\cdots,$ with $c_1>0$. Now, if we fix a parametrisation $t$ of the traces in our domain, we can cut them accordingly, and obtain so a family of univalent functions of the form $$f(t,z)=e^{\gamma(t)}z+c_2(t)z^2+\cdots~,$$ with $\gamma(t)$ a strictly monotone real function reflecting the path along which we are cutting. Further, it follows from the monotonicity that every simple path gives again a simple path in ${\cal M}$. Also, any two different traces in a domain give two different traces in ${\cal M}$, i.e., the map is injective, and unique up to reparametrisation. This can be seen as follows.
The traces are compact subsets of the domain, which is a Hausdorff space. If two traces, with the exception of the endpoints are different, there are at least two interior points which are different. Hence, we can choose open disjoint neighbourhoods. The family of mappings for one of the curves at some instance maps the point onto the boundary, for the other never. Then, it follows from the Open Mapping theorem and the Identity theorem for analytic functions that the corresponding traces in $\aut_+({\OO})$ must have points which are not the same, which in turns shows injectivity.
So, we have that ${\cal M}$ is a subset of the semi-direct product $\R_+\ltimes {\mathrm{Aut}}_+({\OO})$, where $${\mathrm{Aut}}_+({\OO}):=z\left(1+\sum_{k=1}^{\infty} c_k z^k\right)~,$$ and it is enough to study the traces in ${\mathrm{Aut}}_+({\OO})$. We shall think of the spaces as completed with respect to the natural filtration. The respective Lie groups and algebras are given below: $$\begin{aligned}
{\mathrm{Aut}}_+({\OO}) & &\qquad\der_+({\OO})=z^2\C[[z]]\partial_z \\
\cap\qquad & & \qquad\qquad\cap \\
{\mathrm{Aut}}({\OO}) & &\qquad\der_0({\OO})=z\C[[z]]\partial_z \\
& & \qquad\qquad\cap \\
& & \qquad\der({\OO})=\C[[z]]\partial_z\end{aligned}$$ and the exponential map $\exp:\der_+({\OO})\rightarrow{\mathrm{Aut}}_+({\OO})$, being an isomorphism. Further, the group ${\mathrm{Aut}}({\OO})$ acts on itself by composition.
Now, this space has a natural affine structure with co-ordinates $\{c_k\}$, and the identity map corresponding to the origin $0$.
Let us introduce the following spaces, which are in a sense dual, as they represent the power series developments around infinity, namely $$\begin{aligned}
\nonumber
\aut({\cal O_{\infty}}) & := & \{~bz+b_0+\frac{b_1}{z}+\cdots~,\quad b\neq0~\}~. \\
\aut_+({\cal O}_{\infty})& := & \{~z+b_0+\frac{b_1}{z}+\cdots~,~\}~.\end{aligned}$$
Now, the complex Virasoro algebra $\vir_{\C}$ we are considering is spanned by the polynomial vector fields $e_n=-ie^{in\theta}\frac{d}{d\theta}$, $n\in\Z$, and $\mathfrak{c}$, with commutation relations $[\mathfrak{c},e_n]=0$ and $$[e_m, e_n]=[e_m, e_n]+\omega_{c,h}(e_m,e_n)\cdot\mathfrak{c}~,$$ with the extended Gelfand-Fuks cocycle $$\omega_{c,h}(v_1,v_2):=\frac{1}{2\pi}\int_0^{2\pi}\left((2h-\frac{c}{12})v_1'(\theta)-\frac{c}{12}v_1'''(\theta)\right)v_2(\theta)\,d\theta~,$$ and $v_1, v_2$ being complex valued vector fields on $S^1$. It has been shown [@KY], that there exists a two-parameter family of KŠhler metrics on this space, with the form being at the origin $$\label{Kaehler_m}
w_{c,h}:=\sum_{k=1}^{\infty}\left(2hk+\frac{c}{12}(k^3-k)\right)\,dc_k\wedge d\bar{c}_k~,$$ building on the Gelfand-Fuks cocycle. This metrics also generate the Weil-Petersson (WP) metric. So, we can measure “the distance of two SLE curves", i.e., their shapes, and more generally, do Differential Geometry on the path space.
### Analytic line bundles $E_{c,h}$ over ${\cal M}$
The complex vector fields $e_n$ on the circle, which form the Witt algebra, have a representation in terms of the Lie fields ${\cal L}_{e_n}$ which act transitively on $\aut_+(\OO)$.
However, to have an action of ${\vir_{\C}}$, one has to introduce a determinant line bundle. So, on the infinite complex manifold $\aut_+(\OO)$ there exists an analytic line bundle $E_{c,h}$ which carries a transitive action of the Virasoro algebra. The line bundle $E_{c,h}$ is in fact trivial, with total space $E_{c,h}={\aut_+(\OO)}\times\C$. We can parametrise it by pairs $(f,\lambda)$, where $f$ is a univalent function and $\lambda\in\C$.
It carries the following action $$\label{KY2}
L_{v+\tau{\mathfrak c}}(f,\lambda)=({\cal L}_v f, \lambda\cdot\Psi(f,v+\tau{\mathfrak c}))~,$$ where $$\label{Neretin}
\Psi_{c,h}(f,v+\tau{\mathfrak c}):=h\oint\left[\frac{wf'(w)}{f(w)}\right]^2v(w)\frac{dw}{w}+\frac{c}{12}\oint w^2 S(f,w)\frac{dw}{w}+i\tau c~,$$ and where $$S(f,w):=\{f;w\}:=\frac{f'''(w)}{f'(w)}-\frac{3}{2}\left(\frac{f''(w)}{f'(w)}\right)^2~,$$ denotes the Schwarzian derivative of $f$ with respect to $w$. The central element $\mathfrak{c}$ acts fibre-wise linearly by multiplication with $ic$.
We have summarised the geometry in the following commutative diagram $$\begin{CD}
\label{trans_action}
0@>>> \C@>>>\vir_{\C}@>>>\operatorname{Witt}@>>>0\\
@VVV @VVV@VVV@VVV@VVV\\
0 @>>> \C@>>>\Theta_{E_{c,h}}@>>>\Theta_{\cal M}@>>>0\\
@VVV@VVV@VVV@VVV@VVV\\
0@>>> \C@>>>E_{c,h}@>>>{\cal M}@>>>0
\end{CD}$$ where $\Theta_{X}$ denotes the respective tangent sheaf. Generally, to have an action means to have a morphism from a Lie algebra $\mathfrak{g}$ to the tangent sheaf, and transitive means that the map $\mathfrak{g}\otimes{\OO}_{X}\rightarrow\Theta_{X}$ is surjective. Algebraically, the situation corresponds to so-called Harish-Chandra pairs $(\mathfrak{g}, K)$.
### $\vir$-modules and Verma modules $V(c,h)$
One can endow the space of holomorphic sections $|\sigma\rangle\in{\cal O}(E_{c,h})\equiv\Gamma({\aut_+(\OO)}, E_{c,h})$ of the line bundle $E_{c,h}$ with a $\vir_{\C}$-module structure.
Namely, let ${\cal P}$ be the set of (co-ordinate dependent) polynomials on $\cal M$, defined by $$P(c_1,\dots, c_N):{\mathrm{Aut}}({\OO})/\mathfrak{m}^{N+1}\rightarrow \C~,$$ with $\mathfrak{m}$ the unique maximal ideal. In the classical literature on complex variables the above quotient space is called the “coefficient body". ${\cal P}$ corresponds to the sections ${\cal O}({\cal M})$ of the structure sheaf ${\OO}_{\cal M}$ of ${\cal M}$ and it carries an action of the representation of the Witt algebra in terms of the Lie fields ${\cal L}_{n}\equiv{\cal L}_{e_n}$.
In affine co-ordinates $\{c_n\}$, these fields can be written as [@KY], e.g. $$\begin{aligned}
{\cal L}_n & = & \frac{\partial}{\partial c_k}+\sum_{k=1}^{\infty}(k+1) c_k\, \frac{\partial}{\partial c_{n+k}}~\quad n\geq1~.\end{aligned}$$
Now, ${\OO}(E_{c,h})$ can be identified with the polynomial sections of the structure sheaf, i.e. with ${\OO}({\cal M})$, by choice of a free generator, and the previous trivialisation.
The action of $\vir_{\C}$ on sections of $E_{c,h}$ can then be written in co-ordinates, with the notation $\partial_n\equiv\frac{\partial}{\partial c_n}$, according to formulae (\[KY2\], \[Neretin\]) as (cf. [@KY]), $$\begin{aligned}
\label{Vir_coord}
\nonumber
L_n & = & \partial_n+\sum_{k=1}^{\infty}c_k\, \partial_{k+n},\quad n>0 \\\nonumber
L_0 & = & h+\sum_{k=1}^{\infty} k\, c_k\partial_k~,\\ \nonumber
L_{-1} &=& \sum_{k=1}^{\infty}\left((k+2)c_{k+1}-2c_1c_k\right)\partial_k+2hc_1~,\\
L_{-2} &=& \sum_{k=1}^{\infty}\left((k+3)c_{k+2}-(4c_2-c_1^2)c_k-a_k\right)\partial_k+h(4c_2-c_1^2)+\frac{c}{2}(c_2-c_1^2)~,\end{aligned}$$ where the $a_k$ are the Laurent coefficients of $1/f$, and $c$ the central charge.
The conformal invariance will naturally lead to highest-weight modules.\
Now a polynomial $P(c_1,\dots, c_N)\in{\OO}(E_{c,h})$ is a singular vector for $\{L_n\}$, $n\geq1$, if $$\left(\partial_k+\sum_{k\geq1}(k+1)c_k\partial_{k+n}\right)P(c_1,\dots, c_N)=0~.$$ Then, the highest-weight vector is the constant polynomial $1$, and satisfies $L_0.1=h\cdot 1$, $Z.1=c\cdot 1$.
The dual ${\cal O}^*(E_{c,h})$, i.e. the space of linear functionals $\langle\sigma|$ on ${\cal O}(E_{c,h})$, can again be identified with ${\cal O}({\cal M})$ via the pairing, $$\langle P,Q\rangle:=P(\partial_1,\dots,\partial_N)Q(c_1,\dots,c_N)|_{c_1=\dots=c_N=0}~,$$ and the action of $\vir$ can then similarly be given explicitly in co-ordinates as in equations (\[Vir\_coord\]), with the roles of $c_k$ and $\partial_k$ interchanged.
Analogously, the dual space ${\OO}^*(E_{c,h})$ can be endowed with a $\vir_{\C}$ action. The singular vector for the corresponding (irreducible) Verma module $V_{c,h}$ with respect to $\{L_{-n}\}$, $n\geq1$, is again the constant polynomial $1$.
### Hypo-Ellipticity, sub-Riemannian Geometry and Conformal martingales
After this preparatory explanations, we are in position to make the link with SLE.
The L[œ]{}wner equation (\[Loewner-Ito\]) can be seen as a family of ordinary complex stochastic differential equations, labelled by the points of the upper half-plane $\mathbb{H}$. But as the individual processes are not independent, in the sense that at every instant $t$, they constitute a random conformal map for the points, where the individual processes are still defined, the interdependence can be translated into a power series expansion around the point which is the least affected in time, namely the point at infinity, or generally, the point at which the trace is aiming. So, we naturally get a random dynamical system on the infinite coefficient body, i.e. a stochastic process $f_{\infty}(t)$ on univalent maps.
Now, by taking the projective limit we obtain the generator $\hat{A}_{\infty}$ of the flow on $\aut_+({\OO}_{\infty})$, corresponding to the LÏwner equation (6) for some fixed $\kappa$, [@BB]: $$\label{L_generator}
\hat{A}_{\infty}=\varprojlim\left(\frac{\kappa}{2}\frac{\partial^2}{\partial b_1^2}+2\sum_{k=2}^N P_k(b_1,\dots, b_N)\frac{\partial}{\partial b_k}\right)~,$$ which is driven by one-dimensional standard Brownian motion, and where the polynomials $P_k$ in the drift vector are defined on the coefficient body, with the $N\times N$ diffusion matrix $$\left(\begin{array}{cccc}1 & 0 & \dots & 0 \\0 & 0 & & \\\vdots & & \ddots & \vdots \\0 & &\dots & 0\end{array}\right)$$ The generator of the diffusion process can be written in Hšrmander form in terms of the tangent vector fields in the affine co-ordinates $\{b_k\}$, applying the notational conventions for operators acting on polynomial sections of the structure sheaf: $$L_1^{\infty}:=\frac{\partial}{\partial b_1},\qquad\text{and}\qquad L^{\infty}_2:=-\sum_{k=2}^{\infty}P_k(\underline{b})\frac{\partial}{\partial b_k}~.$$ But as they satisfy the commutation relations of the Witt algebra, they span the whole tangent space and since we know that this Lie algebra acts transitively, the strong Hšrmander condition is satisfied. Therefore, the resulting flow is hypo-elliptic [@K; @F], and the corresponding geometry sub-Riemannian.
Let us lift the process $f_{\infty}(t)$ on ${\cal M}$ induced by SLE, to the complex manifold $E_{c,h}$ by using sections $\sigma$; (cf. Figure \[lifted\]). However, there are some constraints this map has to satisfy, which are given by the transformation properties of the partition function under conformal maps. This means, that the sections have to be flat with respect to the Hermitian connection $\nabla_{c,h}$ (cf. eq. (\[Kaehler\_m\])), and the physical connection (\[PFunction\_rel\]), (energy-momentum tensor),
![The stochastic process in the base $\aut_+({\OO})$ is lifted to the total space $E_{c,h}$ via the section $|\sigma\rangle$. The operators acting on the total space, depend on the parameters $\kappa, c, h$, coming from the partition function $Z_{c,h}$. The compatibility conditions, i.e., to be a martingale, require $c,h$ to depend functionally on $\kappa$. $\langle\sigma|$ is the dual vector.[]{data-label="lifted"}](lifted.pdf)
which determines $c$ and $h$, for a given model. Again, we have indicted this in Figure \[lifted\], where the group action which relates different points on the trajectory, corresponds to the transition functions of the L[œ]{}wner chain, and in the total space to the conformal anomaly, which relates the points in the fibre, i.e., the values of the determinants over the respective domains. This is equivalent to the Markov property, which was foundational in the derivation of the driving function for SLE.
Still, there is one more thing. The random trajectories $\sigma_t:=\sigma(f_{\infty}(t))$ should be (local) martingales. This is necessary, if we intend to couple the dynamical picture with the static one, i.e. “ensemble averages should be equal to time averages".
To do so, we have to couple the parameters $\kappa, c, h$, which we shall obtain from the Doob-Getoor $h$-transform. Namely, if we find harmonic sections, $\sigma_{\text{hr.}}$, then $\sigma(f_{\infty}(t))$ is a local martingale, for the lifted, now Virasoro generators $\hat{L}_n$.
But, if we restrict to polynomial sections, generated by the $L_n$, $n<1$, by acting on the constant polynomial $1$, which then successively extends over the coefficient body ${\mathrm{Aut}}({\OO})/\mathfrak{m}^{N+1}$, then as a direct calculation shows, using e.g. (\[Vir\_coord\]), the module contains a null vector, exactly if $$c_{\kappa}=\frac{(6-\kappa)(3\kappa-8)}{2\kappa}\qquad\text{and}\qquad h_{\kappa}=\frac{6-\kappa}{2\kappa}~.$$ But, this is nothing else than the relation in equation (\[c-charge\]), derived from CFT. Therefore, all this polynomials are in the kernel of the lifted generator, $$\frac{\kappa}{2}\hat{L}^2_{1}-2\hat{L}_{2}~,$$ which acts as a differential operator.
Therefore, the representation theoretic notion of “being degenerate at level two", translates in probabilistic language into a generalised Doob $h$-transform.
This is the way, how conformally invariant measures on simple paths and models from Statistical Mechanics couple, as demonstrated for the special case, the disc. But this is also valid in the general situation.
Conclusions {#conclusions .unnumbered}
-----------
We hope that our leisurely promenade has convinced the reader of the beauty and richness of the interplay between the various fields from mathematics and physics.
The connections which emerged with probability theory are certainly one of the more remarkable facts of this decade. Also, the insight that determinant line bundles are a very rewarding starting point to look at SLE, has become, as a glance at the now emerging literature reveals, a ‘sine qua non‘ in the field. However, it is amusing, that although fundamental parts of the material given here have been public for quite some time, all the sudden people start, e.g. to regularise “their determinants".
Finally, it should be immediate that the theory which we presented, builds on an enormous quantity of other theories and mathematical tools, which have been established long before SLE, for other purposes.
Therefore, SLE is, according to our opinion, yet another good example for the restless “globalisation of mathematics" and its evolution into a highly integrated organism.
[99]{} H. Airault, P. Malliavin and A. Thalmaier, [*Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows*]{}, J. de MathŽmatiques Pures et AppliquŽs, Vol. [**83**]{}, Issue 8, (2004). M. Aizenman, [*The geometry of critical percolation and conformal invariance*]{}, StatPhys 19 (Xiamen 1995), (1996). M. Aizenman A. Burchard, [*Hšlder Regularity and Dimension Bounds for Random Curves*]{}, Duke Math. J. [**99**]{}, 419 (1999). S. Albeverio, J. Jost, S. Paycha and S. Scarlatti, [*A mathematical introduction to string theory*]{}, London Math. Soc. LNS [**225**]{}, Cambridge Univ. Press, (1997). O. Alvarez, [*Theory of Strings with Boundaries: Fluctuation, topology and quantum geometry*]{}, Nucl. Phys. B [**216**]{}, (1983). M. Bauer and D. Bernard, [*$2D$ growth processes: SLE and Loewner chains*]{}, Phys. Rep. [**432**]{}, 115 (2006). R. Bauer and R. Friedrich, [*The Correlator Toolbox, Metrics and Moduli*]{}, Nucl. Phys. B [**733**]{}, (2006). A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, [*Infinite conformal symmetry in two-dimensional quantum field theory*]{}, Vol. [**241**]{}, Issue 2, (1984). J. Cardy, [*Boundary Conditions, Fusion Rules and The Verlinde Formula*]{}, Nucl. Phys. B [**324**]{}, (1989). R. Friedrich, preprint, (2008). R. Friedrich, [*On Connections of Conformal Field Theory and Stochastic L[œ]{}wner Evolution*]{}, arXiv, (2004). R. Friedrich and J. Kalkkinen, Ê[*On ConformalÊ Field Theory and Stochastic Loewner Evolution*]{}, Ê arXiv 2003, Nucl. Phys. B, [**687**]{}, 279-302 (2004). R. Friedrich and W. Werner, Ê[*Conformal fields, restriction properties, degenerate representations and SLE*]{}, C.R. Acad. Sci. Paris, Ser. I [**335**]{} (2002). R. Friedrich and W. Werner, [*ConformalÊ restriction, highest-weight representations and SLE*]{}, Comm. Math. Phys., [**243**]{}, (2003). A. Gamsa and J. Cardy, [*Schramm-Loewner evolution in the three-state Potts model–a numerical study*]{}, JSTAT, (2007). A. Hassell and S. Zelditch, [*Determinants of Laplacians in Exterior Domains* ]{}, IMRN, Int. Math. Res. Notices, No.[**18**]{}, (1999). A.A. Kirillov and D. Yur’ev, [*Representation of the Virasoro algebra by the orbit method*]{}, JGP, Vol. 5, n. [**3**]{}, (1988). M. Kontsevich, [*Arbeitstagung 2003–CFT, SLE and phase boundaries*]{}, MPIM2003-60a, (2003). M. Kontsevich and Y. Suhov, [*On Malliavin measures, SLE, and CFT*]{}, Proceedings of the Steklov Institute of Mathematics, Vol. 258([**1**]{}), (2007). R. Langlands, P. Pouliot, and Y. Saint-Aubin, [*Conformal invariance in two-dimensional percolation*]{}. Bul l. Amer. Math. Soc., [**30**]{}(1), (1994). G. Lawler, O. Schramm and W. Werner, [*Conformal restriction. The chordal case* ]{}, J. Amer. Math. Soc. [**16**]{}, (2003). P. Malliavin, [*La diffusion canonique au-dessus du groupe des diffŽomorphismes du cercle*]{}, C.R. Acad. Sci. Paris, Vol. [**329**]{}, Issue 4, (1999). S. Nag and D. Sullivan, [*TeichmŸller theory and the universal period mapping via quantum calculus and the $H^{1Ú2}$ space on the circle*]{}, Osaka J. Math. [**32**]{} (1995). B. Osgood, R. Philips and P. Sarnak, [*Extremals of Determinants of Laplacians*]{}, and, [*Compact Isospectral Sets of Surfaces*]{}, J. Funct. Anal., [**80**]{}, (1988). A. Polyakov, [*Quantum geometry of bosonic strings*]{}, Phys. Lett. B, [**103**]{} (1981). S. Rohde and O. Schramm, [*Basic properties of SLE*]{}, Ann. Math. [**161**]{}, (2005). O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. [**118**]{}, (2000). S. Smirnov, [*Towards conformal invariance of $2D$ lattice models*]{}, Proc. Int. Congr. Mathematicians, vol. 2, Eur. Math. Soc., pp. 1421-51, (2006). A. Tsuchiya, K. Ueno, and Y. Yamada, [*Conformal Field Theory on universal family of stable curves with gauge symmetries, in Integrable systems in quantum field theory and statistical mechanics*]{}, Adv. Stud. Pure Math. [**19**]{}, Academic Press, Boston, (1989). A.B. Zamolodchikov, [*“Irreversibility" of the flux of the renormalization group in a $2D$ field theory*]{}, JETP Lett. [**43**]{}, (1986).
|
---
abstract: |
A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves, are the only minor minimal intrinsically knotted graphs with 21 edges [@BM; @JKM; @LKLO; @M]. This set includes exactly one bipartite graph, the Heawood graph.
In this paper we classify the intrinsically knotted bipartite graphs with at most 22 edges. Previously known examples of intrinsically knotted graphs of size 22 were those with KS graph minor and the 168 graphs in the $K_{3,3,1,1}$ and $E_9+e$ families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the $E_9+e$ family. We show that, in fact, this is a complete listing. That is, there are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110.
address:
- 'Department of Mathematics, Korea University, Anam-dong, Sungbuk-ku, Seoul 136-701, Korea'
- 'Department of Mathematics and Statistics, California State University, Chico, Chico CA 95929-0525, USA'
- 'Department of Mathematics, Korea University, Anam-dong, Sungbuk-ku, Seoul 136-701, Korea'
author:
- Hyoungjun Kim
- Thomas Mattman
- Seungsang Oh
title: Bipartite intrinsically knotted graphs with 22 edges
---
[^1]
Introduction {#sec:intro}
============
Throughout the paper, an embedded graph will mean one embedded in $R^3$. A graph is [*intrinsically knotted*]{} if every embedding contains a non-trivially knotted cycle. Conway and Gordon [@CG] showed that $K_7$, the complete graph with seven vertices, is an intrinsically knotted graph. Foisy [@F] showed that $K_{3,3,1,1}$ is also intrinsically knotted. A graph $H$ is a [*minor*]{} of another graph $G$ if it can be obtained by contracting edges in a subgraph of $G$. If a graph $G$ is intrinsically knotted and has no proper minor that is intrinsically knotted, we say $G$ is [*minor minimal intrinsically knotted*]{}. Robertson and Seymour [@RS] proved that for any property of graphs, there is a finite set of graphs minor minimal with respect to that property. In particular, there are only finitely many minor minimal intrinsically knotted graphs, but finding the complete set is still an open problem. A $\nabla Y$ [*move*]{} is an exchange operation on a graph that removes all edges of a triangle $abc$ and then adds a new vertex $v$ and three new edges $va, vb$ and $vc$. The reverse operation is called a $Y \nabla$ [*move*]{} as follows:
Since the $\nabla Y$ move preserves intrinsic knottedness [@MRS], we will concentrate on triangle-free graphs. We say two graphs $G$ and $G^{\prime}$ are [*cousins*]{} of each other if $G^{\prime}$ is obtained from $G$ by a finite sequence of $\nabla Y$ and $Y \nabla$ moves. The set of all cousins of $G$ is called the $G$ [*family*]{}.
Johnson, Kidwell and Michael [@JKM] and, independently, the second author [@M] showed that intrinsically knotted graphs have at least 21 edges. Hanaki, Nikkuni, Taniyama and Yamazaki [@HNTY] constructed the $K_7$ family which consists of 20 graphs derived from $H_{12}$ and $C_{14}$ by $Y \nabla$ moves as in Figure \[fig11\], and they showed that the six graphs $N_9$, $N_{10}$, $N_{11}$, $N'_{10}$, $N'_{11}$ and $N'_{12}$ are not intrinsically knotted. Goldberg, Mattman and Naimi [@GMN] also proved this, independently.
![The $K_7$ family[]{data-label="fig11"}](fig11.eps)
Recently two groups [@BM; @LKLO], working independently, showed that $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves are the only intrinsically knotted graphs with 21 edges. This gives us the complete set of 14 minor minimal intrinsically knotted graphs with 21 edges, which we call [*the KS graphs*]{} as they were first described by Kohara and Suzuki [@KS].
In this paper, we concentrate on intrinsically knotted graphs with 22 edges. The $K_{3,3,1,1}$ family consists of 58 graphs, of which 26 graphs were previously known to be minor minimal intrinsically knotted. Goldberg et al. [@GMN] showed that the remaining 32 graphs are also minor minimal intrinsically knotted. The graph $E_9+e$ of Figure \[fig12\] is obtained from $N_9$ by adding a new edge $e$ and has a family of 110 graphs. All of these graphs are intrinsically knotted and exactly 33 are minor minimal intrinsically knotted [@GMN]. By combining the $K_{3,3,1,1}$ family and the $E_9+e$ family, all 168 graphs were already known to be intrinsically knotted graphs with 22 edges.
![The graph $E_9+e$[]{data-label="fig12"}](fig12.eps)
A [*bipartite*]{} graph is a graph whose vertices can be divided into two disjoint sets $A$ and $B$ such that every edge connects a vertex in $A$ to one in $B$. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. Among the 14 intrinsically knotted graphs with 21 edges, only $C_{14}$, [*the Heawood graph*]{}, is bipartite.
A bipartite graph formed by adding an edge to the Heawood graph will be bipartite intrinsically knotted. We will show that this is the only way to form such a graph that has a KS graph minor. Among the remaining 168 known examples of intrinsically knotted graphs with 22 edges in the $K_{3,3,1,1}$ and $E_9+e$ families, cousins 89 and 110 of the $E_9+e$ family are the only bipartite graphs.
![Two cousins 89 and 110 of the $E_9+e$ family[]{data-label="fig13"}](fig13.eps)
However, Cousin 89 has the Heawood graph as a subgraph. Our goal in this paper is to show that Cousin 110 completes the list of minor minimal examples. We say that a graph $G$ is [*minor minimal bipartite intrinsically knotted*]{} if $G$ is an intrinsically knotted bipartite graph, but no proper minor of $G$ has this property. Since contracting edges can lead to a bipartite minor for a graph that was not bipartite to begin with, it’s easy to construct examples of graphs that are not themselves bipartite intrinsically knotted even though they have a minor that is minor minimal bipartite intrinsically knotted. Nonetheless, Robertson and Seymour’s [@RS] Graph Minor Theorem guarantees that there are a finite number of minor minimal bipartite intrinsically knotted graphs and every bipartite intrinsically knotted graph must have one as a minor. Our main theorem shows that there are exactly two such of 22 or fewer edges.
\[thm:main\] There are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: The Heawood graph and Cousin 110 of the $E_9+e$ family.
As we show below, the argument quickly reduces to graphs of minimum degree $\delta(G)$ at least three, for which we have:
\[thm:main3\] There are exactly two bipartite intrinsically knotted graphs with 22 edges and minimum degree at least three, the two cousins 89 and 110 of the $E_9+e$ family.
We remark that Cousin 110 was earlier identified as bipartite intrinsically knotted in [@HAMM Theorem 3] as part of a classification of such graphs on ten or fewer vertices. It follows from that classification that Cousin 110 is the only minor minimal bipartite intrinsically knotted graph of order ten or less. It would be interesting to know if there are further examples of order between 11 and 14, which is the order of the Heawood graph. Such examples would have at least 23 edges.
Suppose $G$ is bipartite intrinsically knotted, with $\|G \| \leq 22$. If $\delta(G) \leq 1$, we may delete a vertex (and its edge, if it has one) to obtain a proper minor that also has this property, so $G$ is not minor minimal. If $\delta(G) = 2$, then contracting an edge adjacent to a degree two vertex gives a minor $H$ that remains intrinsically knotted and is of size at most 21. Thus $H$ is one of the KS graphs. In other words $G$ is obtained by a vertex split of the KS graph $H$. Now, a graph obtained in this way from a KS graph will be intrinsically knotted and have 22 edges. However, it’s straightforward to verify that it cannot be bipartite.
So, we can assume $\delta(G) \geq 3$. If $\|G\| = 21$, $G$ must be a KS graph and $C_{14}$, the Heawood graph, is the only bipartite graph in this set. As graphs of 20 edges are not intrinsically knotted [@JKM; @M], $C_{14}$ is minor minimal for intrinsic knotting and, so, also for bipartite intrinsically knotted.
By Theorem \[thm:main3\], if $\|G\| = 22$, $G$ must be one of the two cousins in the $E_9+e$ family. Goldberg et al. [@GMN] showed that all graphs in this family are intrinsically knotted. However, Cousin 89 is formed by adding an edge to the Heawood graph and is not minor minimal. On the other hand, it’s easy to verify that Cousin 110 is minor minimal bipartite intrinsically knotted and, therefore, the only such graph on 22 edges.
The remainder of this paper is a proof of Theorem \[thm:main3\]. In the next section we introduce some terminology and outline the strategy of our proof.
Terminology and strategy {#sec:term}
========================
Henceforth, let $G=(A,B,E)$ denote a bipartite graph with 22 edges whose partition has the parts $A$ and $B$ with $E$ denoting the edges of the graph. Note that $G$ is triangle-free. For any two distinct vertices $a$ and $b$, let $\widehat{G}_{a,b}$ denote the graph obtained from $G$ by deleting $a$ and $b$, and then contracting edges adjacent to vertices of degree 1 or 2, one by one repeatedly, until no vertices of degree 1 or 2 remain. Removing vertices means deleting interiors of all edges adjacent to these vertices and any remaining isolated vertices. Let $\widehat{E}_{a,b}$ denote the set of edges of $\widehat{G}_{a,b}$. The distance, denoted by ${\rm dist}(a,b)$, between $a$ and $b$ is the number of edges in the shortest path connecting them. If $a$ has the distance 1 from $b$, then we say that $a$ and $b$ are [*adjacent*]{}. The degree of $a$ is denoted by $\deg(a)$. Note that $\sum_{a \in A} \deg(a) = \sum_{b \in B} \deg(b) = 22$ by the definition of bipartition. To count the number of edges of $\widehat{G}_{a,b}$, we have some notation.
- $E(a)$ is the set of edges that are adjacent to a vertex $a$.
- $V(a)=\{c \in A \cup B\ |\ {\rm dist}(a,c)=1\}$
- $V_n(a)=\{c \in A \cup B\ |\ {\rm dist}(a,c)=1,\ \deg(c)=n\}$
- $V_n(a,b)=V_n(a) \cap V_n(b)$
- $V_Y(a,b)=\{c \in A \cup B\ |\ \exists \ d \in V_3(a,b) \ \mbox{such that}
\ c \in V_3(d) \setminus \{a,b\}\}$
Obviously in $G \setminus \{a,b\}$ for some distinct vertices $a$ and $b$, each vertex of $V_3(a,b)$ has degree 1. Also each vertex of $V_3(a), V_3(b)$ (but not of $V_3(a,b)$) and $V_4(a,b)$ has degree 2. Therefore to derive $\widehat{G}_{a,b}$ all edges adjacent to $a,b$ and $V_3(a,b)$ are deleted from $G$, followed by contracting one of the remaining two edges adjacent to each vertex of $V_3(a)$, $V_3(b)$, $V_4(a,b)$ and $V_Y(a,b)$ as in Figure \[fig21\](a). Thus we have the following equation counting the number of edges of $\widehat{G}_{a,b}$ which is called the [*count equation*]{}; $$|\widehat{E}_{a,b}| = 22 - |E(a)\cup E(b)| -
(|V_3(a)|+|V_3(b)|-|V_3(a,b)|+|V_4(a,b)|+|V_Y(a,b)|).$$
![Deriving $\widehat{G}_{a,b}$[]{data-label="fig21"}](fig21.eps)
For short, write $NE(a,b) = |E(a)\cup E(b)|$ and $NV_3(a,b) = |V_3(a)|+|V_3(b)|-|V_3(a,b)|$. If $a$ and $b$ are adjacent (i.e. dist$(a,b)=1$), then $V_3(a,b), V_4(a,b)$ and $V_Y(a,b)$ are all empty sets because $G$ is triangle-free. Note that the derivation of $\widehat{G}_{a,b}$ must be handled slightly differently when there is a vertex $c$ in $V$ such that more than one vertex of $V(c)$ is contained in $V_3(a,b)$ as in Figure \[fig21\](b). In this case we usually delete or contract more edges even though $c$ is not in $V_Y(a,b)$.
The following proposition, which was mentioned in [@LKLO], gives two important conditions that ensure a graph fails to be intrinsically knotted. Note that $K_{3,3}$ is a triangle-free graph and every vertex has degree 3.
\[prop:planar\] If $\widehat{G}_{a,b}$ satisfies one of the following two conditions, then $G$ is not intrinsically knotted.
- $|\widehat{E}_{a,b}| \leq 8$, or
- $|\widehat{E}_{a,b}|=9$ and $\widehat{G}_{a,b}$ is not isomorphic to $K_{3,3}$.
If $|\widehat{E}_{a,b}| \leq 8$, then $\widehat{G}_{a,b}$ is a planar graph. Also if $|\widehat{E}_{a,b}|=9$, then $\widehat{G}_{a,b}$ is either a planar graph or isomorphic to $K_{3,3}$. It is known that if $\widehat{G}_{a,b}$ is planar, then $G$ is not intrinsically knotted .
In proving Theorem \[thm:main3\] it is sufficient to consider connected graphs having no vertex of degree 1 or 2. Our process is to construct all possible such bipartite graphs $G$ with 22 edges, delete two vertices $a$ and $b$ of $G$, and then count the number of edges of $\widehat{G}_{a,b}$. If $\widehat{G}_{a,b}$ has 9 edges or less and is not isomorphic to $K_{3,3}$, then we conclude that $G$ is not intrinsically knotted by Proposition \[prop:planar\].
\[prop:deg6\] There is no bipartite intrinsically knotted graph with 22 edges and minimum degree at least three that has a vertex of degree 6 or more.
Suppose that $G$ is an intrinsically knotted graph with 22 edges that has a vertex $a$ in $A$ of degree 6 or more. Since $\sum_{b \in B} \deg(b) = 22$ and each vertex of $B$ has degree at least 3, $B$ consists of at most seven vertices, so the degree of $a$ cannot exceed 7.
If $\deg(a) = 7$, then $B$ consists of seven vertices, and so one vertex $b$ has degree 4 and the others have degree 3. Then $NE(a,b) = 10$ and $|V_3(a)| = 6$. By the count equation, $|\widehat{E}_{a,b}| \leq 6$ in $\widehat{G}_{a,b}$.
Now assume that $\deg(a) = 6$. Since $\sum_{c \in A} \deg(c) = 22$, there is a vertex $c$ of degree at least 4 in $A$. We may assume that $|V_3(a)| + |V_4(a,c)| \leq 3$, otherwise $|\widehat{E}_{a,c}| \leq 8$ because $NE(a,c) \geq 10$. Because $|V_3(a)| \leq 3$, $B$ consists of exactly six vertices and at most three vertices in $B$ have degree 3. Furthermore $c$ is adjacent to at most three vertices of degree 3 or 4 in $B$. Therefore, eventually $c$ has degree 4, and is adjacent to a degree 5 vertex $b$ and three other degree 3 vertices in $B$ as shown in Figure \[fig22\]. If there is another vertex of degree more than 3 in $A$, then, like $c$, it is adjacent to three degree 3 vertices in $B$. Therefore $A$ can have at most three vertices of degree more than 3, including $a$ and $c$. This implies that $V_3(b) \geq 2$, and so $|\widehat{E}_{a,b}| \leq 7$.
![The case of $\deg(a) = 6$[]{data-label="fig22"}](fig22.eps)
Therefore each vertex of $A$ and $B$ has degree 3, 4 or 5 only. Let $A_n$ denote the set of vertices in $A$ of degree $n = 3,4,5$ and $[A] = [|A_5|, |A_4|,|A_3|]$ and similarly for $B$. Then $[A] = [3,1,1]$, $[2,3,0]$, $[2,0,4]$, $[1,2,3]$, $[0,4,2]$ or $[0,1,6]$. Without loss of generality, we may assume that $|A_5| \geq |B_5|$, and if $|A_5| = |B_5|$ then $|A_4| \geq |B_4|$.
This paper relies on the technical machinery developed in [@LKLO]. We divide the proof of Theorem \[thm:main3\] according to the size of $A_5$, always under the assumption that our graphs are of minimum degree at least three. In Section 3, we show that the only bipartite intrinsically knotted graph with two or more degree 5 vertices in $A$ is Cousin 110 of the $E_9+e$ family. In Section 4, we show that there is no bipartite intrinsically knotted graph with exactly one degree 5 vertex in $A$. In Section 5, we show that the only bipartite intrinsically knotted graph with all vertices of degree at most 4 is Cousin 89 of the $E_9+e$ family.
Case of $|A_5| \geq 2$
======================
In this case, $A$ has at least two degree 5 vertices, say $a$ and $b$, and so $[A]$ is one of $[3,1,1]$, $[2,3,0]$ or $[2,0,4]$.
First we consider the case that $[B]$ is $[3,1,1]$, in other words, $([A], [B]) = ([3,1,1], [3,1,1])$. The edges adjacent to each vertex in $A_5$ and $B_5$ are constructed in a unique way because both $A$ and $B$ have exactly five vertices. This determines 21 edges of the graph so that both $A$ and $B$ have three degree 5 vertices and two degree 3 vertices. Now we add a final, dashed edge to connect two degree 3 vertices, one in $A$ and another in $B$. We get the graph shown in Figure \[fig31\] (a) which is Cousin 110 of the $E_9+e$ family and is intrinsically knotted.
In case $[B]$ is $[2,3,0]$, i.e., $([A], [B])$ is either $([3,1,1], [2,3,0])$ or $([2,3,0], [2,3,0])$, we similarly construct the edges adjacent to each vertex in $A_5$ and $B_5$ in a unique way. We add the remaining dashed edges which are also determined uniquely. This gives, for the two cases, the graphs shown in Figure \[fig31\] (b) and (c), respectively. In both cases, $\widehat{G}_{a,b}$ is planar for the vertices $a$ and $b$ shown in the figures.
![The cases that $A$ has at least two degree 5 vertices[]{data-label="fig31"}](fig31.eps)
Now consider the case where $[B]$ is $[2,0,4]$, i.e., $([A], [B])$ is one of $([3,1,1], [2,0,4])$, $([2,3,0], [2,0,4])$ or $([2,0,4], [2,0,4])$. We may assume that $NV_3(a,b) \leq 3$, otherwise $|\widehat{E}_{a,b}| \leq 8$ because $NE(a,b) \geq 10$. Thus $a$ and $b$ are adjacent to the same five vertices in $B$ as shown in Figure \[fig31\] (d). Let $d$ be the remaining degree 3 vertex in $B$, with $d$ adjacent to three vertices other than $a$ and $b$. If $[A]$ is $[3,1,1]$, the remaining vertex in $A_5$ must be adjacent to the same vertices as $a$. This is impossible because this vertex is also adjacent to $d$. If $[A]$ is $[2,3,0]$, the remaining dashed edges can be added in a unique way. Let $c$ be a vertex in $A_4$. Since $NE(a,c) = 9$ and $NV_3(a,c) = 4$, then $|\widehat{E}_{a,c}| \leq 9$. Since $\widehat{G}_{a,c}$ has the degree 4 vertex $b$, it is not isomorphic to $K_{3,3}$. If $[A]$ is $[2,0,4]$, let $a'$ be a vertex in $B_5$, then $NE(a,a') = 9$ and $NV_3(a,a') \geq 6$, so $|\widehat{E}_{a,a'}| \leq 7$.
Consider the case where $[B]$ is $[1,2,3]$, i.e., $([A], [B])$ is one of $([3,1,1], [1,2,3])$, $([2,3,0], [1,2,3])$ or $([2,0,4], [1,2,3])$. Similarly we may assume that $NV_3(a,b) + |V_4(a,b)| \leq 3$, otherwise $|\widehat{E}_{a,b}| \leq 8$. Thus $a$ and $b$ are adjacent to the same four vertices of degree 5 or 3, but different degree 4 vertices in $B$ as shown in Figure \[fig31\] (e). If $[A]$ is $[3,1,1]$, the remaining vertex in $A_5$ must be adjacent to another degree 4 vertex which is not adjacent to $a$ and $b$, but this is impossible. For the remaining two cases, we follow the same argument as the last two cases when $[B]$ was $[2,0,4]$.
It remains to consider $[B] = [0,4,2]$ or $[0,1,6]$. For any two vertices $a$ and $b$ in $A_5$, $NE(a,b) = 10$ and $NV_3(a,b) + |V_4(a,b)| \geq 4$, and so $|\widehat{E}_{a,b}| \leq 8$.
Case of $|A_5| = 1$
===================
In this case, there is only one choice for $[A]$, $[1,2,3]$. Let $a$, $b_1$, $b_2$, $c_1$, $c_2$ and $c_3$ be the degree 5 vertex, the two degree 4 vertices and the three degree 3 vertices in $A$. There are three choices for $[B]$, $[1,2,3]$, $[0,4,2]$ or $[0,1,6]$. We divide into three subsections, one for each case.
$[B] = [1,2,3]$
---------------
Let $a'$, $b'_1$, $b'_2$, $c'_1$, $c'_2$ and $c'_3$ be the degree 5 vertex, the two degree 4 vertices and the three degree 3 vertices in $B$. If $NV_3(a,a') \geq 5$, then $|\widehat{E}_{a,a'}| \leq 8$. Therefore $NV_3(a,a') = 4$ and we get the subgraph shown in Figure \[fig41\] (a). Assume that $c_3$ and $c'_3$ are the remaining unused vertices.
![$[A] = [B] = [1,2,3]$[]{data-label="fig41"}](fig41.eps)
If $V_4(b_1)$ is empty, i.e., $|V_3(b_1)| = 3$, then $|\widehat{E}_{b_1,a'}| \leq 9$ and $\widehat{G}_{b_1,a'}$ has the degree 4 vertex $a$. Thus $|V_4(b_1)| \geq 1$, and similarly $|V_4(b_2)|$, $|V_4(b'_1)|$ and $|V_4(b'_2)| \geq 1$. Without loss of generality, we say that ${\rm dist}(b_1,b'_1) =1$ and ${\rm dist}(b_2,b'_2) =1$. If $V_4(c'_3)$ is empty, i.e., $|V_3(c'_3)| = 3$, then $|\widehat{E}_{a,c'_3}| \leq 9$ and $\widehat{G}_{a,c'_3}$ has the degree 4 vertex $b_1$. Thus we may assume that ${\rm dist}(b_2,c'_3) =1$. If ${\rm dist}(b_2,b'_1) =1$, then $|\widehat{E}_{a,b_2}| \leq 8$. Thus we also assume that ${\rm dist}(b_2,c'_1) =1$. If ${\rm dist}(c_i,c'_1) =1$ for some $i=1,2,3$, then $|\widehat{E}_{a,b_2}| \leq 8$ because $NV_3(a,b_2) + |V_4(a,b_2)| = 4$ and $V_Y(a,b_2) = \{ c_i \}$. Therefore ${\rm dist}(b_1,c'_1) =1$.
Now consider the graph $\widehat{G}_{a,a'}$. Since $|\widehat{E}_{a,a'}| \leq 9$, we only need to consider the case that $\widehat{G}_{a,a'}$ is isomorphic to $K_{3,3}$. Since the three vertices $b_1$, $b'_2$ and $c'_3$ (big black dots in the figure) are adjacent to $b_2$ in $\widehat{G}_{a,a'}$, they’re also adjacent to $b'_1$ and $c_3$ (big white dots) as shown in Figure \[fig41\] (b). Restore the graph $G$ by adding back the two vertices $a$ and $a'$ and their associated nine edges as shown in Figure \[fig41\] (c). The reader can easily check that the graph $\widehat{G}_{a,b_1}$ is planar.
$[B] = [0,4,2]$
---------------
First, we give a very useful lemma. Let $\widetilde{K}_{3,3}$ be the bipartite graph shown in Figure \[fig42\]. The six degree 3 vertices in $\widetilde{K}_{3,3}$ are divided into three big black vertices and three big white vertices. If we ignore the three degree 2 vertices, then we get $K_{3,3}$. The vertex $d_4$ is called the [*$s$-vertex*]{}.
![$\widetilde{K}_{3,3}$[]{data-label="fig42"}](fig42.eps)
\[lem:H\] Let $H$ be a bipartite graph such that one partition of its vertices contains four degree 3 vertices, and the other partition contains two degree 3 vertices and three degree 2 vertices. If $H$ is not planar then $H$ is isomorphic to $\widetilde{K}_{3,3}$.
Let $d_1$, $d_2$, $d_3$ and $d_4$ be the four degree 3 vertices in one partition of $H$, and $d'_1$ and $d'_2$ be the two degree 3 vertices in the other partition, which contains degree 2 vertices. Let $\widehat{H}$ be the graph obtained from $H$ by contracting three edges, one each from the pair adjacent to the three degree 2 vertices. Since $\widehat{H}$ consists of nine edges but is not planar, it must be $K_{3,3}$. Therefore $d'_1$, $d'_2$ and one of the $d_i$’s, say $d_4$, are in the same partition of $K_{3,3}$ since ${\rm dist}(d'_1,d'_2) \geq 2$. Since $H$ is originally a bipartite graph, $d_4$ is connected to $d_i$ for each $i=1,2,3$, by exactly two edges adjacent to each degree 2 vertex of $H$. This gives the graph $\widetilde{K}_{3,3}$ shown in Figure \[fig42\].
Let $b'_1$, $b'_2$, $b'_3$, $b'_4$, $c'_1$ and $c'_2$ be the four degree 4 vertices and the two degree 3 vertices in $B$. First consider the case that $V_3(a)$ has only one vertex, say $c'_1$. Note that ${\rm dist}(b_i,c'_1) = 1$ for each $i=1,2$, otherwise $|\widehat{E}_{a,b_i}| \leq 8$. Now we divide into two cases depending on whether $b_1$ is adjacent to three vertices among the $b'_j$’s (say $b'_1$, $b'_2$ and $b'_3$) or two (say $b'_1$ and $b'_2$) along with $c'_2$. In Figure \[fig43\], the ten non-dashed edges in figures (a) and (b) indicate the first case while the ten non-dashed edges in figures (c)$\sim$(e) indicate the second.
Let $H$ be the bipartite graph $H$ obtained from $G$ by deleting these ten non-dashed edges. Then $H$ has four degree 3 vertices from $A$, and two degree 3 vertices and three degree 2 vertices from $B$. We only need to handle the case that $H$ is not planar because if $H$ is planar, then $\widehat{G}_{a,b_1}$ is also planar. By Lemma \[lem:H\], $H$ is isomorphic to $\widetilde{K}_{3,3}$. In each case, the $s$-vertex is either at $b_2$ as in figures (a) and (c) or at one of $c_i$’s, say $c_1$, as in figures (b), (d) and (e). The big white vertex on the left identifies the $s$-vertex. Indeed these five figures (a)$\sim$(e) represent all the possibilities up to symmetry.
![$[A] = [1,2,3]$ and $[B] = [0,4,2]$ with $|V_3(a)|=1$[]{data-label="fig43"}](fig43.eps)
For the two graphs in the figures (a) and (c), each $\widehat{G}_{b'_1,b'_2}$ has ten edges, but also contains two bi-gons and so is planar. For the other three graphs in the figure, (b), (d) and (e), each $\widehat{G}_{a,b_2}$ has nine edges, but also contains a bi-gon on the vertices $c_2$ and $c_3$. So, these are also planar.
Now consider the case where $V_3(a)$ has two vertices. Thus $a$ is adjacent to three $b'_j$ vertices, say $b'_1$, $b'_2$ and $b'_3$, as well as $c'_1$ and $c'_2$. If $|V_3(b'_4)|=3$, then $NV_3(a,b'_4) = 5$, so $|\widehat{E}_{a,c}| \leq 8$. We may assume that $b'_4$ is adjacent to $b_1$, $b_2$, $c_1$ and $c_2$ as shown in Figure \[fig44\] (a). Furthermore, $|V_3(b'_j)| \leq 2$ for each $j=1,2,3$, otherwise $|\widehat{E}_{a,b'_j}| \leq 9$ and $\widehat{G}_{a,b'_j}$ has the degree 4 vertex $b'_4$. Thus each $b'_j$ is adjacent to either $b_1$ or $b_2$ or both. Without loss of generality, we say that $b_1$ is adjacent to both $b'_1$ and $b'_2$. Also $V_3(b_1)$ must have one vertex, say $c'_1$, and $c'_1$ is adjacent to $b_2$, otherwise $NV_3(a,b_1) + |V_4(a,b_1)| + |V_Y(a,b_1)| \geq 5$, so $|\widehat{E}_{a,b_1}| \leq 8$. The four dashed edges in the figure indicate these new edges.
![$[A] = [1,2,3]$ and $[B] = [0,4,2]$ with $|V_3(a)|=2$[]{data-label="fig44"}](fig44.eps)
Now consider the graph $\widehat{G}_{a,b'_4}$. Since $|\widehat{E}_{a,b'_4}| \leq 9$, we can assume that $\widehat{G}_{a,b'_4}$ is isomorphic to $K_{3,3}$. Since the three vertices $b_2$, $b'_1$ and $b'_2$ (big black dots in the figure) are adjacent to $b_1$ in $\widehat{G}_{a,b'_4}$, they’re also adjacent to $c_3$ and $b'_3$ (big white dots) as shown in Figure \[fig44\] (b). Restore the graph $G$ by adding back the two vertices $a$ and $b'_4$ and their associated nine edges as shown in Figure \[fig44\] (c). The reader can easily check that the graph $\widehat{G}_{a,b_2}$ is planar.
$[B] = [0,1,6]$
---------------
Let $b'$ be the degree 4 vertex in $B$. For given $i=1,2$, $NV_3(a,b_i) + |V_4(a,b_i)| \leq 4$, otherwise $|\widehat{E}_{a,b_i}| \leq 8$. Therefore $V_3(a)$, $V_3(b_1)$ and $V_3(b_2)$ are the same set of four degree 3 vertices in $B$. Since $|V_3(b')|=3$, then $|\widehat{E}_{a,b'}| = 7$.
Case of $|A_5| = 0$
===================
In this case, $([A], [B])$ is one of $([0,4,2], [0,4,2])$, $([0,4,2], [0,1,6])$ or $([0,1,6], [0,1,6])$. We divide into three subsections, one for each case.
$([A], [B]) = ([0,4,2], [0,4,2])$
---------------------------------
We first give a very useful lemma. Let $\widetilde{P}_{10}$ be the bipartite graph shown in Figure \[fig51\]. By deleting the vertex $c'$ and its two edges, we get $\widetilde{K}_{3,3}$. The vertices $d_4$ and $c'$ are called the [*$s$-*]{} and [*$t$-vertex*]{}, respectively.
![$\widetilde{P}_{10}$[]{data-label="fig51"}](fig51.eps)
\[lem:P\] Let $H$ be a bipartite graph such that one partition of its vertices contains two degree 4 vertices and two degree 3 vertices, and the other partition contains two degree 3 vertices and four degree 2 vertices. If $H$ is not planar then $H$ is isomorphic to $\widetilde{P}_{10}$.
Let $d_1$, $d_2$, $d_3$ and $d_4$ be the two degree 4 vertices and the two degree 3 vertices in one partition of $H$, and $d'_1$ and $d'_2$ be the two degree 3 vertices in the other partition, which contains degree 2 vertices. Let $\widehat{H}$ be the graph obtained from $H$ by contracting four edges, one each from the pair adjacent to each of the degree 2 vertices. Since $\widehat{H}$ consists of six vertices and ten edges but is not planar, it must be $K_{3,3}+e$, the graph obtained from $K_{3,3}$ by connecting two vertices in the same partition by an edge $e$. Then $e$ must connect the two degree 4 vertices $d_1$ and $d_2$. Furthermore $d_1$ and $d_2$ and one of $d_3$ or $d_4$, say $d_3$, are in the same partition of $K_{3,3}+e$ containing the edge $e$. Since $H$ was originally a bipartite graph, $d_1$ and $d_2$ are connected by two edges adjacent to a degree 2 vertex, call it $c'$, of $H$, and $d_4$ is connected to $d_i$ for each $i=1,2,3$, by two edges adjacent to each other degree 2 vertex of $H$. So, we get the graph $\widetilde{P}_{10}$ as shown in Figure \[fig51\].
First consider the case that some degree 4 vertex, say $b_1$, in $A$ is adjacent to all four degree 4 vertices in $B$. Let $b_2$ denote another degree 4 vertex in $A$, and $H$ the graph obtained from $G$ by deleting the two vertices $b_1$ and $b_2$ and the adjacent eight edges. If the graph $\widehat{G}_{b_1,b_2}$ is not planar, then $H$ satisfies all assumptions of Lemma \[lem:P\]. Thus $H$ is isomorphic to $\widetilde{P}_{10}$. Now restore the vertex $b_2$ and the associated four dashed edges as shown in Figure \[fig52\] (a). Note that these four edges can be replaced in a unique way because of the assumptions for the vertex $b_1$. Let $b_3$ denote a degree 4 vertex in $A$, other than $b_1$ and $b_2$. The reader can easily check that the graph $\widehat{G}_{b_1,b_3}$ is planar as shown in Figure \[fig52\] (b).
![$[A]=[B] = [0,4,2]$ with $|V_3(b_1)|=0$ or 1[]{data-label="fig52"}](fig52.eps)
Now assume that each degree 4 vertex, say $b_i$ for $i=1,2,3,4$, in $A$ is adjacent to at least one of the two degree 3 vertices, say $c'_1$ and $c'_2$, in $B$. By counting the degrees of each vertex, we may assume that $b_1$ is adjacent to $c'_1$, but not $c'_2$. Also assume that $b_2$ is adjacent to $c'_2$, but not $c'_1$ since at most three vertices among the $b_i$’s can be adjacent to $c'_1$. If $V_4(b_1) = V_4(b_2)$, then $|\widehat{E}_{b_1,b_2}| \leq 9$. Since $\widehat{G}_{b_1,b_2}$ has the remaining degree 4 vertex in $B$ outside of $V_4(b_1)$, it is not isomorphic to $K_{3,3}$. Therefore $V_4(b_1) \cap V_4(b_2)$ has two vertices, say $b'_1$ and $b'_2$, in $B$. As drawn in Figure \[fig52\] (c), the eight non-dashed edges adjacent to the vertices $b_1$ and $b_2$ are settled. Let $H$ denote the graph obtained from $G$ by deleting these two vertices and the associated eight edges. If the graph $\widehat{G}_{b_1,b_2}$ is not planar, then $H$ satisfies all assumptions of Lemma \[lem:P\]. Thus $H$ is isomorphic to $\widetilde{P}_{10}$. There are several choices for the $s$-vertex and the $t$-vertex among $A$ and $B$, respectively. For example, we can choose $c_2$ for the $s$-vertex and $b'_1$ for the $t$-vertex as in figure (c). Whatever choice is made, the graph $\widehat{G}_{b_1,b_3}$ is always planar.
$([A], [B]) = ([0,4,2], [0,1,6])$
---------------------------------
First assume that the degree 4 vertex, say $b'$, in $B$ is adjacent to all four degree 4 vertices, say $b_1$, $b_2$, $b_3$ and $b_4$, in $A$. Let $c'_1$ through $c'_6$ denote the six degree 3 vertices in $B$. If $|V_3(b_j,b_k)| \geq 5$ for some different $j$ and $k$, then $|\widehat{E}_{b_j,b_k}| \leq 8$. Thus we may assume that $|V_3(b_j,b_k)| \leq 4$ for all pairs $j$ and $k$, so $V_3(b_j)$ and $V_3(b_k)$ have at least two common vertices among the $c'_i$’s. Assume that $b_1$ is adjacent to $c'_1$, $c'_2$ and $c'_3$. Note that each $c'_i$ is adjacent to at least one of the $b_j$’s because $c'_i$ has degree 3. We also assume that $c'_4$ is adjacent to $b_2$. Since $|V_3(b_1,b_2)| \leq 4$, assume that $b_2$ is also adjacent to $c'_1$ and $c'_2$. Again, assume that $c'_5$ is adjacent to $b_3$. Since $|V_3(b_j,b_3)| \leq 4$ for $j=1,2$, $b_3$ must be adjacent to $c'_1$ and $c'_2$. Now $c'_6$ is adjacent to $b_4$, and similarly $b_4$ must be adjacent to $c'_1$ and $c'_2$. This is impossible because $c'_1$ and $c'_2$ have degree 3. See Figure \[fig53\] (a).
![$[A]= [0,4,2]$ and $[B] = [0,1,6]$[]{data-label="fig53"}](fig53.eps)
Now consider the case that $b'$ is not adjacent to a degree 4 vertex, say $b_1$, in $A$. Assume that $b_1$ is adjacent to $c'_1$, $c'_2$, $c'_3$ and $c'_4$. If $|V_3(b')| = 2$, then $|\widehat{E}_{b_1,b'}| \leq 8$. Thus we may assume that $b'$ is adjacent to $b_2$, $b_3$, $b_4$ and a degree 3 vertex, say $c_1$. We also assume that $c'_5$ is adjacent to $b_2$ because $c'_5$ has degree 3. If $b_2$ is adjacent to $c'_6$ or $V_Y(b_1,b_2)$ is not empty, then $|\widehat{E}_{b_1,b_2}| \leq 8$. Thus we may assume that $b_2$ is adjacent to $c'_1$ and $c'_2$, and $c'_1$ and $c'_2$ are adjacent to $b_3$ and $b_4$, respectively. These are the non-dashed edges in Figure \[fig53\] (b).
Now consider the graph $\widehat{G}_{b_1,b'}$. Since $|\widehat{E}_{b_1,b'}| \leq 9$, we can assume that $\widehat{G}_{b_1,b'}$ is isomorphic to $K_{3,3}$. Since the three vertices $b_3$, $b_4$ and $c'_5$ (big black dots in the figure) are adjacent to $b_2$ in $\widehat{G}_{b_1,b'}$, they’re also adjacent to $c_2$ and $c'_6$ (big white dots) as shown in Figure \[fig53\] (c). Restore the graph $G$ by adding back the two vertices $b_1$ and $b'$ and their associated nine edges. Then $NV_3(b_2,b_3) + |V_4(b_2,b_3)| = 6$, so $|\widehat{E}_{b_2,b_3}| \leq 8$.
$([A], [B]) = ([0,1,6], [0,1,6])$
---------------------------------
If the degree 4 vertex, say $b$, in $A$ is not adjacent to the degree 4 vertex, say $b'$, in $B$, then $|\widehat{E}_{b,b'}| \leq 6$ because $NV_3(b,b') = 8$. Assume that $b$ is adjacent to $b'$, and let $e$ denote the edge connecting these two vertices.
\[4cycle\] In this case, if $G$ is intrinsically knotted, then every 4-cycle contains the edge $e$.
Suppose that there is a 4-cycle $H$ which does not contain $e$. Then, we may assume that $H$ does not contain $b$. Let $c$ be any vertex in $A$ such that either $b$ or $c$ is adjacent to some vertex of $H$ in $B$, other than $b'$. Since $|V_3(b,c)| = |V_Y(b,c)|$ and $|V_3(c)| + |V_4(b,c)| = 3$, $NV_3(b,c) + |V_4(b,c)| + |V_Y(b,c)| = (|V_3(b)| + |V_3(c)| - |V_3(b,c)|) + |V_4(b,c)| + |V_3(b,c)| = 6$. Therefore $|\widehat{E}_{b,c}| \leq 9$. Furthermore $\widehat{G}_{b,c}$ contains $H$ which is no longer a 4-cycle because at least one vertex of $H$ in $B$ has degree 2. So $\widehat{G}_{b,c}$ is not isomorphic to $K_{3,3}$.
Now consider the subgraph $G \setminus \{e\}$ which consists of fourteen degree 3 vertices and has no 4-cycle by Lemma \[4cycle\]. We name these vertices $c_i$’s and $c'_j$’s as in Figure \[fig54\] (a). Assume that $c_1$ is adjacent to $c'_1$, $c'_2$ and $c'_3$, and $c'_1$ is adjacent to $c_2$ and $c_3$. Since there is no 4-cycle, we can also assume that $c_2$ is adjacent to $c'_4$ and $c'_5$, $c_3$ with $c'_6$ and $c'_7$, $c'_2$ with $c_4$ and $c_5$, and $c'_3$ with $c_6$ and $c_7$ as illustrated by the non-dashed edges in the figure. Without loss of generality, we may assume that $c_4$ is adjacent to $c'_4$ and $c'_6$, and then $c_5$ must be adjacent to $c'_5$ and $c'_7$. Similarly we may assume that $c_6$ is adjacent to $c'_4$, and then $c_6$ must be adjacent to $c'_7$, and $c_7$ to $c'_5$ and $c'_6$. Finally we get the Heawood graph $C_{14}$ as drawn in Figure \[fig54\] (b). Note that $C_{14}$ is symmetric and any pair of vertices $c_i$ and $c'_j$ has distance either 1 or 3. Thus $G$ can be obtained by connecting two such vertices of distance 3 by the edge $e$. This graph is isomorphic to Cousin 89 of the $E_9+e$ family as drawn in Figure \[fig13\].
![The subgraph $G \setminus \{e\}$[]{data-label="fig54"}](fig54.eps)
[AAAAAA]{} J. Barsotti and T. Mattman, [*Intrinsically knotted graphs with 21 edges*]{}, Preprint arXiv:1303.6911. P. Blain, G. Bowlin, T. Fleming, J. Foisy, J. Hendricks, and J. LaCombe, [*Some results on intrinsically knotted graphs*]{}, J. Knot Theory Ramif. **16** (2007) 749–760. J. Conway and C. Gordon, [*Knots and links in spatial graphs*]{}, J. Graph Theory **7** (1985) 445–453. J. Foisy, [*Intrinsically knotted graphs*]{}, J. Graph Theory **39** (2002) 178–187. N. Goldberg, T. Mattman, and R. Naimi, [*Many, many more intrinsically knotted graphs*]{}, Alg. Geom. Top. **14** (2014) 1801–1823. R. Hanaki, R. Nikkuni, K. Taniyama, and A. Yamazaki, [*On intrinsically knotted or completely $3$-linked graphs*]{}, Pacific J. Math. **252** (2011) 407–425. S. Huck, A. Appel, M-A. Manrique, and T. Mattman, [*A sufficient condition for intrinsic knotting of bipartite graphs*]{}, Kobe J. Math. **27** (2010) 47–57. B. Johnson, M. Kidwell, and T. Michael, [*Intrinsically knotted graphs have at least $21$ edges*]{}, J. Knot Theory Ramif. **19** (2010) 1423–1429. T. Kohara and S. Suzuki, [*Some remarks on knots and links in spaital graphs*]{}, Knots 90 (Osaka, 1990) (1992) 435–445. M. Lee, H. Kim, H. J. Lee, and S. Oh, [*Exactly fourteen intrinsically knotted graphs have 21 edges*]{}, Preprint arXiv:1207.7157. T. Mattman, [*Graphs of 20 edges are 2-apex, hence unknotted*]{}, Alg. Geom. Top. **11** (2011) 691–718. R. Motwani, A. Raghunathan, and H. Saran, [*Constructive results from graph minors; linkless embeddings*]{}, Proc. 29th Annual Symposium on Foundations of Computer Science, IEEE (1988) 398–409. M. Ozawa and Y. Tsutsumi, [*Primitive spatial graphs and graph minors*]{}, Rev. Mat. Complut. **20** (2007) 391–406. N. Robertson and P. Seymour, [*Graph minors XX, Wagner’s conjecture*]{}, J. Combin. Theory Ser. B **92** (2004) 325–357.
[^1]: 2010 Mathematics Subject Classification: 57M25, 57M27, 05C10
|
---
abstract: 'Recently, it was argued that the spacetime dynamics can be understood by calculating the difference between the degrees of freedom on the boundary and in the bulk in a region of space. In this Letter, we apply this new idea to braneworld scenarios and extract the corresponding Friedmann equations of $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk with any spacial curvature. We will also extend our study to the more general Gauss-Bonnet braneworld with curvature correction terms on the brane and in the bulk, and derive the dynamical equation in a nonflat Universe.'
address: |
$^1$ Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran\
$^2$ Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
author:
- 'A. Sheykhi $^{1,2}$[^1], M. H. Dehghani $^{1,2}$ [^2] and S. E. Hosseini $^{1}$'
title: Friedmann equations in braneworld scenarios from emergence of cosmic space
---
Introduction\[Intr\]
====================
The emergence properties of gravity has a long history since the original proposal made by Sakharov in 1968 [@Sak]. Recent investigations supports the idea that gravitational field equations in a wide range of theories can be recast as the first law of thermodynamics on the boundary of space [CaiKim,SheyW1,SheyW2,Shey0,Pad0]{}. Among various proposal on the connection between thermodynamics and gravity, the so called entropic origin of gravity proposed by Verlinde [@Ver], has got a lot of attentions [Cai4,Other,newref,sheyECFE,Ling,Modesto,Yi,Sheykhi2]{}. According to Verlinde, gravity can be identified with an entropic force caused by changes in the information associated with the positions of material bodies. Verlinde considers the gravitational field equations as the equations of emergent phenomenon and leaves the spacetime as a background geometric which has already exist.
A new insight to the origin of spacetime dynamics, was recently suggested by Padmanabhan[@Pad1] who claimed that the cosmic space is emergent as the cosmic time progresses. Using this new idea, Padmanabhan [@Pad1] derived the Friedmann equation of a flat Friedmann-Robertson-Walker (FRW) Universe. Following [@Pad1], further investigations have been carried out to extract the Friedmann equations of a FRW Universe in various gravity theories [@Cai1; @Yang; @FQ; @Shey1]. In these investigations ([Cai1,Yang,Shey1,FQ]{}), following [@Pad1], the authors could only derive the Friedmann equations of a flat FRW Universe and they failed to obtain the dynamical equations describing the evolution of the Universe with any spacial curvature in other gravity theories. Very recently, an interesting modification of Padmanabhan’s proposal, which works in a nonflat Universe, was suggested by Sheykhi [@Shey2]. Using this modified proposal one is able to derive the corresponding dynamical equations governing the evolution of the Universe with any spacial curvature not only in Einstein gravity, but also in Gauss-Bonnet and more general Lovelock gravity [@Shey2]. See also [@FF] for some application and extension of [@Shey2]. In this paper, we will address the question on the connection between the degrees of freedom and the spacetime dynamics by investigating whether and how the relation can be found in braneworld models.
Let us briefly review the proposal of [@Shey2]. According to Padmanabhan in an infinitesimal interval $dt$ of cosmic time, the increase $dV$ of the cosmic volume, in a flat Universe, is given by [@Pad1] $$\frac{dV}{dt}=L_{p}^{2}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}). \label{dV1}$$ where $L_{p}$ is the Planck length, $N_{\mathrm{sur}}$ is the number of degrees of freedom on the boundary and $N_{\mathrm{bulk}}$ is the number of degrees of freedom in the bulk. Through this paper we set $k_{B}=1=c=\hbar $ for simplicity. Inspired by (\[dV1\]), an improved extension for $%
n\geq4 $-dimensional Universe with spacial curvature was found as [Shey2]{} $$\beta \frac{dV}{dt}=L_{p}^{n-2}H\tilde{r}_{A}\left(N_{\mathrm{sur}}-N_{%
\mathrm{bulk}}\right), \label{dV21}$$where $H=\dot{a}/a$ is the Hubble parameter, $a$ is the scale factor, $%
\beta ={(n-2)}/{2(n-3)}$ and $\tilde{r}_A$ is the apparent horizon radius of FRW Universe given by $$\label{radius}
\tilde{r}_A=\frac{1}{\sqrt{H^2+k/a^2}}.$$ Motivated by the area law of the entropy, we assume the number of degrees of freedom on the apparent horizon is $$\label{Nsur1}
N_{\mathrm{sur}}=\beta \frac{A}{L_{p}^{n-2}},$$ where $A=(n-1)\Omega _{n-1}\tilde{r}_{A}^{n-2}$ is the area of the apparent horizon with $\Omega_{n-1}$ is the volume of a unit $(n-1)$-sphere. The volume of the $(n-1)$-sphere with radius $\tilde{r}_{A}$ is $V =\Omega_{n-1}
\tilde{r}_{A}^{n-1}$. We assume the energy content inside the $n$-dimensional bulk is in the form of Komar energy [@Cai1] $$E_{\mathrm{Komar}}=\frac{(n-3)\rho+(n-1)p}{n-3}V, \label{Ek2}$$ where $\rho$ and $p$ are the energy density and pressure of the perfect fluid inside the Universe, respectively. Hence according to the equipartition law of energy, the bulk degrees of freedom is obtained as $$\begin{aligned}
\label{Nbulk1}
N_{\mathrm{bulk}}&=&\frac{2\left\vert E_{\mathrm{komar}}\right\vert }{T}
\notag \\
&=&-4 \pi \Omega_{n-1} \tilde{r}^{n}_A \frac{(n-3)\rho+(n-1)p}{n-3},\end{aligned}$$ where $T=1/(2\pi \tilde{r}_{A})$ is the Hawking temperature associated with the apparent horizon. Substituting Eqs. (\[Nsur1\]) and (\[Nbulk1\]) in relation (\[dV2\]), we arrive at $$\label{Fr3}
H^{-1}\dot{r}_{A}\tilde{r}_{A}^{-3}-\tilde{r}_{A}^{-2}=\frac{8\pi
L_{p}^{n-2} }{(n-1)}\times\frac{(n-3)\rho +(n-1)p}{(n-2)}$$Multiplying both hand sides of by factor $2\dot{a}a$, and using the $n$-dimensional continuity equation: $$\dot{\rho}+(n-1)H(\rho +p)=0, \label{cont}$$ we obtain [@Shey2] $$\frac{d}{dt}\left[a^2 \left(H^2+\frac{k}{a^2}\right)\right]=\frac{16 \pi
L_{p}^{n-2}}{(n-1)(n-2)} \frac{d}{dt}(\rho a^2). \label{Fr4}$$ After integrating and setting the constant of integration equal to zero, we find $$H^2+\frac{k}{a^2}=\frac{16 \pi L_{p}^{n-2}}{(n-1)(n-2)}\rho. \label{FrE}$$ This is the Friedmann equation of $n$-dimensional FRW Universe with any spacial curvature [@CaiKim].
Emergence of Friedmann equations in RS II braneworld {#brane}
====================================================
In the remaining part of paper, we want to extend the study to the branworld scenarios. Gravity on the brane does not obey Einstein theory, thus the usual area formula for the holographic boundary get modified on the brane [@SheyW1; @SheyW2]. Two well-known scenarios in braneworld are Randall-Sundrum (RS) II [@RS; @Bin] and Dvali, Gabadadze, Porrati (DGP) [@DGP; @DG] models. In the first scenario an $(n-1)$-dimensional brane embedded in an $(n+1)$-dimensional AdS bulk. In this case, the extra dimension has a finite size and the localization of gravity on the brane occurs due to the negative cosmological constant in the bulk. In the second scenario which is called DGP model, an $(n-1)$-dimensional brane is embedded in a spacetime with an infinite-size extra dimension, with the hope that this picture could shed new light on the standing problem of the cosmological constant as well as on supersymmetry breaking [@DGP]. In the original DGP model the bulk was assumed to be a Minkowskian spacetime with infinite size. In this case the recovery of the usual gravitational laws on the brane is obtained by adding an Einstein-Hilbert term to the action of the brane computed with the brane intrinsic curvature. The so-called warped DGP model corresponds to the case where both the intrinsic curvature term on the brane and the negative cosmological constant in the bulk are taken into account.
In order to apply the proposal (\[dV2\]) to braneworld scenarios, we modify it a little by replacing $L_{p}^{n-2}$ with $G_{n+1}$, namely $$\beta \frac{dV}{dt}=G_{n+1}H\tilde{r}_{A}\left( N_{\mathrm{sur}}-N_{\mathrm{%
bulk}}\right) . \label{dV2}$$First of all, we consider the RS II scenario. The apparent horizon entropy for an $(n-1)$-brane embedded in an $(n+1)$-dimensional bulk in RS II model is given by [@SheyW1] $$S=\frac{2\Omega _{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}\times
{}_{2}F_{1}\left( \frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{\tilde{r}%
_{A}}^{2}}{\ell ^{2}}\right) , \label{entRSAdS1}$$where ${}_{2}F_{1}(a,b,c,z)$ is a hypergeometric function, and $\ell $ is the bulk AdS radius, $$\ell ^{2}=-\frac{n(n-1)}{16\pi G_{n+1}\Lambda _{n+1}}\,,\quad \Omega _{n-1}=%
\frac{\pi ^{(n-1)/2}}{\Gamma ((n+1)/2)}. \label{rela}$$In the above relation, $\Lambda _{n+1}$ represents the $(n+1)$-dimensional bulk cosmological constant. The entropy expression (\[entRSAdS1\]) can be written in the form [@SheyW1] $$S=\frac{(n-1)\ell \Omega _{n-1}}{2G_{n+1}}{\displaystyle\int_{0}^{\tilde{r}%
_{A}}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{\tilde{r}_{A}^{2}+\ell ^{2}}}d\tilde{r}%
_{A}}, \label{entRSAdS2}$$and hence we define the effective area as $$\widetilde{A}=4G_{n+1}S=2(n-1)\ell \Omega _{n-1}\int_{0}^{\tilde{r}_{A}}%
\frac{\widetilde{r}_{A}^{n-2}}{\sqrt{\widetilde{r}_{A}^{2}+\ell ^{2}}}d%
\widetilde{r}_{A}^{{}}$$Now we calculate the increasing in the effective volume as $$\begin{aligned}
\frac{d\widetilde{V}}{dt} &=&\frac{\tilde{r}_{A}}{(n-2)}\frac{d\tilde{A}}{dt}
\notag \\
&=&2\ell \Omega _{n-1}\frac{(n-1)}{(n-2)}\frac{\tilde{r}_{A}^{n-1}}{\sqrt{%
\tilde{r}_{A}^{2}+\ell ^{2}}}\dot{\tilde{r}}_{A} \\
&=&-2\Omega _{n-1}\frac{(n-1)}{(n-2)}\tilde{r}_{A}^{n+1}\frac{d}{dt}\left(
\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}\right) \label{dVt1}\end{aligned}$$Motivated by (\[dVt1\]), we assume the number of degrees of freedom on the boundary is given by $$\begin{aligned}
N_{\mathrm{sur}} &=&\frac{4\beta (n-1)\Omega _{n-1}}{(n-2)G_{n+1}}\tilde{r}%
_{A}^{n}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}\nonumber \\
&=&\frac{2(n-1)\Omega _{n-1}}{(n-3)G_{n+1}}\tilde{r}_{A}^{n}\sqrt{\tilde{r}%
_{A}^{-2}+\frac{1}{\ell ^{2}}}. \label{NsurRS}\end{aligned}$$Inserting Eqs. (\[Nbulk1\]), (\[dVt1\]) and (\[NsurRS\]) in relation (\[dV2\]), after multiplying both hand side by factor $\dot{a}a$, we get $$\begin{aligned}
&&-\frac{\widetilde{r}_{A}^{-3}\dot{\widetilde{r}}_{A}}{\sqrt{\widetilde{r}%
_{A}^{-2}+\frac{1}{\ell ^{2}}}}a^{2}+2\dot{a}a\sqrt{\widetilde{r}_{A}^{-2}+%
\frac{1}{\ell ^{2}}} \notag \\
&=&-4\pi G_{n+1}\dot{a}a\left( \frac{(n-3)\rho +(n-1)p}{(n-1)}\right) .\end{aligned}$$Using the continuity equation (\[cont\]), after some simplification, we arrive at $$\frac{d}{dt}\left( a^{2}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}\right)
=\frac{4\pi G_{n+1}}{(n-1)}\frac{d}{dt}\left( \rho a^{2}\right) .$$Integrating and dividing by $a^{2}$, we find $$\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}=\frac{4\pi G_{n+1}}{(n-1)}\rho
, \label{FRS}$$where we assumed the integration constant to be zero. Substituting the apparent horizon radius from relation (\[radius\]), we get $$\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell ^{2}}}=\frac{4\pi G_{n+1}}{(n-1)}%
\rho . \label{FrRS}$$In this way we derive the Friedmann equation of higher dimensional FRW Universe in RS II braneworld by calculating the difference between the number of degrees of freedom on the boundary and in the bulk. This coincides with the result obtained in [@SheyW1] from the field equations.
Friedmann equations in Warped DGP braneworld
============================================
Next we consider an $(n-1)$-dimensional warped DGP brane embedded in an $%
(n+1)$-dimensional AdS bulk. The entropy associated with the apparent horizon is given by [@SheyW1] $$\begin{aligned}
S&=&\frac{(n-1)\Omega _{n-1}{\tilde{r}_{A}}^{n-2}}{4G_{n}}+\frac{2\Omega
_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}} \notag \\
&&\times {}_{2}F_{1}\left( \frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{%
\tilde{r}_{A}}^{2}}{\ell ^{2}}\right). \label{entDGP}\end{aligned}$$It is important to note that in DGP braneworld, the entropy expression of the apparent horizon consists two terms. The first term which satisfies the area formula on the brane is the contribution from the Einstein-Hilbert term on the brane. The second term is the same as the entropy of RS II braneword and therefore obeys the $(n+1)$-dimensional area law in the bulk [@SheyW1].
One can write the entropy associated with the apparent horizon on the brane as [@SheyW1] $$S=(n-1)\Omega _{n-1}\int_{0}^{\tilde{r}_{A}}\left( \frac{(n-2)\tilde{r}%
_{A}^{n-3}}{4G_{n}}+\frac{\ell }{2G_{n+1}}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{%
\tilde{r}_{A}^{2}+\ell ^{2}}}\right) d\tilde{r}_{A}^{{}}$$We define the effective surface as $$\begin{aligned}
\widetilde{A} &=&4G_{n+1}S=4G_{n+1}(n-1)\Omega _{n-1} \notag \\
&&\times \int_{0}^{\tilde{r}_{A}}\left( \frac{(n-2)\tilde{r}_{A}^{n-3}}{%
4G_{n}}+\frac{\ell }{2G_{n+1}}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{\tilde{r}%
_{A}^{2}+\ell ^{2}}}\right) d\tilde{r}_{A}. \notag \\
&&\end{aligned}$$We also obtain the rate of increase in the effective volume as $$\begin{aligned}
\frac{d\widetilde{V}}{dt} &=&\frac{\tilde{r}_{A}}{(n-2)}\frac{d\tilde{A}}{dt}%
=\Omega _{n-1}\frac{(n-1)}{(n-2)}\dot{\tilde{r}}_{A}\tilde{r}_{A}^{n-2}
\notag \\
&&\times \left( \frac{(n-2)G_{n+1}}{G_{n}}+\frac{2}{\sqrt{\tilde{r}%
_{A}^{-2}+\ell ^{-2}}}\right) \notag \\
&=&-2\Omega _{n-1}\frac{(n-1)}{(n-2)}\tilde{r}_{A}^{n+1} \notag \\
&&\times \frac{d}{dt}\left( \frac{(n-2)G_{n+1}}{4G_{n}}\tilde{r}_{A}^{-2}+%
\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}\right) \label{dVt2}\end{aligned}$$Inspired by (\[dVt2\]), we suppose the number of degrees of freedom on the apparent horizon in warped DGP model is given by $$\begin{aligned}
N_{\mathrm{sur}}=\frac{2\Omega _{n-1}}{G_{n+1}}\frac{(n-1)}{(n-3)}\tilde{r}%
_{A}^{n} &&\left( \frac{G_{n+1}(n-2)\tilde{r}_{A}^{-2}}{4G_{n}}\right.
\notag \\
&&\left. +\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}\right) .
\label{NsurDGP}\end{aligned}$$Combining Eqs. (\[Nbulk1\]), (\[dVt2\]) and (\[NsurDGP\]) with relation (\[dV2\]), it is a matter of calculation to find $$\begin{aligned}
\frac{d}{dt}\left( a^{2}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}\right)
&=&-\frac{G_{n+1}}{4G_{n}}(n-2)\frac{d}{dt}\left( \tilde{r}%
_{A}^{-2}a^{2}\right) \notag \\
&&+\frac{4\pi G_{n+1}}{(n-1)}\frac{d}{dt}\left( \rho a^{2}\right) .\end{aligned}$$Integrating and dividing by $a^{2}$ we obtain $$\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell ^{2}}}=-\frac{G_{n+1}}{4G_{n}}(n-2)%
\tilde{r}_{A}^{-2}+\frac{4\pi G_{n+1}}{(n-1)}\rho .$$Substituting the apparent horizon radius from relation (\[radius\]), we have $$\begin{aligned}
&&\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell ^{2}}}+\frac{G_{n+1}}{4G_{n}}%
(n-2)\left( H^{2}+\frac{k}{a^{2}}\right) \notag \label{FrDGP} \\
&=&\frac{4\pi G_{n+1}}{(n-1)}\rho .\end{aligned}$$This equation is indeed the Friedmann equation of FRW Universe in warped DGP braneworld derived in [@SheyW1] from the field equations. If we define, as usual, the crossover length scale between the small and large distances in DGP braneworld as [@Def] $$r_{c}=\frac{G_{n+1}}{2G_{n}},$$then one can easily check that for $r_{c}\rightarrow \infty $, the standard Friedmann equation in $n$-dimensional FRW Universe presented in (\[FrE\]) is recovered. On the other hand, when $r_{c}\rightarrow 0$, Eq. (\[FrDGP\]) reduces to the Friedmann equation in RS II braneworld obtained in the previous section.
Emergence of spacetime dynamics in Gauss-Bonnet braneworld
==========================================================
Finally, we apply the method developed in the previous sections to investigate the emergence properties of the spacetime dynamics in general braneworld with curvature correction terms including a 4D scalar curvature from induced gravity on the brane, and a 5D Gauss-Bonnet curvature term in the bulk. With these correction terms, especially including a Gauss-Bonnet correction to the 5D action, we have the most general action with second-order field equations in 5D [@lovelock], which provides the most general models for the braneworld scenarios. The entropy of apparent horizon in general Gauss-Bonnet braneworld embedded in a 5D bulk, can be written as [@SheyW2] $$\begin{aligned}
S &=&\frac{3\Omega _{3}{\tilde{r}_{A}}^{2}}{4G_{4}}+\frac{2\Omega _{3}{%
\tilde{r}_{A}}^{3}}{4G_{5}}\times {}_{2}F_{1}\left( \frac{3}{2},\frac{1}{2},%
\frac{5}{2},\Phi _{0}{\tilde{r}_{A}}^{2}\right) \notag \label{ent2} \\
&&+\frac{6{\alpha}\Omega _{3}{\tilde{r}_{A}}^{3}}{G_{5}}\left(
\Phi
_{0}\times {}_{2}F_{1}\left( \frac{3}{2},\frac{1}{2},\frac{5}{2},\Phi _{0}{%
\tilde{r}_{A}}^{2}\right) \right. \notag \\
&&\left. +\frac{\sqrt{1-\Phi _{0}\tilde{r}_{A}^{2}}}{{\tilde{r}_{A}}^{2}}%
\right) ,\end{aligned}$$where $\Phi _{0}=\frac{1}{4{\alpha}}\left( -1+\sqrt{1-\frac{8{%
\alpha}}{\ell ^{2}}}\right) $=constant [@SheyW2], and ${\alpha}$ is the Gauss-Bonnet coefficient with dimension (length)$^{2}$. When ${%
\alpha}\rightarrow 0$ we have $\Phi _{0}=-\ell ^{-2}$ and the above expression reduces to the entropy of warped DGP braneworld presented in ([entDGP]{}) for $n=4$. Expression (\[ent2\]) can be written as [@SheyW2] $$\begin{aligned}
S &=&\frac{3\Omega _{3}}{2G_{4}}\int_{0}^{\tilde{r}_{A}}\tilde{r}_{A}d\tilde{%
r}_{A}+\frac{3\Omega _{3}}{2G_{5}}\int_{0}^{\tilde{r}_{A}}\frac{\tilde{r}%
_{A}^{2}}{\sqrt{1-\Phi _{0}\tilde{r}_{A}^{2}}}d\tilde{r}_{A} \notag \\
&&+\frac{6{\alpha}\Omega _{3}}{G_{5}}\int_{0}^{\tilde{r}_{A}}\frac{%
2-\Phi _{0}\tilde{r}_{A}^{2}}{\sqrt{1-\Phi _{0}\tilde{r}_{A}^{2}}}d\tilde{r}%
_{A},\end{aligned}$$We define the effective area of the apparent horizon corresponding to the above entropy as $$\begin{aligned}
\tilde{A}=4G_{5}S &=&\frac{6G_{5}\Omega _{3}}{G_{4}}\int_{0}^{\tilde{r}_{A}}%
\tilde{r}_{A}d\tilde{r}_{A}+6\Omega _{3}\int_{0}^{\tilde{r}_{A}}\frac{\tilde{%
r}_{A}d\tilde{r}_{A}}{\sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}} \notag \\
&&+24{\alpha}\Omega _{3}\int_{0}^{\tilde{r}_{A}}\frac{2\tilde{r}%
_{A}^{-1}-\Phi _{0}\tilde{r}_{A}}{\sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}}d%
\tilde{r}_{A},\end{aligned}$$and therefore the increase of the effective volume is obtained as $$\begin{aligned}
\frac{d\widetilde{V}}{dt}=\frac{\tilde{r}_{A}}{2}\frac{d\tilde{A}}{dt} &=&%
\frac{3G_{5}\Omega _{3}}{G_{4}}\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}+3\Omega
_{3}\frac{\tilde{r}_{A}^{2}\dot{\tilde{%
r}}_{A} }{\sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}} \notag \\
&&+12{\alpha}\Omega _{3}\frac{2-\Phi _{0}\tilde{r}_{A}^{2}}{\sqrt{%
\tilde{r}_{A}^{-2}-\Phi _{0}}}\dot{\tilde{r}}_{A}. \label{dVt3}\end{aligned}$$Motivated by (\[dVt3\]), we write the number of degrees of freedom on the boundary in general Gauss-Bonnet braneworld as $$\begin{aligned}
N_{\mathrm{sur}} &=&\frac{3\Omega _{3}}{G_{4}}\tilde{r}_{A}^{2}+\frac{%
6\Omega _{3}}{G_{5}}\tilde{r}_{A}^{4}\sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}
\notag \\
&&+\frac{16{\alpha}\Omega _{3}}{G_{5}}\tilde{r}_{A}^{4}\left( \tilde{r}%
_{A}^{-2}+\frac{\Phi _{0}}{2}\right) \sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}.
\label{NsurGB}\end{aligned}$$Substituting Eqs. (\[Nbulk1\]), (\[dVt3\]) and (\[NsurGB\]) into (\[dV2\]) and setting $n=4$, after some mathematic simplification, one obtains $$\begin{aligned}
&&\frac{3G_{5}}{G_{4}}\frac{d}{dt}\left( a^{2}\tilde{r}_{A}^{-2}\right) +6%
\frac{d}{dt}\left( a^{2}\sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}\right) \notag
\\
&&+\frac{d}{dt}\Bigg{\{}16{\alpha}a^{2}\left( \tilde{r}_{A}^{-2}+\frac{%
\Phi _{0}}{2}\right) \sqrt{\tilde{r}_{A}^{-2}-\Phi _{0}}\Bigg{\}} \notag \\
&=&8\pi G_{5}\frac{d}{dt}\left( \rho a^{2}\right) .\end{aligned}$$Integrating, dividing by $a^{2}$ and then using the definition (\[radius\]), we find $$\begin{aligned}
&&\left[ 1+\frac{8}{3}{\alpha}\left( H^{2}+\frac{k}{a^{2}}-\frac{1}{%
2\ell ^{2}}\right) \right] \sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell ^{2}}}
\notag \\
&=&\frac{4\pi G_{5}}{3}\rho -\frac{G_{5}}{2G_{4}}\left( H^{2}+\frac{k}{a^{2}}%
\right) .\end{aligned}$$This is the Friedmann equation governing the evolution of the Universe in general Gauss-Bonnet braneworld with curvature correction terms on the brane and in the bulk. This result is exactly the same as one obtains from the field equation of Gauss-Bonnet braneworld [@kofin]. Here we arrived at the same result by using the novel proposal of [@Shey2]. When $\alpha =0$, the above result reduces to the Friedmann equation of warped DGP model obtained in Eq. (\[FrDGP\]) for $n=4$.
Summery and discussion\[Con\]
=============================
Recently, Padmanabhan [@Pad1] argued that the spacetime dynamics can be considered as an emergent phenomena and the cosmic space is emergent as the cosmic time progresses. An improved version of Padmanabhan proposal which is applicable to a nonflat Universe was found by one of the present author [@Shey2]. In this paper, we extended the study to other gravity theory such as braneworld scenarios. Gravity on the brane does not obey the Einstein theory and therefore the usual area formula for the entropy does not hold on the brane. We have discussed several cases including whether there is or not a Gauss-Bonnet curvature correction term in the bulk and whether there is or not an intrinsic curvature term on the brane. We found that one can always derive the Friedmann equations of FRW Universe with any spacial curvature, by calculating the difference between the horizon degrees of freedom and the bulk degrees of freedom regardless of the existence of the intrinsic curvature term on the brane and the Gauss-Bonnet correction term in the bulk.
The result obtained here in RS II, warped DGP and the general Gauss-Bonnet braneworld scenarios further supports the novel idea of Padmanabhan ([dV1]{}) and its extension as (\[dV2\]), and show that this approach is powerful enough to extract the dynamical equations describing the evolution of the Universe in other gravity theories with any spacial curvature.
.
[99]{} A. D. Sakharov, Sov. Phys. Dokl. **12** (1968) 1040 \[Dokl. Akad. Nauk Ser. Fiz. 177 (1967) 70\] \[Sov. Phys. Usp. 34 (1991) 394\] \[Gen. Rel. Grav. 32 (2000) 365\].
R. G. Cai and S. P. Kim, JHEP **0502**, 050 (2005).
A. Sheykhi, B. Wang and R. G. Cai, Nucl. Phys. B **779** (2007)1.
A. Sheykhi, B. Wang and R. G. Cai, Phys. Rev. D **76** (2007) 023515;A. Sheykhi, JCAP **05**, 019 (2009).
A. Sheykhi, Class. Quantum Gravit. **27**, 025007 (2010);A. Sheykhi, Eur. Phys. J. C **69**, 265 (2010).
T. Padmanabhan, Rep. Prog. Phys. **73**, 046901 (2010)
E. Verlinde, JHEP **1104**, 029 (2011).
R.G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D **81**, 061501 (2010).
R.G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D **81**, 084012 (2010); Y.S. Myung, Y.W Kim, Phys. Rev. D **81**, 105012 (2010); R. Banerjee, B. R. Majhi, Phys. Rev. D **81**, 124006 (2010); S.W. Wei, Y. X. Liu, Y. Q. Wang, Commun. Theor. Phys. **56**, 455 (2011); Y. X. Liu, Y. Q. Wang, S. W. Wei, Class. Quantum Gravit. **27**, 185002 (2010); R.A. Konoplya, Eur. Phys. J. C **69**, 555 (2010); H. Wei, Phys. Lett. B **692**, 167 (2010).
C. M. Ho, D. Minic and Y. J. Ng, Phys. Lett. B **693**, 567 (2010);V.V. Kiselev, S.A. Timofeev Mod. Phys. Lett. A **26**, (2011) 109;W. Gu, M. Li and R. X. Miao, Sci.China G **54**, 1915 (2011), arXiv:1011.3419;R. X. Miao, J. Meng and M. Li, Sci. China G **55**, 375 (2012), arXiv:1102.1166.
A. Sheykhi, Phys. Rev. D **81**, 104011 (2010).
Y. Ling and J.P. Wu, JCAP **1008**, (2010), 017.
L. Modesto, A. Randono, arXiv:1003.1998; L. Smolin, arXiv:1001.3668;X. Li, Z. Chang, arXiv:1005.1169.
Y.F. Cai, J. Liu, H. Li, Phys. Lett. B **690**, (2010) 213; M. Li and Y. Wang, Phys. Lett. B **687**, 243 (2010).
S. H. Hendi and A. Sheykhi, Phys. Rev. D **83**, 084012 (2011); A. Sheykhi and S. H. Hendi, Phys. Rev. D **84**, 044023 (2011); S. H. Hendi and A. Sheykhi, Int. J. Theor. Phys. **51**, 1125 (2012) ;A. Sheykhi and Z. Teimoori, Gen Relativ Gravit. **44**, 1129 (2012);A. Sheykhi, Int. J. Theor. Phys. **51**, 185 (2012);A. Sheykhi, K. Rezazadeh Sarab, JCAP **10**, 012 (2012).
T. Padmanabhan, arXiv:1206.4916.
R. G. Cai, JHEP **11**, 016 (2012).
K. Yang, Y. X. Liu and Y. Q. Wang, Phys. Rev. D **86**, 104013 (2012).
A. Sheykhi, M.H. Dehghani, S.E. Hosseini , JCAP **04**, 038 (2013).
F. Q. Tu and Y. X. Chen, JCAP, 05 (2013)024 ;Y. Ling and W. J. Pan, arXiv:1304.0220. A. Sheykhi, Phys. Rev. D 87, 061501(R) (2013)
F. F. Yuan, Y. C. Huang, arXiv:1304.7949 ;M. Eune and W. Kim, arXiv:1305.6688. L. Randall, R. Sundrum, Phys. Rev. Lett. **83**, 3370 (1999);L. Randall, R. Sundrum, Phys. Rev. Lett. **83**, 4690 (1999).
P. Binetruy, C. Deffayet, and D. Langlois, Nucl. Phys. B **565**, 269 (2000).
G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B **485**, 208 (2000).
G. Dvali, G. Gabadadze, Phys.Rev. D **63**, 065007 (2001).
C. Deffayet, Phys. Lett. B **502**, 199 (2001).
D. Lovelock, J. Math. Phys. **12**, 498 (1971).
G. Kofinas, R. Maartens, E. Papantonopoulos, JHEP **0310**, 066 (2003).
[^1]: asheykhi@shirazu.ac.ir
[^2]: mhd@shirazu.ac.ir
|
---
abstract: 'In this paper we present the fit to the recent high-statistical KLOE data on the $\phi\to\eta\pi^0\gamma$ decay. This decay mainly goes through the $a_0\gamma$ intermediate state. The obtained results differ from those of the previous fits: data prefer a high $a_0$ mass and a considerably large $a_0$ coupling to the $K\bar{K}$.'
address: ' Laboratory of Theoretical Physics, Sobolev Institute for Mathematics, Novosibirsk, 630090'
author:
- 'N.N. Achasov [^1] and A.V. Kiselev [^2]'
title: 'The new analysis of the KLOE data on the $\phi\to\eta\pi^0\gamma$ decay'
---
$ $\
Introduction
============
The lightest scalar mesons $a_0(980)$ and $f_0(980)$, discovered more than thirty years ago, became the hard problem for the naive quark-antiquark ($q\bar q$) model from the outset. Really, on the one hand the almost exact degeneration of the masses of the isovector $a_0(980)$ and isoscalar $f_0(980)$ states revealed seemingly the structure similar to the structure of the vector $\rho$ and $\omega$ mesons , and on the other hand the strong coupling of $f_0(980)$ with the $K\bar K$ channel pointed unambiguously to a considerable part of the strange quark pair $s\bar s$ in the wave function of $f_0(980)$. It was noted late in the 1970s that in the MIT bag model there are light four-quark scalar states and suggested that $a_0(980)$ and $f_0(980)$ might be these states [@jaffe]. From that time $a_0(980)$ and $f_0(980)$ resonances came into beloved children of the light quark spectroscopy, see, for example, Refs. [@montanet; @achasov-84; @conf].
Ten years later there was proposed in Ref.[@achasov-89] to study radiative $\phi$ decays $\phi\to
a_0\gamma\to\eta\pi^0\gamma$ and $\phi\to
f_0\gamma\to\pi^0\pi^0\gamma$ to solve the puzzle of the lightest scalar mesons. Over the next ten years before the experiments (1998), this question was examined from different points of view [@bramon; @achasov-97; @achasov-97a; @achasov-97b; @achasov-98].
Now these decays have been studied not only theoretically but also experimentally. The first measurements have been reported by the SND [@snd-1; @snd-2; @snd-fit; @snd-ivan] and CMD-2 [@cmd] Collaborations which obtain the following branching ratios $$Br(\phi\to\gamma\pi^0\eta)=(8.8\pm 1.4\pm0.9)\times 10^{-5}\
\mbox{\cite{snd-fit}},$$ $$Br(\phi\to\gamma\pi^0\pi^0)=
(12.21\pm0.98\pm0.61)\times 10^{-5}\ \mbox{\cite{snd-ivan}}$$ $$Br(\phi\to\gamma\pi^0\eta)=(9.0\pm 2.4\pm 1.0)\times 10^{-5},$$ $$Br(\phi\to\gamma\pi^0\pi^0)=(9.2\pm0.8\pm0.6)\times 10^{-5}\
\mbox{\cite{cmd}}.$$ More recently the KLOE Collaboration has measured [@publ; @pi0publ] $$Br(\phi\to\gamma\pi^0\eta)=(8.51\pm0.51\pm0.57)\times 10^{-5}\
\mbox{in}\ \eta\to\gamma\gamma\ \mbox{\cite{publ}},$$ $$Br(\phi\to\gamma\pi^0\eta)=(7.96\pm0.60\pm0.40)\times 10^{-5}\
\mbox{in}\ \eta\to\pi^+\pi^-\pi^0\ \mbox{\cite{publ}},$$ $$Br(\phi\to\gamma\pi^0\pi^0)= (10.9\pm0.3\pm0.5)\times 10^{-5}\
\mbox{\cite{pi0publ}},$$ in agreement with the Novosibirsk data [@snd-fit; @snd-ivan; @cmd] but with a considerably smaller error.
In this work we present the new analysis of the recent KLOE data on the $\phi\to\eta\pi^0\gamma$ decay [@publ; @kloe]. In contradistinction to [@publ], we
1\) treat the $a_0$ mass $m_{a_0}$ as a free parameter of the fit;
2\) fit the phase $\delta $ of the interference between $\phi\to
a_0\gamma\to\eta\pi^0\gamma$ (signal) and $\phi\to\rho^0\pi^0\to\eta \pi^0\gamma$ (background) reactions;
3\) use new more precise experimental values of the input parameters.
All formulas for the $\phi\to(a_0\gamma+\rho^0\pi^0)\to\eta
\pi^0\gamma$ reaction taking the background into account are shown in Sec.\[sf\]. The results of the 4 different fits are presented in Sec.\[sr\]. A brief summary is given in Sec.\[sc\].
The formalism of the and reactions {#sf}
===================================
In Ref.[@a0f0] was shown that the process $\phi\to
a_0\gamma \to\eta\pi^0\gamma$ dominates in the $\phi\to
\eta\pi^0\gamma$ decay (see also [@achasov-89; @achasov-97], where it was predicted in four-quark model). This was confirmed in [@publ; @kloe]. Nevertheless, the main background process $\phi\to\rho\pi^0\to\eta\pi^0\gamma$ should be taken into account also (see [@a0f0; @publ]).
The amplitude of the background process $\phi(p)\to\pi^0\rho^0\to\gamma(q)\pi^0(k_1)\eta(k_2)$ is [@a0f0]: $$M_B=\frac{g_{\phi\rho\pi}g_{\rho\eta\gamma}}{D_{\rho}(p-k_1)}
\phi_{\alpha}k_{1\mu}p_{\nu}\epsilon_{\delta}(p-k_1)_{\omega}q_{\epsilon}
\epsilon_{\alpha\beta\mu\nu}\epsilon_{\beta\delta\omega\epsilon}.$$
According to the one-loop mechanism of the decay $\phi\to
K^+K^-\to\gamma a_0$, suggested in Ref.[@achasov-89], the amplitude of the signal $\phi\to\gamma a_0\to\gamma\pi^0\eta$ has the form:
$$M_a=g(m)\frac{g_{a_0K^+K^-}g_{a_0\pi\eta}}{D_{a_0}(m)}\bigg(
(\phi\epsilon)- \frac{(\phi q)(\epsilon p)}{(pq)} \bigg)
\label{a0signal}\,,$$
where $m^2=(k_1+k_2)^2$, $\phi_{\alpha}$ and $\epsilon_{\mu}$ are the polarization vectors of $\phi$ meson and photon, the forms of $g_R(m)$ and $g(m)=g_R(m)/g_{RK^+K^-}$ everywhere over the $m$ region are in Refs. [@achasov-89] and [@achasov-01a] respectively:
For $m<2m_{K^+}$
$$\begin{aligned}
&&g(m)=\frac{e}{2(2\pi)^2}g_{\phi K^+K^-}\Biggl\{
1+\frac{1-\rho^2(m^2)}{\rho^2(m^2_{\phi})-\rho^2(m^2)}\times\nonumber\\
&&\Biggl[2|\rho(m^2)|\arctan\frac{1}{|\rho(m^2)|}
-\rho(m^2_{\phi})\lambda(m^2_{\phi})+i\pi\rho(m^2_{\phi})-\nonumber\\
&&-(1-\rho^2(m^2_{\phi}))\Biggl(\frac{1}{4}(\pi+
i\lambda(m^2_{\phi}))^2- \nonumber\\
&&-\Biggl(\arctan\frac{1}{|\rho(m^2)|}\Biggr)^2
\Biggr)\Biggr]\Biggr\},\end{aligned}$$
where $$\rho(m^2)=\sqrt{1-\frac{4m_{K^+}^2}{m^2}}\,\,;\qquad
\lambda(m^2)=\ln\frac{1+\rho(m^2)}{1-\rho(m^2)}\,\,;\qquad
\frac{e^2}{4\pi}=\alpha=\frac{1}{137}\,\,.$$
For $m\geq 2m_{K^+}$ $$\begin{aligned}
&&g(m)=\frac{e}{2(2\pi)^2}g_{\phi K^+K^-}\Biggl\{
1+\frac{1-\rho^2(m^2)}{\rho^2(m^2_{\phi})-\rho^2(m^2)}\times\nonumber\\
&&\times\Biggl[\rho(m^2)(\lambda(m^2)-i\pi)-
\rho(m^2_{\phi})(\lambda(m^2_{\phi})-i\pi)-\nonumber\\
&&\frac{1}{4}(1-\rho^2(m^2_{\phi}))
\Biggl((\pi+i\lambda(m^2_{\phi}))^2-
(\pi+i\lambda(m^2))^2\Biggr)\Biggr]\Biggr\}.\end{aligned}$$
The mass spectrum is $$\frac{d\Gamma(\phi\to\gamma\pi^0\eta,m)}{dm}=\frac{d\Gamma_{a_0}(m)}{dm}+
\frac{d\Gamma_{back}(m)}{dm}+ \frac{d\Gamma_{int}(m)}{dm}\,,$$ where the mass spectrum for the signal is $$\frac{d\Gamma_{a_0}(m)}{dm}=\frac{2}{\pi}\frac{m^2\Gamma(\phi\to\gamma
a_0,m)\Gamma(a_0\to\pi^0\eta,m)}{|D_{a_0}(m)|^2}=
\frac{2|g(m)|^2p_{\eta\pi}(m_{\phi}^2-m^2)}
{3(4\pi)^3m_{\phi}^3}\bigg|\frac{g_{a_0K^+K^-}g_{a_0\pi\eta}}{D_{a_0}(m)}\bigg|^2\,.
\label{spectruma0}$$
The mass spectrum for the background process $\phi\to\pi^0\rho\to\gamma\pi^0\eta$ is [@a0f0]:
$$\frac{d\Gamma_{back}(m)}{dm}=\frac{(m_{\phi}^2-m^2)p_{\pi\eta}
}{128\pi^3m_{\phi}^3}\int_{-1}^{1}dxA_{back}(m,x)\,,$$
where $$\begin{aligned}
&&A_{back}(m,x)=\frac{1}{3}\sum|M_B|^2= \nonumber \\
&&=\frac{1}{24}(m_{\eta}^4m_{\pi}^4+2m^2m_{\eta}^2m_{\pi}^2
\tilde{m_{\rho}}^2-2m_{\eta}^4m_{\pi}^2\tilde{m_{\rho}}^2-2m_{\eta}^2m_{\pi}^4
\tilde{m_{\rho}}^2+\nonumber \\ &&2m^4\tilde{m_{\rho}}^4-
2m^2m_{\eta}^2\tilde{m_{\rho}}^4+m_{\eta}^4\tilde{m_{\rho}}^4
-2m^2m_{\pi}^2\tilde{m_{\rho}}^4+4m_{\eta}^2m_{\pi}^2\tilde{m_{\rho}}^4
+m_{\pi}^4\tilde{m_{\rho}}^4+\nonumber \\
&&2m^2\tilde{m_{\rho}}^6-
2m_{\eta}^2\tilde{m_{\rho}}^6-2m_{\pi}^2\tilde{m_{\rho}}^6+
\tilde{m_{\rho}}^8-2m_{\eta}^4m_{\pi}^2m_{\phi}^2-
2m^2m_{\eta}^2m_{\phi}^2\tilde{m_{\rho}}^2+\nonumber \\
&&2m_{\eta}^2m_{\pi}^2m_{\phi}^2\tilde{m_{\rho}}^2-
2m^2m_{\phi}^2\tilde{m_{\rho}}^4+
2m_{\eta}^2m_{\phi}^2\tilde{m_{\rho}}^4-
2m_{\phi}^2\tilde{m_{\rho}}^6+ m_{\eta}^4m_{\phi}^4+
m_{\phi}^4\tilde{m_{\rho}}^4)\times\nonumber \\
&&\bigg|\frac{g_{\phi\rho\pi}g_{\rho\eta\gamma}}
{D_{\rho}(\tilde{m_{\rho}})}\bigg|^2\,, \label{Aback}\end{aligned}$$ and $$\begin{aligned}
&&\tilde{m_{\rho}}^2=m_{\eta}^2+\frac{(m^2+m_{\eta}^2-m_{\pi}^2)(m_{\phi}^2-
m^2)}{2m^2}-\frac{(m_{\phi}^2-m^2)x}{m}p_{\pi\eta}\nonumber \\
&&p_{\pi\eta}=\frac{\sqrt{(m^2-(m_{\eta}-m_{\pi})^2)
(m^2-(m_{\eta}+m_{\pi})^2)}}{2m}\,.\end{aligned}$$
Note that there is a misprint in Eq.(6) of Ref.[@a0f0], which describes $A_{back}(m,x)$: the 7th term in the brackets “$+2m_{\eta}^4\tilde{m_{\rho}}^4$” should be replaced by “$+m_{\eta}^4\tilde{m_{\rho}}^4$”, as above in Eq.(\[Aback\]). Emphasize that all evaluations in Ref.[@a0f0] were done with the correct formula.
The term of the interference between the signal and the background processes is written in the following way:
$$\frac{d\Gamma_{int}(m)}{dm}=\frac{(m_{\phi}^2-m^2)p_{\pi\eta}
}{128\pi^3m_{\phi}^3} \int_{-1}^{1}dxA_{int}(m,x)\,,$$
where $$\begin{aligned}
&&A_{int}(m,x)=\frac{2}{3}Re\sum M_aM_B^*=
\frac{1}{3}\left((m^2-m_{\phi}^2)\tilde{m_{\rho}}^2+
\frac{m_{\phi}^2(\tilde{m_{\rho}}^2-m_{\eta}^2)^2}{m_{\phi}^2-m^2}\right)
\times\nonumber\\ &&Re\{\frac{e^{i\delta
}g(m)g_{a_0K^+K^-}g_{a_0\pi\eta}g_{\phi\rho\pi}g_{\rho\eta\gamma}}
{D^*_{\rho}(\tilde{m_{\rho}})D_{a_0}(m)}\}\,.\end{aligned}$$
Note that the phase $\delta$ isn’t taken into account in [@a0f0]. The inverse propagator of the scalar meson R ($a_0$ in our case), is presented in Refs. [@achasov-80; @z_phys; @achasov-89; @achasov-97]:
$$\label{propagator} D_R(m)=m_R^2-m^2+\sum_{ab}[Re
\Pi_R^{ab}(m_R^2)-\Pi_R^{ab}(m^2)],$$
where $\sum_{ab}[Re \Pi_R^{ab}(m_R^2)-
\Pi_R^{ab}(m^2)]=Re\Pi_R(m_R^2)- \Pi_R(m^2)$ takes into account the finite width corrections of the resonance which are the one loop contribution to the self-energy of the $R$ resonance from the two-particle intermediate $ab$ states.
For the pseudoscalar $ab$ mesons and $m_a\geq m_b,\ m\geq m_+$ one has [@achasov-80; @z_phys; @achasov-97b; @achasov-84; @achasov-95][^3]:
$$\begin{aligned}
\label{polarisator}
&&\Pi^{ab}_R(m^2)=\frac{g^2_{Rab}}{16\pi}\left[\frac{m_+m_-}{\pi
m^2}\ln \frac{m_b}{m_a}+\right.\nonumber\\
&&\left.+\rho_{ab}\left(i+\frac{1}{\pi}\ln\frac{\sqrt{m^2-m_-^2}-
\sqrt{m^2-m_+^2}}{\sqrt{m^2-m_-^2}+\sqrt{m^2-m_+^2}}\right)\right]\end{aligned}$$
For $m_-\leq m<m_+$ $$\begin{aligned}
&&\Pi^{ab}_{R}(m^2)=\frac{g^2_{Rab}}{16\pi}\left[\frac{m_+m_-}{\pi
m^2}\ln \frac{m_b}{m_a}-|\rho_{ab}(m)|+\right.\nonumber\\
&&\left.+\frac{2}{\pi}|\rho_{ab}(m)
|\arctan\frac{\sqrt{m_+^2-m^2}}{\sqrt{m^2-m_-^2}}\right].\end{aligned}$$ For $m<m_-$ $$\begin{aligned}
&&\Pi^{ab}_{R}(m^2)=\frac{g^2_{Rab}}{16\pi}\left[\frac{m_+m_-}{\pi
m^2}\ln \frac{m_b}{m_a}-\right.\nonumber\\
&&\left.-\frac{1}{\pi}\rho_{ab}(m)\ln\frac{\sqrt{m_+^2-m^2}-
\sqrt{m_-^2-m^2}}{\sqrt{m_+^2-m^2}+\sqrt{m_-^2-m^2}}\right].\end{aligned}$$ and $$\label{rho-ab}
\rho_{ab}(m)=\sqrt{(1-\frac{m_+^2}{m^2})(1-\frac{m_-^2}{m^2})}\,\,,\qquad
m_{\pm}=m_a\pm m_b$$ The constants $g_{Rab}$ are related to the width $$\Gamma(R\to ab,m)=\frac{g_{Rab}^2}{16\pi m}\rho_{ab}(m).
\label{f0pipi}$$
In our case we take into account intermediate states $ab=\eta\pi^0,\ K\bar{K}$ and $\eta '\pi^0$:
$$\Pi_{a_0}=\Pi_{a_0}^{\eta\pi^0}+\Pi_{a_0}^{K^+K^-}+
\Pi_{a_0}^{K^0\bar{K^0}}+\Pi_{a_0}^{\eta' \pi^0},$$
$g_{a_0K^+K^-}=-g_{a_0K^0\bar{K^0}}$. Note that the $\eta '\pi^0$ contribution is of small importance due to the high threshold. Even fitting with $|g_{a_0\eta '\pi^0}| =0$ changes the results less than 10% of their errors. We set $|g_{a_0\eta
'\pi^0}| =|1.13\,g_{a_0K^+K^-}|$ according to the four-quark model, see [@achasov-89], but this is practically the same as the 2-quark model prediction $|g_{a_0\eta '\pi^0}|
=|1.2\,g_{a_0K^+K^-}|$, see [@achasov-89].
The inverse propagator of the $\rho$ meson has the following expression $$D_{\rho}(m)=m_{\rho}^2-m^2-im^2\frac{g^2_{\rho\pi\pi}}{48\pi}
\bigg(1-\frac{4m_{\pi}^2}{m^2}\bigg)^{3/2}\,.$$
The coupling constants $g_{\phi K^+K^-}=4.376\pm 0.074$ and $g_{\phi\rho\pi}=0.814\pm0.018$ GeV$^{-1}$ are taken from the new most precise measurement Ref.[@sndphi]. Note that in Ref. [@publ; @a0f0] the value $g_{\phi K^+K^-}=4.59$ was obtained using the [@pdg] data. The coupling constant $g_{\rho\eta\gamma}=0.56\pm 0.05$ GeV$^{-1}$ is obtained from the data of Ref.[@pdg-2002] with the help of the expression $$\Gamma(\rho\to\eta\gamma)=\frac{g_{\rho\eta\gamma}^2}{96\pi
m_{\rho}^3} (m_{\rho}^2-m_{\eta}^2)^3.$$
=12 cm
Results {#sr}
=======
The KLOE data on the $\phi\to\eta\pi^0\gamma$ decay may be found in Table 5 of Ref.[@kloe] (see also Fig.\[fig1\]). Note that as in Refs.[@publ; @kloe], we don’t fit 1st,10th and 27th points of this table (cross points in the Fig. \[fig1\]). Emphasize that the 10th (1.014 GeV) and 27th (1.019 GeV) points are obvious artifacts because the mass spectrum behaviour on the right slope of the resonance has the form the (photon energy)$^3$ according to gauge invariance.
In the experiment the whole mass region ($m_{\eta}+m_{\pi^0},m_{\phi}$) is divided into some number of bins. Experimenters measure the average value ${\bar{B}_i}$ (“i” is the number of bin) of $dBr(\phi\to\eta\pi^0\gamma)/dm$ around each i-th bin:
$$\bar{B}_i=\frac{1}{m_{i+1}-m_i }\int ^{m_{i+1}}_{m_i}
dBr(\phi\to\eta\pi^0\gamma)/dm,$$
In this case one should define $\chi^2$ function as:
$$\chi^2=\sum_i \frac{(\bar{B}_i^{th}-\bar{B}_i^{exp})^2}{\sigma
_i^2},$$
where $\bar{B}_i^{exp}$ are the experimental results, $\sigma _i$ are the experimental errors, and
$$\bar{B}_i^{th}=\frac{1}{m_{i+1}-m_i }\int ^{m_{i+1}}_{m_i}
dBr^{th}(\phi\to\eta\pi^0\gamma)/dm$$
($dBr^{th}(\phi\to\eta\pi^0\gamma)/dm$ is the theoretical curve).
The free parameters of the fit are $m_{a_0},
g_{a_0K^+K^-}^2/4\pi$, the phase $\delta $ (we assume it is constant) and the ratio $g_{a_0\eta\pi}/g_{a_0K^+K^-}$. The results are shown in Table 1 (fit 1) [^4].
The quality of the fit is good. The phase $\delta $ is consistent with zero, so we make a fit with $\delta =0$ (fit 2 in Table 1).
To check the correctness of treating the phase $\delta $ as a constant, we have done a fit with $\delta$ taken in the form $\delta(m)=bp_{\eta\pi}(m)$ (the phase of the elastic background in $\eta\pi^0$ scattering may have such behaviour), and found out that the constant b=$(2.8\pm 3.2)$ GeV$^{-1}$ is also consistent with zero. Change of the other values is not principal.
[|c|c|c|c|c|c|]{}\
fit & $m_{a_0}$, MeV & $\frac{g_{a_0K^+K^-}^2}{4\pi}$, GeV$^2$ & $g_{a_0\eta\pi}/g_{a_0K^+K^-}$ & $\delta $, $^{\circ}$ & $\chi^2/n.d.f.$\
1 & $1003^{+32}_{-13}$ & $0.82^{+0.81}_{-0.27}$ & $1.06^{+0.20}_{-0.27}$ & $27 \pm 29$ & 24.2/20\
2 & $995^{+22}_{-8}$ & $ \hspace{1mm} 0.65^{+0.42}_{-0.18}
\hspace{1mm}$ & $1.17^{+0.17}_{-0.24}$ & 0 & 25.2/21\
3 & $994^{+22}_{-8}$ & $0.62^{+0.4}_{-0.17}$ & $1.21^{+0.17}_{-0.24}$ & $21 \pm 30$ & 16.3/19\
4 & $992^{+14}_{-7}$ & $0.55^{+0.27}_{-0.13}$ & $1.26^{+0.16}_{-0.2}$ & $ 0 $ & 16.9/20\
Since the discrepancy between fits and the experimental point number 26 (0.999 GeV) in the Table 5 of Ref.[@kloe] (the box point in Fig.1) is about 3 standard deviations (i.e. this point may be an artifact also), we make another fit without this point (fit 3). The phase $\delta$ is again consistent with zero, so we make a fit without it (fit 4).
In Table 2 we present the results on the total branching ratio $\mbox{Br}(\phi\to(a_0\gamma+\rho\pi^0)\to\eta\pi^0\gamma)$, the signal contribution $\mbox{Br}(\phi\to
a_0\gamma\to\eta\pi^0\gamma)$, $\Gamma_{a_0\eta\pi^0}\equiv
\Gamma(a_0\to\eta\pi^0,m_{a_0})=g_{a_0\eta\pi}^2 \rho_{\eta\pi^0 }
(m_{a_0})/(16\pi m_{a_0})$ and the ratio $R=g^2_{f_0K^+K^-}/g^2_{a_0K^+K^-}$. The last is obtained using the Ref.[@pi0publ] value $g^2_{f_0K^+K^-}/(4\pi)=2.79\pm0.12$ GeV$^2$. The branching ratio of the background $\mbox{Br}(\phi\to\rho\pi^0\to\eta\pi^0\gamma)$ accounts for $(0.5\pm 0.1) \times 10^{-5}$.
[|c|c|c|c|c|]{}\
fit & $\Gamma_{a_0\eta\pi^0}$, MeV & R & $10^5\times
\mbox{Br}(\phi\to (a_0\gamma + \rho\pi ^0) \to\eta\pi ^0\gamma)$ & $10^5\times \mbox{Br}(\phi\to a_0\gamma\to\eta\pi^0\gamma)$\
1 & $153^{+22}_{-17}$ & $3.4\pm 1.7$ & $7.6\pm 0.4$ & $7.3\pm 0.4$\
2 & $148^{+17}_{-15}$ & $4.3\pm 1.7$ & $7.6\pm 0.4$ & $7.1\pm 0.4$\
3 & $149^{+19}_{-16}$ & $4.5\pm 1.7$ & $7.6\pm 0.4$ & $7.2\pm 0.4$\
4 &$146^{+17}_{-15}$& $5.0\pm 1.6$ & $7.6\pm 0.4$ & $7.1\pm 0.4$\
Conclusion {#sc}
==========
Note that the obtained value of the ratio $g_{a_0\eta\pi}/g_{a_0K^+K^-}$ doesn’t contradict the first predictions based on the four-quark model of the $a_0$: $g_{a_0\eta\pi}/g_{a_0K^+K^-}\approx 0.85$ [@achasov-89][^5]. But even if $g_{a_0\eta\pi}/g_{a_0K^+K^-}$ deviates from $0.85$, there is no tragedy, because those variant of the four-quark model is rather rough, it is considered as a guide.
For all fits the obtained value R differs from the [@publ] value $R=7.0\pm0.7$. So the conclusion that the constant $g^2_{a_0K^+K^-}/(4\pi)$ is small, obtained in [@publ; @kloe] ($g^2_{a_0K^+K^-}/(4\pi)=0.4\pm 0.04$ GeV$^2$), is the result of the parameters restrictions, especially fixing $m_{a_0}$ at the PDG-2000 value 984.8 MeV. Note that a high $a_0$ mass is also needed to describe $\gamma\gamma\to\eta\pi^0$ experiment, see [@lmas].
There should be no confusion due to the large $a_0$ width. In the peripheral production of the $a_0$ (for example, in the reaction $\pi^-p\to\eta\pi^0 n$) the mass spectrum is given by the relation
$$\frac{dN_{\eta\pi^0}}{dm}\sim S_{per}(m)=\frac{2m^2}{\pi}
\frac{\Gamma(a_0\to\eta\pi^0,m)}{|D_{a_0}(m)|^2}.$$
The effective (visible) width of this distribution is much less then the nominal width $\Gamma_{a_0\eta\pi^0}$. For example, for the fit 1 results (Table 1) the effective width is $\sim 50$ MeV, see Fig. \[fig2\].
=12 cm
As it is noted in Ref.[@our], there is no tragedy with the relation between branching ratios of $a_0$ and $f_0$ production in $\phi$ radiative decays. The early predictions [@achasov-89] are based on the one-loop mechanism $\phi\to K^+K^-\to
a_0\gamma\to\eta\pi^0\gamma$ and $\phi\to K^+K^-\to
f_0\gamma\to\pi\pi\gamma$ at $m_{a_0}=980$ MeV, $m_{f_0}=975$ MeV and $g_{a_0K^+K^-}=g_{f_0K^+K^-}$[^6], that leads to $\mbox{Br}(\phi\to
a_0\gamma\to\eta\pi^0\gamma)\approx \mbox{Br}(\phi\to
f_0\gamma\to\pi\pi\gamma)$. But it is shown in Ref. [@achasov-97], that the relation between branching ratios of $a_0$ and $f_0$ production in $\phi$ radiative decays essentially depends on a $a_0-f_0$ mass splitting. This strong mass dependence is the result of gauge invariance, the (photon energy)$^3$ law on the right slope of the resonance. Our present analysis confirms this conclusion. Note that a noticeable deviation from the naive four-quark model equality $g_{a_0K^+K^-}=g_{f_0K^+K^-}$ is not crucial. What is more important is the mechanism of the production of the $a_0$ and $f_0$ through the charged kaon loop, i.e. the four-quark transition. As is shown in Ref.[@conf], this gives strong evidence in favor of the four-quark model of the $a_0(f_0)$.
Note that the constant $g^2_{f_0K^+K^-}/(4\pi)$ also can differ a lot from those obtained in [@pi0publ]. The point is that the extraction of this constant is very model dependent. For example, fitting with taking into account the mixing of the resonances can decrease the value of $g^2_{f_0K^+K^-}/(4\pi)$ considerably. For instance, fitting the data of [@snd-ivan] without mixing one has $g^2_{f_0K^+K^-}/(4\pi)=2.47^{+0.37}_{-0.51}$ GeV$^2$ [@snd-ivan], while fitting with taking the mixing into account gives $g^2_{f_0K^+K^-}/(4\pi)=1.29\pm 0.017$ GeV$^2$ [@a0f0]. Note also that in [@pi0publ] the phase $\delta_B$ of the background is taken from the work [@a0f0], where it is obtained by the simultaneous fitting the $m_{\pi^0\pi^0}$ spectrum and the phase $\delta_0$ of the $\pi\pi$ scattering, taking into account the mixing of the resonances. In [@pi0publ] the mixing isn’t taken into account, so additional phase dealing with it is omitted.
Acknowledgements
================
We thank C. Bini very much for providing the useful information, discussions and kind contacts. This work was supported in part by RFBR, Grant No 02-02-16061.
[99]{} R.L. Jaffe, Phys. Rev. D [**15**]{}, 267, 281 (1977). L. Montanet, Rep. Prog. Phys. [**46**]{}, 337 (1983); F.E. Close, Rep. Prog. Phys. [**51**]{}, 833 (1988); L.G. Landsberg, Usp. Fiz. Nauk. [**160**]{}, 3 (1990) \[Sov. Phys. Usp. [**33**]{}, 1 (1990)\]; M.R. Pennington, HADRON ’95, Proceedings of the 6h International Conference on Hadron Spectroscopy, Manchester, UK, 10th-14th July 1995, Eds. M.C. Birse, G.D. Lafferty, and J.A. McGovern, page 3, World Scientific, Singapore; T. Barnes, HADRON SPECTROSCOPY, VII International Conference, Upton, NY August 1997, Eds. S.U. Chung, H.J. Willutzki, AIP CONFERENCE PROCEEDINGS 432, page 3; C. Amsler, Rev.Mod.Phys. [**70**]{}, 1293 (1998); Muneyuki Ishida, Shin Ishida, and Taku Ishida, Prog.Theor.Phys. [**99**]{}, 1031 (1998); S. Godfrey and J. Napolitano, Rev. Mod. Phys. [**71**]{}, 1411 (1999); P. Minkowski and W. Ochs, Eur. Phys. J. C [**9**]{}, 283 (1999); O. Black, A. Fariborz, F. Sannino, and J. Schechter, Phys.Rev. D [**59**]{}, 074026 (1999); S. Spanier and N.A. Törnqvist, Eur. Phys. J. C [**15**]{}, 437 (2000).
N.N. Achasov, S.A. Devyanin and G.N. Shestakov, Usp. Fiz. Nauk. [**142**]{}, 361 (1984) \[Sov. Phys. Usp. [**27**]{}, 161 (1984)\]; N.N. Achasov and G.N. Shestakov, Usp. Fiz. Nauk. [**161**]{} (6), 53 (1991)\[Sov. Phys. Usp. [**34**]{} (6), 471 (1991)\]; N.N. Achasov, Nucl. Phys. B (Proc. Suppl.) [**21**]{}, 189 (1991); N.N. Achasov, Usp. Fiz. Nauk. [**168**]{}, 1257 (1998) \[ Phys. Usp. [**41**]{}, 1149 (1998)\]. N.N. Achasov, Nucl. Phys. A [**675**]{}, 279c (2000). N.N. Achasov (Plenary session talk), HADRON SPECTROSCOPY, Ninth International Conference on Hadron Spectroscopy, Protvino, Russia 25 August – 1 September 2001, HADRON 2001, edited by D. Amelin and A.M. Zaitsev, AIP CONFERENCE PROCEEDINGS, Vol. 619, p. 112; N.N. Achasov, hep-ph/0201299.
N.N. Achasov and V.N. Ivanchenko, Nucl. Phys. B [**315**]{}, 465 (1989); Preprint INP 87-129 (1987) Novosibirsk.
A. Bramon, A. Grau, and G. Pancheri, Phys.Lett. B [**289**]{}, 97 (1992); F.E. Close, N. Isgur, and S. Kumano, Nucl. Phys. B [**389**]{}, 513 (1993); J.L. Lucio, M. Napsuciale, Phys. Lett. B [**331**]{}, 418 (1994).
N.N. Achasov and V.V. Gubin, Phys. Rev. D [**56**]{}, 4084 (1997); Yad. Fiz. [**61**]{}, 274 (1998) \[Phys. At. Nucl. [**61**]{}, 224 (1998)\]. N.N. Achasov, V.V. Gubin, and V.I. Shevchenko, Phys. Rev. D [**56**]{}, 203 (1997); Int. J. Mod. Phys. A [**12**]{}, 5019 (1997); Yad. Fiz. [**60**]{}, 89 (1997) \[Phys. At. Nucl. [**60**]{}, 81 (1997)\]. N.N. Achasov, V.V. Gubin, and E.P. Solodov, Phys. Rev. D [**55**]{}, 2672 (1997); Yad. Fiz. [**60**]{}, 1279 (1997) \[Phys. At. Nucl. [**60**]{}, 1152 (1997)\]. N.N. Achasov and V.V. Gubin, Phys. Rev. D [**57**]{}, 1987 (1998); Yad. Fiz. [**61**]{}, 1473 (1998) \[Phys. At. Nucl. [**61**]{}, 1367 (1998)\]. M.N. Achasov et al., Phys. Lett. B438, 441 (1998). M.N. Achasov et al., Phys. Lett. B440, 442 (1998).
M.N. Achasov et al., Phys. Lett. B479, 53 (2000). M.N.Achasov et al., Phys. Lett. B485, 349 (2000). R.R. Akhmetshin et al., Phys. Lett. B462, 380 (1999).
KLOE Collaboration, A.Aloisio et al., Phys. Lett. B536 (2002). KLOE Collaboration, A.Aloisio et al., Phys. Lett. B537 (2002). C. Bini, S. Giovanella,D. Leone, and S. Miscetti, KlOE Note 173 06/02, http://www.lnf.infn.it/kloe/. N.N. Achasov, V.V. Gubin, Phys.Rev.D63,094007 (2001); Yad. Fiz. [**65**]{}, 1566 (2002) \[Phys. At. Nucl. [**65**]{}, 1528 (2002)\]. N.N. Achasov and V.V. Gubin, Phys. Rev. D [**64**]{}, 094016 (2001); Yad. Fiz. [**65**]{}, 1939 (2002) \[Phys. At. Nucl. [**65**]{}, 1887 (2002)\]. N.N. Achasov, S.A. Devyanin, and G.N. Shestakov, Yad. Fiz. [**32**]{}, 1098 (1980)\[Sov. J. Nucl. Phys. [**32**]{}, 566 (1980)\]. N.N. Achasov, S.A. Devyanin, and G.N. Shestakov, Z. Phys. C22(1984), 56-61. N.N. Achasov, [*THE SECOND DA$\Phi$NE PHYSICS HANDBOOK*]{}, Vol. II, edited by L. Maiani, G. Pancheri, N. Paver, dei Laboratory Nazionali di Frascati, Frascati, Italy ( May 1995), p. 671. M.N. Achasov et al, Phys. Rev D63, 072002 (2001). Particle Data Group-2002: K. Hagiwara et al., Phys. Rev. D66 010001 (2002) Particle Data Group-2000: D. E. Groom et al., Eur. Phys. J. [**C15**]{}, 1 (2000).
N.N. Achasov, S.A. Devyanin, and G.N. Shestakov, Phys. Lett. B108, 134(1982); N.N. Achasov, S.A. Devyanin and G.N. Shestakov, Z. Phys. C16 (1982), 55. N.N. Achasov and G.N. Shestakov, Z. Phys. C41 (1988), 309. N.N. Achasov and A.V. Kiselev, Phys. Lett. B534 (2002), 83.
[^1]: achasov@math.nsc.ru
[^2]: kiselev@math.nsc.ru
[^3]: Note that in Ref.[@achasov-80] $\Pi_R^{ab}(m^2)$ differs by a real constant from those determined in other enumerated works in the case of $m_a\neq m_b$, but obviously it has no effect on Eq.(\[propagator\]).
[^4]: Note that fitting without averaging the theoretical curve $\bigg($changing $\bar{B}_i^{th}\to dBr^{th}(\phi\to\eta\pi^0\gamma)/dm\bigg|
_{m=(m_{i+1}+m_i)/2}\bigg)$ results to worse $\chi^2/n.d.f.=28.8/20$. The results in this case are consistent within errors with those obtained with averaging the theoretical curve.
[^5]: Note that the prediction $g_{a_0\eta\pi}/g_{a_0K^+K^-}\approx 0.93$, made in [@jaffe], was corrected in [@mistake]
[^6]: Emphasize that the isotopic invariance doesn’t require $g_{a_0K^+K^-}=g_{f_0K^+K^-}$.
|
---
abstract: |
The phase diagrams of isotropic and anisotropic triangular lattices with local Coulomb interactions are evaluated within cluster dynamical mean field theory. As a result of partial geometric frustration in the anisotropic lattice, short range correlations are shown to give rise to reentrant behavior which is absent in the fully frustrated isotropic limit. The qualitative features of the phase diagrams including the critical temperatures are in good agreement with experimental data for the layered organic charge transfer salts $\kappa$-(BEDT-TTF)$_2$Cu\[N(CN)$_2$\]Cl and $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$.\
\
DOI: PACS numbers: 71.10.Fd, 71.15.-m,71.27.+a
author:
- 'A. Liebsch$^1$, H. Ishida$^2$, and J. Merino$^3$'
title: 'Mott transition in two-dimensional frustrated compounds'
---
Introduction
============
The influence of spatial quantum fluctuations on the nature of the Mott transition in strongly correlated systems is currently of great interest. A class of materials in which these effects can be studied in detail are the layered charge transfer salts of the $\kappa$-(BEDT-TTF)$_2 X$ family, where $X$ denotes an inorganic monovalent anion such as Cu\[N(CN)$_2$\]Cl or Cu$_2$(CN)$_3$. The electronic properties of these compounds have been shown to be highly sensitive functions of hydrostatic pressure. [@komatsu; @ito; @lefebre; @limelette; @shimizu; @kagawa; @kurosaki; @ohira] As a result, the temperature versus pressure phase diagram is remarkably rich, exhibiting Fermi-liquid and bad-metallic behavior, superconductivity, as well as paramagnetic and antiferromagnetic insulating phases. These observations suggest fascinating connections to analogous phenomena in various transition metal oxides.[@imada]
A feature of particular interest in the organic salts is magnetic frustration. Since the geometric structure corresponds to an anisotropic triangular lattice, with inequivalent nearest neighbor hopping interactions $t$ and $t'$,[@kino; @mckenzie] long-range magnetic ordering becomes increasingly frustrated if the lattice is nearly isotropic, giving rise to an exotic spin-liquid phase in the absence of symmetry breaking. [@anderson] Such a spin-liquid phase [@shimizu; @kurosaki; @yamashita] is realised in the organic insulator $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ (denoted below as $\kappa$-CN) which corresponds to $t'\approx 1.06t$, whereas $\kappa$-(BEDT-TTF)$_2$Cu\[N(CN)$_2$\]Cl (denoted as $\kappa$-Cl) with $t'\approx 0.75t$ is an antiferromagnetic (AF) insulator. [@lefebre; @limelette] AF order is also found in those Pd(dimt)$_2$ salts for which $0.55< t'/t < 0.85$. In contrast, C$_2$H$_5$(CH$_3$)$_3$P\[Pd(dimt)$_2$\]$_2$ with $t'=1.05t$ is a valence bond insulator at ambient pressure. [@kato] Experiments on these kinds of two-dimensional frustrated systems have greatly stimulated theoretical investigations of the electronic and magnetic properties of anisotropic triangular lattices. The focus of the present study is the band width controlled finite temperature phase diagram of the Hubbard model for isotropic and anisotropic triangular lattices. The key result is that small changes in the ratio $t'/t$ can give rise to fundamental changes of the phase diagram. Thus, partial and full magnetic frustration reveal strikingly different metal-insulator coexistence regions, in qualitative agreement with the experimental phase disgrams for $\kappa$-Cl [@limelette] and $\kappa$-CN. [@kurosaki]
The anisotropic triangular lattice has recently been studied also by Ohashi [*et al.*]{} [@ohashi] who used dynamical mean field theory[@dmft] (DMFT) with a cluster extension to account for spatial fluctuations. Although at $t'\approx0.8t$ reentrant behavior was found as observed for $\kappa$-Cl, the calculated $T_c$ was much larger than the measured value. Moreover, only the lower boundary of the metal-insulator coexistence region was determined. Here, we investigate both the isotropic and anisotropic triangular lattices and use exact diagonalization[@ed] (ED) combined with cluster DMFT [@cdmft] to evaluate the upper and lower phase boundaries, $U_{c1}(T)$ and $U_{c2}(T)$, of the coexistence region. As shown below, the shape of these boundaries, as well as the critical temperatures, are consistent with the experimental data for $\kappa$-Cl and $\kappa$-CN.
Theory and Results
==================
The minimal model Hamiltonian that captures the interplay between geometrical frustration and strong Coulomb interaction present in the conducting layers of organic salts such as $\kappa$-Cl and $\kappa$-CN is $$H = - \sum_{ ij\sigma} t_{ij} ( c^+_{i\sigma} c_{j\sigma} + {\rm H.c.})
+ U \sum_i n_{i\uparrow} n_{i\downarrow} -
\mu\sum_{i \sigma} c^+_{i\sigma} c_{i\sigma},$$ where the sum in the first term is limited to nearest neighbor sites. The hopping integrals in a unit cell consisting of three sites are $t_{13}=t_{23}=t$ and $t_{12}=t'$. The band width is $W=9t$ for $t'=t$ and $W=8.5t$ for $t'=0.8t$. The chemical potential $\mu$ is fixed to give half-filling. Within cluster DMFT the interacting lattice Green’s function in the cluster site basis is defined as $$G_{ij}(i\omega_n) = \sum_{\vec k} \left( i\omega_n + \mu - t(\vec k) -
\Sigma(i\omega_n)\right)^{-1}_{ij} , \label{G}$$ where $\vec k$ extends over the reduced Brillouin Zone and $\omega_n=(2n+1)\pi T$ are Matsubara frequencies. $t(\vec k)$ denotes the hopping matrix for the superlattice and $\Sigma(i\omega_n)$ represents the non-diagonal cluster self-energy matrix. This self-energy is calculated within ED where the environment of the three-site cluster is replaced via a bath consisting of 6 or 9 levels, i.e., for a total cluster size $n_s=9$ or $n_s=12$. The calculations are carried out in a site basis and in a mixed site/molecular orbital basis. Due to ED finite-size effects, these treatments give results that differ quantitatively. Nevertheless, the qualitative features of the phase diagrams are consistently reproduced by these ED versions. Details of the cluster ED/DMFT formalism can be found in Ref. [@lie2008].
-3mm
Figure \[fig1\](a) shows the calculated phase diagram for the anisotropic lattice in the region below the critical temperature. To facilitate the comparison with the experimental data for $\kappa$-Cl, [@limelette] the hopping matrix elements are chosen as $t=0.04$ eV and $t'=0.8t$ to reproduce the single particle band width, $W=0.34$ eV. [@LDA] A similar value was used in the numerical renormalization group (NRG) DMFT analysis of the high-$T$ data in Ref.[@limelette]. Since the data were plotted in a $T / P$ phase diagram, we show the transition temperatures as functions of the inverse Coulomb energy. Increasing pressure $P$ implies increasing electronic band width, so that this measurement is equivalent to keeping $W$ fixed and reducing $U$ in the calculation. The phase boundaries of the coexistence region are obtained by carefully increasing or decreasing $U$ from the metallic or insulating domains, respectively. Fig. 1(b) shows the phase diagram for the isotropic case corresponding to $\kappa$-CN.
The critical temperatures for $t'=0.8t$ and $t'=t$, $T_c\approx 50$ K $\approx 0.11t$, are consistent with the measured values $T_c\approx40$ K for $\kappa$-Cl [@limelette] and $T_c\approx50$ K for $\kappa$-CN.[@kurosaki] $T_c\approx0.1t$ was recently obtained also for the fully unfrustrated square lattice. [@park] On the other hand, within Quantum Monte Carlo (QMC) DMFT at temperatures $T = 0.1t \ldots 1.0t$, Ohashi [*et al.*]{} [@ohashi] found a much larger value, $T_c\approx 0.3t\approx140$ K. The experimental data and the present ED/DMFT results suggest that the metal-insulator coexistence region is located at temperatures below those considered in Ref.[@ohashi].
For $t'=0.8t$, the first-order phase boundaries separating the Fermi liquid from the Mott insulator in Fig. 1(a) show the same kind of reentrant behavior as measured for $\kappa$-Cl. For instance, at $U=1/3$ eV and $T\approx 50$ K the system is a Mott insulator which turns into a Fermi liquid if $T$ is lowered to about $20$ K. Further reduction of $T$ reverts the system to a Mott insulator, just as seen in the data. (We do not consider here the antiferromagnetic insulating phase which is detected at even lower temperature.) Ohashi [*et al.*]{} [@ohashi] found reentrant behavior at considerably higher temperatures.
At present the origin of differences between the phase diagram for $t'=0.8t$ shown in Fig. 1 and the one found by Ohashi [*et al.*]{} is not clear. One reason might be that we consider a triangular lattice (3 sites per cluster) while in Ref. [@ohashi] a square lattice with one diagonal was used (4 sites per cluster). However, since the experimentally observed critical temperature is much lower than the range treated in Ref. [@ohashi], it would be interesting to apply continuous-time QMC to this problem in order to reach lower temperatures.
The reentrant behavior for $t'=0.8t$ is in striking contrast to the phase diagram obtained for the isotropic triangular lattice shown in Fig. 1(b). This limit resembles more closely the phase diagram derived within single-site DMFT. [@dmft] The main effect of short-range fluctuations in the isotropic case is a significant lowering of the critical Coulomb energy. Here, $U_{c2}\approx 1/2.63$ eV $\approx9.5t$, whereas $U_{c2} \approx 12t\ldots15t$ in local DMFT for the triangular lattice. [@merino06; @aryanpour] Comparing Figs. 1(a) and (b), it is evident that anisotropy causes a further lowering of the critical Coulomb energies. This trend is consistent with $U_c\approx 6t$ for the fully unfrustrated square lattice [@park; @white] which is topologically equivalent to the triangular lattice in the limit $t'=0$.
It is interesting also to analyze the width of the metal-insulator coexistence region obtained by increasing vs. decreasing pressure. For $\kappa$-Cl, it is observed at $P\approx 200\ldots400$bar, which according to the high-$T$ NRG analysis corresponds to band width changes of about 2%. [@limelette] The calculated coexistence region shown in Fig. 1(a) is only slightly wider than this experimental range.
-5mm
The qualitative change in the phase diagram caused by reduced geometrical frustration can be understood by analyzing the magnetic properties of the frustrated lattice. In the $U\rightarrow \infty$ limit, the Hubbard model can be mapped onto the anisotropic Heisenberg model with $J'/J=0.64$ and $J'/J=1$ for $t'=0.8t$ and $t'=t$, respectively. At $T=0$, $t'=t$ yields long range antiferromagnetic (AF) order of the 120$^0$ type, whereas $t'=0.8t$ gives rise to row-wise AF Néel order. [@zheng] However, in the Heisenberg model the temperature scale for AF order in the isotropic triangular lattice is expected to be strongly suppressed relative to the square lattice. For the Hubbard model, the cluster DMFT provides information on how the magnetic correlations $<S_{iz}S_{jz}>$ vary across the Mott transition in the isotropic case compared with $t'=0.8t$. The results shown in Fig. \[fig2\] demonstrate that spin correlations are strongly enhanced as the geometrical frustration is suppressed. The isotropic lattice displays weak AF coupling for any $U$. This is in contrast to $t'=0.8t$, for which the weaker hopping amplitude displays ferromagnetic correlations whereas spins with the larger hopping amplitude are antiferromagnetically coupled, indicating a row-wise AF Neel arrangement of spins. Thus, $t'=0.8t$ induces a much stronger tendency towards magnetic order than $t'=t$, which explains why the reentrant behavior occurs for $t'=0.8t$, but not for $t'=t$ (see Fig. \[fig1\]). At low $T$, the electron entropy is suppressed for $t'=0.8t$ as compared to $t'=t$. As $T$ is increased for $t'=0.8t$, the system lowers its free energy by transforming to a metal since the entropy of the metal exceeds that of the ordered insulator. At even higher temperatures the system gains entropy of $\log(2)$ by transforming back into a paramagnetic insulator. This result is analogous to the one found for the unfrustrated square lattice. [@park] In the isotropic lattice magnetic ordering is suppressed and the reentrant behavior disappears.
-5mm
-5mm
To illustrate the first-order nature of the metal-insulator transition we show in the upper panel of Fig. \[fig3\] the spectral weights of the cluster sites at $E_F=0$ as functions of $U$. The lower panel shows the average double occupancy $d_{\rm occ}=\sum_i\langle n_{i\uparrow}n_{i\downarrow}\rangle/3$. Both quantities exhibit hysteresis for increasing and decreasing $U$, indicating coexistence of metallic and insulating solutions.
Finally, Fig. \[fig4\] shows the spectral densities at Coulomb energies below and above the Mott transition for $T=0.02t$ and $t'=0.8t$. Plotted is the average over the three inequivalent sites within the unit cell. Since we are here concerned with the metal-insulator transition we give the ED cluster spectra which can be evaluated without requiring extrapolation from Matsubara to real frequencies. In the metallic phase the spectra reveal large quasi-particle weight at low frequencies as well as upper and lower Hubbard bands at high frequencies. The insulating phase exhibits a Mott gap, as well as pronounced spectral weight in the region of the Hubbard bands. Qualitatively similar features are also seen for the unfrustrated square lattice. [@park; @zhang]
Conclusion
==========
In conclusion, the phase diagrams of the Hubbard model for the isotropic and anisotropic triangular lattices have been determined within cluster DMFT and exact diagonalization. For moderate frustration, $t'=0.8t$, reentrant behavior is found and the phase boundaries of the metal-insulator coexistence region are in qualitative agreement with the $T / P$ phase diagram observed experimentally for the anisotropic organic salt $\kappa$-Cl. The reentrant behavior disappears in the fully frustrated limit, $t'=t$, in agreement with measurements on the nearly isotropic compound $\kappa$-CN. The phase diagram then bears overall resemblance to the one obtained within local DMFT, i.e., in the absence of inter-site correlations. The critical temperatures, $T_c\approx 50$ K for the isotropic and anisotropic lattices, are consistent with the data for $\kappa$-CN and $\kappa$-Cl. These results should also be relevant for the phase diagram of \[Pd(dimt)$_2$\]$_2$ salts exhibiting small deviations from the isotropic lattice. [@kato2]
Computational work (A.L.) was carried out on the Juelich JUMP. J.M. thanks MCI for financial support: CTQ2008-06720-C02-02/BQU.
[99]{}
T. Komatsu [*et al.*]{}, J. Phys. Soc. Jpn. [**65**]{}, 1340 (1996).
H. Ito, T. Ishiguro, M. Kubota, and G. Saito, J. Phys. Soc. Jpn. [**65**]{}, 2987 (1996).
S. Lefebvre, P. Wzietek, S. Brown, C. Bourbonnais, D. Jérome, C. Mézière, M. Fourmigué and P. Batail, Phys. Rev. Lett. [**85**]{}, 5420 (2000).
P. Limelette, P. Wzietek, S. Florens, A. Georges, T. A. Costi, C. Pasquier, D. Jérome, C. Mézière, and P. Batail, Phys. Rev. Lett. [**91**]{}, 016401 (2003).
Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, Phys. Rev. Lett. [**91**]{}, 107001 (2003).
F. Kagawa, T. Itou, K. Miyagawa, and K. Kanoda, Phys. Rev. B [**69**]{}, 064511 (2004).
Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, and G. Saito, Phys. Rev. Lett. [**95**]{}, 177001 (2005).
S. Ohira [*et al.*]{}, J. Low Temp. Phys. [**142**]{}, 153 (2006).
M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. [**70**]{}, 1039 (1998).
H. Kino and H. Fukuyama, J. Phys. Soc. Jpn. [**65**]{}, 2158 (1996).
R. H. McKenzie, Science [**278**]{}, 820 (1997); Comments Cond. Matt. Phys. [**18**]{}, 309 (1998).
P. W. Anderson, Mater. Res. Bull. [**8**]{}, 153 (1973).
M. Yamashita, [*et al.*]{}, Nat. Phys. (23 Nov 2008); doi:10.1038/nphys1134.
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. [**99**]{}, 256403 (2007).
T. Kashima and M. Imada, J. Phys. Soc. Jpn. [**70**]{}, 3052 (2001); H. Morita, S. Watanabe, and M. Imada, J. Phys. Soc. Jpn. [**71**]{}, 2109 (2002); T. Mizusaki and M. Imada, Phys. Rev. B [**74**]{}, 014421 (2006).
Y. Imai and N. Kawakami, Phys. Rev. B [**65**]{}, 233103 (2002).
S. Onoda and M. Imada, Phys. Rev. B [**67**]{}, R161102 (2003); S. Onoda and N. Nagaosa, J. Phys. Soc. Jpn. [**72**]{}, 2445 (2003).
O. Parcollet, G. Biroli, and G. Kotliar Phys. Rev. Lett. [**92**]{}, 226402 (2004).
S. Watanabe and M. Imada, J. Phys. Soc. Jpn. [**73**]{}, 3341 (2004); S. Watanabe [*et al.*]{}, [*ibid.*]{} [**75**]{}, 074707 (2006).
A. Singh, Phys. Rev. B [**71**]{}, 214406 (2005).
H. Yokoyama [*et al.*]{}, J. Phys. Soc. Jpn. [**75**]{}, 114706 (2006).
B. Kyung and A.-M. S. Tremblay, Phys. Rev. Lett. [**97**]{}, 046402 (2006); B. Kyung, Phys. Rev. B [**75**]{}, 033102 (2007).
P. Sahebsara and D. Senechal, Phys. Rev. Lett. [**97**]{}, 257004 (2006); Phys. Rev. Lett. [**100**]{}, 136402 (2008).
T. Koretsune [*et al.*]{}, J. Phys. Soc. Jpn. [**76**]{}, 074719 (2007).
T. Senthil, Phys. Rev. B [**78**]{}, 045109 (2008).
T. Ohashi, T. Momoi, H. Tsunetsugu, and N. Kawakami, Phys. Rev. Lett. [**100**]{}, 076402 (2008).
A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, Rev. Mod. Phys. [**68**]{}, 13 (1996).
M. Caffarel and W. Krauth, Phys. Rev. Lett. [**72**]{}, 1545 (1994).
A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B [**62**]{}, R9283 (2000); G. Kotliar, S. Y. Savrasov, G. Palsson, and G. Biroli, Phys. Rev. Lett. [**87**]{}, 186401 (2001).
A. Liebsch, H. Ishida, and J. Merino, Phys. Rev. B [**78**]{}, 165123 (2008); see also: C. A. Perroni, H. Ishida, and A. Liebsch, Phys. Rev. B [**75**]{}, 045125 (2007).
W. Y. Ching, Y. N. Xu, Y. C. Jean, and Y. Lou, Phys. Rev. B [**55**]{}, 2780 (1997).
H. Park, K. Haule, and G. Kotliar, Phys. Rev. Lett. [**101**]{}, 186403 (2008).
J. Merino, B. J. Powell, and R. H. McKenzie, Phys. Rev. B [**73**]{}, 235107 (2006).
K. Aryanpour, W. E. Pickett, and R. T. Scalettar, Phys. Rev. B [**74**]{}, 085117 (2006).
See also: M. Vekic and S. R. White, Phys. Rev. B [**47**]{}, R1160 (1193).
A. E. Trumper, Phys. Rev. B [**60**]{}, 2987 (1999); J. Merino, R. H. McKenzie, J. B. Marston, and C.-H.Chung, J. Phys. Condens. Matter [**11**]{}, 2965 (1999); W. Zheng, R. R. P. Singh, R. H. McKenzie, and R. Coldea, Phys. Rev. B [**71**]{}, 134422 (2006).
Y. Z. Zhang and M. Imada, Phys. Rev. B [**76**]{}, 045108 (2008).
R. Kato, A. Tajima, A. Nakao, and M. Tamura, J. Am. Chem. Soc. [**128**]{}, 10016 (2006).
|
---
abstract: 'We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin–La–Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.'
address:
- 'Departamento de Matemáticas, Universidade de Santiago de Compostela, Spain.'
- 'Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain.'
- 'Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain.'
author:
- 'Miguel Domínguez-Vázquez'
- Alberto Enciso
- 'Daniel Peralta-Salas'
title: Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems
---
Introduction
============
In two dimensions, it is easy to construct stationary solutions with compact support to the incompressible Euler equation $$\label{Euler}
{\partial}_t u + u\cdot\nabla u+\nabla p=0\,,\qquad \operatorname{div}u=0\,.$$ For instance, an explicit $C^\infty$ stationary Euler flow supported in the unit disk ${\mathbb D}$ is given by the stream function $\psi:= e^{1/(|x|^2-1)}\,
{1}_{{\mathbb D}}(x)$, which determines the velocity field as $u=\nabla^\perp \psi$. The question of whether there are stationary Euler flows of compact support in three dimensions is much harder, and has attracted much recent attention. In the important particular case that the stationary Euler flow is a generalized Beltrami field, Nadirashvili [@N] and Chae–Constantin [@CC] have shown that no compactly supported solutions exist, and that in fact there are not even any generalized Beltrami fields with finite energy. It is also known that axisymmetric stationary Euler flows of compact support without swirl do not exist [@JX]. In contrast, it has long been known that there exist $C^{1,{\alpha}}$ stationary Euler flows whose vorticity is compactly supported [@FB]. Weak stationary Euler flows in $L^\infty$ of compact support can also be constructed with the convex integration technique developed in [@CS].
A major recent breakthrough was Gavrilov’s construction of compactly supported stationary Euler flows in three dimensions [@Gavrilov], which are axisymmetric and of class $C^\infty$. More concretely, these solutions are of the form $$\label{uGavrilov}
u= G(p_R)\, u_R\,, \qquad p=\int_0^{p_R}G(q)^2\, dq\,,$$ where $(u_R,p_R)$ are certain concrete functions depending on a positive parameter $R$ that solve the stationary Euler equation in a toroidal domain, and $G$ is an essentially arbitrary function of one variable (compactly supported). Slightly more general solutions can be constructed using the same method.
These solutions have been revisited and put in a broader context using the perspective of the Grad–Shafranov equation by Constantin, La and Vicol [@CV], which allowed them to develop nontrivial applications to other equations of fluid mechanics as well. As stressed by these authors, the key property of these solutions is that they are [ *localizable*]{}, meaning that the pressure is constant along the stream lines of the flow: $u_R\cdot \nabla p_R=0$ (which in turn implies that $u\cdot \nabla p=0$).
Our objective in this paper is to derive a different approach to the construction of stationary Euler flows with compact support. The solutions we construct are very different from those obtained by Gavrilov and Constantin–La–Vicol: they are piecewise smooth stationary weak solutions with axial symmetry, and they are not localizable. Each stationary solution we construct is bounded (with bounded vorticity) and supported on a toroidal domain with a smooth boundary; the flow is smooth in the interior of this domain (up to the boundary) and possibly discontinuous across the boundary. As we will make precise below, this approach yields a wide range of axisymmetric Euler flows of compact support.
Our construction of stationary Euler flows with compact support is based on showing the existence of nontrivial solutions to a boundary value problem for an elliptic equation where both Dirichlet and Neumann data are prescribed. This kind of boundary problems are usually called [*overdetermined*]{}.
To see how overdetermined boundary problems appear in this context, let us start by recalling the Grad–Shafranov formulation of the axisymmetric Euler equation in three dimensions. This consists in writing an axisymmetric solution to the Euler equation in cylindrical coordinates (in terms of the orthonormal basis $\{e_z,e_r,e_{\varphi}\}$) as $$\label{defu}
u= \frac1r\left[ {\partial}_r\psi\, e_z-{\partial}_z\psi\, e_r +F(\psi)\, e_{\varphi}\right]\,,$$ where the function $\psi(r,z)$ satisfies the equation $$\label{elliptic}
L\psi=r^2 H'(\psi) -\frac12 (F^2)'(\psi)$$ for some function $H$. The pressure is then given by $$\label{defp}
p= H(\psi)-\frac1{2r^2}\left[|\nabla \psi|^2 +F(\psi)^2\right]$$ and we have set $$\label{defL}
L\psi:={\partial}_{rr}\psi+{\partial}_{zz}\psi - \frac1r{\partial}_r\psi\,.$$ The functions $F$ and $H$ can be picked freely.
The first observation of this paper, which explains why we are interested in overdetermined boundary problems in this context, is the following:
\[L.weak\] Let ${\Omega}$ be a $C^2$ bounded domain whose closure is contained in the half-space $\{(r,z)\in\RR^2: r>0\}$. Suppose that there is a function $\psi\in
C^2({\Omega})$ satisfying Equation in ${\Omega}$ and the boundary conditions $$\begin{aligned}
\psi|_{{\partial}{\Omega}}=0\,,\label{Dirichlet}\\
\left.\frac{({\partial}_\nu\psi)^2+ F(0)^2}{r^2}\right|_{{\partial}{\Omega}}=c\,,\label{Neumann}\end{aligned}$$ where $c$ is a constant, and $\nu$ is a unit normal field on $\partial\Omega$. Then the vector field $u$ defined by inside ${\Omega}$ and as $u:=0$ outside ${\Omega}$ is a weak solution of the Euler equation, the pressure being given by inside ${\Omega}$ and by $p:=H(0)-c/2$ outside ${\Omega}$.
Therefore, the key step of our construction of stationary Euler flows with compact support is to show the existence of nontrivial solutions to a (non-standard) overdetermined boundary problem for a certain semilinear elliptic equation in two variables. While the investigation of overdetermined boundary value problems traces back to Serrin’s seminal paper [@Serrin] in 1971, until very recently the literature on overdetermined boundary problems was essentially limited to proofs that solutions need to be radial in cases that could be handled using the method of moving planes.
However, in two surprising papers, Pacard and Sicbaldi [@PS:AIF] and Delay and Sicbaldi [@DS:DCDS] proved the existence of extremal domains with small volume for the first eigenvalue $\lambda_1$ of the Laplacian in any compact Riemannian manifold, which guarantees the existence of solutions to a certain overdetermined problem for the linear elliptic operator ${\Delta}+{\lambda}_1$ in a domain with both zero Dirichlet data and constant Neumann data. Very recently we managed to show the existence of nontrivial solutions, still with the same Dirichlet and Neumann data, for fairly general semilinear elliptic equations of second order with possibly nonconstant coefficients [@overdet]. Our strategy here is to start by tweaking the proof of this result to accomodate to the non-standard boundary condition we must impose.
The main difficulty to solve Equation with the overdetermined boundary conditions and is that the Neumann data depend on the point of the boundary $\partial{\Omega}$, a situation that was not considered in our paper [@overdet]. The technique used there is based on a variational technique that relates the existence of overdetermined solutions with the critical points of certain functional, a strategy that is successful only for constant boundary data. Roughly speaking, the gist of the argument is that the overdetermined condition with constant data is connected with the local extrema for a natural energy functional, restricted to a specific class of functions labeled by points in the physical space. This fact ultimately permits to derive the existence of solutions from the fact that a continuous function attains its maximum on a compact manifold. We do not know of an analog of this fact for the nonconstant boundary conditions considered in this paper. Instead, we rely on a new, hands-on approach to the problem that results in a flexible, very explicit existence theorem:
\[T.main\] Consider any functions ${\widetilde{F}},H\in C^s((-1,0])$ satisfying $${\widetilde{F}}(0)=0\,,\qquad {\widetilde{F}}'(0)<0\,,\qquad H'(0)>0\,,$$ where $s>2$ is not an integer. Then the following statements hold:
1. For each small enough ${\varepsilon}>0$ and any large enough $R>0$, there exists a nontrivial, piecewise $C^s$, axisymmetric stationary Euler flow of compact support $u$ of the form described in Lemma \[L.weak\] for a suitable $C^{s+1}$ planar domain ${\Omega}_{R,{\varepsilon}}$.
2. The domain ${\Omega}_{R,{\varepsilon}}$ is a small deformation of a disk of radius ${\varepsilon}$ centered at the point $(R,0)$ of the $(r,z)$-plane.
3. The functions that define the solution are $$\label{FH}
F(\psi) := [{\varepsilon}^2 F_R+ {\widetilde{F}}(\psi)]^{1/2}$$ and $H(\psi)$, where $F_R$ is a positive constant depending on $R$.
4. The function $\psi$, which is of class $C^{s+1}$ in ${\Omega}_{R,{\varepsilon}}$ up to the boundary, is approximately radial. Moreover, ${\varepsilon}^2F_R+{\widetilde{F}}\circ\psi>0$, $F\circ \psi>0$ and $H\circ\psi$ are of class $C^s$ in ${\Omega}_{R,{\varepsilon}}$.
In fact, the value of the constants and the structure of the solutions is completely explicit:
1. $R$ must be larger than $[-\frac32{\widetilde{F}}'(0)/H'(0)]^{1/2}$.
2. The boundary of ${\partial}{\Omega}_{R,{\varepsilon}}$ is the curve defined by an equation of the form $z^2+(r-R)^2-{\varepsilon}^2 = O({\varepsilon}^3)$.
3. The constant $F_R$ is $$F_R:=\frac{1}{16}\left[H'(0)R^2-\frac12{\widetilde{F}}'(0)\right]\left[H'(0)R^2+\frac32{\widetilde{F}}'(0)\right]>0\,,$$ and the constant $c$ in the Neumann condition is of the form $$c=\frac{{\varepsilon}^2}{16R^2}\left[H'(0)R^2-\frac12{\widetilde{F}}'(0)\right]\left[5H'(0)R^2-\frac12{\widetilde{F}}'(0)\right]+O({\varepsilon}^3)\,.$$
4. The function $\psi$ is of the form $$\psi=\frac14 \left[H'(0)R^2-\frac12{\widetilde{F}}'(0)\right]\left[(r-R)^2+z^2-{\varepsilon}^2\right]+O({\varepsilon}^3)\,.$$
5. The vorticity, $${\omega}= \frac{F'(\psi)}r\left( {\partial}_r\psi\, e_z-{\partial}_z\psi\, e_r\right) +\left[-rH'(\psi)+\frac{(F^2)'(\psi)}{2r}\right]\, e_{\varphi}\,,$$ is also of class $C^{s-1}$ up to the boundary.
Several comments are in order. First, let us emphasize that the solutions we construct are piecewise smooth but discontinuous across a smooth surface; hence, from the point of view of their regularity, they stand between the smooth solutions of [@Gavrilov; @CV] and the rough weak solutions of [@CS]. Concerning the flexibility of the construction, it is apparent that the range of solutions we obtain is much larger than that of [@Gavrilov; @CV]. Indeed, our solutions are not localizable in general and are of class $C^s$ in the toroidal domain of $\RR^3$ defined by the planar domain ${\Omega}$. In contrast, the basic vector field $u_R$ that appears in is not defined at an inside point (“the origin”), so the function $G$ in that equation is chosen so that it vanishes to infinite order there; in fact, the supports of the solutions constructed so far are toroidal shells instead of solid tori. Actually, the functions $H$ and $F$ in [@Gavrilov; @CV] defining the vector field $u_R$ are precisely $H(\psi)=a\psi$ and a certain function of the form $F(\psi)=Rb\sqrt\psi\,[1+O(\psi)]$, where again the positive constants $a$ and $b$ are fixed.
It should be noted that the philosophy that underlies the proof of Theorem \[T.main\] has, in fact, a wider range of applicability. To illustrate this fact, we include in Section \[S.final\] a brief discussion on the existence of compactly supported solutions for a class of functions $F$ and $H$ different from that considered above.
The paper is organized as follows. In Section \[S.weak\] we will prove Lemma \[L.weak\] as a corollary of a more general result about piecewise smooth weak solutions to the stationary Euler equation. The rest of the paper is devoted to the proof of Theorem \[T.main\]. In Section \[S.Dirichlet\] we construct solutions to the Dirichlet problem for the Equation , and subsequently compute their asymptotic expansion in the parameter ${\varepsilon}$ (Section \[S.B=0\]). The way these solutions change when the domain is perturbed a little is discussed in Section \[S.variations\]. This is key for the proof that there exists a domain which is ${\varepsilon}^2$-close to a disk of radius ${\varepsilon}$ where the Dirichlet solution also satisfies the additional Neumann condition . This result, which we prove in Section \[S.Neumann\], allows us to complete the proof of Theorem \[T.main\]. To conclude, in Section \[S.final\] we briefly discuss the existence of compactly supported solutions defined by functions $F$ and $H$ different from those considered in Theorem \[T.main\].
Stationary Euler flows via an overdetermined boundary problem {#S.weak}
=============================================================
In this short section we prove that if one has a stationary Euler flow on a bounded domain ${\Omega}$ which is tangent to the boundary and whose pressure is constant on ${\partial}{\Omega}$, then one can trivially extend it to a stationary Euler flow on the whole space with compact support. In the particular case when the initial flow is axisymmetric, and hence described by the Grad–Shafranov formulation (Equation ), this reduces to Lemma \[L.weak\] stated in the Introduction.
Let us start by recalling the definition of a weak stationary Euler flow. We say that a pair $(u,p)$ of class, say, $L^2{_{\mathrm{loc}}}(\RR^3)$ is a [*weak solution to the stationary Euler equation*]{} if $$\int_{\RR^3} \left[(u\otimes u)\cdot \nabla w+ p\operatorname{div}w\right]\, dx=0\quad
\text{and}\quad \int_{\RR^3}u\cdot \nabla\phi\,dx=0$$ for any vector field $w\in C^\infty_c(\RR^3)$ and any scalar function $\phi\in C^\infty_c(\RR^3)$. Of course, if $u$ and $p$ are smooth enough, this is equivalent to saying that they satisfy Equation in $\RR^3$.
\[L.Euler\] Given a bounded domain ${\Omega}$ in $\RR^3$ with a $C^2$ boundary, suppose that $v\in C^1({\Omega})\cap L^2({\Omega})$ is a solution to the stationary Euler equation in ${\Omega}$ with pressure ${\widetilde{p}}\in C^1({\Omega})\cap L^1({\Omega})$. Assume that $\nu\cdot v|_{{\partial}{\Omega}}=0$ and ${\widetilde{p}}|_{{\partial}{\Omega}}=c$, where $c$ is a constant. Then $$u(x):=\begin{cases}
v(x) &\text{ if }\; x\in{\Omega}\,,\\
0 &\text{ if }\; x\not\in{\Omega}\,,
\end{cases}$$ is a weak solution of the stationary Euler equation on $\RR^3$ with pressure $$p(x):=\begin{cases}
{\widetilde{p}}(x) &\text{ if }\; x\in{\Omega}\,,\\
c &\text{ if }\; x\not\in{\Omega}\,.
\end{cases}$$
We start by noticing that, for all $\phi\in C^\infty_c(\RR^3)$, $$\begin{aligned}
\int_{\RR^3}u\cdot \nabla \phi\, dx=\int_{\Omega}v\cdot \nabla \phi\,
dx = -\int_{\Omega}\phi\,\operatorname{div}v\, dx + \int_{{\partial}{\Omega}} \phi\, \nu\cdot v\, dS=0\,,\end{aligned}$$ where we have used that $\operatorname{div}v=0$ in ${\Omega}$ and $\nu \cdot v=0$ on ${\partial}{\Omega}$. Hence $\operatorname{div}u=0$ in the sense of distributions.
Let us now take an arbitrary vector field $w\in
C^\infty_c(\RR^3)$. As $\int_{\RR^3} \operatorname{div}w\,dx =0$, we can write $$\begin{aligned}
\int_{\RR^3} [(u&\otimes u)\cdot \nabla w+ p\operatorname{div}w]\,
dx
=
\int_{{\Omega}} (u\otimes u)\cdot \nabla w\, dx + \int_{\RR^3}(p-c)\, \operatorname{div}w\,
dx\\
&=\int_{\Omega}\left[(v\otimes v)\cdot \nabla w- w\cdot \nabla {\widetilde{p}}\right]\, dx
+ \int_{{\partial}{\Omega}} ({\widetilde{p}}-c) \, w\cdot
\nu\, dS \\
&=-\int_{{\Omega}} \left[\operatorname{div}(v\otimes v) + \nabla {\widetilde{p}}\right]\cdot w\, dx + \int_{{\partial}{\Omega}} \left[ (v\cdot w)(v\cdot \nu) + ({\widetilde{p}}-c) \, w\cdot
\nu\right]\, dS\,.\end{aligned}$$ The volume integral is zero because $v$ satisfies the stationary Euler equation in ${\Omega}$. Since $v\cdot \nu={\widetilde{p}}-c=0$ on ${\partial}{\Omega}$, the surface integral vanishes too. It then follows that $u$ is a weak solution of the Euler equation in $\RR^3$, as claimed.
With $u$ given by , and using the notation in Lemma \[L.weak\], the conditions that $u$ be tangent to the axisymmetric domain defined by ${\Omega}$ and that the pressure be constant on the boundary amount to saying that $\psi$ takes a constant value $c_0$ on ${\partial}{\Omega}$ and that $[({\partial}_\nu\psi)^2+ F(c_0)^2]/r^2$ is also constant on ${\partial}{\Omega}$. Setting $c_0:=0$ without loss of generality, Lemma \[L.Euler\] results in the statement of Lemma \[L.weak\].
Solutions to the Dirichlet boundary problem {#S.Dirichlet}
===========================================
Let us take any $R>0$ that will be fixed during the whole construction and introduce suitably translated and rescaled variables $(x,y)\in\RR^2$ as $$r=:R+{\varepsilon}x\,,\qquad z=: {\varepsilon}y\,.$$ Throughout, ${\varepsilon}>0$ denotes a small parameter. We will also consider the polar coordinates $(\rho,{\theta})\in (0,\infty)\times \TT$, where $\TT:=\RR/2\pi\ZZ$, that are defined in terms of $(x,y)$ through the formulas $$x= \rho\cos{\theta}\,,\qquad y=\rho\sin{\theta}\,.$$ In these variables, the Grad–Shafranov equation reads as $$\label{GS2}
{\Delta}\psi -\frac{\varepsilon}{R+{\varepsilon}x}{\partial}_x \psi= {\varepsilon}^2(R+{\varepsilon}x)^2 H'(\psi) -\frac12{\varepsilon}^2 (F^2)'(\psi)\,,$$ where $F$ and $H$ are of the form described in Theorem \[T.main\].
We look for solutions to Equation in domains of the form $${\Omega}_{{\varepsilon}B}:=\{ \rho<1+{\varepsilon}B({\theta})\}$$ for some function $B\in C^{s+1}(\TT)$; notice that ${\Omega}_{{\varepsilon}B}$ only depends on the product ${\varepsilon}B$ and not on ${\varepsilon}$ and $B$ separately. To keep track of the size of $\psi$, we will set $$\psi=:{\varepsilon}^{2}\phi\,.$$ For ${\varepsilon}\neq0$, Equation can be written in terms of $\phi$ as $$\label{eq}
{\Delta}\phi -\frac{\varepsilon}{R+{\varepsilon}x}{\partial}_x \phi= aR^2+b + 2aR{\varepsilon}x + {{\mathcal R}}(x,\phi)\,,$$ where we have defined the positive constants $$a:= H'(0) \qquad b:=-\frac12{\widetilde{F}}'(0)\,,$$ and where $${{\mathcal R}}(x,\phi):= {\varepsilon}^2 ax^2 + (R+{\varepsilon}x)^2 H_1'({\varepsilon}^2\phi) -\frac12 {\widetilde{F}}_1'({\varepsilon}^2\phi)\,.$$ Here $$H_1(\psi):= H(\psi)-a\psi-H(0)\,,\qquad {\widetilde{F}}_1(\psi):= {\widetilde{F}}(\psi)+2b\psi$$ are functions that vanish to second order at $\psi=0$, so we have the obvious bound $$\sup_{\|\phi\|_{C^{s-1}({\Omega})}<C}\|{{\mathcal R}}\|_{C^{s-1}({\Omega})}{\leqslant}C'{\varepsilon}^2\,.$$ The Dirichlet boundary condition $\psi=0$ on ${\partial}{\Omega}_{{\varepsilon}B}$ can then be rewritten in terms of $\phi(\rho,{\theta})$ as $$\label{DBC}
\phi(1+{\varepsilon}B({\theta}),{\theta})=0\,.$$
Henceforth we will say that a function $f(\rho,{\theta})$ is [ *even*]{} if $$f(\rho,{\theta})=f(\rho,-{\theta})\,,$$ and similarly for a function $g({\theta})$. Equivalently, a function is even if it is invariant under the reflection $y\mapsto -y$.
Since the function $$\phi_0:= \frac{(aR^2+b)(\rho^2-1)}{4}$$ satisfies Equation and the Dirichlet condition when ${\varepsilon}=0$, it is straightforward to show that there are solutions to the problem for small ${\varepsilon}$ using the implicit function theorem in Banach spaces:
\[P.existence\] For any small enough ${\varepsilon}$ and any function $B$ with $\|B\|_{C^{s+1}}<1$ there is a unique function $\phi={\phi_{{\varepsilon},B}}$ in a small neighborhood of $\phi_0$ in $C^{s+1}({\Omega}_{{\varepsilon}B})$ that satisfies Equation and the Dirichlet boundary condition . Furthermore, ${\phi_{{\varepsilon},B}}<0$ in ${\Omega}_{{\varepsilon}B}$ and ${\phi_{{\varepsilon},B}}$ is even if $B$ is.
For small enough ${\varepsilon}\neq0$ and $\|B\|_{C^{s+1}}<1$, let $\chi_{{\varepsilon}B}:{\mathbb D}\to {\Omega}_{{\varepsilon}B}$ be the diffeomorphism defined in polar coordinates as $$(\rho,{\theta})\mapsto \big([1+{\varepsilon}\chi(\rho)\, B({\theta})]\rho,{\theta}\big)\,,$$ where $\chi(\rho)$ is a smooth cutoff function that is zero for $\rho<1/4$ and $1$ for $\rho>1/2$, and where ${\mathbb D}:=\{\rho<1\}$ is the unit disk . Then one can define a map $${{\mathcal H}}: (-{\varepsilon}_0,{\varepsilon}_0)\times C^{s+1}{_{\mathrm{Dir}}}({\mathbb D})\to C^{s-1}({\mathbb D})$$ as $$\begin{gathered}
{{\mathcal H}}({\varepsilon},\bar\phi):=\Big[{\Delta}(\bar\phi\circ \chi_{{\varepsilon}B}^{-1}) -\frac{\varepsilon}{R+{\varepsilon}x}{\partial}_x (\bar\phi\circ \chi_{{\varepsilon}B}^{-1})\\
-[ aR^2+b + 2aR{\varepsilon}x +
{{\mathcal R}}(x,\bar\phi\circ \chi_{{\varepsilon}B}^{-1})\Big]\circ \chi_{{\varepsilon}B}\,.\end{gathered}$$ Here $$C^{s+1}{_{\mathrm{Dir}}}({\mathbb D}):= \{\bar\phi\in C^{s+1}({\mathbb D}): \bar\phi|_{{\partial}{\mathbb D}}=0\}\,.$$ Notice that $\phi:=\bar\phi\circ \chi_{{\varepsilon}B}^{-1}\in C^{s+1}({\Omega}_{{\varepsilon}B})$ and $\phi=0$ on ${\partial}{\Omega}_{{\varepsilon}B}$. It is obvious that if $\overline \phi$ solves ${{\mathcal H}}({\varepsilon},\bar\phi)=0$, then $\phi$ is a solution to the Dirichlet problem –.
Since ${{\mathcal H}}(0,\phi_0)=0$ and the Fréchet derivative $$D_{\bar\phi} {{\mathcal H}}(0,\phi_0)={\Delta}$$ is an invertible mapping $C^{s+1}{_{\mathrm{Dir}}}({\mathbb D})\to C^{s-1}({\mathbb D})$, it follows from the implicit function theorem that for any small enough ${\varepsilon}$ and $B$ there is a unique solution $\bar\phi_{{\varepsilon},B}$ to the equation ${{\mathcal H}}({\varepsilon},\bar\phi_{{\varepsilon},B})=0$ in a neighborhood of $\phi_0$. As $B$ only appears in the problem through the product ${\varepsilon}B$, this is equivalent to the first part of the statement. Also, this uniqueness property immediately implies that $\phi_{{\varepsilon},B}:=\bar\phi_{{\varepsilon},B}\circ \chi_{{\varepsilon}B}^{-1}$ is even when $B$ is. The property that $\phi_{{\varepsilon},B}<0$ in ${\Omega}_{{\varepsilon}B}$ follows from Equation , i.e., $$\Delta \phi_{{\varepsilon},B}=aR^2+b+O({\varepsilon})>0\,,$$ via the maximum principle.
Analysis of the solution {#S.B=0}
========================
In the following proposition we compute an asymptotic expansion for the function ${\phi_{{\varepsilon},B}}$ for small ${\varepsilon}$. The constants $A_0$ and $A_1$ appearing in this expansion, which depend on $R$ but not on ${\varepsilon}$, will play a major role in the rest of the paper.
Throughout, we will denote by ${\mathbb{P}_{{\varepsilon}B}}$ the Poisson integral operator of the domain ${\Omega}_{{\varepsilon}B}$ in the coordinates $(\rho,{\theta})$. That is, $v(\rho,{\theta}):= {\mathbb{P}_{{\varepsilon}B}}f$ denotes the only harmonic function in ${\Omega}_{{\varepsilon}B}$ satisfying the boundary condition $$v(1+{\varepsilon}B({\theta}),{\theta})=f({\theta})\,.$$ Note that ${\mathbb{P}_{{\varepsilon}B}}$ only depends on the product ${\varepsilon}B$, not on ${\varepsilon}$ and $B$ separately. It is standard that ${\mathbb{P}_{{\varepsilon}B}}$ defines a map $C^{s+1}(\TT)\to C^{s+1}({\Omega}_{{\varepsilon}B})$. A convenient explicit formula for the Poisson operator in the case of the disk is $$\PP f(\rho,{\theta}):= \sum_{n\in\ZZ} f_n\,\rho^{|n|}\, e^{in{\theta}}\quad
\text{if} \quad f({\theta})= \sum_{n\in\ZZ} f_n\, e^{in{\theta}}\,.$$
\[P.phi0\] For small enough ${\varepsilon}$, the function ${\phi_{{\varepsilon},B}}$ has the asymptotic form $${\phi_{{\varepsilon},B}}= A_0(\rho^2-1) + {\varepsilon}[A_1 (\rho^3-\rho) \cos{\theta}-2A_0\, {\mathbb{P}_{{\varepsilon}B}}B] + O({\varepsilon}^2)\,,$$ with the constants $$A_0:=\frac{aR^2+b}4\,,\qquad A_1:= \frac{5aR^2+b}{16R}\,.$$
As it is clear that $\phi_0:=
A_0(\rho^2-1)$ is the solution to the boundary value problem under consideration when ${\varepsilon}=0$, let us assume ${\varepsilon}$ is nonzero. Note that the equation for $\phi={\phi_{{\varepsilon},B}}$ is of the form $${\Delta}\phi -\frac {\varepsilon}R{\partial}_x\phi- (aR^2+b+2aR{\varepsilon}x)=O({\varepsilon}^2)\,,\qquad
\phi(1+{\varepsilon}B({\theta}),{\theta})=0\,.$$ One can then set $\phi_1:=(\phi-\phi_0)/{\varepsilon}$ and arrive at the equation $${\Delta}\phi_1 = 8A_1 x+O({\varepsilon})\,,\qquad \phi_1(1+{\varepsilon}B({\theta}),{\theta}) =
-2A_0B({\theta})+ O({\varepsilon})\,.$$ A short computation then shows that $h:= \phi_1-\frac43A_1x^3$ satisfies $${\Delta}h=O({\varepsilon})\,,\qquad h(1+{\varepsilon}B({\theta}),{\theta})=-2A_0B({\theta}) -\frac43A_1\cos^3{\theta}+O({\varepsilon})\,.$$ Hence $$\begin{aligned}
h&=-2A_0{\mathbb{P}_{{\varepsilon}B}}B-\frac43A_1{\mathbb{P}_{{\varepsilon}B}}(\cos^3{\theta})+O({\varepsilon})\\
&=-2A_0{\mathbb{P}_{{\varepsilon}B}}B-\frac43A_1\PP(\cos^3{\theta})+O({\varepsilon})\,.
\end{aligned}$$ As $\cos^3{\theta}= \frac14\cos 3{\theta}+ \frac34\cos{\theta}$, we then have $$\begin{aligned}
\phi_1&= -2A_0{\mathbb{P}_{{\varepsilon}B}}B+\frac{4A_1}3\left[\rho^3\cos^3{\theta}- \PP(\cos^3{\theta})\right]+O({\varepsilon})\\
&=-2A_0{\mathbb{P}_{{\varepsilon}B}}B+ \frac{A_1}3\left[\rho^3(\cos 3{\theta}+ 3\cos{\theta}) - (\rho^3\cos3{\theta}+ 3\rho\cos{\theta})\right]+O({\varepsilon})\\
&= -2A_0{\mathbb{P}_{{\varepsilon}B}}B+ A_1(\rho^3-\rho)\cos{\theta}+O({\varepsilon})\,.
\end{aligned}$$ This is the desired expression for $\phi$.
Note that we cannot replace ${\mathbb{P}_{{\varepsilon}B}}$ by $\PP$ in the formula presented in Proposition \[P.phi0\] because, generally, $\PP B$ would only be defined on the unit disk, not on the possibly larger domain $\{\rho<1+{\varepsilon}B({\theta})\}$.
For future reference, we record some formulas that stem from Proposition \[P.phi0\] and will be useful later on:
\[formulaphi1\] $$\begin{aligned}
\nabla {\phi_{{\varepsilon},B}}&= [2A_0\rho +{\varepsilon}A_1(3\rho^2-1)\cos{\theta}]\,
e_\rho-2A_0{\varepsilon}\,\nabla{\mathbb{P}_{{\varepsilon}B}}B \notag\\
&\qquad \qquad \qquad \qquad \qquad \qquad\qquad+ {\varepsilon}A_1(\rho^3-\rho)\,\nabla\cos{\theta}+ O({\varepsilon}^2)\,,\\
|\nabla {\phi_{{\varepsilon},B}}|^2 &= 4A_0^2\rho^2+ 4{\varepsilon}A_0 [
A_1(3\rho^3-\rho)\cos{\theta}-2A_0\rho\, {\partial}_\rho{\mathbb{P}_{{\varepsilon}B}}B]+
O({\varepsilon}^2)\,.
$$ Here $e_\rho:=(x/\rho,y/\rho)$ is the unit vector field in the radial direction and we have used that $e_\rho\cdot\nabla \cos{\theta}=0$.
Eventually we will need to evaluate the above formulas on the boundary of the domain, that is, at $\rho=1+{\varepsilon}B({\theta})$. In this direction, recall that the Dirichlet–Neumann map of the disk, defined as $${\Lambda_0}f({\theta})= {\partial}_\rho\PP f(1,{\theta})\,,$$ is the operator $C^{s+1}(\TT)\to C^s(\TT)$ given by $${\Lambda_0}f({\theta}):= \sum_{n\in\ZZ} f_n\,|n|\, e^{in{\theta}}\quad
\text{if} \quad f({\theta})= \sum_{n\in\ZZ} f_n\, e^{in{\theta}}\,.$$ Also, note that the Dirichlet–Neumann map of the domain ${\Omega}_{{\varepsilon}B}$ is an elliptic pseudodifferential operator of first order of the form $$\Lambda_{{\varepsilon}B}f:={\partial}_\rho{\mathbb{P}_{{\varepsilon}B}}f|_{\rho={1+{\varepsilon}B}}= {\Lambda_0}f+O({\varepsilon})\,,$$ where the above notation can be taken to mean that the $C^s$ norm of the error is bounded by $C{\varepsilon}\|f\|_{C^{s+1}}$.
Computing the variations with respect to the domain {#S.variations}
===================================================
Our next objective is to compute how $\phi$ changes as we change the domain by perturbing the function $B$. More precisely, we aim to compute the derivative of $\phi$ with respect to $B$, which we will denote as $${\Phi}_{{\varepsilon},B,{\mathbf B}}:= \frac{{\partial}}{{\partial}t}\bigg|_{t=0}\phi_{{\varepsilon},B+ t{\mathbf B}}\,,$$ where ${\mathbf B}({\theta})$ is a function defined on $\TT$. In this section, we are mainly interested in the derivative at $B=0$, ${\Phi_{{\varepsilon},0,{\mathbf B}}}$.
In the statement of the next proposition, we will need the operator $T:C^{s+1}(\TT)\to C^{s+1}({\mathbb D})$ defined as $$Tf(\rho,{\theta}):= \sum_{n\in\ZZ\backslash\{0\}} f_n\, (\rho^{|n|-1}-\rho^{|n|+1})\,
e^{i[n-\operatorname{sign}(n)]{\theta}}$$ for $f({\theta})=\sum_{n\in\ZZ} f_n\, e^{in{\theta}}$. Here and in what follows, $\operatorname{sign}(n):=n/|n|$ is the sign of the nonzero integer $n$.
\[P.dphi\] For ${\mathbf B}\in C^{s+1}(\TT)$ and any small enough ${\varepsilon}$, $${\Phi_{{\varepsilon},0,{\mathbf B}}}= -2{\varepsilon}A_0\,\PP {\mathbf B}+{\varepsilon}^2 \left[ \frac{A_0}{2R} \, T{\mathbf B}-2A_1\, \PP(\cos{\theta}\, {\mathbf B})\right] + O({\varepsilon}^3)\,.$$
Differentiating Equation with respect to $B$ at $B=0$, we obtain that ${\Phi}\equiv {\Phi_{{\varepsilon},0,{\mathbf B}}}$ satisfies the equation $$\label{Lapdp}
{\Delta}{\Phi}-\frac{\varepsilon}{R+{\varepsilon}x}{\partial}_x{\Phi}= {\varepsilon}^2\Big[(R+{\varepsilon}x)^2H_1''({\varepsilon}^2\phi_{{\varepsilon},0})-\frac{{\varepsilon}^2}{2}{\widetilde{F}}_1''({\varepsilon}^2\phi_{{\varepsilon},0})\Big]\, {\Phi}$$ in ${\mathbb D}$. Likewise, differentiating the boundary condition we obtain that $${\Phi}(1,{\theta})= -{\varepsilon}{\partial}_\rho{\phi_{{\varepsilon},0}}(1,{\theta})\, {\mathbf B}({\theta})\,.$$ In view of the asymptotics for ${\phi_{{\varepsilon},0}}$ computed in Proposition \[P.phi0\], this boundary condition can be rewritten as $${\Phi}(1,{\theta})= -2{\varepsilon}A_0 {\mathbf B}({\theta}) -2 {\varepsilon}^2A_1 {\mathbf B}({\theta})\, \cos{\theta}+ O({\varepsilon}^3)\,,$$ and ${\Phi}$ has the expression $${\Phi}=-2{\varepsilon}A_0 \PP{\mathbf B}+O({\varepsilon}^2)\,.$$ Equation is then of the form $${\Delta}{\Phi}-\frac{\varepsilon}{R}{\partial}_x{\Phi}=O({\varepsilon}^2){\Phi}+ O({\varepsilon}^2){\partial}_x{\Phi}=O({\varepsilon}^3)\,.$$
Assuming that ${\varepsilon}\neq0$ (since otherwise $\Phi=0$), let us set $$\Phi_1:= (\Phi-{\varepsilon}\Phi_0)/{\varepsilon}^2\,,$$ with $\Phi_0:= -2A_0\PP{\mathbf B}$. A short calculation shows that $\Phi_1$ must solve the equation $${\Delta}\Phi_1=\frac 1R{\partial}_x\Phi_0+ O({\varepsilon})$$ with the boundary condition $$\Phi_1(1,{\theta})=-2A_1{\mathbf B}({\theta}) \cos{\theta}+ O({\varepsilon})\,.$$
Since $${\partial}_x= \cos{\theta}\,{\partial}_\rho - \frac1\rho\sin{\theta}\,{\partial}_{\theta}=\frac12
e^{i{\theta}} \left({\partial}_\rho +\frac i{\rho}{\partial}_{\theta}\right) +
\frac12e^{-i{\theta}}\left({\partial}_\rho -\frac i\rho {\partial}_{\theta}\right)\,,$$ one readily finds that $${\partial}_x\Phi_0 = -2A_0 \sum_{n\in\ZZ\backslash\{0\}} |n|{\mathbf B}_n \,\rho^{|n|-1} e^{i[n-{\operatorname{sign}}(n)]{\theta}}\,,$$ if ${\mathbf B}= \sum_{n\in\ZZ} {\mathbf B}_n\, e^{in{\theta}}$. Let us now note that if we set $$\Phi_2:= -\frac{A_0}{2R}\sum_{n\in\ZZ\backslash\{0\}} {\mathbf B}_n\rho^{|n|+1}e^{i[n-\operatorname{sign}(n)]{\theta}}\,,$$ then ${\Delta}\Phi_2=\frac{{\partial}_x\Phi_0}{R}$. Consequently, the function $\Phi_3:=
\Phi_1-\Phi_2$ satisfies the equation $${\Delta}\Phi_3=O({\varepsilon})$$ and the boundary condition $$\begin{aligned}
\Phi_3(1,{\theta})&= -2A_1{\mathbf B}({\theta}) \cos{\theta}-\Phi_2(1,{\theta})+O({\varepsilon})\\
&=
-2A_1{\mathbf B}({\theta}) \cos{\theta}+ \frac{A_0}{2R}\sum_{n\in\ZZ\backslash\{0\}}
{\mathbf B}_ne^{i[n-\operatorname{sign}(n)]{\theta}} + O({\varepsilon})\,.\end{aligned}$$ This shows that $$\begin{aligned}
\Phi_3&= -2A_1\PP({\mathbf B}\cos{\theta}) + \frac{A_0}{2R}\sum_{n\in\ZZ\backslash\{0\}}
{\mathbf B}_n\PP(e^{i[n-\operatorname{sign}(n)]{\theta}})+ O({\varepsilon})\\
&= -2A_1\PP({\mathbf B}\cos{\theta}) + \frac{A_0}{2R}\sum_{n\in\ZZ\backslash\{0\}}
{\mathbf B}_n\rho^{|n|-1} e^{i[n-\operatorname{sign}(n)]{\theta}}+ O({\varepsilon})\,,\end{aligned}$$ which results in $$\begin{aligned}
\Phi_1&= \Phi_2+\Phi_3= \frac{A_0}{2R} \, T{\mathbf B}-2A_1\, \PP(\cos{\theta}\, {\mathbf B}) + O({\varepsilon})\,,\end{aligned}$$ as claimed.
As a consequence of Proposition \[P.dphi\], we record that $$\begin{aligned}
\label{formulas2}
{\partial}_\rho{\Phi_{{\varepsilon},0,{\mathbf B}}}|_{\rho=1} = -2{\varepsilon}A_0\, {\Lambda_0}{\mathbf B}- 2{\varepsilon}^2\, \left[\frac{A_0}R\,
T'{\mathbf B}+ A_1\,
{\Lambda_0}(\cos{\theta}\, {\mathbf B})\right] + O({\varepsilon}^3)\,,\end{aligned}$$ where $T':C^{s+1}(\TT)\to C^{s+1}(\TT)$ is the operator defined as $$\label{T'}
T' f({\theta}) := \frac12\sum_{n\in\ZZ\backslash\{0\}}f_n\, e^{i[n-\operatorname{sign}(n)]{\theta}}$$ for $f({\theta})= \sum_{n\in\ZZ} f_n\, e^{in{\theta}}$.
Analysis of the Neumann condition and conclusion of the proof {#S.Neumann}
=============================================================
Let us now set $$\label{cF}
{{\mathcal F}}({\varepsilon},B)({\theta}):= |\nabla{\phi_{{\varepsilon},B}}(1+{\varepsilon}B({\theta}),{\theta})|^2 - {c_{{\varepsilon},B}}\,
[R+{\varepsilon}(1+{\varepsilon}B({\theta}))\cos{\theta}]^2\,,$$ where the constant ${c_{{\varepsilon},B}}$ will be defined later. In view of the definition of the function $F$ (Equation ), one should notice that the Neumann condition holds with a constant $c={\varepsilon}^2 {c_{{\varepsilon},B}}$ if and only if ${{\mathcal F}}({\varepsilon},B)+F_R$ is the zero function.
Next we pick the constant ${c_{{\varepsilon},B}}$ so that ${{\mathcal F}}({\varepsilon},B)$ is $L^2$-orthogonal to $\cos{\theta}$. The reason for which we do so will be clear later. This amounts to setting $$\label{ceB}
{c_{{\varepsilon},B}}:= \frac{\int_0^{2\pi} |\nabla{\phi_{{\varepsilon},B}}(1+{\varepsilon}B({\theta}),{\theta})|^2\cos{\theta}\,
d{\theta}}{\int_0^{2\pi} [R+{\varepsilon}(1+{\varepsilon}B({\theta}))\cos{\theta}]^2\cos{\theta}\, d{\theta}}\,.$$
The following result guarantees that this choice of ${c_{{\varepsilon},B}}$ makes sense for all small enough ${\varepsilon}$, including ${\varepsilon}=0$, and shows that ${{\mathcal F}}(0,B)$ is in fact the constant $${\kappa}:=4A_0(A_0 -A_1R)\,,$$ which depends on $R$ but not on $B$. In what follows, we employ the notation $$\langle f,g\rangle:= \int_0^{2\pi} f({\theta})\, g({\theta})\, d{\theta}$$ for the $L^2$ product on $\TT$.
\[P.ceB\] For small enough ${\varepsilon}$ and any $B$, $$\begin{aligned}
{c_{{\varepsilon},B}}&= \frac{4A_0A_1}R + O({\varepsilon})\,,\\
{{\mathcal F}}({\varepsilon},B)&={\kappa}+O({\varepsilon})\,,\\
{{\mathcal F}}({\varepsilon},0)&={\kappa}+O({\varepsilon}^2)\,.
\end{aligned}$$
Let us assume that ${\varepsilon}\neq0$. In view of Equation , let us write ${c_{{\varepsilon},B}}= c_1/c_2$, with
\[cj\] $$\begin{aligned}
c_1&:= \int_0^{2\pi} |\nabla{\phi_{{\varepsilon},B}}(1+{\varepsilon}B({\theta}),{\theta})|^2\cos{\theta}\,
d{\theta}\,,\\
c_2&:= \int_0^{2\pi} [R+{\varepsilon}(1+{\varepsilon}B({\theta}))\cos{\theta}]^2\cos{\theta}\, d{\theta}\,.
\end{aligned}$$
It follows from the formula for $|\nabla{\phi_{{\varepsilon},B}}|^2$ derived in that $$\begin{aligned}
c_1 &= \int_0^{2\pi} \left[ 4A_0^2 + 8A_0{\varepsilon}\big( A_0(B - {\Lambda_0}B) +
A_1\cos{\theta}\big)\right] \cos{\theta}\, d{\theta}+O({\varepsilon}^2)\notag\\
&= 8{\varepsilon}A_0^2 \langle B-{\Lambda_0}B, \cos{\theta}\rangle
+8{\varepsilon}A_0A_1\int_0^{2\pi}\cos^2{\theta}\, d{\theta}+O({\varepsilon}^2) \notag\\
&= 8\pi{\varepsilon}A_0A_1+O({\varepsilon}^2)\,,\label{c1}
\end{aligned}$$ where we have used that $$\langle B-{\Lambda_0}B, \cos{\theta}\rangle = \langle B, (1-{\Lambda_0})\cos{\theta}\rangle=0$$ for any $B$ because ${\Lambda_0}$ is self-adjoint and ${\Lambda_0}(\cos{\theta})= \cos{\theta}$.
The computation of $c_2$ is straightforward: $$\label{c2}
c_2= \int_0^{2\pi}[R^2+2{\varepsilon}R\cos{\theta}]\cos{\theta}\, d{\theta}+ O({\varepsilon}^2)= 2\pi
{\varepsilon}R+ O({\varepsilon}^2)\,.$$ This readily implies that ${c_{{\varepsilon},B}}$ can be defined at ${\varepsilon}=0$ by continuity and yields the formula for ${c_{{\varepsilon},B}}$ presented in the statement. Also, the above formulas immediately imply that $$\begin{aligned}
{{\mathcal F}}({\varepsilon},B)&= |\nabla{\phi_{{\varepsilon},B}}(1+{\varepsilon}B({\theta}),{\theta})|^2- {c_{{\varepsilon},B}}[R+{\varepsilon}(1+{\varepsilon}B({\theta}))\cos{\theta}]^2\\
&= 4A_0(A_0 -A_1R)+ O({\varepsilon})\,,\end{aligned}$$ as claimed.
To prove that ${{\mathcal F}}({\varepsilon},0)={\kappa}+O({\varepsilon}^2)$, it is convenient to define the ($R$-dependent) constant $$\begin{aligned}
c_3:=\lim_{{\varepsilon}\to 0}\frac{c_{{\varepsilon},0}-\frac{4A_0A_1}{R}}{{\varepsilon}}\,.\end{aligned}$$ A straightforward computation using Equations and shows that $${{\mathcal F}}({\varepsilon},0)={\kappa}-R^2c_3{\varepsilon}+ O({\varepsilon}^2)\,.$$ We claim that $c_3=0$. Indeed, noticing that the previous results imply that $$\label{ce0}
2\pi{\varepsilon}Rc_{{\varepsilon},0}=\int_0^{2\pi}|\nabla\phi_{{\varepsilon},0}(1,{\theta})|^2\cos {\theta}\,d{\theta}=8\pi{\varepsilon}A_0A_1+2\pi {\varepsilon}^2Rc_3+O({\varepsilon}^3)\,,$$ to compute $c_3$ it is enough to obtain the ${\varepsilon}^2$-term of $|\nabla\phi_{{\varepsilon},0}(1,{\theta})|^2$. Recall that the function $\phi_{{\varepsilon},0}$ is the solution to the Equation with Dirichlet boundary condition $\phi_{{\varepsilon},0}(1,{\theta})=0$. According to Proposition \[P.phi0\], it is easy to check that the (${\varepsilon}$-dependent) function $\phi_2$ defined as $$\phi_{{\varepsilon},0}=:A_0(\rho^2-1)+{\varepsilon}A_1(\rho^2-1)\rho\cos{\theta}+{\varepsilon}^2\phi_2\,,$$ satisfies the boundary value problem $$\Delta \phi_2=A_2+A_3x^2+A_4y^2+O({\varepsilon})\,, \qquad \phi_2(1,{\theta})=0\,,$$ for some explicit constants $A_2,A_3,A_4$ (depending on $R$ but not on ${\varepsilon}$) that are not be relevant for our purposes. The solution to this problem is therefore of the form $$\phi_2=\frac{A_2}{4}(\rho^2-1)+\frac{A_3+A_4}{32}(\rho^4-1)+\frac{A_3-A_4}{24}\rho^2(\rho^2-1)\cos 2{\theta}+O({\varepsilon})\,.$$ Using again Equation we obtain that $$|\nabla \phi_{{\varepsilon},0}(1,{\theta})|^2=4A_0^2+8{\varepsilon}A_0A_1\cos{\theta}+4{\varepsilon}^2[A_1^2\cos^2{\theta}+A_0\nabla\phi_2(1,{\theta})\cdot e_\rho] +O({\varepsilon}^3)\,,$$ where the scalar product $\nabla\phi_2(1,{\theta})\cdot e_\rho$ is given by $$\nabla\phi_2(1,{\theta})\cdot e_\rho=\frac{4A_2+A_3+A_4}{8}+\frac{A_3-A_4}{12}\cos 2{\theta}+O({\varepsilon})\,.$$ It is then immediate that the ${\varepsilon}^2$-term of $|\nabla\phi_{{\varepsilon},0}(1,{\theta})|^2$ does not contribute to the integral in Equation , thus proving that $c_3=0$ as claimed.
It stems from Proposition \[P.ceB\] that the function $$\label{cG}
{{\mathcal G}}({\varepsilon},B):= \frac1{\varepsilon}\left[{{\mathcal F}}({\varepsilon},B)-{\kappa}\right]$$ can be defined at ${\varepsilon}=0$ by continuity, so that ${{\mathcal G}}(0,0)=0$, resulting in a map defined for all $|{\varepsilon}|<{\varepsilon}_0$, where ${\varepsilon}_0$ is some positive constant. A more convenient way of looking at this map, however, is by restricting our attention to those variations of the domain that are even and orthogonal to $\cos{\theta}$. Hence, let us now define, for each non-integer $s>2$, the space $$\begin{aligned}
X_s :=\left\{ f\in C^s(\TT): f({\theta})= f(-{\theta})\,,\; \langle f,\cos{\theta}\rangle=0\right\}\,,\end{aligned}$$ and its ball of radius $1$, $$\begin{aligned}
X_{s}^1 :=\left\{ f\in C^s(\TT): \|f\|_{C^s}<1,\; f({\theta})= f(-{\theta})\,,\; \langle f,\cos{\theta}\rangle=0\right\}\,.\end{aligned}$$ As $\langle {{\mathcal F}}({\varepsilon}, B),\cos{\theta}\rangle=0$ by the definition of ${c_{{\varepsilon},B}}$, and ${\phi_{{\varepsilon},B}}$ is an even function if $B$ is (cf. Proposition \[P.existence\]), our previous results then immediately imply the following:
\[P.cG\] Given any $R>0$, there is some ${\varepsilon}_0>0$ such that the formula defines a map $${{\mathcal G}}: (-{\varepsilon}_0,{\varepsilon}_0)\times X_{s+1}^1\to X_s\,.$$
In the following theorem we derive the key property of the map ${{\mathcal G}}$: as its domain consists of the even functions orthogonal to $\cos{\theta}$, we can show that its derivative with respect to $B$ at certain points is an invertible map:
\[T.IFT\] For any $R>0$ such that $aR^2-3b\neq 0$, the Fréchet derivative $$D_B{{\mathcal G}}(0,0): X_{s+1}\to X_s$$ is one-to-one.
It follows from the definition of ${{\mathcal F}}$ (Equation ) and of ${\Phi_{{\varepsilon},B,{\mathbf{B}}}}$ that $$\begin{gathered}
D_B{{\mathcal F}}({\varepsilon},0){\mathbf B}= (2\nabla{\phi_{{\varepsilon},0}}\cdot \nabla{\Phi_{{\varepsilon},0,{\mathbf B}}}+ {\varepsilon}{\mathbf B}\, {\partial}_\rho|\nabla{\phi_{{\varepsilon},0}}|^2)|_{\rho=1} \\
-{{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}(R+{\varepsilon}\cos{\theta})^2 -2{c_{{\varepsilon},0}}{\varepsilon}^2(R+{\varepsilon}\cos{\theta}) {\mathbf B}\cos{\theta}\,,
\end{gathered}$$ where the constant ${{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}$ is given by the derivative $${{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}:= \left.\frac{\partial}{{\partial}t}\right|_{t=0} c_{{\varepsilon}, t{\mathbf B}}\,.$$
We readily obtain from formulas and that $$\begin{gathered}
(2\nabla{\phi_{{\varepsilon},0}}\cdot \nabla{\Phi_{{\varepsilon},0,{\mathbf B}}}+{\varepsilon}{\mathbf B}{\partial}_\rho
|\nabla{\phi_{{\varepsilon},0}}|^2)|_{\rho=1} =
8{\varepsilon}A_0^2({\mathbf B}-{\Lambda_0}{\mathbf B}) \\
+8{\varepsilon}^2A_0
\left[4 A_1\, {\mathbf B}\cos{\theta}-A_1\cos{\theta}\,{\Lambda_0}{\mathbf B}-
A_1{\Lambda_0}({\mathbf B}\cos{\theta})-\frac{A_0}R
T'{\mathbf B}\right] +O({\varepsilon}^3)\,.\label{rollo1}
\end{gathered}$$ Since ${{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}$ is obviously of order $O({\varepsilon})$, cf. Proposition \[P.ceB\], it suffices to employ the leading order terms of this expression to arrive at $$D_B{{\mathcal F}}({\varepsilon},0){\mathbf B}= 8{\varepsilon}A_0^2({\mathbf B}-{\Lambda_0}{\mathbf B})-{{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}R^2+O({\varepsilon}^2) \,.$$
Hence, in order to compute this derivative modulo an error of order ${\varepsilon}^2$ we only need to derive asymptotics for ${{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}$. To do so, we write $$c_{{\varepsilon}, t{\mathbf B}}=c_1/c_2$$ as in (where now $B:= t{\mathbf B}$) and compute $${{\mathcal C}}_j:= \left.\frac{\partial}{{\partial}t}\right|_{t=0} c_j\,.$$ Notice that, as we showed in the proof of Proposition \[P.ceB\] that $c_j=O({\varepsilon})$, we will need to compute ${{\mathcal C}}_j$ to order $O({\varepsilon}^2)$.
Let us start with ${{\mathcal C}}_2$. Since $c_2:= \langle [R+{\varepsilon}(1+t{\varepsilon}{\mathbf B})\cos{\theta}]^2,\cos{\theta}\rangle$, it is immediate that $$\begin{aligned}
{{\mathcal C}}_2=2{\varepsilon}^2R\langle{\mathbf B},\cos^2{\theta}\rangle + O({\varepsilon}^3)\,.\end{aligned}$$ To compute ${{\mathcal C}}_1$, we again employ the formula , now to second order: $$\begin{aligned}
{{\mathcal C}}_1&= \int_0^{2\pi} \left.\left( 2\nabla{\phi_{{\varepsilon},0}}\cdot \nabla{\Phi_{{\varepsilon},0,{\mathbf B}}}+{\varepsilon}{\mathbf B}{\partial}_\rho
|\nabla{\phi_{{\varepsilon},0}}|^2\right)\right|_{\rho=1}\cos{\theta}\, d{\theta}\\
&=8{\varepsilon}A_0^2 \langle {\mathbf B}-{\Lambda_0}{\mathbf B},\cos{\theta}\rangle +8{\varepsilon}^2A_0
\bigg[4 A_1 \langle{\mathbf B},\cos^2{\theta}\rangle\\
&\qquad \qquad -A_1 \langle{\Lambda_0}{\mathbf B},\cos^2{\theta}\rangle -
A_1\langle{\Lambda_0}({\mathbf B}\cos{\theta}),\cos{\theta}\rangle-\frac{A_0}R
\langle T'{\mathbf B},\cos{\theta}\rangle \bigg] +O({\varepsilon}^3)\\
&= 8{\varepsilon}^2A_0
\bigg(3 A_1 \langle{\mathbf B},\cos^2{\theta}\rangle
-A_1 \langle{\Lambda_0}{\mathbf B},\cos^2{\theta}\rangle -\frac{A_0}R
\langle T'{\mathbf B},\cos{\theta}\rangle \bigg) +O({\varepsilon}^3)\,.\end{aligned}$$ Here we have used that ${\Lambda_0}$ is self-adjoint and that ${\Lambda_0}(\cos{\theta})=\cos{\theta}$.
Now we need to compute the scalar products appearing in the two previous formulas in terms of the Fourier coefficients ${\mathbf B}_n$ (note that ${\mathbf B}_{-n}= {\mathbf B}_n$ because ${\mathbf B}$ is even): $$\begin{aligned}
\langle{\mathbf B},\cos^2{\theta}\rangle&= \frac14\langle{\mathbf B},e^{2i{\theta}}+e^{-2i{\theta}}+2\rangle=\pi ({\mathbf B}_2+{\mathbf B}_0)\,,\\
\langle{\Lambda_0}{\mathbf B},\cos^2{\theta}\rangle&=
\frac14\langle{\mathbf B},{\Lambda_0}(e^{2i{\theta}}+e^{-2i{\theta}}+2)\rangle=
2\pi {\mathbf B}_2\,,\\
\langle T'{\mathbf B},\cos{\theta}\rangle&= \frac14\sum_{n\in\ZZ\backslash\{0\}} \left\langle
{\mathbf B}_n \, e^{i[n-\operatorname{sign}(n)]{\theta}} ,
{e^{i{\theta}}+e^{-i{\theta}}}\right\rangle = \pi{\mathbf B}_2\,.\end{aligned}$$ Using the formulas for $c_1$ and $c_2$ derived in the proof of Proposition \[P.ceB\], this immediately yields $$\begin{aligned}
{{{\mathcal C}}_{{\varepsilon},{\mathbf B}}}&=\frac{{{\mathcal C}}_1}{c_2}-{c_{{\varepsilon},B}}\frac{{{\mathcal C}}_2}{c_2}\\
&= -{\varepsilon}\frac{8A_0^2}{R^2}\left(
\frac12{\mathbf B}_2-\frac{A_1R}{A_0}{\mathbf B}_0\right) + O({\varepsilon}^2)\,,\end{aligned}$$ which results in $$D_B{{\mathcal F}}({\varepsilon},0){\mathbf B}= 8{\varepsilon}A_0^2\left(
{\mathbf B}-{\Lambda_0}{\mathbf B}+\frac12{\mathbf B}_2-\frac{A_1R}{A_0}{\mathbf B}_0\right)+ O({\varepsilon}^2)\,.$$
We are now ready to analyze the differential of ${{\mathcal G}}$, which we have shown to be given by the formula $$D_B{{\mathcal G}}(0,0){\mathbf B}= 8 A_0^2\left(
{\mathbf B}-{\Lambda_0}{\mathbf B}+\frac12{\mathbf B}_2-\frac{A_1R}{A_0}{\mathbf B}_0\right)\,,$$ understood as a map $X_{s+1}\to X_s$. We recall that, as ${\mathbf B}$ is an even function ortogonal to $\cos{\theta}$, the Fourier series ${\mathbf B}=\sum_{n\in\ZZ} {\mathbf B}_n e^{in{\theta}}$ can be equivalently written as $${\mathbf B}({\theta})= {\mathbf B}_0 + 2\sum_{n=2}^\infty {\mathbf B}_n\cos n{\theta}\,.$$ Therefore, the action of the linear elliptic operator $D_B{{\mathcal G}}(0,0)$ is given by $$\begin{aligned}
D_B{{\mathcal G}}(0,0){\mathbf B}= 8 A_0^2\left[
\left(1-\frac{A_1R}{A_0}\right){\mathbf B}_0+\frac12{\mathbf B}_2 -2\sum_{n=2}^\infty (n-1){\mathbf B}_n\cos n{\theta}\right]\,.\end{aligned}$$ Note that $A_1R\neq A_0$ for all $a,b,R>0$ such that $aR^2-3b\neq 0$ because $$A_0-A_1R=\frac{3b-aR^2}{16}\,.$$ This implies that the kernel of the map $D_B{{\mathcal G}}(0,0): X_{s+1}\to
X_s$ is trivial, and that its range is the whole space $X_s$, as claimed.
In the following corollary we show that, by the implicit function theorem for Banach spaces, Theorem \[T.IFT\] yields the existence of solutions to the overdetermined boundary value problem – for all small enough ${\varepsilon}$ and all $R$ such that $aR^2-3b>0$. In turn, these define piecewise smooth stationary Euler flows of compact support via Lemma \[L.weak\], thereby completing the proof of the main result of the paper (Theorem \[T.main\]). Recall that the constant $F_R$ appears in the definition of the function $F$, cf. Equation .
\[C.IFT\] Fix any $R>0$ such that $aR^2-3b>0$. Then, for any small enough ${\varepsilon}$ there is a unique $B\in X_{s+1}$ in a $C^{s+1}$ neighborhood of $0$ such that $\psi:= {\varepsilon}^2{\phi_{{\varepsilon},B}}$ satisfies Equation in ${\Omega}_{R,{\varepsilon}}:={\Omega}_{{\varepsilon}B}$ and the overdetermined boundary conditions - with $F_R:=-{\kappa}>0$ and $c={\varepsilon}^2 c_{{\varepsilon},B}$.
Since ${{\mathcal G}}(0,0)=0$, in view of Theorem \[T.IFT\], the implicit function theorem guarantees that if $|{\varepsilon}|$ is small enough, there is a unique function $B$ in a small neighborhood of $0$ in $X_{s+1}^1$ such that $${{\mathcal G}}({\varepsilon},B)=0\,.$$ This is equivalent to saying that $$|\nabla\psi|^2-{\varepsilon}^2{c_{{\varepsilon},B}}r^2-{\varepsilon}^2 {\kappa}=0$$ on ${\partial}{\Omega}_{{\varepsilon}B}$, with $\psi:={\varepsilon}^2{\phi_{{\varepsilon},B}}$. The assumption that $F^2(0)={\varepsilon}^2F_R =-{\varepsilon}^2{\kappa}$ then ensures that we have a solution to the overdetermined boundary problem –, as claimed. Observe that the condition $aR^2-3b>0$ implies that $${\kappa}=\frac{(aR^2+b)(3b-aR^2)}{16}<0$$ and hence $F_R>0$. Accordingly, the function $F(\psi)$ is well defined: $$F(\psi)=\Big({\varepsilon}^2F_R-2b\psi+O(\psi^2)\Big)^{1/2}$$ because $\psi=O({\varepsilon}^2)$ and $\psi<0$ in ${\Omega}_{{\varepsilon}B}$ (cf. Proposition \[P.existence\]).
Different choices for the functions $F$ and $H$ {#S.final}
===============================================
As we mentioned in the Introduction, for the sake of concreteness we have chosen the functions $H$ and $F$ as described in Theorem \[T.main\]. However, the method introduced in this paper is flexible enough to construct compactly supported stationary Euler flows with other choices for the functions $H$ and $F$. To illustrate this additional flexibility, in this section we show how a straightforward modification of the previous computations allows us to prove the following:
Take any non-integer $s>2$ and any functions ${\widetilde{F}},H\in C^s((-1,0])$ with $${\widetilde{F}}(0)={\widetilde{F}}'(0)=0\,,\qquad H'(0)> 0\,.$$ Then the following statements hold:
1. For each small enough ${\varepsilon}>0$ and any $R>0$, there exists a nontrivial, piecewise $C^s$, axisymmetric stationary Euler flow of compact support $u$ of the form described in Lemma \[L.weak\] for a suitable $C^{s+1}$ planar domain ${\Omega}_{R,{\varepsilon}}$.
2. The boundary of ${\Omega}_{R,{\varepsilon}}$ is a small deformation of a disk of radius ${\varepsilon}$, given by an equation of the form $z^2+(r-R)^2-{\varepsilon}^2 = O({\varepsilon}^3)$.
3. The functions that define the solution are $$F(\psi) := {\varepsilon}F_R+ {\widetilde{F}}(\psi)$$ and $H(\psi)$, where $F_R$ is the positive constant $$\label{FR}
F_R:=\frac{R^2H'(0)}{4}\,.$$
4. The function $\psi$ is of class $C^{s+1}$ in ${\Omega}_{R,{\varepsilon}}$ up to the boundary, and has the form $$\psi=\frac14 H'(0)R^2\Big[(r-R)^2+z^2-{\varepsilon}^2\Big]+O({\varepsilon}^3)\,.$$ Moreover, $F\circ \psi>0$ and $H\circ\psi$ are of class $C^s$ in ${\Omega}_{R,{\varepsilon}}$. In particular, the vorticity is of class $C^{s-1}$ up the boundary.
Indeed, using the same notation as in Section \[S.Dirichlet\], and noticing that $$(F^2)'(\psi)={\varepsilon}O(\psi)\,,$$ Equation takes the form $${\Delta}\phi -\frac{\varepsilon}{R+{\varepsilon}x}{\partial}_x \phi= aR^2+ 2aR{\varepsilon}x + O({\varepsilon}^2)\,,$$ where we have defined the constant $a:=H'(0)$. Notice that this is exactly the same as Equation with $b=0$. Repeating all the arguments in Sections \[S.Dirichlet\]–\[S.Neumann\], we obtain the same equations and results as in these sections with $b=0$. In particular, the constant ${\kappa}$ in Proposition \[P.ceB\] is given by $${\kappa}=-\frac{a^2R^4}{16}<0\,,$$ ${{\mathcal F}}({\varepsilon},0)={\kappa}+O({\varepsilon}^2)$, and the invertibility condition in Theorem \[T.IFT\] is simply $a\neq0$. The Neumann boundary condition is then satisfied taking $F(0)={\varepsilon}F_R$, with $F_R$ as in Equation . Notice that $F(\psi)={\varepsilon}F_R+O(\psi^2)={\varepsilon}F_R+O({\varepsilon}^4)>0$ in ${\Omega}_{R,{\varepsilon}}$.
Acknowledgements {#acknowledgements .unnumbered}
================
M.D.-V. is supported by the grants MTM2016-75897-P (AEI/FEDER) and ED431F 2017/03 (Xunta de Galicia), and by the Ramón y Cajal program of the Spanish Ministry of Science. A.E. is supported by the ERC Starting Grant 633152. D.P.-S. is supported by the grants MTM2016-76702-P (MINECO/FEDER) and Europa Excelencia EUR2019-103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001.
[99]{}
D. Chae, P. Constantin, Remarks on a Liouville-type theorem for Beltrami flows, Int. Math. Res. Not. 2015, 10012–10016.
A. Choffrut, L. Székelyhidi, Weak solutions to the stationary incompressible Euler equations, SIAM J. Math. Anal. 46 (2014) 4060–4074.
P. Constantin, J. La, V. Vicol, Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications, Geom. Funct. Anal. 29 (2019) 1773–1793.
E. Delay, P. Sicbaldi, Extremal domains for the first eigenvalue in a general compact Riemannian manifold, [Discrete Contin. Dyn. Syst.]{} [35]{} (2015) 5799–5825.
M. Domínguez-Vázquez, A. Enciso, D. Peralta-Salas, Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities, Adv. Math. 351 (2019) 718–760.
L.E. Fraenkel, M.S. Berger, A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974) 14–51.
A.V. Gavrilov, A steady Euler flow with compact support, Geom. Funct. Anal. 29 (2019) 190–197.
Q. Jiu, Z. Xin, Smooth approximations and exact solutions of the 3D steady axisymmetric Euler equations, Comm. Math. Phys. 287 (2009) 323–350.
N. Nadirashvili, Liouville theorem for Beltrami flow, Geom. Funct. Anal. 24 (2014) 916–921.
F. Pacard, P. Sicbaldi, Extremal domains for the first eigenvalue of the Laplace-Beltrami operator, [Ann. Inst. Fourier]{} [59]{} (2009) 515–542.
J. Serrin, A symmetry problem in potential theory, [Arch. Rational Mech. Anal.]{} [43]{} (1971) 304–318.
|
---
abstract: 'We study the detailed evolution of the fine-structure constant $\alpha$ in the string-inspired runaway dilaton class of models of Damour, Piazza and Veneziano. We provide constraints on this scenario using the most recent $\alpha$ measurements and discuss ways to distinguish it from alternative models for varying $\alpha$. For model parameters which saturate bounds from current observations, the redshift drift signal can differ considerably from that of the canonical $\Lambda$CDM paradigm at high redshifts. Measurements of this signal by the forthcoming European Extremely Large Telescope (E-ELT), together with more sensitive $\alpha$ measurements, will thus dramatically constrain these scenarios.'
address:
- 'Centro de Astrofísica, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal'
- 'Instituto de Astrofísica e Ciências do Espaço, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal'
- 'Institut de Física d’Altes Energies, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain'
- 'Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 16, 69120, Heidelberg, Germany'
- 'Sub-department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK'
- 'Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark'
author:
- 'C. J. A. P. Martins'
- 'P. E. Vielzeuf'
- 'M. Martinelli'
- 'E. Calabrese'
- 'S. Pandolfi'
bibliography:
- 'dilaton.bib'
title: 'Evolution of the fine-structure constant in runaway dilaton models'
---
Cosmology ,Dynamical dark energy ,Fine-structure constant ,Astrophysical observations
Introduction
============
The observational evidence for cosmic acceleration, first inferred from the luminosity distance of type Ia supernovae in 1998 [@SN1; @SN2], opened a new avenue in cosmological research. The most obvious task in this endeavor is to identify the source of this acceleration—the so-called Dark Energy—and in particular to ascertain whether it is due to a cosmological constant or to a new dynamical degree of freedom. While the former option, corresponding to the canonical $\Lambda$CDM paradigm, is arguably the simplest, many alternative models have been proposed and still have to be tested [@Dark].
The most natural way to model dynamical energy is through a scalar field, of which the recently discovered Higgs is the obvious example [@ATLAS; @CMS]. String theory predicts the presence of a scalar partner of the spin-2 graviton, the dilaton, hereafter denoted $\phi$. Here, we will study the cosmological consequences of a particular class of string-inspired models, the runaway dilaton scenario of Damour, Piazza and Veneziano [@DPV1; @DPV2]. In this scenario, which among other things provides a way to reconcile a massless dilaton with experimental data, the dilaton decouples while cosmologically attracted towards infinite bare coupling, and the coupling functions have a smooth finite limit $$\label{eq:coupfunc}
B_i(\phi)=c_i+{\cal O}(e^{-\phi})\,.$$ As discussed in [@DPV2], provided there’s a significant (order unity) coupling to the dark sector, the runaway of the dilaton towards strong coupling may yield violations of the Equivalence Principle and variations of the fine-structure constant $\alpha$ that are potentially measurable.
More than a decade after the original analysis the available $\alpha$ measurements have improved substantially [@LP1; @LP3], and it’s therefore timely to revisit these models. Additional gains in sensitivity will be provided by forthcoming facilities such as the E-ELT: its high-resolution ultra-stable spectrograph (HIRES) will significantly improve tests of the stability of fundamental couplings and will also be sensitive enough to carry out a first measurement of the redshift drift deep in the matter-dominated era [@Liske; @Hires]. The combination of both types of measurements is a powerful probe of dynamical dark energy, as it can distinguish between models that are indistinguishable at low redshifts [@Codex]. In what follows we obtain constraints on this runaway dilaton scenario using current $\alpha$ data, and also discuss how they may be further improved.
Runaway dilaton cosmology
=========================
As discussed in [@DPV1; @DPV2], the Einstein frame Lagrangian for this class of models is $$\label{eq:lagr}
{\cal L}=\frac{R}{16\pi G}-\frac{1}{8\pi G}\left(\nabla\phi\right)^2-\frac{1}{4}B_F(\phi)F^2+... \,.$$ where $R$ is the Ricci scalar and $B_F$ is the gauge coupling function. From this one can show [@DPV2] that the corresponding Friedmann equation, relating the Hubble parameter, $H$, to the dilaton and the other components of the universe is as follows $$\label{eq:friedmann}
3H^2=8\pi G\sum_i \rho_i+H^2\phi'^2\,,$$ where the sum is over the components of the universe, except the kinetic part of the dilaton field which is described by the last term (where the prime is the derivative with respect to the logarithm of the scale factor). The sum does include the potential part of the scalar field; the total energy density and pressure of the field are $$\label{phidens}
\rho_\phi=\rho_k+\rho_v=\frac{(H\phi')^2}{8\pi G}+V(\phi)$$ $$\label{phipres}
p_\phi=p_k+p_v=\frac{(H\phi')^2}{8\pi G}-V(\phi)\,;$$ here $k$ and $v$ correspond to the kinetic and potential parts of the field, with the latter providing the dark energy. On the other hand, the evolution equation for the scalar field is $$\label{eq:field}
\frac{2}{3-\phi'^2}\phi''+\left(1-\frac{p}{\rho}\right)\phi'=-\sum_i\alpha_i(\phi)\frac{\rho_i-3p_i}{\rho}\,.$$ Here $p=\sum_ip_i$, $\rho=\sum_i\rho_i$, and sums are again over all components except the kinetic part of the scalar field.
The $\alpha_i(\phi)$ are the couplings of the dilaton with each component $i$, so they characterize the effect of the various components of the universe in the dynamics of the field. One may generically expect that the dilaton has different couplings to different components [@DPV2]. Experimental constraints impose a tiny coupling to baryonic matter, as we will discuss presently. In these models, this small coupling could naturally emerge due to a Damour-Polyakov type screening of the dilaton [@Polyakov].
The relevant parameter here is the coupling of the dilaton field to hadronic matter. As discussed in [@Polyakov], to a good approximation this is given by the logarithmic derivative of the QCD scale, since hadron masses are proportional to it (modulo small corrections). Assuming that all gauge fields couple, near the string cutoff, to the same $B_F(\phi)$, and in accordance with Eq. (\[eq:coupfunc\]) which yields $$\label{bfhere}
B_F^{-1}(\phi)\propto (1-b_Fe^{-c\phi})\,,$$ we can write $$\label{alphadefn}
\alpha_{had}(\phi)\sim 40 \frac{\partial\ln B_F^{-1}(\phi)}{\partial\phi}\,,$$ (where the numerical coefficient is further described in [@DPV2]) and we finally obtain $$\label{alphahad}
\alpha_{had}(\phi)\sim 40\, b_F c\,e^{-c\phi}\,.$$ Note that $c$ and $b_F$ are constant free parameters: the former one is expected to be of order unity and the latter one much smaller. Moreover, if we set $c=1$ (which we will do henceforth) we can also eliminate $b_F$ by writing $$\label{alphahadrel}
\frac{\alpha_{had}(\phi)}{\alpha_{had,0}}=e^{-(\phi-\phi_0)}\,,$$ (where $\phi_0$ is the value of the field today) and simultaneously writing the field equation in terms of $(\phi-\phi_0)$.
There are two local constraints. Firstly the Eddington parameter $\gamma$, which quantifies the amount of deflection of light by a gravitational source, has the value $$\label{eddingt}
\gamma-1=-2\alpha_{had,0}^2\,,$$ and is constrained by the Cassini bound, $\gamma-1=(2.1\pm2.3)\times10^{-5}$ [@Cassini]. Secondly the dimensionless Eötvös parameter, quantifying violations to the Weak Equivalence Principle, has the value $$\label{eotvos}
\eta_{AB}\sim5.2\times10^{-5}\alpha_{had,0}^2\,,$$ and recent torsion balance tests lead to $\eta_{AB}=(-0.7\pm1.3)\times10^{-13}$ [@Torsion], while from lunar laser ranging one finds $\eta_{AB}=(-0.8\pm1.2)\times10^{-13}$ [@Lunar]. From these we conservatively obtain the bound $$\label{boundhad}
|\alpha_{had,0}|\le 10^{-4}\,.$$ Using Eq. (\[alphahad\]), and still assuming that $c\sim1$, this yields a bound on the product of $b_F$ and (the exponent of) $\phi_0$, namely $\phi_0\ge\ln{(|b_F|/2\times10^{-6})}$. Nevertheless, this is not explicitly needed: the evolution of the system will be determined by $\alpha_{had}$ rather than by $b_F$ or $\phi_0$.
These constraints do not apply to the dark sector (*i.e.* dark matter and/or dark energy) whose couplings may be stronger. There are two possible scenarios to consider. A first possibility is that the dark sector couplings (which we will denote $\alpha_m$ and $\alpha_v$ for the dark matter and dark energy respectively) are also much smaller than unity, that is $\alpha_m,\alpha_v\ll1$. In this case the small field velocity leads to violations of the Equivalence Principle and variations of the fine-structure constant that are quite small. Indeed, for this case to be observationally realistic the fractions of the critical density of the universe in the kinetic and potential parts of the scalar field must be $$\label{lambdadil}
\Omega_k=\frac{1}{3}{\phi'}^2\ll1\,,\qquad \Omega_v\sim0.7;$$ note that if one assumes a flat universe, then $\Omega_m+\Omega_k+\Omega_v=1$ (do not confuse the index $k$, which refers to the kinetic part of the scalar field, with the curvature term in standard cosmology, which we are setting to zero throughout). A more interesting possibility is that the dark couplings ($\alpha_m$ and/or $\alpha_v$) are of order unity. If so, violations of the Equivalence Principle and variations of the fine-structure constant are typically larger. In this case $\Omega_k$ may be more significant, and $\Omega_v$ should be correspondingly smaller [@quint]. Nevertheless the dark matter coupling is also constrained: during matter-domination the equation of state has the form $$w_m(\phi)=\frac{1}{3} {\phi'}^2 \sim\frac{1}{3}\alpha_m^2\,.$$
The present value of the field derivative is also constrained if one assumes a spatially flat universe; in that case the deceleration parameter $$\label{deccel}
q=-\frac{a{\ddot a}}{{\dot a}^2}=-1-\frac{\dot H}{H^2}\,$$ can be written as $$\label{deccelhere}
{\phi_0'}^2=(1+q_0)-\frac{3}{2}\Omega_{m0} \,$$ and using a reasonable upper limit for the deceleration parameter [@Neben] and a lower limit for the matter density (say, from the Planck mission [@XVI]) we obtain $$\label{phibound}
|\phi_0'|\le 0.3 \,,$$ almost three times tighter than the one available at the time of [@DPV2]. Thus in this scenario both the hadronic coupling and the field speed today are constrained.
Moreover, we can use the field equation, Eq. (\[eq:field\]), to set a consistency condition for $\phi_0'$. For this we only need to assume that the field is moving slowly today (a good approximation given the bounds on its speed) and therefore the $\phi''$ term should be subdominant in comparison with the other two. Then we easily obtain $$\label{phicons}
\phi_0'=\, - \, \frac{\alpha_{had}\Omega_b+\alpha_m\Omega_c+4\alpha_v\Omega_v}{\Omega_b+\Omega_c+2\Omega_v} \,,$$ with all quantities being evaluated at redshift $z=0$. To avoid confusion we have denoted baryonic and cold dark matter by $\Omega_b$ and $\Omega_c$ respectively; naturally $\Omega_m=\Omega_b+\Omega_c$. We choose the cosmological parameters in agreement with recent Planck data [@XVI], specifically setting the current fractions of baryons, dark matter and dark energy to be respectively $\Omega_{b}\sim0.04$, $\Omega_{c}\sim0.27$ and $\Omega_{\phi}=\Omega_{k}+\Omega_{v}\sim0.69$. Noting that $|\alpha_{had,0}|\le 10^{-4}$, that $|\phi_0'|\le 0.3$ and that $\Omega_k={\phi_0'}^2/3$ is necessarily small, we can consider three particular cases of this relation
- The [**dark coupling**]{} case, where $\alpha_m=\alpha_v$ (and both are assumed to be constant), leads to $$\label{alphadark}
|\alpha_v|\lesssim 0.3 \frac{\Omega_m+2\Omega_v}{\Omega_c+4\Omega_v}\sim0.17 \,;$$
- The [**matter coupling**]{} case, where $\alpha_m=\alpha_{had}$ (and both are field-dependent, as in Eq. \[alphahadrel\]), leads to $$\label{alphamat}
|\alpha_v|\lesssim 0.3 \frac{\Omega_m+2\Omega_v}{4\Omega_v}\sim0.18 \,;$$
- The [**field coupling**]{} case, where $\alpha_m=-\phi'$, leads to $$\label{alphafield}
|\alpha_v|\lesssim 0.3 \frac{\Omega_b+2\Omega_v}{4\Omega_v}\sim0.15 \,.$$
Note that in all cases $\alpha_v$ is a constant (field-independent) parameter. Naturally these are back-of-the-envelope constraints that need to be improved by a more robust analysis, but they are enough to show that order unity couplings $\alpha_v$ will be strongly constrained. An additional constraint will come from atomic clock measurements, as we will now discuss.
![Evolution of $H(z)$ in runaway dilaton models (left), compared to the measurements in [@Farooq], for $|\alpha_{had,0}|=10^{-4}$ and $\phi_0'$ spanning the observationally allowed range. The right-side panel depicts the evolution of the field for the same parameter choices.[]{data-label="fig1"}](figure1a.eps "fig:"){width="2.6in"} ![Evolution of $H(z)$ in runaway dilaton models (left), compared to the measurements in [@Farooq], for $|\alpha_{had,0}|=10^{-4}$ and $\phi_0'$ spanning the observationally allowed range. The right-side panel depicts the evolution of the field for the same parameter choices.[]{data-label="fig1"}](figure1b.eps "fig:"){width="2.6in"}
Varying fine-structure constant
===============================
Given some recent evidence, from archival Keck and VLT data, of space-time variations of the fine-structure constant $\alpha$ [@Webb], it’s interesting to study its behavior in this class of models. Consistently with our previous assumption that all gauge fields couple to the same $B_F$, here $\alpha$ will be proportional to $B_F^{-1}(\phi)$, as given by Eq. (\[bfhere\]). Note that this will also imply that $\alpha$ will be related to the hadronic coupling, as further discussed below.
The original work of Damour *et al.* [@DPV2] shows (under the same assumptions as we are using here) that the evolution of $\alpha$ is given by $$\label{alphazero}
\frac{1}{H}\frac{\dot\alpha}{\alpha}=\frac{b_Fce^{-c\phi}}{1-b_Fce^{-c\phi}}\,{\phi'}\sim b_Fce^{-c\phi}{\phi'}\sim\frac{\alpha_{had}}{40}{\phi'}\,.$$ In particular this equation applies at the present day (describing the current running of $\alpha$) and this variation is constrained by the Rosenband bound [@Rosenband] $$\left(\frac{1}{\alpha}\frac{{\rm d}\alpha}{{\rm d}t}\right)_0=(-1.6\pm2.3)\times 10^{-17}{\rm yr}^{-1}\,;$$ assuming the Planck value for the Hubble constant $H_0=(67.4\pm1.4)\, {\rm km/s/Mpc}$, we find $$\label{jointphibound}
|\alpha_{had,0}{\phi'}_0| \sim |b_Fce^{-c\phi_0}{\phi'}_0| \le 3\times 10^{-5}\,.$$ Thus atomic clock experiments constrain the product of the hadronic coupling and the field speed today. It is interesting to note that this constraint—which stems from microphysics—is comparable to the one obtained by multiplying the individual constraints on each of them, which are given respectively by Eq. \[boundhad\] and Eq. \[phibound\] and come from macrophysics (Solar System or torsion balance tests, plus a cosmology bound).
![Evolution of $\alpha$, plotted with the same conventions as in Fig. \[fig1\]. The data of [@Webb] is plotted in the left panel (VLT data as black asterisks, Keck data as green circles) while the data of Table \[tablealpha\] is shown as red circles in the right panel. One-sigma uncertainties are shown in all cases.[]{data-label="fig2"}](figure2a.eps "fig:"){width="2.6in"} ![Evolution of $\alpha$, plotted with the same conventions as in Fig. \[fig1\]. The data of [@Webb] is plotted in the left panel (VLT data as black asterisks, Keck data as green circles) while the data of Table \[tablealpha\] is shown as red circles in the right panel. One-sigma uncertainties are shown in all cases.[]{data-label="fig2"}](figure2b.eps "fig:"){width="2.6in"}
In [@DPV2] the authors obtain approximate solutions for the evolution of $\alpha$ by assuming that $\phi'=const.$ in both the matter and the dark energy eras (naturally the two constants are different). However, by integrating Eq. (\[alphazero\]) or by directly using the relation between $\alpha$ and $B_F(\phi)$ we can express the redshift dependence of $\alpha$ in the general form $$\label{evolfull0}
\frac{\Delta\alpha}{\alpha}(z)\equiv\frac{\alpha(z)-\alpha_0}{\alpha_0}=B_F^{-1}(\phi(z))-1 =b_F\left(e^{-\phi_0}-e^{-\phi(z)}\right)\,,$$ where for simplicity we have again set $c\sim1$. This can also be recast in the more suggestive form $$\label{evolfull}
\frac{\Delta\alpha}{\alpha}(z)=\frac{1}{40}\alpha_{had,0}\left[1-e^{-(\phi(z)-\phi_0)}\right]\,.$$ Thus the behaviour of $\Delta\alpha/\alpha$ close to the present day depends both on $\alpha_{had,0}$ (which provides an overall normalization) and on the speed of the field, ${\phi_0'}$, which can also be related to the values of the couplings as in Eq. (\[phicons\]).
In our analysis we will use both the data of Webb [*et al.*]{} [@Webb] (which is a large dataset of archival data measurements) and the smaller and more recent dataset of dedicated measurements listed in Table \[tablealpha\]. The latter include the recent first result of the UVES Large Program for Testing Fundamental Physics [@LP1; @LP3], which is expected to be the one with a better control of possible systematics. The source of the data in this Table is also further discussed in [@Ferreira]. We emphasize that all the data we use comes from high-resolution spectroscopy comparisons of optical/UV fine-structure atomic doublets, which are only sensitive to the value of $\alpha$—and not, say, to the values of particle masses (ratios of which can be probed by other means) [@Kozlov].
Object z ${ \Delta\alpha}/{\alpha}$ (ppm) Spectrograph Ref.
---------------- ------ ---------------------------------- ---------------- -------------------
3 sources 1.08 $4.3\pm3.4$ HIRES [@Songaila]
HS1549$+$1919 1.14 $-7.5\pm5.5$ UVES/HIRES/HDS [@LP3]
HE0515$-$4414 1.15 $-0.1\pm1.8$ UVES [@alphaMolaro]
HE0515$-$4414 1.15 $0.5\pm2.4$ HARPS/UVES [@alphaChand]
HS1549$+$1919 1.34 $-0.7\pm6.6$ UVES/HIRES/HDS [@LP3]
HE0001$-$2340 1.58 $-1.5\pm2.6$ UVES [@alphaAgafonova]
HE1104$-$1805A 1.66 $-4.7\pm5.3$ HIRES [@Songaila]
HE2217$-$2818 1.69 $1.3\pm2.6$ UVES [@LP1]
HS1946$+$7658 1.74 $-7.9\pm6.2$ HIRES [@Songaila]
HS1549$+$1919 1.80 $-6.4\pm7.2$ UVES/HIRES/HDS [@LP3]
Q1101$-$264 1.84 $5.7\pm2.7$ UVES [@alphaMolaro]
: \[tablealpha\]Recent dedicated measurements of $\alpha$. Listed are, respectively, the object along each line of sight, the redshift of the measurement, the measurement itself (in parts per million), the spectrograph, and the original reference. The recent UVES Large Program measurements are [@LP1; @LP3]. The first measurement is the weighted average from 8 absorbers in the redshift range $0.73<z<1.53$ along the lines of sight of HE1104-1805A, HS1700+6416 and HS1946+7658, reported in [@Songaila] without the values for individual systems. The UVES, HARPS, HIRES and HDS spectrographs are respectively in the VLT, ESO 3.6m, Keck and Subaru telescopes.
Note that since in the current work we will be interested in the evolution of $\alpha$ at relatively low redshifts, one could think of linearizing the field evolution $$\label{evollinear}
\phi\sim\phi_0+{\phi_0'}\ln{a}\,,$$ in which case Eq. (\[evolfull\]) takes the simpler form $$\label{evolslow}
\frac{\Delta\alpha}{\alpha}(z)\approx\, -\frac{1}{40}\alpha_{had,0} {\phi_0'}\ln{(1+z)}\,;$$ this is indeed what is obtained with the simplifying assumptions of [@DPV1; @DPV2]. Nevertheless, as shown in the second panel of Fig. \[fig1\], $\phi-\phi_0$ can still be of order unity by redshift $z=5$ for values of the coupling that saturate the current bounds, and therefore in what follows the evolution of $\alpha$ will be calculated using the full equations.
Current constraints
===================
By numerically solving the previously discussed Friedmann and scalar field equations we can study the cosmological dynamics of this model. We will start by assuming that the value of $\alpha_{had,0}$ is the maximal one allowed by Eq. (\[alphahad\])—we will relax this assumption later on. We allow $\phi_0'$ to vary in the whole range allowed by Eq. (\[phibound\]), and we further assume the dark coupling case, where $\alpha_m=\alpha_v$; it then follows from in Eq. (\[phicons\]) that $\phi_0'\approx -1.79\alpha_v$.
![Constraints in the $\alpha_{had,0}$–$\phi_0'$ space, with other parameters as described in the text; in the left panel only the $\alpha$ measurements were used, while the right one the $H(z)$ measurements were also included. The colormap shows the reduced chi-square, while the black solid, dashed and dotted lines identify the one, two and three-sigma confidence regions in this parameter space.[]{data-label="fig3"}](figure3a.eps "fig:"){width="2.6in"} ![Constraints in the $\alpha_{had,0}$–$\phi_0'$ space, with other parameters as described in the text; in the left panel only the $\alpha$ measurements were used, while the right one the $H(z)$ measurements were also included. The colormap shows the reduced chi-square, while the black solid, dashed and dotted lines identify the one, two and three-sigma confidence regions in this parameter space.[]{data-label="fig3"}](figure3b.eps "fig:"){width="2.6in"}
We choose the same cosmological parameters as previously discussed. Note that in this model the dark energy equation of state is $$\label{darkenergyeos}
1+w_0=\frac{2\Omega_{k}}{\Omega_{k}+\Omega_{v}}=\frac{2}{3}\frac{{\phi_0'}^2}{\Omega_{k}+\Omega_{v}}\,,$$ and the range of allowed values for ${\phi_0'}$ (specifically, $|\phi_0'|\le 0.3$) leads to $-1\le w_0<-0.91$, which is perfectly compatible with current observational bounds [@XVI]. We then numerically integrate the dynamical equations of this model backwards in time. The evolution of the Hubble parameter for this set of models is plotted in Fig. \[fig1\], and compared to the available measurements, as compiled in [@Farooq]. As expected the sign of the coupling $\alpha_{had,0}$ has a negligible effect on $H(z)$ (since the coupling itself is very small), while that of the field speed is more noticeable.
We then calculate the evolution of $\alpha$ in these models; this is shown in Fig. \[fig2\], again for the maximally allowed $|\alpha_{had,0}|=10^{-4}$. With these parameter choices the typical variations are at the parts per million level, comparable to the sensitivity of the current measurements [@LP1; @LP3]. The value of $\alpha$ also depends on the present speed of the field (and not only on its absolute value), which can be understood from Eq. \[evolfull\].
As a second step in our analysis, we now relax the assumption of $\alpha_{had,0}$ fixed to its maximum allowed value and let it vary freely. We use the available data to constrain it, together with the field speed. The results of this analysis are shown in Fig. \[fig3\]. Using all available $\alpha$ data (both that of [@Webb] and the dedicated measurements of Table \[tablealpha\]) one finds no significant evidence for a non-zero coupling $\alpha_{had,0}$. While the weighted mean of the data in Table \[tablealpha\] is consistent with no variations, that of [@Webb] is slightly negative; this explains why in the first panel of Fig. \[fig3\] there is a slight preference for similar signs for the field speed and the coupling (however, this is not statistically significant). We thus see that with the $\alpha$ data alone the constraints are not that much stronger than we already discussed above. The addition of Hubble parameter measurements does constrain the current speed of the field to be small, and the combination of the two datasets yields the constraints in the second panel of Fig. \[fig3\]. In both cases the model is compatible with the current data.
We caution the reader that this analysis assumed fixed values of the cosmological parameters $\Omega_{b}$, $\Omega_{c}$ and $\Omega_{\phi}$, but we expect the results not to change significantly had we allowed them to vary and marginalized over them. Perhaps more relevant are our ‘maximal’ assumptions for the dark sector couplings, which can be justified in the context of a preliminary assessment of the feasibility of the model. Thus our present results suggest that a more thorough exploration of this parameter space is justified, but we leave it for a more detailed follow-up publication.
![Redshift drift signal for allowed runaway dilaton models (plotted with the same conventions as in Fig. \[fig1\]) compared to the standard $\Lambda$CDM model (red curve, shown with errorbars expected for E-ELT measurements).[]{data-label="fig4"}](figure4.eps){width="3in"}
Outlook
=======
While current astrophysical and laboratory constraints on $\alpha$ provide (together with Equivalence Principle tests) interesting constraints on string theory inspiredscenarios, prospects for further improvements are excellent in the context of European Extremely Large Telescope (E-ELT): this will not only enable much more sensitive measurements of the fine-structure constant but it will also open a new and complementary observational window into these models.
Redshifts of cosmologically distant objects drift slowly with time [@Sandage]. This provides a direct measurement of the Universe’s expansion history, with the advantage of being a non-geometric, completely model-independent test, uniquely probing the global dynamics of the metric [@Liske]. Rather than mapping our past light-cone, it directly measures evolution by comparing past light-cones at different times. While plans are being developed to carry out these measurements at low redshift (with the SKA [@SKA] and intensity mapping experiments [@Yu]), the E-ELT offers the unique advantage of probing deep in the matter era and thus a much larger redshift lever arm. The precision needed for these measurements, a few cm/s, will be reached with the E-ELT’s high-resolution ultra-stable spectrograph currently dubbed ELT-HIRES. A Phase A study [@Liske] led to the following estimate for the spectroscopic velocity precision $$\label{eq:hiresprec}
\sigma_v=1.35\left(\frac{S/N}{2370}\right)^{-1}\left(\frac{N_{QSO}}{30}\right)^{-1/2}\left(\frac{1+z_{QSO}}{5}\right)^{-1.7}\,;$$ this depends on the signal-to-noise of the spectra, as well as on the number and the redshift of the quasar absorption systems used. The signal for a given model can be derived from the definition of redshift and expressed in a model independent way in terms of the spectroscopic velocity (which is the actual observable) as $$\label{eq:slsignal}
\frac{\Delta v}{c}=\frac{\Delta t}{(1+z)} \left[H_0(1+z)-H(z)\right]\,,$$ where $\Delta t$ is the timespan of the measurements.
The drift signal for our range of models is plotted in Fig. \[fig4\] and compared to $\Lambda$CDM, for $\Delta t=30$ years. The error bars depict the expected accuracy of ELT- HIRES, assuming 40 sources with $S/N=2000$. As with several other alternatives to $\Lambda$CDM studied in the literature [@Quercellini], it is clear that the drift signal in runaway dilaton models can differ significantly from that of $\Lambda$CDM, and ELT-HIRES will thus be able to distinguish the two paradigms and set tighter constraints both on $\alpha_{had,0}$ and on the dark sector couplings.
In conclusion, the runaway dilaton scenario is compatible with current data. It (and many other models) will be subject to much more stringent tests as the next generation of high-resolution, ultra-stable spectrographs becomes available. A roadmap for these tests is further discussed in [@grg]. Meanwhile, the Eötvös parameter sensitivity is also expected to improve to $2\times10^{-15}$ with a hypothetical STE-QUEST [@stequest] and to $10^{-18}$ with STEP [@step], and these will provide complementary constraints. Thus quantitative astrophysical tests of string-inspired scenarios will soon become possible.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was done in the context of project PTDC/FIS/111725/2009 (FCT, Portugal). CJM is also supported by an FCT Research Professorship, contract reference IF/00064/2012, funded by FCT/MCTES (Portugal) and POPH/FSE. MM acknowledges the DFG TransRegio TRR33 grant on The Dark Universe. EC acknowledges funding from ERC grant 259505. The Dark Cosmology Centre is funded by the Danish National Research Foundation.
|
---
abstract: 'The influence of time-dependent fitnesses on the infinite population dynamics of simple genetic algorithms (without crossover) is analyzed. Based on general arguments, a schematic phase diagram is constructed that allows one to characterize the asymptotic states in dependence on the mutation rate and the time scale of changes. Furthermore, the notion of [*regular*]{} changes is raised for which the population can be shown to converge towards a [*generalized*]{} quasispecies. Based on this, error thresholds and an optimal mutation rate are approximately calculated for a generational genetic algorithm with a moving needle-in-the-haystack landscape. The so found phase diagram is fully consistent with our general considerations.'
author:
- 'Christopher Ronnewinkel,\'
- 'Claus O. Wilke and Thomas Martinetz\'
date: |
-1.2emContact: [ronne@neuroinformatik.ruhr-uni-bochum.de]{}\
(to be published in the [*Proceedings of the 2nd EvoNet Summerschool*]{},\
Natural Computing Series, Springer)\
November 2, 1999
title: 'Genetic Algorithms in Time-Dependent Environments'
---
Genetic algorithms (s) as special instances of evolutionary algorithms have been established during the last three decades as optimization procedures, but mostly for static problems (see [@baeck] for an overview and [@baeckhb] for an in-depth presentation of the field). In view of real-world applications, such as routing in data-nets, scheduling, robotics etc., which include essentially dynamic optimization problems, there are two alternative optimization strategies. On the one hand, one can take snapshots of the system and search “offline” for the optimal solutions of the static situation represented by each of these snapshots. In this approach, the algorithm is restarted for every snapshot and solves the new problem from scratch. On the other hand, the optimization algorithm might reevaluate the real, current situation in order to reuse information gained in the past. In this case, the algorithm works “online”. As can be argued from the analogies to natural evolution, evolutionary algorithms seem to be promising candidates for “online” optimization [@baeck; @brankereview]. The reevaluation of the situation or environment then introduces a [*time-dependency*]{} of the fitness landscape. This time-dependency occurs as external to the algorithm’s population and does not emerge from coevolutive interactions. Coevolutive interactions as an alternative source of time dependency in the fitness landscape are not within the scope of this work.
In the last years, many different methods and extensions of standard evolutionary algorithms for the case of time-dependent fitnesses have been analyzed on the basis of experiments (see [@brankereview] for a review) but only seldom on the basis of [*theoretical*]{} arguments (see [@rowegecco; @nehaniv]). To take a step into the direction of a better theoretical understanding of “online” evolutionary algorithms, we will study the effects of simple time dependencies of the fitness landscape on the dynamics of s (without crossover), or more generally saying, of populations under mutation and probabilistic selection. As we will see, it is possible to characterize the asymptotic states of such a system for a particular class of dynamic fitness landscapes that is introduced below. The asymptotic state forms the basis on which it can be decided whether the population is able to adapt to, or track, the changes in the fitness landscape. Our mathematical formalism applies to s as well as to biological self-replicating systems, since the analyzed model and Eigen’s quasispecies model [@eigen71; @eigen79; @eigen89] in the molecular evolution theory (see [@baake] for a recent review) are very similar. Hence, all introduced concepts for s are valid and relevant in analogous form for molecular evolutionary systems.
In the following section, we will introduce the model to be analyzed and show the correspondence to the quasispecies model. Then, we will introduce the mathematical framework, based on which we will formally characterize the asymptotic state as fixed point. After presenting the main concepts, we will proceed with the construction of a phase diagram that allows to characterize the order found in the asymptotic state for different parameter settings. Finally, a moving needle-in-the-haystack () landscape is analyzed and its phase diagram, including the optimal mutation rate, is calculated.
Mathematical Framework
======================
In order to study the influence of a time-dependent fitness landscape on the dynamics of a genetic algorithm (), we consider s to be discrete dynamical systems. A detailed introduction to the resulting dynamical systems model is given by Rowe [@rowe] (in this book). Here, we will only shortly introduce the basic concepts and the notations we use within the present work.
The is represented as a generation operator $\smash{G_{t}^{(m)}}$ acting on the space $\Lambda_{m}$ of all populations of size $m$ for some given encoding of the population members. If we choose the members $i$ to be encoded as bit-strings of length $l$, this state space is given by $$\Lambda_{m}=\{(n_{0},\ldots,n_{2^{l}-1})/m \mid \textstyle\sum_{i}n_{i}=m,
n_{i}\in\NN_{0}\},$$ where $n_{i}$ denotes the number of bit-strings in the population, which are equal to the binary representation of $i\in\{0,\ldots,2^{l}-1\}$.
The generation operator maps the present population onto the next generation, $$\xx(t+1)=G_{t}^{(m)}[\xx(t)].$$ This is achieved by applying a sampling procedure that draws the members of the next generation’s population $\xx(t+1)$ according to their expected concentrations $\langle
\xx(t+1)\rangle\in\Lambda_{\infty}$ which are defined by the mixing [@rowe; @vose] and the selection scheme. For an infinite population size, the sampling acts like the identity resulting in $$G_{t}^{(\infty)}\xx(t)=\xx(t+1)=\langle\xx(t+1)\rangle.$$ Hence, $G_{t}:=G_{t}^{(\infty)}$ represents in fact the mixing and selection scheme. For finite population size, $\langle\xx(t+1)\rangle\in\Lambda_{\infty}$ is approximated by using the sampling process to obtain $\xx(t+1)\in\Lambda_{m}$. The deviations thereby possible become larger with decreasing $m$ and distort the finite population dynamics as compared to the infinite population case. This results in fluctuations and epoch formation as shown in [@rowe; @vose; @nimw97]. In the following, we will consider the infinite population limit, because it reflects the exact flow of probabilities for a particular fitness landscape. In a second step, the fluctuations and epoch formation introduced by the finiteness of a real population can be studied on the basis of that underlying probability flow.
The generation operator is assumed to decompose into a separate mutation and a separate selection operator, like $$G_{t}=M\cdot S(t),
\label{eq:genga}$$ where the selection operator $S(t)$ contains the time dependency of the fitness landscape. Crossover is not considered in this work.
Inspired by molecular evolution, and also by common usage, we assume that the mutation acts like flipping each bit with probability $\mu$. If we set the duration of one generation to $1$, $\mu$ equals to the mutation rate. The mutation operator then takes on the form $$M=\left(\begin{matrix}1-\mu &\mu\\\mu & 1-\mu\end{matrix}
\right)^{\otimes l}\mbox{,\quad i.\ e.}\quad
M_{ij}=\mu^{d_{\rm H}(i,j)}(1-\mu)^{l-d_{\rm H}(i,j)},$$ where $\otimes$ denotes the Kronecker (or canonical tensor) product and $d_{\rm H}(i,j)$ denotes the Hamming distance of $i$ and $j$.
To keep the description analytically tractable, we will focus on fitness-proportionate selection, $$\begin{aligned}
S(t)\cdot\xx=F(t)\cdot\xx\big/\langle f(t)\rangle_{\xx},\quad
\mbox{where } F(t)&={\rm diag}\big(f_{0}(t),\ldots,f_{2^{l}-1}(t)\big)\\
\mbox{ and
$\langle f(t)\rangle_{\xx}$}&=\textstyle\sum_{i}f_{i}(t)x_{i}
=\|F(t)\cdot\xx\|_{1}.\end{aligned}$$ This will already provide us with some insight into the general behavior of a in time-dependent fitness landscapes.
Since the corresponding to Eq. \[eq:genga\] applies mutation to the current population and selects the new population with complete replacement of the current one, it is called a [*generational*]{} (). In addition to s, [*steady-state*]{} s (s) with a two step reproduction process are also in common use: First, a small fraction $\gamma$ of the current population is chosen to produce $m\gamma$ mutants according to some heuristics. Second, another fraction $\gamma$ of the current population is chosen to get replaced by those mutants according to some other heuristics (see [@dejong; @rogers; @branke] and references therein). We can include s into our description in an approximate fashion by simply bypassing a fraction $(1-\gamma)$ of the population into the selection process without mutation, whereas the remaining fraction $\gamma$ gets mutated before it enters the selection process. The generation operator then reads $$G_{t}=\left[(1-\gamma)\Id+\gamma M\right]S(t).
\label{eq:ssga}$$ By varying $\gamma$ within the interval $]0,1]$, we can interpolate between steady-state behavior () for $\gamma\ll 1$ and generational behavior () for $\gamma=1$. Equation \[eq:ssga\] is only an approximation of the true generation operator for s because the heuristics involved in the choice of the mutated and replaced members are neglected. But in the next section, the heuristics are expected to play a minor role for our general conclusion on an inertia of s against time-variations.
At this point, we want to review shortly the correspondence of our model with the quasispecies model, extensively studied by Eigen and coworkers [@eigen71; @eigen79; @eigen89] in the context of molecular evolution theory (see also [@roweqs] in this book). The quasispecies model describes a system of self-replicating entities $i$ (e. g. RNA-, DNA-strands) with replication rates $f_{i}$ and an imperfect copying procedure such that mutations occur. For simplicity reasons, the overall concentration of molecules in the system is held constant by an excess flow $\Phi(t)$. In the above notation, the continuous model reads $$\dot{\xx}(t)=\left[M\cdot F(t)-\Phi(t)\right]\xx(t),
\label{eq:conteig}$$ where the flux needs to equal the average replication, $\Phi(t)=\langle f(t)\rangle_{\xx(t)}$, in order to keep the concentration vector $\xx(t)$ normalized. This model might then be discretized via $t\to t/\delta t$, which unveils the similarity to a : $$\xx(t+1)=\left[(1-\delta t\,\langle f(t)\rangle_{\xx(t)})\Id+\delta t\,
M\cdot F(t)\right]\xx(t)\quad\mbox{for $\delta t\ll 1$.}
\label{eq:disceig}$$ By comparison with Eq. \[eq:ssga\], we can easily read off that $\gamma=\delta t\,\langle f(t)\rangle_{\xx(t)}=:\gamma_{\xx(t)}$. This means a low (resp. high) average fitness leads to a small (resp. large) replacement – a property that is not wanted in the context of optimization problems, which s are usually used for, because one does not want to remain in a region of low fitness for a long time. Another difference to s is the fact that in the continuous Eigen model, selection acts only on the mutated fraction of the population – although this leads only to subtle differences in the dynamics of s and the Eigen model.
Equation \[eq:conteig\] is commonly referred to as ‘continuous Eigen model’ in the literature, because of the continuous time, and Eq. \[eq:disceig\] is simply its discretized form which can be used for numerical calculations. Nonetheless, the notion ‘discrete Eigen model’ is seldom used for Eq. \[eq:disceig\] but it is often used for the , $$\xx(t+1)=\left[M\cdot S(t)\right]\xx(t),
\label{eq:geneig}$$ in the literature. This stems from the identical asymptotic behavior of Eqs.\[eq:disceig\] and \[eq:geneig\] for static fitness landscapes. However, there are differences for time-dependent fitness landscapes, as we will see in the following two sections.
Regular Changes and Generalized Quasispecies {#sec:fp}
============================================
In the case of a static landscape, the fixed points of the generation operator, which are in fact stationary states of the evolving system (if contained within $\Lambda_{m}$, see [@rowe]), can be found by solving an eigenvalue problem, because of $$\xx=G\xx\quad\Longleftrightarrow\quad MF\,\xx=\langle
f\rangle_{\xx}\xx\ .
\label{eq:statfp}$$ Let $\lambda_{i}$ and $\vv_{i}$ denote the eigenvalues and eigenvectors of $MF$ with descending order $\lambda_{0}\ge\cdots\ge\lambda_{2^{l}-1}$ and $\|\vv_{i}\|_{1}=1$. For $\mu\not=0,1$ the Perron-Frobenius theorem assures the non-degeneracy of the eigenvector $\vv_{0}$ to the largest eigenvalue and moreover it assures $\vv_{0}\in\Lambda_{\infty}$. Often, $\vv_{0}$ is called Perron vector. After a transformation to the basis of the eigenvectors $\{\vv_{i}\}$ it can be straightforwardly shown that $\xx(t)$ converges to $\vv_{0}$ for $t\to\infty$. The population represented by $\vv_{0}$ was called the ‘quasispecies’ by Eigen, because this population does not consist of only a single dominant genotype, or string, but it consists of a particular stable mixture of different genotypes.
Let us now consider time-dependent landscapes. If the time dependency is introduced simply by a single scalar factor, like $$F(t)=F\,\rho(t)\quad\mbox{with $\rho(t)\ge0$ for all $t$,}$$ it immediately drops out of the selection operator for s. For the continuous Eigen model, we note that the eigenvectors of $F(t)$ and $F$ are the same and that $\lambda_{i}(t)=\lambda_{i}\,\rho(t)$. Since $\rho(t)\ge0$, which is necessary to keep the fitness values positive, the order of the eigenvalues remains, such that $MF(t)$ will show the same quasispecies $\vv_{0}$ as $MF$. Contrasting to that special case, a general, individual time dependency of the string’s fitnesses does indeed change the eigenvalues and eigenvectors of $MF(t)$ compared to $MF$. For an arbitrary time dependency the Perron vector is constantly changing, and therefore, we cannot even define a unique asymptotic state. However, this problem disappears for what we call [*regular*]{} changes. After having established a theory for such changes, we can then take into account more and more non-regular ingredients. What do we mean by “[*regular*]{} change”? We define it heuristically in the following way: a regular change is a change that happens with fixed duration $\tau$ and obeys some deterministic rule that is the same for all change cycles. Let us express the latter more formally and make it more clear what we mean by “same rule of change”. Within a change cycle, we allow for an arbitrary time dependency of the fitness, up to the restriction that two different change cycles must be connected by a permutation of the sequence space. Thus, if the time dependency is chosen for one change cycle, e. g. the first change cycle starting at $t=0$, it is already fixed for all other cycles, apart from the permutations. We will represent permutations $\pi$ from the permutation group $\Perm_{2^{l}}$ of the sequence space as matrices $$(P_{\pi})_{ij}=\delta_{\pi(i),j}\quad\mbox{for $i,j\in\{0,\ldots,2^{l}-1\}$.}$$ The permutations of vectors $\xx$ and matrices $A$ are obtained by $$\begin{gathered}
(P_{\pi}\xx)_{i}=x_{\pi(i)}\quad\mbox{and}\quad
(P_{\pi}AP_{\pi}^{\rm T})_{i,j}=A_{\pi(i),\pi(j)},\end{gathered}$$ where $P_{\pi}^{\rm T}$ denotes the transpose of $P_{\pi}$ with the property $P_{\pi}^{\rm T}=P_{\pi^{-1}}=P_{\pi}^{-1}$.
In reference to the first change cycle, we define the fitness landscape $F(t)$ as being [*single-time-dependent*]{}, if and only if for each change cycle $n\in\NN_{0}$ there exists a permutation $\pi_{n}\in\Perm_{2^{l}}$, such that for all cycle phases $\pha\in\{0,\dots,\tau-1\}$ $$P_{n}\,F(\pha+n\tau)\,P_{n}^{\rm T}=
F(\pha)\qquad\mbox{(abbreviatory $P_{n}:=P_{\pi_{n}}$).}$$ We will call each permutation $P_{n}$ a [*jump-rule*]{}, or simply [*rule*]{}, which connects $F(\pha+n\tau)$ and $F(\pha)$. To make predictions about the asymptotic state of the system, we need to relate the generation operators of different change cycles to each other. This is readily achieved if the permutations $P_{n}$ [*commute*]{} with the mutation operator $M$. The condition for this being the case is that for all $i,j$, $$M_{ij}=M_{\pi_{n}(i),\pi_{n}(j)}\quad\mbox{or equivalently}\quad
d_{\rm H}(i,j)=d_{\rm H}\big(\pi_{n}(i),\pi_{n}(j)\big).$$ Thus, the Hamming distances $d_{\rm H}(i,j)$ need to be [*invariant*]{} under the permutations $P_{n}$. Geometrically this means that the fitness landscape gets “translated” or “rotated” by those permutations without changing the neighborhood relations. Then, we find for arbitrary $n\in\NN$ and $\pha\in\{0,\ldots,\tau-1\}$, $$G_{\pha+n\tau}=P_{n}^{\rm T}G_{\pha}P_{n}.
\label{eq:gperm}$$ To study the asymptotic behavior of the system, it is useful to accumulate the time dependency of a change cycle by introducing the $\tau$-generation operators, $$\Gamma_{n}:=G_{\tau-1+n\tau}\cdots G_{n\tau}\quad
\mbox{for all $n\in\NN_{0}$}.$$ Because of Eq. \[eq:gperm\], all these operators are related to $\Gamma_{0}$ by $$\Gamma_{n}=P_{n}^{\rm T}\Gamma_{0}P_{n},$$ This property allows us to write the time evolution of the system in the form $$\xx(\pha+n\tau)=P_{n-1}^{\rm T}\Gamma_{0}P_{n-1}\,\,\cdots\,\,
P_{1}^{\rm T}\Gamma_{0}P_{1}\,\Gamma_{0}\,\,\xx(\pha),
\label{eq:gtev}$$ where $\pha\in\{0,\ldots,\tau-1\}$ denotes in the following always the phase within a cycle.
Let us consider the special case of a single rule $P$ being applied at the end of each change cycle, which results in $P_{n}=(P)^{n}$, e.g. imagine a fitness peak that moves at a constant “velocity” through the string space. We will see below that for those cases it is possible to identify the asymptotic state with a quasispecies in analogy to static fitness landscapes. Because of that, we can now define the notion of [*regularity*]{} of a fitness landscape formally in the following manner:
A time-dependent fitness landscape $F(t)$ is [*regular*]{}, if and only if: (i) the fitness landscape is [*single-time-dependent*]{}, (ii) there exists some rule $P\in\Perm_{2^{l}}$ which is applied at the end of each cycle such that $P_{n}=(P)^{n}$, and (iii) the rule $P$ [*commutes*]{} with the mutation operator $M$.
In this case, we get with $P P^{\rm T}=\Id$ the time evolution $$\xx(\pha+n\tau)=\big(P^{\rm T}\big)^{n}\big(P\Gamma_{0}\big)^{n}\,\xx(\pha).
\label{eq:xevol}$$ To proceed, it is useful to permute the concentrations compatible to the rule of the fitness landscape. By this, concentrations are measured in reference to the fitness landscape structure of the start cycle $n=0$. We will denote those concentrations by $\xx'(t)$ and they are related to the concentrations $\xx(t)$ by $$\begin{aligned}
\xx'(\pha+n\tau)&=(P)^{n}\,\xx(\pha+n\tau)\\
&=(P\Gamma_{0})^{n}\,\xx(\pha)
\qquad\mbox{and}\quad \xx'(\pha)=\xx(\pha).\end{aligned}
\label{eq:xxp}$$ For example, if there is no time-dependency within the cycles, some $x'_{i}$ will for all cycles measure the concentration of the highest fitness string, independent of its current position in string space. Thus, $\xx'(t)$ evolves in a fitness landscape with periodic change, which can also be seen from the second line of Eq. \[eq:xxp\]. In analogy to the static case Eq. \[eq:statfp\], the calculation of fixed points of $\xx'(t)$ is equivalent to an eigenvalue problem, $$\xx'(t+\tau)=\xx'(t)\quad\Longleftrightarrow\quad
P\widetilde{\Gamma}_{0}\,\xx'(t)=
\|P\widetilde{\Gamma}_{0}\,\xx'(t)\|_{1}\,\,\xx'(t),$$ where $\widetilde{\Gamma}_{0}$ is the [*unnormalized*]{} $\tau$-generation operator obtained from the accumulation of the [*unnormalized*]{} generation operators $\widetilde{G}_{\pha}=MF(\pha)$.
The corresponding periodic quasispecies $\vv_{0}$ can be calculated for all phases $\pha$ of the change cycle from the Perron vector $\vv_{0}$ of $P\Gamma_{0}$ in the following way, $$\xx'(\pha+n\tau)\xrightarrow{n\to\infty}\vv_{0}(\pha)=
G_{\pha-1}\cdots G_{0}\,\vv_{0}\quad\mbox{for $\pha\in\{0,\ldots,\tau-1\}$}.
\label{eq:pqs}$$ To find the asymptotic states of the concentrations $\xx(t)$, we simply need to invert Eq. \[eq:xxp\], $$\xx(\pha+\nu\tau)=\big(P^{\rm T}\big)^{\nu}\xx'(\pha+\nu\tau)\quad
\mbox{for $\nu\in\{0,\ldots, \eta-1\}$},
\label{eq:xqs}$$ where $\eta:={\rm ord}\, P$ is the order of the group element $P\in\Perm_{2^{l}}$.
The essential reason for the existence of asymptotic states for $\xx(t)$ lies in the finiteness of the permutation group $\Perm_{2^{l}}$. Because of $P^{\eta}=\Id$, we find directly from Eq. \[eq:xevol\] the asymptotic state $$\xx(\pha+\tilde{n}\eta\,\tau)=(P\Gamma_{0})^{\eta\,\tilde{n}}\,\xx(t)
\xrightarrow{\tilde{n}\to\infty}\vv_{0}(\pha),$$ where $\vv_{0}(\pha)$ is the same as in Eq. \[eq:pqs\], because $(P\Gamma_{0})^{\eta}$ and $P\Gamma_{0}$ have the same eigenvectors, in particular the same Perron vector. Moreover, we get $$\xx\big(\pha+(\nu+\tilde{n}\eta)\tau\big)\xrightarrow{\tilde{n}\to\infty}
\big(P^{\rm T}\big)^{\nu}\vv_{0}(\pha)\quad\mbox{for $\nu\in\{0,\ldots,\eta-1\}$},
\label{eq:pqsx}$$ which is the same result as Eqs. \[eq:pqs\] and \[eq:xqs\] yield. In the limit of long strings $l\to\infty$, ${\rm ord}\,P$ is not necessarily finite anymore. If ${\rm
ord}\,P\smash{\xrightarrow{l\to\infty}}\infty$, then the asymptotic states Eq. \[eq:pqsx\] for $\xx(t)$ do not exist, but Eq. \[eq:pqs\] still holds. Hence, a quasispecies exists even in the limit $l\to\infty$ if measured in reference to the structure of the fitness landscape.
In conclusion, Eqs. \[eq:pqs\] and \[eq:pqsx\] represent the [*generalized*]{} quasispecies for the class of [*regular*]{} fitness landscapes which includes as special cases static and periodic fitness landscapes. In fact, the simplest case of a [*regular*]{} change is a periodic variation of the fitness values $f_{i}(t)=f_{i}(t+\tau)$ because [*no*]{} permutations are involved ($P=\Id$) and hence $\xx'(t)=\xx(t)$ for all $t$. The quasispecies was generalized for this case already in [@wilke99a] and – using a slightly different formalism – in [@rowegecco]. In Section \[sec:genga\], we will study a more complicated example.
Schematic Phase Diagram {#sec:spd}
=======================
To get an intuitive feeling for the typical behavior of s and s, let us consider some special lines in the plane spanned by the mutation rate $\mu$ and the time scale for changes $\tau$, as shown in Fig. \[fig:phase\]. The mutation operator represents only for $\mu<1/2$ a copying procedure with occurring errors, whereas for $\mu>1/2$ it systematically tends to invert strings, i. e. it resembles an inverter with occurring errors. Since mutation should introduce [*weak*]{} modifications to the strings, we will consider only $\mu\le1/2$.
-0.07{width=".9\textwidth"}
Disorder line:
: For $\mu=1/2$, the Perron vector of $MF(t)$ is always $\vv_{0}^{T}=(1,\ldots,1)/2^{l}$. The population will therefore converge towards the disordered state. Because of the continuity of $M$ in $\mu$, we already enter a disordered phase for $\mu\approx1/2$.
Time-average region:
: For $\mu=0$, the mutation operator is the identity. We find as time evolution simply the product average over the fitness of the evolved time steps: $$\begin{aligned}
\xx(t+\tau)&=\left[\prod_{\pha=t}^{t+\tau-1}S(\pha)\right]\xx(t)\\
&=\tilde{F}(t+\tau,t)\,\xx(t)\big/\|\ldots\|_{1}\mbox{\ with
$\tilde{F}(t+\tau,t)=\prod_{\pha=t}^{t+\tau-1}F(\pha)$}.
\end{aligned}$$ Since diagonal operators commute, the order in which the $F(\pha)$ get multiplicated does not make any difference. For the case of a $\tau$-periodic landscape, $\tilde{F}=\tilde{F}(t+\tau,t)=\tilde{F}(\tau,0)$ is [ *independent*]{} of $t$. The quasispecies is then a linear superposition of the eigenvectors of the largest eigenvalue of the product averaged fitness landscape $\tilde{F}$ – there might be more then one such eigenvector, since $\tilde{F}$ is diagonal and the Perron-Frobenius theorem does not apply. Because of the continuity of $M$ in $\mu$ the dynamics are governed already for $0<\mu\ll 1$ by the product average $\tilde{F}$. Analogous conclusions apply to those non-periodic landscapes for which by choosing a suitable time scale $\tau$ a meaningful average $\tilde{F}(t+\tau,t)$ can be defined.
For s, $\gamma$ is small and we find to first order in $\tau\gamma$: $$\begin{gathered}
\xx(t+\tau)=(1-\tau\gamma)\tilde{F}(t+\tau,t)\\
+\tau\gamma\biggl(\frac{1}{\tau}\sum_{\pha=0}^{\tau-1}S(t+\tau)\cdots
\underbrace{M}_{\hbox to 1pt {\scriptsize $\pha$th factor from left}}
\cdots S(t)\biggr)
+\mathcal{O}\big((\tau\gamma)^{2}\big).
\end{gathered}$$ If $\tau\gamma\ll1$ holds, the time evolution is governed by $\tilde{F}(t,t+\tau)$. For changes on a time scale $\tau$, we find time-averaged behavior if $\tau\ll 1/\gamma$. Thus, the width of the time-average region is proportional to $1/\gamma$. A detailed analysis of the effect of the different positions of the mutation operator $M$ within the $\tau\gamma$-term, which is otherwise an arithmetic time-average, has not yet been carried out.
Quasi-static region:
: If the changes happen on a time scale $\tau$ very large compared to the average relaxation time ($\sim
1/\langle\lambda_{0}-\lambda_{1}\rangle$) the quasispecies grows nearly without noticing the changes. Thus, in the quasi-static region all quasispecies that might be expected from the static landscapes $\tilde{F}=F(t)$ will occur at some time during one cycle $\tau$.
-.12{width=".9\textwidth"}
Wilke [*et al. *]{}raise in [@wilke99b] the schematic phase diagram of the continuous Eigen model, which exhibits the same time-average phases as that for s. Their result is in perfect agreement with two recently, explicitly studied time-dependent landscapes. First, Wilke [*et al. *]{}studied in [@wilke99a] a needle-in-the-haystack () landscape with oscillating, $\tau$-periodic fitness of the needle, i. e.$$f_{0}(t)>f_{1}=\cdots=f_{2^{l}-1}=1\quad\mbox{and}\quad
f_{0}(t)=\sigma\exp\left\{\varepsilon\sin(2\pi\,t/\tau)\right\}.$$ The continuous model was represented for $\delta t\to 0$ as Eq. \[eq:disceig\] and the periodic quasispecies Eq. \[eq:pqs\] was calculated. Figure \[fig:ss-phase\] [ *(left)*]{} shows the resulting phase diagram. For small $\tau$, the error threshold is given by the one of the time-averaged landscape, whereas for large $\tau$, the error threshold oscillates between minimum and maximum values corresponding to $\min_{t}f_{0}(t)$ and $\max_{t}f_{0}(t)$, as expected in the quasi-static regime. Second, Nilsson and Snoad studied in [@nilsson] a moving that jumps randomly to one of its nearest neighbor strings every $\tau$ time steps. The time-average of this landscape over many jump cycles is a totally flat or neutral landscape, which explains the extension of the disordered phase to small $\mu$ and small $\tau$ as it is shown in Fig. \[fig:ss-phase\] [ *(right)*]{}. In the quasi-static region, order is expected because the needle stays long enough at each position for a quasispecies to grow. Hence, we can understand the existence of the observed and calculated phase diagrams in Fig. \[fig:ss-phase\] from simple arguments. In fact, they are special instances of the general schematic phase diagram depicted in Fig. \[fig:phase\].
In the following, we will consider regularly moving s and derive the infinite population behavior of a in such landscapes. This is interesting, since s should be considered to adapt faster to changes compared to s, as the missing time-average region of s for small $\tau$ suggests. To clarify whether a different phase diagram compared to Fig. \[fig:ss-phase\] [*(right)*]{} emerges for s with moving , we will calculate the phase diagram including the optimal mutation rate that maximizes a lower bound for the concentration of the needle string in the population.
{width="55.00000%"}
Generational and a moving {#sec:genga}
==========================
In this section, we want to analyze quantitatively the asymptotic behavior of a with that moves [*regularly*]{} in the sense of Section \[sec:fp\] to one of its $l$ nearest neighbors every $\tau$ time steps. At the end, we will also be able to comment on the case of a that jumps [*randomly*]{} to one of its nearest neighbors.
A simple example of a that moves regularly to nearest neighbors is shown in Fig. \[fig:peak\] [*(left)*]{}. Each jump corresponds to a $\pi/2$-rotation of the four-dimensional hypercube $\{0,1\}^{4}$ along the $1100$ axis, i. e. the lower two bits are rotated as shown in Fig. \[fig:peak\] [*(right)*]{}. We will call the set of strings $\{P^{n}\,i\mid n\in\NN\}$ which is obtained by applying the same rule $P\in\Perm_{2^{l}}$ over and over to some initial string $i\in\{0,1\}^{l}$, the [*orbit of $i$ under $P$*]{}. The period length $4$ of the orbit shown in Fig. \[fig:peak\] [*(left)*]{} originates from the rotation angle $\pi/2$ and hence is independent of the string length $l$. The orbits of such rotations will always be restricted to only four different strings. For reasons that will become clear below, we are looking for [*regular*]{} movements of the needle that are [*not*]{} restricted to such a small subspace of the string space. Instead, the needle is supposed to move ‘straight away’ from previous positions in string space. Since a complete classification and analysis of all possible [*regular*]{} movements for given string length $l$ and jump distance $d$ is out of the scope of this work, we will simply give an example of a rule $P\in\Perm_{2^{l}}$ that generates such movements: the composition of a cyclic 1-bit left-shift, which we denote by $P_{\ll}$, and an exclusive-or with $0\cdots01$, which we denote by $P_{\oplus}$.
{width="35.00000%"}
For string length $l\le3$, $P_{\ll}$ corresponds to a $2\pi/l$ rotation along the $1\cdots1$ axis as can be seen in Fig. \[fig:crot\]. Moreover, the orbit of $0\cdots0$ under $P_{\oplus\ll}=P_{\oplus}\circ P_{\ll}$ is shown in Fig. \[fig:corb\] also for $l=3$. For arbitrary string length $l$, it is more difficult to visualize the action of $P_{\ll}$ and hence of $P_{\oplus\ll}$. But, it is easily verified that starting from all zeros $0\cdots0$, the string with $n\le l$ ones $0\cdots01\cdots1$ will be reached after exactly $n$ jumps. Moreover, the orbit of $0\cdots0$ under $P_{\oplus\ll}$ has the period length $2l$. In the limit of long strings $l\to\infty$, this periodicity is broken because the needle never (i. e. after $\infty$ many jumps) returns to all zeros $0\cdots0$, but – as we have shown in Eq. \[eq:pqs\] using Eq. \[eq:xxp\] – there still exists an asymptotic quasispecies.
{width="35.00000%"}
How does our simple behave with a that moves according to $P_{\oplus\ll}$? In Fig. \[fig:regular\], two typical runs of a with a like that are depicted. The setting $(m,l,f_{0},\tau)$ was kept fixed but two different mutation rates $\mu$ were chosen. In the case of Fig. \[fig:regular\] [*(right)*]{}, the mutation rate is ‘too high’ to allow the population to track the movement. The concentration of the future needle string (solid line) cannot grow much within one jump cycle resulting in a decreasing initial condition (bullet) for the growth of the needle concentration (dotted line) in the next cycle. The population looses the peak – in this case after $\approx 90$ generations. It might happen that the population finds the needle again by chance (or better saying the moving needle jumps into the population), but the population will not be able to stably track the movement. Contrasting to that, the mutation rate was chosen to maximize the concentration of the future needle string at the end of each jump cycle (bullets) in Fig. \[fig:regular\] [*(left)*]{}.
-0.08{width="120.00000%"}
Since in that case, the best achievable initial condition is given to each jump cycle, the movement of the needle is tracked with the highest possible stability for the given setting $(m,l,f_{0},\tau)$. As can be expected from Fig. \[fig:regular\] and is affirmed by further experiments, the bullets keep on fluctuating around an average value for $n\to\infty$ which is for the infinite population given by the quasispecies Eq. \[eq:pqs\]. In the following, we are going to model that system with some idealizations and we will calculate a lower boundary for this average value.
We adopt the viewpoint of permuting the concentration vector compatible to the movement of the needle as we have done implicitly in Fig. \[fig:regular\] and formally in the definition of $\xx'(t)$ in Eq. \[eq:xxp\], but we drop the primes henceforth. The concentration of the needle string within jump cycle $n$ is denoted by $x_{0}(n,\pha)$ and the concentration of the string the needle will move to with the $(n+1)$th jump (i. e. the future needle string in jump cycle $n$) is denoted by $x_{1}(n,\pha)$. The initial cycle prior to which [*no*]{} jump has occurred is $n=0$. Within a cycle, the time or generation is counted as phase $\pha\in\{0,\ldots,\tau\}$. Two succeeding cycles are connected by the (approximated) rule of change $$\label{eq:approx}
x_{0}(n+1,0)=x_{1}(n,\tau)\quad\mbox{and}\quad x_{1}(n+1,0)\approx 0.$$ The second relation is an approximation which is made to simplify the coming calculations, but it holds only if the needle jumps onto a string which has not been close to one of the previous needle positions. Otherwise, the future needle string could already be present with a concentration significantly larger than $1/2^{l}\approx0$. In Fig. \[fig:regular\], we have chosen the rule $P_{\oplus\ll}$ to get experimental data for a case in which this assumption is fulfilled. Later on we will see that we can still make useful comments about cases in which that approximation is partly broken.
If we plot $x_{0}(n+1,0)=x_{1}(n,\tau)$ against $x_{0}(n,0)$, we get an intuitive picture for the system’s evolution towards the quasispecies. The concentration $x_{0}(n,0)$ converges for $n\to\infty$ towards a fixed point, $$x_{\rm fix}:=\lim_{n\to\infty}x_{0}(n,0),$$ as shown in Fig. \[fig:iterfix\] for a finite value of $x_{\rm fix}$. Obviously, this fixed point depends on the full setting $x_{\rm fix}=x_{\rm fix}(m,l,f_{0},\tau,\mu)$. Since we are especially interested in the effects of various cycle lengths $\tau$ and mutation rates $\mu$, we keep $(m,l,f_{0})$ fixed, such that $x_{\rm fix}=x_{\rm fix}(\tau,\mu)$.
In the remaining of this section, we will calculate $x_{0}(n+1,0)=x_{1}(n,\tau)$ in dependence on $x_{0}(n,0)$, which is the solid curve in Fig. \[fig:iterfix\], for arbitrary parameter settings. From this knowledge, we will construct the phase diagram. Since we stay within one jump cycle, we drop $n$ to take off some notational load.
Derivation of the Fixed Point Concentrations
--------------------------------------------
To calculate $x_{1}(\tau)$, it is sufficient to take only $x_{0}$ and $x_{1}$ into account, because the assumed initial condition is $x_{1}(0)\approx 0$, such that the main growth of $x_{1}$ is produced by the mutational flow from the needle. Moreover, we assume $\mu$ to be small enough such that terms proportional to $\mu^{2}$ can be neglected. This means we restrict ourselves to the case in which the system is mainly driven by one-bit mutations. Without normalization, the evolution equations then read $$\begin{aligned}
y_{0}(t+1)&=\phantom{\mu}(1-\mu)^{l\phantom{-1}}f_{0}\,y_{0}(t)
+\big\{\mu(1-\mu)^{l-1}\,y_{1}(t)\big\},\\
y_{1}(t+1)&=\mu(1-\mu)^{l-1}f_{0}\,y_{0}(t)
+\phantom{\big\{\mu}(1-\mu)^{l\phantom{-1}}\,y_{1}(t),
\label{eq:tevol}\end{aligned}$$ where $y_{i}$ denote unnormalized concentrations in contrast to the normalized concentrations $x_{i}$.
For $f_{0}(1-\mu)\gg \mu$, which is always the case for large enough $f_{0}$, we can further neglect the back-flow $\{\cdots\}$ from the future needle string compared to the self-replication of the current needle string. The solution of Eq. \[eq:tevol\] is then given by $$\begin{aligned}
y_{0}(t)&=\left[(1-\mu)^{l}f_{0}\right]^{t}y_{0}(0),\\
y_{1}(t)&=\kappa_{t}(\mu)\,y_{0}(0)+(1-\mu)^{lt}y_{1}(0),\\[1.5ex]
&\hskip2cm\mbox{with $\left\{\begin{aligned}
\kappa_{t}(\mu)&=\mu(1-\mu)^{lt-1}\alpha_{t}\\
\alpha_{t}&=\textstyle\sum_{\nu=1}^{t}f_{0}^{\nu}=f_{0}
\frac{f_{0}^{t}-1}{f_{0}-1}.
\end{aligned}\right.$}
\end{aligned}$$ The coefficient $\kappa_{t}(\mu)$ measures the growth of $y_{1}(t)$ starting from the initial condition $y_{1}(0)\approx0,
y_{0}(0)\not=0$. As long as $y_{0}(t)+y_{1}(t)\ll1$, this gives already a good approximation for the concentrations $x_{0}(t)$ and $x_{1}(t)$. But in general, this approximation breaks down for large $t$, because of the exponential growth of $y_{0}(t)$. We need to normalize our solution, which can be done by $$\xx(t)=\yy(t)\big/\langle f\rangle_{0}\cdots\langle f\rangle_{t-1},
\quad\mbox{where $\langle f\rangle_{t}=(f_{0}-1)x_{0}(t)+1$.}
\label{eq:norm}$$ By expressing the fitness averages in terms of $y_{0}(t)$, we find, after solving a simple recursion, $$\begin{aligned}
\langle f\rangle_{0}\cdots\langle f\rangle_{t-1}&= \textstyle 1+(f_{0}-1)\left[
\sum_{\nu=0}^{t-1}(1-\mu)^{l\nu}f_{0}^{\nu}\right]x_{0}(0)\\
&=1+(f_{0}-1)\beta_{t}(\mu)x_{0}(0),\\[1.5ex]
&\hskip2cm\mbox{where
$\beta_{t}(\mu)=\frac{\tilde{f}^{t}-1}{\tilde{f}-1}\mbox{ and }
\tilde{f}=(1-\mu)^{l}f_{0}$}.
\end{aligned}$$ Finally, we arrive at the normalized concentrations $$\begin{aligned}
x_{0}(t)&=\left[(1-\mu)^{l}f_{0}\right]^{t}x_{0}(0)\Big/
\left[1+(f_{0}-1)\beta_{t}(\mu)x_{0}(0)\right],\\
x_{1}(t)&=\left[\kappa_{t}(\mu)\,x_{0}(0)+(1-\mu)^{lt}x_{1}(0)\right]\Big/
\left[1+(f_{0}-1)\beta_{t}(\mu)x_{0}(0)\right].
\end{aligned}$$ The asymptotic state can now be calculated by using the initial condition $x_{1}(0)\approx 0, x_{0}(0)\not=0$ and demanding $x_{1}(\tau)=x_{0}(0)$. It is easily verified that for the fixed point follows $$x_{\rm fix}(\tau, \mu)=\frac{\kappa_{\tau}(\mu)-1}{(f_{0}-1)\beta_{\tau}(\mu)}.
\label{eq:fix}$$
Consistency in the Quasi-Static Limit {#sec:cons}
-------------------------------------
How can we test the quality of the approximate result Eq. \[eq:fix\]? For large cycle lengths $\tau$, we enter the quasi-static regime, where we can approximate the population at the end of each cycle by the quasispecies of the corresponding static landscape. Figure \[fig:approx\] shows a comparison of the exact numerical calculations of the quasispecies ($\tau\to\infty$) and the $\mathcal{O}(\mu^{2})$ calculations ($\tau=100$). In the numerical $\mathcal{O}(\mu^{2})$ calculation, the back-flow from the first error class to the needle string is included. Overall, we find the error threshold and the maximum of the fixed point concentration well represented. This also suggests that the deviation of the $\mathcal{O}(\mu^{2})$ approximation from the exact values should be small for smaller $\tau$, because those deviations add up for $\tau\to\infty$ by the iterative procedure.
How do the calculated fixed point concentrations compare to simulations with (large) finite population? In Fig. \[fig:regular\], the values of $x_{\rm fix}(\infty,\mu)$ and $x_{\rm fix}(4,\mu)$ are shown. For $\tau\to\infty$, the deviation from the average $\langle
x_{1}(n,\pha)\rangle$ (in generations $0-20$) is in fact the same as what can be read off in Fig. \[fig:approx\]. The deviation of $x_{\rm
fix}(4,\mu)$ from the average value $\langle x_{0}(n,0)\rangle$ in generations $24,28,\ldots,100$ is significantly larger. This is caused by the neglect of all other strings’ contributions apart from the current needle string’s contribution to the flow onto the future needle string. These neglected contributions increase the average fixed point concentration measured in the experiment in comparison to the calculated value $x_{\rm fix}(\tau,\mu)$. But even though there are deviations, we conclude that the approximately calculated value is always a lower bound for the exact value. In the next section, we will use this observation to derive an expression for the mutation rate that maximizes the average fixed point concentration.
Phase Diagram {#sec:pd}
-------------
In Fig. \[fig:taudep\], the fixed point values $x_{\rm fix}(\tau,\mu)$ are shown for small cycle lengths $\tau$. For the shown parameter setting, the region with $x_{\rm fix}(2,\mu)>0$ is extremely small. We notice that there are two error thresholds, one for ‘too low’ mutation rates, $\mu_{{\rm th}<}$, and one for ‘too high’ mutation rates, $\mu_{{\rm th}>}$. The intuition behind that was already given in Section \[sec:spd\]. For too low mutation rates the population becomes slow and evolves in the averaged, flat landscape, whereas for too high mutation rates the usual transition to the disordered phase takes place. In the following we will calculate the phase diagram starting from Eq. \[eq:fix\].
#### Error Thresholds:
The error thresholds are given by $$x_{\rm fix}(\tau,\mu)=0\quad\Longleftrightarrow\quad
\kappa_{\tau}(\mu)=1.
\label{eq:et}$$ This is the same condition as one would get using only unnormalized concentrations $y_{i}(t)$. Since $y_{i}(t)\approx0$ near the error thresholds, the neglect of the normalization is not critical for the calculation of the error thresholds themselves, whereas it is important for the optimal mutation rate and of course for the fixed point concentration. Since Eq. \[eq:et\] cannot be solved for $\mu$ in closed form, we write down the corresponding recursion relation that converges, for a suitable starting value of $\mu$, to the solution of Eq. \[eq:et\] in the limit $k\to\infty$, $$\begin{aligned}
\mu_{{\rm th}<}^{(k)}&=1\Big/\alpha_{\tau}\left(1-\mu_{{\rm th}<}^{(k-1)}\right), &
\mu_{{\rm th}<}^{(0)}&=0,\\
\mu_{{\rm th}>}^{(k)}&=1-\left(1\Big/\alpha_{\tau}\mu_{{\rm th}>}^{(k-1)}\right)^{1/(l\tau-1)},
&\quad \mu_{{\rm th}>}^{(0)}&=1-f_{0}^{-1/l}=:\mu_{{\rm th}}^{\infty}.
\end{aligned}$$ For $\mu_{{\rm th}<}$, a good starting value is $0$, since $\mu_{{\rm th}<}\approx0$ anyway. For $\mu_{{\rm th}>}$, the approximate value for the error threshold of the static (i. e. $\tau\to\infty$) landscape $\mu_{{\rm th}}^{\infty}$ can be chosen, which is obtained by calculating the fixed point \[using Eq. \[eq:tevol\] and \[eq:norm\]\], $$x_{0}(t+1)=x_{0}(t)\quad\Longleftrightarrow\quad
x_{\rm fix}^{\infty}=\frac{(1-\mu)^{l}f_{0}-1}{f_{0}-1},$$ setting it to zero and solving for $\mu$.
#### Optimal Mutation Rate:
In order to track changes with the best achievable stability for a given setting $(m,l,f_{0},\tau)$, the lowest possible concentration (infimum of) $x_{0}(n,\pha)$ needs to be maximized, because a low concentration might result in the loss of the needle string in a finite population. Since for infinite populations $x_{0}(n,\pha)$ is monotonously increasing with $\pha$ it is sufficient to maximize $x_{0}(n,0)$. Moreover, we derived above that $x_{0}(n,0)$ approaches the fixed point value $x_{\rm fix}(\tau,\mu)$ for $n\to\infty$. For finite populations, we expect similar behavior but the strict monotony of $x_{0}(x,\pha)$ in $\pha$ will be destroyed by fluctuations and also the fixed point value itself will fluctuate around some average value $\langle
x_{\rm fix}\rangle$ as can be seen in Fig. \[fig:regular\]. However, the [*safest*]{} way to avoid any loss of the needle string is still to maximize the average fixed point value $\langle x_{\rm fix}\rangle$. In this sense, we define the [*optimal mutation rate*]{} $\mu_{\rm opt}$ as the one that maximizes $\langle x_{\rm fix}\rangle$. In the previous Section \[sec:cons\], we noted that our approximated infinite population value $x_{\rm fix}(\tau,\mu)$ represents a lower bound for $\langle
x_{\rm fix}\rangle$, where the maxima of the two curves are expected to coincide for fixed $\tau$. Thus, $\mu_{\rm opt}$ can be obtained by maximization of $x_{\rm fix}(\tau,\mu)$.
We can derive an expression for the optimal mutation rate $\mu_{\rm opt}$ from $$\frac{\partial x_{\rm fix}}{\partial \mu}(\tau,\mu_{\rm opt})=0$$ If we neglect the $\mu$ dependence of $\beta_{\tau}(\mu)$ in Eq. \[eq:fix\], which corresponds to the approach in [@nilsson], we simply find $\mu^{\rm NS}_{\rm opt}(\tau,l)=1/l\tau$. Because of $\smash{\mu^{\rm NS}_{\rm opt}\xrightarrow{\tau\to\infty} 0}$, this result is inconsistent with the quasi-static limit, because $\mu_{\rm opt}$ should approach the value for which the concentration of 1-mutants in the quasispecies of the corresponding static landscape is maximized. We conclude that the $\mu$ dependence of $\beta_{\tau}(\mu)$ cannot be neglected for the correct optimal mutation rate, which we are now going to calculate.
For $\alpha_{\tau}\gg1$, which is the case for $\tau\gg 1$ and $f_{0}>1$, or $\tau\approx 1$ and $f_{0}\gg 1$, we can neglect the $-1$ in the numerator of $x_{\rm fix}(\tau,\mu)$ and take only $\alpha_{\tau}$ into account for the calculation of $\partial x_{\rm fix}
/\partial \mu$. After some algebra, we find $$\mu_{\rm opt}=\frac{(\tilde{f}^{\tau}-1)(\tilde{f}-1)}{l(\tilde{f}^{\tau+1}
-(\tau+1)\tilde{f}+\tau)},\quad\mbox{where $\tilde{f}=f_{0}(1-\mu_{\rm opt})^{l}$}.$$ Since $\tilde{f}=\tilde{f}(\mu_{\rm opt})$, this equation cannot be solved in a closed form for $\mu_{\rm opt}$. However, for $\tau\to\infty$ the equation simplifies to $$\mu^{\infty}_{\rm opt}=\begin{cases}
\displaystyle(\tilde{f}-1)/l\tilde{f}&: \tilde{f}>1\\
\phantom{f}0&:\tilde{f}\le 1.
\end{cases}$$ In the case $\tilde{f}>1$, we find $$(1-l\mu^{\infty}_{\rm opt})(1-\mu^{\infty}_{\rm opt})^{l}=1\big/f_{0}.$$ By approximating $(1-\mu)^{l}\approx (1-l\mu)^{2}$, we get a cubic equation. The real root of that equation is approximately [@ropt] given by (see also Fig. \[fig:optmut\]) $$\begin{gathered}
\mu_{\rm opt}^{\infty}(f_{0},l)\approx \mu_{+}
\left[1+\frac{(l-1)\mu_{+}(1-l\mu_{+})}{3l(l-1)\mu_{+}^{2}-2\mu_{+}(3l-1)+4}
\right]\\[1.5ex]
\mbox{with $\mu_{+}=\displaystyle\frac{1}{l}\left[1+f_{0}^{-1/2}\right]$.}
\label{eq:optmut}\end{gathered}$$
#### Resulting Phase Diagram:
From the above, we are able to plot the phase diagram for our model as shown in Fig. \[fig:pd-genga\]. Two settings are plotted. For $f_{0}=2\ (\mbox{resp.\ }10)$ the diamonds (resp. circles) are the numerically obtained error thresholds. The solid and dash-dotted lines are $\smash{\mu_{{\rm th}<}^{\smash{(5)}}}$ and $\smash{\mu_{{\rm th}>}^{\smash{(5)}}}$. To show the convergence property of $\smash{\mu_{{\rm th}<,>}^{\smash{(k)}}}$, $\smash{\mu_{{\rm th}<,>}^{\smash{(0)}}}$ are plotted for $f_{0}=10$ as dashed lines. Obviously, the needed corrections to the chosen starting values increase for smaller $\tau$, such that more iterations are needed to describe the error thresholds correctly for small $\tau$. The expressions $\smash{\mu^{\smash{(5)}}_{{\rm th}<,>}}$ are already a good approximation for the given settings. Representing the quasi-static limit, $\mu_{\rm
th}^{\infty}$ is plotted as dotted line and gets consistently approached by $\mu_{{\rm th}>}(\tau)$ for $\tau\to\infty$. Furthermore, $\mu_{\rm opt}^{\infty}$ is plotted as dash-dot-dotted line. The numerically measured values for $\mu_{\rm opt}(\tau)$ are shown for $f_{0}=2\ (\mbox{resp.\ }10)$ as triangle (resp. squares). They approach $\mu_{\rm opt}^{\infty}$ very quickly already for $\tau\approx 20\
(\mbox{resp.\ }10)$.
We conclude that the above quantitative description is in good agreement with the numerical observations and approaches the quasi-static region in a consistent way. Moreover, the phase diagram fits well into the general one raised in Section \[sec:spd\]. Even in the considered case of a , we find – depending on the parameter setting – a time-averaged phase for very small $\tau$. The time-averaged phase broadens for small $f_{0}$.
Stochastically moving
----------------------
Up to now, we analyzed a regularly moving , for example with the rule $P_{\oplus\ll}$. What happens if the is allowed to move to a [*randomly*]{} picked nearest neighbor, as it is shown in Fig. \[fig:stochpeak\] for $l=4$?
{width="55.00000%"}
-0.08{width="120.00000%"}
Two typical runs of a with this fitness landscape are depicted in Fig. \[fig:stoch\]. The setting $(m,l,f_{0},\tau)$ was chosen the same as in Fig. \[fig:regular\] which allows for a direct comparison of the ’s behavior for regularly and stochastically moving s. The overall behavior is similar. For large mutation rates, the population looses the needle string, whereas the moving needle is tracked stably for mutation rates close to the above defined optimal mutation rate. In addition, strong fluctuations in the values of $x_{1}(n,0)$ (lower ends of solid lines) as well as $x_{0}(n+1,0)=x_{1}(n,\tau)$ (bullets) occur in the [*stochastic*]{} case. These result from [*back-jumps*]{}. If, at the end of the current cycle, the needle jumps back to the string it has been to in the previous cycle, then $x_{1}(n,0)=x_{0}(n-1,\tau)$ is significantly larger than zero. This can be seen in Fig. \[fig:stoch\] [*(right)*]{} at generations $36,40$ and $64$ and also in Fig. \[fig:stoch\] [*(left)*]{} at generations $72$ and $88$ (the gaps in Fig. \[fig:stoch\] [*(left)*]{} correspond to $x_{1}, x_{0}$ being much larger than $0.025$). If no back-jumps occur, as in generations $24-72$ in Fig. \[fig:stoch\] [*(left)*]{}, the system with stochastic behaves nearly indistinguishable from the one with regularly moving . Since back-jumps always increase the concentrations of the needle string in the very next occurring jumps, the above calculated fixed point $x_{\rm fix}(\tau,\mu)$ is still a lower bound. Thus, our previous notion of optimal mutation rate remains applicable to the stochastically moving although the assumption $x_{1}(n,0)\approx0$ from Eq. \[eq:approx\] is not always fulfilled.
Nilsson and Snoad [@nilsson] did their analysis of the continuous Eigen model Eq. \[eq:conteig\] with stochastic in a similar way as we did above. In analogy to their calculation for the continuous Eigen model, we find for a the optimal mutation rate $\mu^{\rm NS}_{\rm opt}(\tau,l)=1/l\tau$ which is inconsistent with the quasi-static limit (see Section \[sec:pd\]). The reason is the missing normalization in the work of Nilsson and Snoad. Furthermore, they could not derive an expression for the fixed point concentration $x_{\rm fix}(\tau,\mu)$ because of that same reason.
Jumps of larger Distance
------------------------
To conclude this section about the behavior of s with different kinds of s that move to [*nearest*]{} neighbors, let us shortly discuss jumps of Hamming distance $d$ [*larger*]{} than one. Obviously, the analytical calculations get more complicated, because the $\mathcal{O}(\mu^{2})$-approximation is not sufficient anymore as it connects only nearest neighbors. To describe jumps of a larger distance, the concentrations of some intermediate sequences need to be taken into account, so that we have to solve a time evolution much more complicated than Eq. \[eq:tevol\]. Hence, we cannot make simple statements for finite $\tau$. On the other hand, the system approaches the quasi-static region for large $\tau$ and it is characterized by $\mu_{{\rm th}<,>}^{\infty}$ and $\mu_{\rm
opt}^{\infty}$ as we have seen in Fig. \[fig:pd-genga\].
The exact quasispecies for $\tau\to\infty$ is shown in Fig. \[fig:all-x-r\]. The plotted values are error class concentrations, in order to make the higher error classes visible at all. Each $k$-mutant has a concentration of $\tilde{x}_{k}/\smash{\left({l \atop k}\right)}$ in the quasispecies state, because for a the mutant’s fitness depends only on its Hamming distance to the needle and therefore all $\smash{\left({l \atop k}\right)}$ $k$-mutants have the same concentration in the quasispecies. For finite populations, this is only true on average, because the asymptotic state is distorted by fluctuations. But in the following, we assume that the quasispecies is still representative for the average distribution of the population in the asymptotic state. Then, the optimal mutation rate in the sense of Section \[sec:pd\] for jumps of distance $d$ is by definition the position of the maximum of $\tilde{x}_{d}$. For $d\ge l/2$, optimal mutation rate and error threshold become identical. Although $\tilde{x}_{d}$ is maximized for mutation rates close to the error threshold it amounts, as do all other concentrations to only $\approx 1/2^{l}$, which leads to an approximately random drift for finite populations. On the other hand, the chance of tracking the needle decreases even further for small mutation rates because then the concentration $\tilde{x}_{d}$ becomes even smaller. In this sense, the quasispecies distribution, which is centered on the needle string, is useless for tracking the next jump if $d\ge l/2$. This also suggests – in agreement with the experimental findings of Rowe [@roweqs] (in this book) – that finite populations are for low mutation rates unable to track large jumps – in particular in the extreme case $d=l$. Only for jumps of $d<l/2$ the corresponding error class concentration $\tilde{x}_{d}$ shows a concentration maximum significantly above $1/2^{l}$. From the heights of the concentration maxima, we see that the difficulty of tracking the changes increases with the Hamming distance $d$ of the jumps. Vice versa, the advantage a population gets after a jump from its structure prior to the jump decreases with increasing jump distance $d$. In addition, a mutation rate which is simultaneously optimal for more than one distance cannot be found.
Conclusions and Future Work
===========================
On the basis of general arguments, the phase diagrams of population-based mutation and probabilistic selection systems like the above , and Eigen model in time-dependent fitness landscape can be easily understood. The notion of regular changes allows for an exact calculation of the asymptotic state in the sense of a generalized, time-dependent quasispecies. For a with that moves regularly to nearest neighbors, the quasispecies can be straightforwardly calculated under simplifying assumptions. The result is a lower bound for the exact quasispecies. With that lower bound, we have constructed the phase diagram in the infinite population limit. This phase diagram is in agreement with the one raised from general arguments.
In order to improve our analysis, we need to weaken our assumptions. In particular, we have to overcome the restriction of taking into account only the flow from the current towards the future needle string. The presence of other contributions to the flow has to be modeled in some way. Another future step could be an investigation of the fluctuations that are introduced by the finiteness of realistic populations (discreteness of $\Lambda_{m}$) around the quasispecies. This would lead to a lower boundary for the population size above which the needle string is not lost due to those fluctuations.
An extension of our analysis to non-regularities like the occurrence of more than a single jump rule, can be achieved by averaging the time evolution Eq. \[eq:gtev\] for $n\to\infty$ according to each rule’s probability of being applied. A similar averaging procedure will be necessary if fluctuations of the cycle length $\tau$ are present. Finally, an extension of the description to broader, more realistic peaks, as well as models including crossover and other selection schemes, are important topics for future work.
[xxx]{}
T. Bäck, U. Hammel and H.-P. Schwefel. [*Evolutionary Computation: Comments on the History and Current State*]{}. IEEE Transactions on Evol. Comp. 1(1), p. 3, 1997.
T. Bäck, D. B. Fogel and Z. Michalewicz, editors. [*Handbook of Evolutionary Computation*]{}. IOP Publishing, Bristol, 1997.
J. Branke. [*Evolutionary Algorithms for Dynamic Optimization Problems, A Survey*]{}. Technical Report 387, AIFB University Karlsruhe, 1999.
J. E. Rowe. [*Finding attractors for periodic fitness functions*]{}. In W. Banzhaf [*et al.*]{}, editors, [*Proceedings to GECCO 1999*]{}, Morgan Kaufmann, San Mateo, p. 557, 1999.
L. M. Schmitt, C. L. Nehaniv and R. H. Fujii. [ *Linear analysis of genetic algorithms*]{}. Theoretical Computer Science 200, p. 101, 1998.
M. Eigen. [*Selforganization of matter and the evolution of biological macromolecules*]{}. Naturwissenschaften 58, p. 465, 1971.
M. Eigen and P. Schuster. [*The Hypercycle – A Principle of Natural Self-Organization*]{}. Springer-Verlag, Berlin, 1979.
M. Eigen, J. McCaskill and P. Schuster. [*The molecular quasispecies*]{}. Adv. Chem. Phys. 75, p. 149, 1989.
E. Baake and W. Gabriel. [*Biological evolution through mutation, selection, and drift: An introductory review*]{}. Ann. Rev. Comp. Phys. 7, in press, 1999.
J. E. Rowe. [*The dynamical systems model of the simple Genetic Algorithm*]{}. this issue, p. XXX, 1999. M. D. Vose. [*The simple Genetic Algorithm – Foundations and Theory*]{}. MIT Press, Cambridge, 1999.
E. van Nimwegen, J. P. Crutchfield and M. Mitchell. [*Statistical Dynamics of the Royal-Road genetic algorithms*]{}. Theoretical Computer Science, special issue on Evolutionary Computation, A. Eiben, G. Rudolph, editors, in press, 1998.
J. E. Rowe. [*Cyclic Attractors and Quasispecies Adaptability*]{}. this issue, p. XXX, 1999. K. DeJong and J. Sarma. [*Generation Gaps Revisited*]{}. In L. D. Whitley, editor, [*Foundations of Genetic Algorithms 2*]{}, Morgan Kaufmann, San Mateo, p. 19, 1993.
A. Rogers and A. Prügel-Bennett. [*Modeling the Dynamics of a Steady State Genetic Algorithm*]{}. In W. Banzhaf and C. Reeves, editors, [*Foundations of Genetic Algorithms 5*]{}, Morgan Kaufmann, San Mateo, p. 57, 1998.
J. Branke, M. Cutaia and H. Dold. [*Reducing Genetic Drift in Steady State Evolutionary Algorithms*]{}. In W. Banzhaf [*et al.*]{}, editors, [*Proceedings to GECCO 1999*]{}, Morgan Kaufmann, San Mateo, p. 68, 1999.
C. O. Wilke, C. Ronnewinkel and T. Martinetz. [ *Molecular Evolution in time-dependent Environments*]{}. In D. Floreano, J.-D. Nicoud and F. Mondada, editors, [ *Proceedings to European Conference on Artificial Life 1999*]{}, Springer, Berlin, p. 417, 1999.
C. O. Wilke and C. Ronnewinkel. [ *Dynamic Fitness landscapes in the Quasispecies model*]{}. in preparation.
M. Nilsson and N. Snoad. [*Error Thresholds on dynamic Fitness-Landscapes*]{}. Working Paper 99-04-030, Santa Fe Institute, 1999.
A more detailed explanation and analysis of the used approximation will be presented elsewhere.
|
---
abstract: 'Electron antineutrinos stream freely from rapidly decaying fission products within nuclear reactors and from long-lived radioactivity within Earth. Those with energy greater than 1.8 MeV are regularly observed by several kiloton-scale underground detectors. These observations estimate the amount of terrestrial radiogenic heating, monitor the operation of nuclear reactors, and measure the fundamental properties of neutrinos. The analysis of antineutrino observations at operating detectors or the planning of projects with new detectors requires information on the expected signal and background rates. We present a web application for modeling global antineutrino energy spectra and detection rates for any surface location. Antineutrino sources include all registered nuclear reactors as well as the crust and mantle of Earth. Visitors to the website may model the location and power of a hypothetical nuclear reactor, copy energy spectra, and analyze the significance of a selected signal relative to background.'
author:
- 'A.M. Barna'
- 'S.T. Dye'
bibliography:
- 'Nugeo\_bib.bib'
- 'revtex-custom.bib'
nocite: '[@apsrev41control]'
title: Web Application for Modeling Global Antineutrinos
---
introduction
============
The flux and energy spectrum of electron antineutrinos from $\beta^-$ decay measured at the surface of Earth provides important information on the location and operation of nuclear reactors, the quantity and distribution of planetary radioactivity, and the mass states of neutrinos. The monitoring of antineutrinos from nuclear reactors at long [@nudar13; @snif10] and short [@nucifer15; @songs07] distances has important applications for nuclear non-proliferation [@adam10], as well as significant capabilities for fundamental physics [@reines53; @reines76; @jgl08]. Knowledge of the reactor antineutrino flux is essential for the observation of geological antineutrinos [@kl05]. Geo-neutrino observations apprise terrestrial radiogenic heating [@gando13; @agostini15], guiding basic understanding of the composition, structure, and thermal evolution of Earth [@dye_etal15]. Recent modeling projects provide static maps the surface flux of antineutrinos [@baldoncini; @agm15] with planned revisions and additions as the inventory of nuclear reactors changes, geo-neutrino observational exposures increase, and geological modeling becomes more sophisticated. The web application, described herein and available at http://geoneutrinos.org/reactors, complements the mapping projects by estimating and displaying the energy spectrum and detection rate of electron antineutrinos at any surface location. This dynamic display allows users to download energy spectra and to analyze the significance of a selected signal relative to background, providing an estimate of the required detector exposure.
Reactor Antineutrinos
=====================
Nuclear power reactors are intense sources of antineutrinos, which emerge during the beta decay of short-lived fission products. An empirical fit estimates the energy spectrum of un-oscillated electron antineutrinos detected at a distance $R$ (km) from a reactor with a thermal power $P$ (GW) with the formula $$\label{reacspec}
N(E_{\overline{\nu}_e}) = 2075 (E_{\overline{\nu}_e} - e_2)^2 exp [-(\frac{E_{\overline{\nu}_e} + e_1}{e_3})^2] \frac{P} {R^2},$$ which is valid for $E_{\overline{\nu}_e}$ greater than the inverse beta decay interaction threshold energy. The values of the parameters are $e_1=0.8$, $e_2=1.43$, and $e_3=3.2$ all in units of MeV. In , the decaying exponential describes antineutrino production and the rising quadratic represents the interaction cross section.
Mixing of neutrino mass states along the flight path from source to detector converts some of the electron antineutrinos to a flavor ($\mu$, $\tau$) that does not participate in the inverse beta decay interaction at MeV-scale energy. This flavor oscillation suppresses the detection rate and introduces conspicuous features to the energy spectrum. With three active neutrinos the survival probability is $$\label{posc}
\begin{array} [b] {r}
P_{\overline{\nu}_{e}\rightarrow\overline{\nu}_{e}} = 1- \cos^4 \theta_{13} \sin^2(2\theta_{12}) \sin^2 (\Delta_{21} L/E_{\overline{\nu}_e})\\
- \sin^2(2\theta_{13}) \cos^2 \theta_{12} \sin^2 (\Delta_{31} L/E_{\overline{\nu}_e})\\
- \sin^2(2\theta_{13}) \sin^2 \theta_{12} \sin^2 (\Delta_{32} L/E_{\overline{\nu}_e}),
\end{array}$$ where $\Delta_{ij}=1.27\delta m^2_{ij}$, $L$ is in meters, and $E_{\overline{\nu}_e}$ is in MeV. Using the identities $\delta m^2_{31} = \delta m^2_{21} + \delta m^2_{32}$, $\cos^2 u = 1 - \sin^2 u$, and $\sin (2u) = 2 \sin u \cos u$, is determined by the values $\delta m^2_{31} = (2.457 \pm 0.047) \times 10^{-3}$ eV$^2$, $\delta m^2_{21} = (7.50 \pm 0.19) \times 10^{-5}$ eV$^2$, $\sin^2 \theta_{12} = 0.304 \pm 0.013$, [@gonz14] and $\sin^2 \theta_{13} = 0.0215 \pm 0.0013$ [@dayabay15]. The given value of $\delta m^2_{31}$ applies to normal mass hierarchy. Inverted mass hierarchy is approximated by using $\delta m^2_{32} = (2.457 \pm 0.047) \times 10^{-3}$ eV$^2$ and $\delta m^2_{32} = \delta m^2_{21} + \delta m^2_{31}$.
Geo-neutrinos
=============
The sources of antineutrinos from Earth are assumed to be crust and mantle. These geological reservoirs of the silicate portion of Earth contain thorium and uranium [@mcd95], which through a series of decays transmute to stable isotopes of lead. Several beta decay spectra in these series have endpoint energy greater than the 1.8 MeV detection threshold energy, providing an observable geo-neutrino signal [@dye12]. The probability density distributions as a function of energy for uranium and thorium geo-neutrino interactions are calculated and displayed in Figure \[fig:gnupdf\]. Due to the different shapes of the distributions a geological reservoir with a given ratio of thorium to uranium produces a geo-neutrino interaction spectrum characteristic of the given ratio. Geo-neutrino observations to date lack sufficient exposure to measure the thorium to uranium ratio of the recorded interactions.
The surface geo-neutrino signal is observed with significance by two underground detectors, one in Japan [@gando13] and one in Italy [@agostini15]. Although geological models of the crust predict distinct variation in the surface geo-neutrino signal [@huang13], the observed signals are consistent with each other. It remains the task of future observations, perhaps with detectors at locations informed by this web application, to independently test this fundamental geological model.
The web application displays at each Earth surface location the energy spectrum and detection rate of geo-neutrinos. The contribution from the crust is calculated from a model prediction of the non-oscillated geo-neutrino fluxes from thorium and uranium [@huang13]. Fluxes are converted to detection rates using standard methods [@dye12]. An average survival probability, given by $$\label{avgposc}
\overline{P}_{\overline{\nu}_{e}\rightarrow\overline{\nu}_{e}} = \frac{1} {2} \cos^4 \theta_{13} \big (1 - \sin^2(2\theta_{12}) \big) + sin^4 \theta_{13},$$ accounts for the effects of neutrino oscillations. A constant suppression factor of $0.55$ follows from the identities and values given in the previous section. Due to uncertainties in the model-dependent predictions for the mantle geo-neutrino rate and spectrum, these quantities are specified by the user of the web application.
Signal Sigificance
==================
An important function of this web application is the assessment of the significance of observing a signal in the presence of background. The assessment statistic is $$\label{stats}
N\sigma = \frac{S \sqrt {t}} {\sqrt {S + 2B}},$$ with $S$ the signal, $B$ the background, and $t$ the dwell time. With $S$ and $B$ in TNU, $t$ is given in years of exposure of a perfect detector with $10^{32}$ free proton targets. Presently, the web application does not include background due to cosmogenic sources or accidental coincidences.
Web Application
===============
The web application consists of four pages: “Detector,” “Reactors,” “GeoNu,” and “Output & Stats.” Each page has a pane displaying a Mercator projection of Earth. On the map, land areas are shown in tan with country borders in thin black lines and water areas are shown in blue.
The “Detector” page has two additional panes: “Spectrum” and “Location.” The “Spectrum” pane shows the energy spectrum of electron antineutrino detection rates, some output text on rates and distances, and the “Invert Neutrino Mass Hierarchy” checkbox. The spectrum plot displays the total of all rates (black line), the contribution from all IAEA-registered reactors (shaded green), the contribution from the closest reactor core (shaded gray), the contribution from geo-neutrinos (shaded yellow), the contribution from uranium geo-neutrinos (blue line), and the contribution from thorium geo-neutrinos (red line). The “Location” pane has boxes showing the detector “Latitude” and “Longitude” values, the “Follow Cursor On Map” checkbox, and the “Location Presets” box.
The “Reactors” page has two additional panes: “Reactor Load Factors” and “Custom Reactor.” The “Reactor Load Factors” pane simply contains the box for selecting either “Mean LF” or “2013 LF.” The “Custom Reactor” pane has the “Power” box for setting the thermal power of the custom reactor, the “Use Custom Reactor” checkbox, and the “Location” sub-pane with boxes showing the custom reactor “Latitude” and “Longitude” values and the “Place Reactor” button.
The “GeoNu” page has two additional panes: “Mantle” and “Crust.” The “Mantle” pane has the “Mantle Signal” box for setting the interaction rate in TNU and the “Th/U Ratio” box for setting the spectral shape of the mantle signal. The “Crust” pane simple displays some text, referencing the source of the crust flux values.
The “Output & Stats” page has the additional “Calculator” pane and an untitled output data box for inspecting, or copying, the plotted energy spectrum values. The “Calculator” pane has the “Signal” box for selecting the source with background of the antineutrino signal, the “Solve For” box for selecting whether the calculation returns the “Exposure Time” or the “Significance,” the “$E_{min}$ box for setting a lower energy cut (in MeV) on the spectrum to be analyzed, the ”$E_{max}$ box for setting the upper energy cuts (in MeV) on the spectrum to be analyzed, the “Time (years)” box for setting the exposure time if the “Solve For” box is set to “Significance,” and the “Sigma” box or setting the significance if the “Solve For” box is set to “Exposure Time.” The untitled output data box contains the antineutrino rate energy spectrum and its components at the selected location. The data, which range from $0$ to $10$ MeV, are in units of TNU per $10$ keV. There are six columns of data separated by commas, which correspond to: total rate, rate from all known IAEA reactor cores, rate from closest reactor core, rate from user-defined core (0 if not using a custom reactor), rate from uranium geo-neutrinos, and rate from thorium geo-neutrinos. There are a total of $1000$ rows of data under each column. The first $180$ data rows have value $0$ due to the $1.8$ MeV energy threshold of the electron antineutrino inverse beta decay interaction on a free proton. For plotting or further analysis, users simply copy and paste the contents of this box into a text file or spreadsheet program.
Development Framework
---------------------
Our web application is built using Flask [@flask], a micro-framework written in Python [@python]. Flask is utilized for Uniform Resource Locator (URL) routing and rendering web page templates. Only calculations required to route and render templates are performed on the web server. The computations for rendering the user interface, output figures, and output text data are handled by the client-side web browser. All client-side programs are written in JavaScript (ECMAScript 5.1). The components of the client-side programs are described in this section.
Request Routing
---------------
When the client-side web browser requests data from a web server using Hypertext Transfer Protocol (HTTP), the server needs to determine the response. While some web server software is configured with a set of rules to determine what, if anything, should be returned to the requesting agent, Flask uses a routing table. The routing table contains a list of routes (e.g. /model) and the call methods of the routes. Query parameters and URL fragments are ignored when determining a matching route.
The call method returns a suitable response, usually some text body and appropriate HTTP header fields. This can be as simple as returning a preset string or as complicated as performing a set of complex calculations to generate the response. Most of the methods employed by our web application simply compose and return web page templates. Irrespective of purpose, it is important for the response time to be fast, usually less than one second.
Web Page Template Rendering
---------------------------
A templating system minimizes code repetition and speeds development. Specifically, we use jinja2 [@jinja2]. A base template is defined (see: webnu/templates/base.html in the source), which contains basic elements displayed on all pages. This would include the Hypertext Markup Language (HTML) headers, the navigation bar appearing on every page, the JavaScript script, and the web style sheets, which are used by all pages. The base page template includes several placeholders where content from other page templates can be placed.
Other page templates are inherent and extend the base page template. When, for example, the /reactors page is requested, the reactor page template is found and starts to be rendered. The reactor page template has a declaration to extend the base page template in it and contains only the content for the placeholders in the base page template. This content is then placed inside the placeholders and the entire result is returned to the user.
Rendering a Page
----------------
The events, which occur when the reactors page is requested, are as follows (in this order):
1. The page template is rendered and returned.
2. When the returned HTML element is parsed, the browser starts rendering the page.
3. When the links to style sheets are encountered, the rendering is blocked until the style sheets are fetched.
4. References to external JavaScript files are loaded in the order they appear in the document.
5. The reactor data is loaded as part of these external files.
6. Since the main application JavaScript is inline and at the bottom of the document, it executes as soon as it is encountered.
7. The spectrum plot is initialized (though empty) and the methods used to dispatch events (event listeners) are attached to elements on the webpage.
8. The method for updating the spectrum plot is called once for the default display.
Once the above has finished, the page is ready for user input and will wait for an event needing a response.
User Input Events
-----------------
The interaction between the browser rendered document and any JavaScript is usually by what are called Events. Events are fired automatically by the web browser when the user performs certain actions. The reactor page is listening for the following events:
- The cursor has moved while over the map.
- The cursor has clicked the map.
- The Detector latitude has changed.
- The Detector longitude has changed.
- The ÒFollow Cursor On MapÓ checkbox has changed.
- The Detector preset selection has changed.
- The Reactor power has changed.
- The ÒUse Custom ReactorÓ checkbox has changed.
- The Custom Reactor latitude has changed.
- The Custom Reactor longitude has changed.
- The ÒPlace ReactorÓ button has been clicked.
- The ÒInvert Neutrino Mass HierarchyÓ checkbox has changed.
Each of these events can cause a state change in the application. The events may also be in conflict with each other, prompting a brief discussion of how each event is handled.
### Cursor Move Over Map
An event listener for the JavaScript mouse-move is attached to the map image. When the cursor (mouse) moves while over the map, the attached function is called with the ÒEventÓ passed into it. If the ÒFollow Cursor On MapÓ checkbox is selected, the coordinates from the event are translated into latitude and longitude. The latitude and longitude values are placed in the appropriate box in the detector ÒLocationÓ panel. The ÒLocation PresetsÓ selection input is set to no selection. Finally the Òupdate spectrumÓ function is called.
If the ÒFollow Cursor On MapÓ checkbox is not selected, the function returns immediately without doing anything.
### Cursor Click on Map
The JavaScript click event is listened for on the map. When clicked, usually the ÒFollow Cursor On MapÓ checkbox is toggled. The exception is if the user has clicked the ÒPlace ReactorÓ button. If this is the case, the next click will set the latitude and longitude of the custom reactor.
### Text Input of Checkbox Change
When any of the text input boxes has a value change, the Òupdate spectrumÓ routine is simply called. These boxes include the reactor power box, the reactor latitude and longitude boxes, and the detector latitude and longitude boxes. If the state of either the ÒInvert Neutrino Mass HierarchyÓ or the ÒUse Custom ReactorÓ checkbox has changed, the update spectrum routine is called.
### Preset Selected
When the user selects an option from the ÒLocation PresetsÓ selection input the latitude and longitude for that detector are placed in the text inputs. The ÒFollow Cursor On MapÓ checkbox is set to off. The update spectrum function is then called.
Spectrum Update
---------------
Most user actions require an updated energy spectrum, making the spectrum update function central to the web application. The function has several important tasks:
- Get the new user input values;
- Calculate the distances to the detector from all the reactors;
- Calculate the neutrino oscillation survival probability distribution for each distance;
- Multiply the reactor output spectrum by the appropriate survival probability distribution;
- Sum all the oscillated reactor output spectra;
- Draw the new spectrum plot;
- Locate the detector and reactor icons on the map;
- Update the spectrum text output.
When the spectrum update function is called it performs the following actions. First, the detector and user reactor icon locations are set. To do this, the latitude and longitude of the detector and user reactor are taken directly from the text input fields on the webpage. Then the geographic coordinates are converted to the image coordinates of the map with the origin in the upper left corner. The detector and user reactor image positions are then set with the image coordinates. If the user does not want the custom reactor to be used, the reactor image display attribute is set to Ònone.Ó
Next the distance and spectral contribution of each reactor are calculated. For computational simplicity, reactor positions are stored as Cartesian coordinates in a three dimensional array. The distance between each reactor and the user provided detector location is given by the Euclidean distance. The neutrino spectrum function is called for each reactor, the returning spectra are stored separately temporarily. The distance loop records which reactor is the closest to the detector so its contribution may be plotted separately on the output figure.
The spectrum update function then calculates all the ancillary output parameters: the distances to the user reactor and the closest reactor, and the TNU outputs. It also updates the text spectrum output box.
Finally, the line plot figure is updated. Since the D3.js [@d3js] data binding library is used, this is done simply by instructing D3 to use the newly calculated values. The y-axis domain is updated. The entire figure does not need to be redrawn, only what has changed.
Survival Probability
--------------------
The neutrino oscillation survival probability function calculates the probability distribution spectrum for an input distance. Due to multiple calls to computationally expensive trigonometric functions specified by , the computed spectrum for any given input distance is cached for future use.
Additional Calculation
----------------------
The resulting signal components are cached for additional analysis provided by the “Calculator” pane. On the Calculator pane, we provide the ability to calculate the exposure time required to have a user input signal significance or the signal significance provided by a given exposure time. The additional analysis does not require recalculation of the component signals so the results are nearly instantaneous. The calculator pane is implemented as an independent “web app” which only has the results of the reactor, and geo-neutrino calculations as the input.
Discussion
==========
We demonstrate this web application with several examples relevant to nuclear non-proliferation and observational neutrino geoscience. Readers who visit http://geoneutrinos.org/reactors can follow along.
Detecting the diversion from the reactor core of a significant quantity of material (e.g. 8 kg Pu), which could be directly used in manufacturing a nuclear explosive device, with a high probability ($>95$%) within $30$ days is an established nuclear safeguard [@iaea]. The capability of meeting the criteria of this safeguard by the observation of reactor antineutrinos is an area of active research. Electron antineutrino detectors of cubic meter size are capable of monitoring the operation of nuclear reactors when deployed about $10$ meters from the reactor core and at a depth equivalent only to about $10$ meters of water. Successful demonstrations include the monitoring of both 3-GW commercial [@songs07] and 70-MW research [@nucifer15] reactors. Demonstrations at greater distances require larger detectors, probably operating under greater overburden.
A nuclear monitoring demonstration project under discussion [@watch15] involves deploying a $1000$ cubic meter detector $13$ km from the Perry Nuclear Power Plant east of Cleveland, Ohio and in a salt mine at a depth equivalent to about $1.5$ km of water. An existing excavation in the mine, which formerly housed the IMB nucleon decay detector [@IMB], would accommodate the nuclear monitoring demonstration detector. The rate and energy spectrum of antineutrinos estimated by this web application for this site is displayed by selecting “IMB” under the “North America” category in the “Location Presets” box in the “Location” pane of the “Detector” page. The total rate is dominated by the closest core ($94-95$%, depending on the load factor), which is 13 km away. Moving to the “Output & Stats” and selecting “Closest Core (geonu + other reactors background)” in the “Signal” box in the “Calculator” pane shows that a $3 \sigma$ detection would result from a perfect detector of $10^{32}$ free proton targets exposed for $0.15$% of a year, which is about $13$ hours. This level of significance would take the nuclear monitoring demonstration detector a considerably longer time to achieve due to a smaller number of targets, a less than perfect detection efficiency primarily due to relying solely on Cherenkov light for the detectable signal, and cosmogenic background, which is presently not accounted for by the web application. However, the exposure time would still be much less than the $30$ days established by the IAEA safeguard.
While $13$ km is much farther than the $\sim10$ meter distances demonstrated by the meter cube projects, it is still closer than $100$ km, which is considered by the intelligence community to be the distance beyond which remote monitoring occurs. In our second example, enter latitude $34.75$ N and longitude $-121.89$ E in the “Location” pane of the “Detector” page. This places a detector $107$ km offshore from the Diablo Canyon Nuclear Plant at San Luis Obispo, CA. and near the Santa Lucia Escarpment in $3.6$ km of water. The energy spectrum plot shows the closest core contributes $42-43$% of the total rate. The reduction compared with the previous example is because the Diablo Canyon Nuclear Power Plant has two reactor cores rather than the single core at the Perry Nuclear Power Plant. Note the wiggled structure in the spectrum due to neutrino oscillations, in particular the prominent dip at about $2.3$ MeV. Shifting to the “Output & Stats” page shows that a $3 \sigma$ detection would result from a perfect detector of $10^{32}$ free proton targets exposed for about $45$% of a year, which is about $163$ days. Cutting off the energy spectrum below $2.3$ MeV removes more background than signal, thereby reducing the exposure time to about $41$% of a year, which is about $150$ days. Decreasing this exposure time to the $30$ day safeguard level could be accomplished by increasing the size of the detector to greater than $5 \times 10^{32}$ free proton targets. Cosmogenic background would be significantly reduced by the deep ocean water compared with the relatively shallow salt mine site in the previous example. Moreover, if scintillating liquid (oil- or water-based) were used instead of water, the detection efficiency could be $>80$% [@gando13; @agostini15]. The feasibility of deploying an electron antineutrino observatory in the deep ocean is addressed by a previous study [@hano]. This example indicates that remote monitoring of nuclear reactor operation is a realistic possibility off the coast of California.
One goal of nuclear monitoring is the discovery of a clandestine reactor. The web application aids in specifying the observational requirements to make such a discovery. The following example is considerably contrived and fully hypothetical. Nonetheless it serves to illustrate the function of the web application. Use the same detector location as in the previous example ($34.75$ N and $-121.89$ E). In the “Reactors page set the ”Use Custom Reactor“ checkbox and dial up the ”Power“ box to 40 MW. In the ”Location“ sub-pane enter ”Latitude“ $34.98$ N and ”Longitude“ $-121.37$ E. This places a custom reactor 54 km from the detector on a line directly to the Diablo Canyon Nuclear Power Plant. The energy spectrum plot on the ”Detector“ page shows the rate from the closest reactor is only about $1$% of the total rate. Selecting ”Closest Core (geonu + other reactors background)“ in the ”Signal“ box of the ”Calculator“ pane on the ”Output & Stats“ page with the ”Solve For“ box set to ”Exposure Time" finds that 836 years of exposure of a perfect detector with $10^{32}$ free proton targets is required for a $3 \sigma$ detection. A very rough estimate of the exposure is $1000$ TNU$^{-1}$. If the positive detection is required in $N$ years, then the detector size is given by $1000/N \times 10^{32}$ free protons. A positive detection in five years would require a detector of about $250$ kT of scintillating liquid. The largest scintillating liquid antineutrino detector under discussion tips the scales at $50$ kT [@lena], making the present example within the realm of possibility.
As a final example, this one relevant to neutrino geoscience, select“ACO” under the “Pacific Ocean” category in the “Location Presets” box in the “Location” pane of the “Detector” page. This places a detector at the Aloha Cabled Observatory [@aco] site $100$ km north of Oahu, Hawaii. With the closest reactor over $3500$ km distant the energy spectrum plot clearly shows that the estimated signal is dominated by geo-neutrinos. Moreover, about one-half of the total estimated signal originates in the mantle. Moving to the “Output & Stats” and selecting “Geoneutrino (reactor background)” in the “Signal” box in the “Calculator” pane shows that a $3 \sigma$ detection would result from a perfect detector of $10^{32}$ free proton targets exposed for $1.19$% years. At a depth of almost $5$ km cosmogenic background would be negligible. This example reinforces the previously stated conclusion that the measurement of mantle geo-neutrinos is greatly facilitated by observation from the deep ocean [@roth98; @dye06; @enomoto07; @dye08].
Revisions and Additions
=======================
As with the static mapping projects [@baldoncini; @agm15] revisions and additions can be implemented to improve this web application. Perhaps the most prominent deficiency is the omission of uncertainties in the modeled antineutrino rates. Although these uncertainties are relatively small ($<5$%) for the rate from a reactor of known power output, uncertainties associated with the geological antineutrino rates are large ($\sim20$% for the crust and $>50$% for the mantle). Including information on the uncertainties in the modeled rates would be useful. There remains a very interesting possibility of enhanced geo-neutrino emission from two seismically resolved large structures at the base of the mantle [@sramek], the so-called large low-seismic velocity provinces (LLSVPs). Incorporating the seismic models and the ability to specify radiogenic element concentrations within a volume bounded by a certain seismic speed deficit would allow calculation of corresponding surface rate enhancement. As the capability to resolve the direction of antineutrino signals develops the angular distribution of sources at a given detection site becomes beneficial. Adding the means to plot anisotropic crust, reactor, and LLSVP distributions to the web application would be helpful. Modeling of radiogenic element concentrations in continental crust is maturing and refinements seem inevitable. Besides improving the estimate of the crust rate, potential revisions include upgrading the geo-neutrino and reactor spectra with those resulting from more precise calculations.
Conclusions
===========
We present a web application for modeling global antineutrino energy spectra and detection rates for any surface location. Sources include all registered nuclear reactors as well as the crust and mantle of Earth. Visitors to the website may model the location and power of a hypothetical nuclear reactor, copy energy spectra, and analyze the significance of a selected signal relative to background.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the Lawrence Livermore National Laboratory. The authors thank W.F. McDonough for providing the crust flux model in a useful format, G.R. Jocher for furnishing a listing of registered nuclear reactor information, and A. Bernstein for useful comments and suggestions during the development of the web application.
|
---
abstract: 'We estimate the He [ii]{} to H [i]{} column density ratio, $\eta = N({\mbox{He\,{\sc ii}}})/N({\mbox{H\,{\sc i}}})$, in the intergalactic medium towards the high redshift ($z_{\rm em}$ = 2.885) bright quasar QSO HE 2347$-$4342 using Voigt-profile fitting of the H [i]{} transitions in the Lyman series and the He [ii]{} Lyman-$\alpha$ transition as observed by the [*FUSE*]{} satellite. In agreement with previous studies, we find that $\eta > 50$ in most of the Lyman-$\alpha$ forest except in four regions where it is much smaller ($\eta \sim 10-20$) and therefore inconsistent with photo-ionization by the UV background flux. We detect O [vi]{} and C [iv]{} absorption lines associated with two of these regions ($z_{\rm abs}$ = 2.6346 and 2.6498). We show that if we constrain the fit of the H [i]{} and/or He [ii]{} absorption profiles with the presence of metal components, we can accommodate $\eta$ values in the range 15-100 in these systems assuming broadening is intermediate between pure thermal and pure turbulent. While simple photo-ionization models reproduce the observed $N$(O [vi]{})/$N$(C [iv]{}) ratio, they fail to produce low $\eta$ values contrary to models with high temperature (i.e T $\ge 10^5$ K). The Doppler parameters measured for different species suggest a multiphase nature of the absorbing regions. Therefore, if low $\eta$ values were to be confirmed, we would favor a multi-phase model in which most of the gas is at high temperature ($>$ 10$^5$ K) but the metals and in particular C [iv]{} are due to lower temperature ($\sim$ few $10^4$ K) photo-ionized gas.'
author:
- |
S. Muzahid$^{1}$[^1], R. Srianand$^{1}$, P. Petitjean$^{2}$\
$^{1}$ Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India\
$^{2}$ Université Paris 6, UMR 7095, Institut d’Astrophysique de Paris-CNRS, 98bis Boulevard Arago, 75014 Paris, France\
bibliography:
- 'mybib.bib'
date: 'Accepted. Received; in original form '
title: 'Revisiting the He [ii]{} to H [i]{} ratio in the Intergalactic Medium '
---
[$z_{\rm abs}$]{}[$z_{\rm abs}$]{}
\[firstpage\]
galaxies: quasar: absorption line – quasar: individual(HE 2347$-$4342) – galaxies: intergalactic medium
Introduction
============
The presence of metals in the H [i]{} Lyman-$\alpha$ forest at optical depths $\tau_{\rm Ly\alpha}\ge 1$, detected through C [iv]{} and O [vi]{} absorption lines seen in QSO spectra, is now well established [see @Songaila96; @Bergeron02; @Simcoe04]. Observations are consistent with an average carbon metallicity relative to solar of \[C/H\] $\sim$ $-2.8$ with no sign of redshift evolution over the range $1.8\le z \le 4.1$ but a significant trend with over-densities [@Schaye03; @Aguirre08]. Given the expected low metallicities and the high ionization state of the gas, direct detection of metal absorption lines from underdense regions of the intergalactic medium (IGM) is beyond the scope of present day large telescopes. Statistical methods like pixel analysis are used instead [@Ellison00; @Schaye03; @Aracil04; @Scannapieco06; @Pieri10] and show that metals must be present in the low-density regions. Even in regions where C [iv]{} absorption is detected directly, it is not clear however what is the main physical process that is maintaining the ionization state of the gas. In general, it is believed that photo-ionization keeps the gas ionized. However, it is probable that mechanical inputs from galactic winds can influence the ionization state of part of the IGM gas through collisional ionization at least in the proximity of galaxies. Therefore, it is important to simultaneously study different species covering a wide range of ionization states to get a better understanding of the metal enrichment and the different ionizing mechanisms at play. Recent hydrodynamical simulations [@Dave01; @Fang01; @Kang05; @Bouche06; @Bouche07] suggest that the missing baryons at low redshift, $z\sim 0-0.5$, and the missing metals at high redshift, $z\sim 2.5$, could reside in the warm-hot phase of the intergalactic medium (called WHIM) with $T \approx 10^{5} - 10^{7}$ K). Highly ionized species of oxygen such as O [vi]{}, O [vii]{} and O [viii]{} can be useful probes of the WHIM. While the strongest transitions of the latter two species have rest-wavelengths in the soft X-ray range, the spectral doublet O [vi]{}$\lambda\lambda$1032,1037 is seen in the near UV range and is therefore a useful probe of the gas at a temperature of $\sim$3$\times$10$^{5}$ K, temperature at which O$^{5+}$/O is maximum. It has been suggested that a large fraction of the conspicuous phase seen to be associated to high-$z$ damped Lyman-$\alpha$ systems may originate from collisionally ionized gas [@Fox07b]. However, photo-ionization can also maintain oxygen in a high ionization state and at relatively low temperature [$T$ $\sim$ a few 10$^{4}$ K, see @Oppenheimer09]. Actually a large fraction of the O [vi]{} absorption seen at $z>2.5$ in quasars show Doppler broadening consistent with photo-ionization in the vicinity of the QSOs [for example @Srianand00apm] but also in the diffuse intergalactic medium [for example, @Bergeron02; @Simcoe04]. While it is expected that the intrinsic ionizing spectrum of QSOs is hard enough to maintain a high degree of ionization of oxygen in their vicinity, in the IGM, the hardness of the ionizing spectrum will depend on the intrinsic spectral shape of the ionizing sources and the IGM opacity at the H [i]{} and He [ii]{} Lyman Limit [@Haardt96; @Fardal98]. An additional piece of information comes from QSO lines of sight transparent in the Lyman continuum \[i.e a high-$z$ QSO line of sight without any intervening Lyman limit system (LLS) blocking the UV end of the spectrum\]. It is then possible to observe the rest wavelength ranges of the H [i]{} and He [ii]{} Lyman-$\alpha$ forests and to compute the ratio of the to the optical depth (i.e the $\eta$ parameter). The bright QSO HE 2347$-$4342 at = 2.885 [@Reimers97] is one such targets that attracted a lot of attention in the past years. It was shown that the opacity is “patchy” in nature [@Reimers97; @Smette02] and that $\eta$ decreases gradually from higher to lower redshift possibly due to a change in the slope of the ionizing spectrum [@Zheng04]. @Shull04 discussed the small scale variations (over $\Delta z \approx 10^{-3}$) of $\eta$ and found an apparent correlation between high $\eta$ (less ionized He [ii]{}) and low column density. They ascribed these small scale $\eta$ variations to “local ionization effects” in the proximity of QSOs located close to the line of sight and having spectral indices ranging from $\alpha_{\rm s}$ = 0 to 3. @Worseck07 reported the detection of 14 foreground QSOs in the field located close to the line of sight and could not find any convincing evidence for any transverse proximity effect from a decrease in the absorption, although they did claim that the local UV spectrum inferred in the vicinity of three foreground QSOs appeared harder than expected, which is an indication of a transverse proximity effect. In turn these fluctuations could be due to an appreciable contribution of thermal broadening to the velocity width of absorption lines at high $N$(H [i]{}) [@Fechner07a]. In this paper and after a description of the observations (Section 2), we use a different approach involving Voigt profile fitting analysis of the H [i]{} and He [ii]{} absorption lines to measure $\eta$ (Section 3). We then report new detections of O [vi]{} absorption associated with regions with low $\eta$ values (Section 4) and construct models of these regions (Section 5) before concluding in Section 6.
Observations
============
The optical spectrum of HE 2347$-$4342 ($z_{\rm em}$ = 2.885) used in this study was obtained with the VLT UV Echelle Spectrograph (UVES) [@Dekker00] mounted on the ESO Kueyen 8.2-m telescope at the Paranal observatory in the course of the ESO-VLT large programme ‘The Cosmic Evolution of the IGM’ [@Bergeron04]. HE 2347$-$4342 was observed through a 1-arcsec slit (with a typical seeing of 0.8 arcsec) for 12 h with central wavelengths adjusted to 3460 and 5800 Å in the blue and red arms, respectively, using dichroic \#1 and for another 14 h with central wavelengths at 4370 and 8600Å in the blue and red arms with dichroic \#2. The raw data were reduced using the latest version of the UVES pipeline [@Ballester00] which is available as a dedicated context of the MIDAS data reduction software. The main function of the pipeline is to perform a precise inter-order background subtraction for science frames and master flat fields, and to apply an optimal extraction to retrieve the object signal, rejecting cosmic ray impacts and performing sky subtraction at the same time. The reduction is checked step-by-step. Wavelengths are corrected to vacuum-heliocentric values and individual one-dimensional spectra are combined. Air-vacuum conversions and heliocentric corrections were done using standard conversion equations [@Edlen66; @Stumpff80]. Addition of individual exposures is performed by adjusting the flux in individual exposures to the same level and inverse variance weighting the signal in each pixel. Great care was taken in computing the error spectrum while combining the individual exposures. Our final error in each pixel is the quadratic sum of the weighted mean of errors in the different spectra and the scatter in the individual flux measurements. Errors in individual pixels obtained by this method are consistent with the rms dispersion around the best fitted continuum in regions free of absorption lines. The final combined spectrum covers the wavelength range of 3000 to $10,000$ [Å]{}. A typical SNR$\sim$60 per pixel (of 0.035 Å) is achieved over the whole wavelength range of interest for a spectral resolution of $R \sim 45,000$. The detailed quantitative description of data calibration is presented in @Aracil04 and @Chand04. We use the continuum normalized FUSE data provided by Dr. Zheng. The details of the data reduction and continuum normalization can be found in @Zheng04 [^2]. The original data have typical resolution of $R=20,000$ and signal-to-noise ratio $\sim$5 in the long wavelength range ($\lambda$ $>$ 1050 Å). Following @Zheng04, we have re-binned this data to 0.1 Å, which leads to an effective resolution of $R\sim4000$. We restrict ourselves to the wavelength range with SNR $>$ 10. This corresponds to a redshift range 2.58$\le$ z$\le$2.70 or a velocity range of $\sim$$10,000$ around a central redshift of $z=2.6346$ (see Fig. \[total\_vplot\]).
$N$()/$N$() ratio
=================
In this Section we concentrate on the column density ratio $\eta$ = $N$(He [ii]{})/$N$(H [i]{}) over the redshift range 2.58$\le z\le 2.70$ where the FUSE data show relatively good signal to noise ratio. This range roughly corresponds to a relative velocity range of $-$4000 to $+$ 5000 around the strong absorber seen at [$z_{\rm abs}$]{} = 2.6346 (see Fig. \[total\_vplot\]).
As a first step we fitted simultaneously the Lyman-$\alpha$ to Lyman-$\gamma$ profiles when possible, e.g. when the Lyman-$\beta$ and/or Lyman-$\gamma$ lines are not blended with another intervening Lyman-$\alpha$ line. Then we compare the He [ii]{} absorption profile with a model with the same components as the H [i]{} model, scaling the fitted H [i]{} column densities by the parameter $\eta$. We consider two alternatives: In the first case we use the same Doppler parameter for H [i]{} and He [ii]{} (assuming turbulent broadening); in the second case we give the He [ii]{} $b$-parameter the value expected from thermal broadening (i.e $b$(He [ii]{}) = 0.5$\times$$b$(H [i]{})). The best fitted values of $\eta$ is obtained by $\chi^2$ minimization. While fitting the He [ii]{} profiles we use a Gaussian convolving function to correctly represent the FUSE spectral resolution. For Voigt-profile decomposition we have used the fitting code developed by @Khare97.
As the FUSE data are of much lower resolution and SNR than the UVES data, we cannot estimate $\eta$ for individual H [i]{} components. Instead, we have singled out 15 small regions named as A, B, C etc., in Fig. \[total\_vplot\] and we derive the best $\eta$ value over each region. We wish to point out that the approach we have taken here is very different from previous studies. Indeed, @Shull04 used apparent optical depth in Lyman-$\alpha$ only (AOD) method, whereas @Kriss01 and @Zheng04 used Gaussian decomposition and @Fechner07a scale the whole spectrum by $\eta$ = $4\times$$\tau_{\rm HeII}/\tau_{\rm HI}$ to fit the data. In all these studies only H [i]{} Lyman-$\alpha$ is used [^3]. This is the use of the H [i]{} optical depths in all available Lyman series lines that allows us to discriminate between thermal and turbulent broadening.
\[lgeta\_lgNH\]
The best fitted Voigt profiles to the Lyman-$\alpha$ and Lyman-$\beta$ absorption lines are shown in the bottom and middle panels of Fig. \[total\_vplot\]. The top panel shows the best fitted He [ii]{} Lyman-$\alpha$ line with the two assumptions on the Doppler parameter discussed above. The $\chi^2$ curves as a function of $\eta$ for the different regions singled out in Fig. \[total\_vplot\] are shown in Fig. \[chisq\_plot\]. The solid and dashed lines in these plots represent the cases of thermal and turbulent broadening respectively. In most cases the curve shows a clear minimum thereby allowing us to discriminate between the turbulent and thermal cases, and to derive the best fitted value of $\eta$. Errors are estimated from the range of $\eta$ values corresponding to $\Delta \chi^2$ = $\pm$1 around the minimum. There are regions, especially when the He [ii]{} Lyman-$\alpha$ line is saturated, for which the $\chi^2$ curve flattens (e.g. regions [**E**]{} and [**G**]{}), we have only one-sided limit. In these cases we define the 2$\sigma$ lower limit of $\eta$ as the value corresponding to a $\chi^2$ equal to $\chi^2$ of the flat part of the curve plus four. The shapes of the $\chi^2$ curves are not symmetric which is a natural consequence of line saturation. It is clear from the Figure that, apart from region [**I**]{}, the $\chi^{2}$ values are smaller in case of turbulent broadening and that minima are reached only in that case. In the case of thermal broadening, the $\chi^2$ curves seem to saturate to some asymptotic value probably because the observed He [ii]{} profiles are too broad to be reproduced by the model. Thus the exercise presented here shows that the width of He [ii]{} Lyman-$\alpha$ lines are consistent with the $b$-parameter derived from H [i]{} lines. If the gas is optically thin and photo-ionized by a UV background dominated by QSOs, we would expect $\eta$ to be in the range 40$-$400 depending on the exact spectral index and the IGM opacity. In the case of self-shielded optically thick gas, $\eta$ could be even higher [@Fardal98]. Four regions ([**D**]{}, [**F**]{}, [**H**]{} and [**N**]{}) in Fig. \[chisq\_plot\] have $\eta\le40$. These regions are associated with large H [i]{} column densities as can be seen on Fig. \[lgeta\_lgNH\] where log $\eta$ is plotted against log $N$(H [i]{}) as measured in the different regions. This correlation was already noted in earlier works. Fechner & Reimers (2007) argued that this can be explained if the thermal broadening of lines are also important. In the following, we will use additional information on metal lines observed in the UVES spectrum to discuss further the ionization state of the gas in these regions.
Regions with low $\eta$ values
==============================
In the previous Section, we have shown that the $N$(He [ii]{})/$N$(H [i]{}) ratio can be explained over most of the observed spectrum by ionization of the gas by the UV background except in four regions: [**D**]{} ([$z_{\rm abs}$]{}= 2.6346), [**F**]{} ([$z_{\rm abs}$]{}= 2.6498), [**H**]{} ([$z_{\rm abs}$]{}= 2.6624) and [**N**]{} ([$z_{\rm abs}$]{}= 2.6910). The presence of and absorption in systems showing low values of $\eta$ may yield interesting clues about (i) the nature of the ionizing radiation, (ii) the effect of thermal/turbulent broadening and (iii) the possible mechanical feedback from winds. Regions [**D**]{} and [**F**]{} are associated with and strong absorption lines. These are the only two systems in the redshift range $2.58\le z\le 2.70$ (see top panels in Fig. \[vplot\]) and we discuss them in detail below. For region [**H**]{}, the wavelength range where possible $\lambda$1031 absorption is redshifted is strongly blended and only a possible weak line is present at the expected location of $\lambda$1037 (see bottom-left hand-side panel in Fig. \[vplot\]). As no other metal line is detected at this redshift we are unable to confirm if this feature is indeed due to absorption. For region [**N**]{}, while both regions are strongly blended, the optical depth constraints are satisfied at two velocity positions (see Fig. \[vplot\]). However, the possible O [vi]{}$\lambda$1031 feature is also consistent with being C [iii]{} absorption at [$z_{\rm abs}$]{}$\sim$2.8972. Similarly, the possible O [vi]{}$\lambda$1037 line is blended with Lyman-$\beta$ at [$z_{\rm abs}$]{} = 2.7306 and O [vi]{}$\lambda$1031 at [$z_{\rm abs}$]{} = 2.7121. Hence, we cannot confirm the presence of absorption in this region. Note that in region [**N**]{} (i.e. [$z_{\rm abs}$]{}= 2.6910) $\eta$ is probably affected by transverse proximity effect from QSO J23495-4338 located at redshift $z=2.690$$\pm$0.006, 15 arcmin away from the line of sight of interest [@Worseck07].
Note that we detect absorption at [$z_{\rm abs}$]{}= 2.7121, 2.7356 and 2.7456 as well. The opacity is high at [$z_{\rm abs}$]{} = 2.7121 and 2.7456 which makes $\eta$ difficult to estimate. If we scale the Voigt-profile fits to the absorption to reproduce the profile, we find $\eta$ to be in the range 10$-$100 and $>$100 for, respectively, the systems at [$z_{\rm abs}$]{} = 2.7121 and 2.7456. The wavelength range in which the absorption at [$z_{\rm abs}$]{} = 2.7356 is expected to be redshifted has been removed by @Zheng04 because of the strong airglow lines so that we cannot estimate $\eta$ for this system. The system at [$z_{\rm abs}$]{} = 2.7356 is a known Lyman-limit system. A Voigt-profile fit to the absorption gives log $N({\mbox{H\,{\sc i}}})$ = 16.50$\pm$0.28.
System at [$z_{\rm abs}$]{} = 2.6498
------------------------------------
A velocity plot of high ionization metal lines and H [i]{} lines from this system is shown in the top-right panel of Fig. \[vplot\]. Clearly the metal lines are off-centered with respect to the absorption. In addition, there is a velocity off-set of 2 to 10 km s$^{-1}$ between the centroids of the and absorption profiles. Interestingly, all the shifts are in the same direction as would be expected in a flow ionized from the same side. The best fit of the profiles is obtained when we allow for component redshifts to be independent of that of the components (see Table \[tab1\]). Doppler parameters are larger for compared to which supports neither pure thermal nor pure turbulent broadening. The upper limits on the kinetic temperature of the gas measured from the $b$-parameters of components is 1.4$\times10^{6}$, 8$\times10^4$, 1.2$\times10^5$ and 4$\times10^4$ K respectively for components at $-76.2$, $-48.6$, $-20.2$ and $+4.2$ . Therefore within the allowed error in $b$-parameters, the profile allows for the existence of high temperature ($T> 10^5$ K) at least in part of the associated gas. The top panel of Fig. \[aod2p64\] shows the apparent column densities of (in blue) and (in red) per unit velocity interval versus relative velocity. Since $\lambda$1037 is heavily blended we use only the $\lambda$1031 line. For , we have used the oscillator strength weighted mean of the column densities per unit velocity measured from both absorption lines of the doublet. For clarity, we have multiplied the apparent column density profile by a factor of 10. Vertical dashed and dotted lines show the positions of peaks in the optical depth of and respectively. It is apparent that the peaks are shifted compared to the ones.
In the lower panel of Fig. \[aod2p64\] we plot the ratio of to apparent column densities per unit velocity against the relative velocity and find that the ratio varies between 10 and 20 through the absorption profile. The fact that the absorption profile is broader suggests the existence of gas outside the profile with to column density ratio higher than 20. The component at $\sim +4.2$ has virtually no detectable H [i]{} absorption associated. From the Lyman-$\alpha$ line we derive an upper limit of log $N$(H [i]{}) = 12.80 suggesting that metallicity is probably high in this component. Indeed, given the low $b$ value of the component, it is probable that the gas is photo-ionized in which case metallicity has to be close to solar. For the other three components that coincide with a strong H [i]{} absorption and it is impossible to quantify the amount of H [i]{} absorption associated with them individually such that useful metallicity limits can be established.
[lclcc]{} $ v$ () & Ion & lines used & $b$ () & log($N$ in [cm$^{-2}$]{})\
$-$214.6 $\pm$ 1.0 & & [Ly-$\alpha$]{}& 16.3 $\pm$ 1.5 & 12.59 $\pm$ 0.03\
$-$150.8 $\pm$ 0.3 & & [Ly-$\alpha$]{}, [Ly-$\beta$]{}& 23.6 $\pm$ 0.3 & 14.63 $\pm$ 0.01\
$-$76.2 $\pm$ 0.0 & & [Ly-$\alpha$]{}, [Ly-$\beta$]{}& 37.6 $\pm$ 0.9 & 14.91 $\pm$ 0.01\
$-$48.6 $\pm$ 0.0 & & [Ly-$\alpha$]{}, [Ly-$\beta$]{}& 23.1 $\pm$ 0.7 & 14.68 $\pm$ 0.02\
$-$20.2 $\pm$ 0.0 & & [Ly-$\alpha$]{}, [Ly-$\beta$]{}& 26.4 $\pm$ 1.1 & 13.53 $\pm$ 0.06\
\
$-$76.2 $\pm$ 3.3 & & & 38.6 $\pm$ 3.9 & 13.56 $\pm$ 0.05\
\
$-$58.8 $\pm$ 1.0 & & & 14.5 $\pm$ 1.5 & 12.30 $\pm$ 0.03\
$-$48.6 $\pm$ 0.3 & & & 9.4 $\pm$ 0.9 & 13.16 $\pm$ 0.06\
\
$-$24.1 $\pm$ 0.6 & & & 5.9 $\pm$ 1.0 & 12.06 $\pm$ 0.04\
$-$20.2 $\pm$ 0.2 & & & 10.9 $\pm$ 0.4 & 13.55 $\pm$ 0.02\
\
$+$1.9 $\pm$ 0.4 & & & 4.8 $\pm$ 0.7 & 12.16 $\pm$ 0.03\
$+$4.2 $\pm$ 0.1 & & & 6.5 $\pm$ 0.2 & 13.49 $\pm$ 0.01\
\[tab1\]
We have seen before (Fig. \[chisq\_plot\]) that the $\chi^{2}$ curve corresponding to the fit of the He [ii]{} absorption shows a marked minimum for $\eta$ = 12 in the case of turbulent broadening (i.e $\xi$ = $b_{{\mbox{He\,{\sc ii}}}}/ b_{{\mbox{H\,{\sc i}}}}$ = 1) and no minimum for pure thermal broadening ($\xi$ = 0.5). In Fig. \[heprof1\] we show in the left hand-side row the simulated profiles for $\eta$$\sim$130 and $\xi$ = 0.5 (solid curve) and $\eta \sim$ 12 and $\xi$ = 1.0 (dashed curve). Remember that for these fits we have used the minimum number of Voigt profile components without any constraint from the O [vi]{} profile. It is apparent that the red solid profile (obtained assuming pure thermal broadening) is missing several pixels in the red wing of the region of interest around 0 km/s. This is because the $b$ value of the corresponding component (fixed by the H [i]{} profile) must be much larger to reach this position. If we now add the constraint that H [i]{} should be associated with the three O [vi]{} components, we can reproduce this profile better. Indeed, because of the extra component at $v\sim -20$ km/s, pure thermal broadened profile with higher $\eta$ ($\sim$ 100) gives an equally good fit (right panel of Fig. 6). It seems therefore that if we add a He [ii]{} component at the position of the redder O [vi]{} component, any value of $\eta$ between $\sim$15 and 100 is acceptable. Thus it seems that the possible presence of unresolved narrow H [i]{} components could be one of causes of low $\eta$ measurements. It is a fact however that the main H [i]{} components have large $b$ values, corresponding to temperatures in excess of 10$^{5}$ K. Therefore it is not impossible that part of the gas is at high temperature.
System at [$z_{\rm abs}$]{} = 2.6346
------------------------------------
[llllc]{} $ v_0$ () & Ion & lines used & $b$ () & log($N$ in [cm$^{-2}$]{})\
$-$140.6 $\pm$ 0.0 & & [Ly-$\alpha$]{}& 27.8 $\pm$ 0.6 & 13.41 $\pm$ 0.01\
$-$91.6 $\pm$ 0.0 & & [Ly-$\alpha$]{},[Ly-$\beta$]{},[Ly-$\gamma$]{}& 22.5 $\pm$ 0.3 & 13.80 $\pm$ 0.01\
\
$-$16.4 $\pm$ 0.2 & & [Ly-$\alpha$]{},[Ly-$\beta$]{},[Ly-$\gamma$]{}& 38.2 $\pm$ 0.9 & 14.27 $\pm$ 0.03\
& & & 20.3 $\pm$ 8.3 & 12.76 $\pm$ 0.28\
& & & 11.5 $\pm$ 2.4 & 12.55 $\pm$ 0.12\
\
$-$0.6 $\pm$ 0.7 & & [Ly-$\alpha$]{},[Ly-$\beta$]{},[Ly-$\gamma$]{}& 28.3 $\pm$ 2.2 & 14.14 $\pm$ 0.06\
& & & 18.9 $\pm$ 0.5 & 14.05 $\pm$ 0.02\
& & & 8.1 $\pm$ 0.6 & 12.83 $\pm$ 0.06\
\
$+$36.2 $\pm$ 0.7 & & [Ly-$\alpha$]{},[Ly-$\beta$]{},[Ly-$\gamma$]{}& 25.7 $\pm$10.4 & 13.24 $\pm$ 0.37\
& & & 9.6 $\pm$ 0.8 & 13.37 $\pm$ 0.04\
& & & 7.5 $\pm$ 1.0 & 12.19 $\pm$ 0.04\
\
$+$48.5 $\pm$ 0.0 & & [Ly-$\alpha$]{},[Ly-$\beta$]{},[Ly-$\gamma$]{}& 46.6 $\pm$ 1.5 & 14.32 $\pm$ 0.01\
& & & 25.2 $\pm$ 1.6 & 13.61 $\pm$ 0.03\
\
$+$101.4 $\pm$ 0.0 & & [Ly-$\alpha$]{},[Ly-$\beta$]{}& 69.5 $\pm$ 2.0 & 13.69 $\pm$ 0.02\
& & & 32.8 $\pm$ 3.5 & 13.35 $\pm$ 0.04\
\[tab2p6346\]
Absorption profiles from this system are shown on a velocity scale in Fig. \[vplot\]. Unlike in the previous system the velocity range of metal lines falls well within the Lyman-$\alpha$ profile. The $\lambda$1037 line is blended with Lyman-$\gamma$ at $z$ = 2.8781 and Lyman-$\beta$ at $z$ = 2.6765. Because of this contamination we use the well measured redshifts of components to fit the doublet. The contributions of the contaminating lines are self-consistently computed using other available transitions. In addition to the C [iv]{} counterparts, we need two components in the red part of the profile to fit the doublet where there is no absorption. H [i]{} Lyman-$\alpha$, Lyman-$\beta$, and Lyman-$\gamma$ lines have been fitted simultaneously imposing components at the redshifts of five O [vi]{} components. Two extra components are required in the blue ($\sim -100$ km/s) to cover the total absorption. The details of the fit results are given in Table \[tab2p6346\]. As in the previous system, for the components with both and the $b$-parameters are larger than the ones and the column density ratio of to is as high as $\sim 15$. The $b$-parameters correspond to upper limits on the kinetic temperature of $4\times 10^5$, $3\times 10^5$ and $9\times 10^4$ K respectively for the components at $-16.4$, $-0.6$ and $+36.2$ . In the components where we find only the ratio of to column densities can be higher than 20. These components have broad lines with $b$-parameters corresponding to upper limits of $6\times10^5$ and $10^6$ K respectively for the components at $+48$ and $+101$ . The corresponding H [i]{} components also have high $b$ values allowing for high temperature ($\sim 10^5$ K) in the gas associated with these two components. All this suggests a multiphase structure in this absorbing gas with the possible existence of a hot phase contributing to most of the absorption. Indeed, the O [vi]{} profile is suggestively broad.
We fitted the H [i]{} and He [ii]{} profiles in the two extreme cases of pure turbulent and pure thermal broadening, considering both components from the fit of the H [i]{} profile only and from the fit of metal lines. Results are given in Fig \[heprof2\] and Table 2. We notice from right panel of Fig \[heprof2\] that even when we tie the H [i]{} components to components the best fitted $\chi^2$ is obtained for the pure turbulent case with low $\eta$. However, reality probably corresponds to an intermediate case with $\xi$ between 0.5 and 1. In the bottom panel of Fig. \[xichisq1\] we plot the minimum $\chi^2$ value obtained for different values of $\xi$. Even though the best fitted $\chi^2$ value is obtained for $\xi$ = 1 the curve is flat and the 1$\sigma$ range is $\xi\ge0.6$. As can be seen in the top panel of the figure, this can accommodate a wide range of $\eta$. Therefore, in this system also high $\eta$ values are acceptable although H [i]{} and O [vi]{} absorption profiles are broad and highly suggestive of a gas with temperature higher than the typical photo-ionization temperature (i.e few $10^4$ K).
Models
======
Given the particularities of the systems singled out by the presence of O [vi]{} absorption, possible low $\eta$ values and high O [vi]{}/C [iv]{} ratios, we have constructed models to test the different mechanisms that could induce such properties. It is well known that photo-ionization by a power-law spectrum with appropriate slope can yield low $\eta$ values. This would require the presence of local sources of hard photons [see @Shull04]. Observations by @Worseck07 seem to show however that there is no QSO present in the vicinity of the two absorbers considered in the previous Section. While this observation does not rule out a QSO emission highly beamed perpendicular to the line of sight or a short lived QSO emission in the vicinity of the absorbers, we explore alternate explanations for low $\eta$ in the absorbers. Therefore, in the following we present the results of models of a hot gas embedded in the meta-galactic UV background. We use the photo-ionization code Cloudy (v07.02; @Ferland98) to derive the ionization structure in a gas with fixed temperature (therefore [*not*]{} controlled by photo-ionization). This will make it possible to discuss at the same time both extreme situations (collisional ionization and photo-ionization) but also the intermediate situation of high-temperature gas with a contribution of photo-ionization. For comparison, we also show results from the model in which the temperature is the consequence of thermal equilibrium under photo-ionization. The calculations are made in the optically thin case. We use the Haardt and Madau (2005) background spectrum dominated by QSOs. We assume relative solar abundances and \[C/H\] = $-1.0$. In the top panel of Fig. \[constT\_model\], we plot the variation of the to ratio with hydrogen density. The solid black line is the result of model calculations where temperature is calculated by CLOUDY assuming photo-ionization equilibrium. Other lines are for temperatures in the range 5$\times$10$^{4}$$-$5$\times$10$^{5}$ K. It is to be remembered that when pure collisional excitation is considered the fraction of He [ii]{} is maximum when 4.5 $\le log~{\rm T(K)} \le$ 4.9 and in the case of it is $T\sim$ 3$\times$$10^5$ K [@Gnat07]. At low temperature (say $T\le5\times 10^{4}$ K) the ionization is dominated by photo-ionization. As expected the transition between photo-ionization dominated and collisional ionization dominated regimes happens at $T\sim 10^{5}$ K. The horizontal dotted lines show the range of observed to column density ratios (between 10 and 20) seen in the components of the two systems discussed above. This range is well reproduced by models with T$\le 10^5$ K for a typical density of 10$^{-4}$ cm$^{-3}$. However the higher to ratio inferred in the velocity range (or Voigt profile components) where only is detected needs either low density (and low temperature) photo-ionized gas or high density (i.e $\ge 10^{-3}$ cm$^{-3}$) hot gas (T$> 10^5$ K) where collisions begin to play a role. Interestingly such high temperatures are not ruled out by the $b$-parameters of components (see discussions in the previous Section).
In the bottom panel of Fig. \[constT\_model\] we plot $\eta$ as predicted by the models versus the hydrogen density. It is apparent that low $\eta$ values (i.e $\le 60$) are only possible for $T>10^{5}$ K. Available data on H [i]{} and He [ii]{} profiles allow for the existence of such hot gas that would also produce the component with high to column density ratio (i.e $N$()/$N$()$\ge$20). It is apparent from Fig. 4 that the absorption profiles indicate higher Doppler parameters going from C [iv]{} to O [vi]{} to H [i]{}. This has already been noted for C [iv]{} and O [vi]{} by Fox et al. (2007) and interpreted as the existence of a hot phase. We note that the $b$ values measured for the strongest H [i]{} components in the two systems (38.6 km/s at $z_{\rm abs}$ = 2.6498 and 46.6 and 69.5 km s$^{-1}$ at $z_{\rm abs}$ = 2.6346, see Tables 1 and 2) are consistent with a temperature, $T\ge10^{5}$ K and it is apparent from the absorption profiles that larger $b$ values could be accommodated. If the low $\eta$ values were to be confirmed, we would favor a scenario where the absorbing gas is a multiphase medium in which photo-ionized gas components coexist with a wide range of density and temperature. While most of the metal absorption traced by comes from relatively cold (i.e T$\le 10^5$ K) gas part of and predominant contributions of H [i]{} and He [ii]{} could be due to a hot phase ($T>10^{5}$ K). There is evidence for the existence of multiphase media in the low-z absorbers [@Tripp08] and absorption associated with high-$z$ DLAs [@Fox07b].
Conclusions
===========
We have reanalyzed the line of sight towards the high redshift ($z_{\rm em}$ = 2.885) bright quasar QSO HE 2347$-$4342 and measured the parameter $\eta$ = $N$(He [ii]{})/$N$(H [i]{}) in the Lyman-$\alpha$ forest using Voigt-profile fitting of the H [i]{} transitions in the Lyman series. As in previous studies, we find that $\eta > 50$ in most of the Lyman-$\alpha$ forest except in four regions where it is much smaller ($\eta \sim 10-20$). We detect O [vi]{} absorption associated with two of these regions (at $z_{\rm abs}$ = 2.6346 and 2.6498). The corresponding wavelength ranges for the two other regions are too blended to reach any firm conclusion on the presence of associated O [vi]{} absorption. We observe that the $z_{\rm abs}$ = 2.6346 system is a usual system with the metals located at the center of the H [i]{} profile whereas the $z_{\rm abs}$ = 2.6498 system has the metals displaced in the red wing of the H [i]{} absorption but moreover, with the C [iv]{} profile systematically shifted compared to O [vi]{}. Doppler parameters of the well-defined C [iv]{} components rule out the fact that the associated gas is hot and favor the idea that this gas is photo-ionized. We show that if we constrain the fit of the H [i]{} and/or He [ii]{} absorption profiles with the presence of metal components, we can accommodate $\eta$ values in the range 15–100 in these systems assuming broadening is intermediate between pure thermal and pure collisional. We construct constant density photo-ionized models and show that while simple photo-ionization models reproduce the observed $N$(O [vi]{})/$N$(C [iv]{}) ratio for a range of density, they fail to produce low $\eta$ values. On the contrary, models with high temperature (i.e T $\ge 10^5$ K) can produce low values of $\eta$. The Doppler parameters of the strongest H [i]{} components are consistent with such a temperature. In addition, the higher $b$ values observed for compared to C [iv]{} and the existence of alone components suggest a multiphase nature of the absorbing region. Therefore, if low $\eta$ values were to be confirmed, we would favor a multi-phase model in which most of the gas in the regions of low $\eta$ is at high temperature ($>$10$^5$ K) but the metals and in particular C [iv]{} are located in lower temperature photo-ionized and probably transient regions. As the high temperature gas can not be produced by photo-ionization, we expect the systems with low $\eta$ to be associated with galaxies. Therefore, deep search for Lyman break galaxies at these redshifts may be interesting to perform in these fields.
Acknowledgment {#acknowledgment .unnumbered}
==============
We wish to thank Dr. Zheng for providing the FUSE data and the referee Dr. Williger for useful comments. SM thanks CSIR for providing support for this work. RS thanks University Paris 6 and IAP for an invitation as Professeur Associé.
====
[^1]: E-mail: sowgat@iucaa.ernet.in
[^2]: We have obtained individual spectra reduced using Calfuse 3.2.1 version from http://fuse.iap.fr/interface.php. We combine LIF spectra after correcting for the background by demanding zero flux in the core of strong saturated absorptions in the wavelength range 1130–1185 Å. When we follow the same continuum fitting and re-binning procedures, we find the new data follow the structures (both in wavelength and flux) as seen in the data of @Zheng04 very well and fitting results are not changed. So, results presented in this paper will not change when one uses the new pipeline for the data reduction. Whereas this work was already completed, new COS data on this object were reported by @Shull10. As the COS spectrum is found to be consistent (see their Fig. 3) with the FUSE spectra used here, this has no consequence on the results of this paper.
[^3]: limited amount of analysis of Lyman-$\beta$ have been done by @Zheng04.
|
---
abstract: 'We consider the finite-temperature dynamical structure factor (DSF) of gapped quantum spin chains such as the spin one Heisenberg model and the disordered transverse field Ising model. At zero temperature the DSF in these models is dominated by a delta-function line arising from the coherent propagation of single particle modes. Using methods of integrable quantum field theory we determine the evolution of the lineshape at low temperatures. We show that the line shape is in general asymmetric in energy and we discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as ${\rm CsNiCl_3}$ and ${\rm YBaNiO_5}$.'
author:
- 'Fabian H.L. Essler$^{(a)}$ and Robert M. Konik$^{(b)}$'
title: 'Finite-temperature lineshapes in gapped quantum spin chains'
---
Quasi one-dimensional spin chains are materials where quantum fluctuations give rise to striking strongly correlated phenomena. An exemplar of such behavior is the distinction, first identified by Haldane [@haldane] over 20 years ago, between integer and half-integer isotropic spin chains. The former are generically gapped while the latter are generically gapless. This effect is topological in origin and arises from the presence of a quantized Berry’s phase in the effective model describing the chains. Contemporary examples of strong correlations in spin chains revolve around the role the multi-excitation continuum play in their physics. This continuum has been studied both experimentally[@igor; @kenz1; @kenz2] and theoretically [@3p] and is understood to be the origin of a process known as spectrum termination, where coherent excitations cross into a multi-excitation continuum and then experience rapid decay.[@igor1]
In order to probe the dynamical behavior of spin chains, inelastic neutron scattering is the premier tool. In particular, using inelastic neutron scattering it is possible to determine with impressive accuracy the spectrum of spin excitations together with their lifetimes.[@igor2] The theoretical counterpart of such measurements is a computation of the dynamical structure factor (DSF). For theoretical models that admit an integrable continuum field theoretic description, the computation of the zero temperature DSF is possible with impressive accuracy.[@smirnov; @review] However at finite temperatures, while important progress has been made,[@muss; @rmk; @alt] a general theoretical framework has yet to be settled upon. It is one aim of this letter to outline a promising approach.
We do so against an experimental mise en scène of particular relevance. In a system that supports a coherent, gapped, magnetic single-particle excitation at $T=0$, the question arises of how the corresponding delta-function in the DSF broadens at non zero-temperatures. [@young; @damle; @xu; @kenz3; @kenz4] A partial answer to this question has been given by Sachdev and collaborators:[@young; @damle] they demonstrated that at temperatures far below the gap the broadening in the immediate vicinity of the $T=0$ gap is Lorentzian in form. In the present work, using our approach to computing finite temperature DSFs, we determine the [*entire*]{} lineshape. As our central finding, we demonstrate that the lineshape is always asymmetric in energy, a feature that becomes more pronounced as the temperature increases. While we focus upon the lineshape in gapped quantum spin chains, we stress that this approach is applicable to the computation of general response functions in generic integrable continuum models, such as those considered in Ref.().
[**General Theoretical Framework:**]{} The systems we study all have representations as general Heisenberg models: $$\label{ei}
H = \sum_i J_\perp{\bf S_\perp}_i\cdot {\bf S_\perp}_{i+1} + J_zS_{zi}S_{zi+1} + {\bf H}\cdot{\bf S}.$$ Here ${\bf S_i}=({\bf S_{\perp i}},S_{zi})$ is a quantum spin (either integer or half-integer) at chain site $i$. We allow both the spin-chain to have anisotropic couplings $(J_\perp,J_z)$ and for a magnetic field, ${\bf H}$, to be potentially present. We are interested in computing the DSF $$\label{eii}
\chi(\omega,q)\!=\!-\!\int\! d\tau dx
e^{i\tilde\omega \tau-iqx}
\langle {\bf S}(\tau,x) {\bf S}(0)\rangle\bigg|_{\tilde\omega\rightarrow-i\omega+\delta}.$$ To compute this quantity, we expand $C(\tau ,x) \equiv\langle {\bf S}(\tau ,x) {\bf S}(0)\rangle$ in a basis, $\{|l\rangle\}$, of exact eigenstates of $H$, $$\label{eiii}
C(\tau ,x)=\frac{1}{\cal Z}
\sum_{l,m} e^{-\beta E_l}
\langle l|S_z(\tau ,x)|m\rangle\langle m|S_z(0)|l\rangle,$$ where $E_l$ is the energy of eigenstate, $|l\rangle$, and ${\cal Z} = \sum_le^{-\beta E_l}$ is the partition function of the theory. By virtue of the gap, $\Delta$, in this system, the Fourier transform of $C(\tau ,x)$ has a well defined low temperature expansion.
This representation of the DSF finds its virtue when we employ a continuum, integrable reduction of the (lattice) model in Eqn. (\[ei\]). In such cases the matrix elements $\langle l|S_z(0)| m\rangle$ can readily be computed exactly. At $T=0$ this permits the exact computation of ${\rm Im} \chi(\omega ,q)$ at energies, $\omega$, in the vicinity of the gap through the computation of a small number of matrix elements. At finite temperatures, this approach, for the problem at hand, breaks down in two fashions: i) the needed matrix elements (as well as ${\cal Z}$) become highly singular objects; and ii) to obtain the finite temperature broadening of the coherent mode, an infinite number of matrix elements are needed. We solve these problems in a two step fashion. The singularities of the matrix elements are intimately associated with treating the spin chain as infinite in length. While it is possible in certain circumstances to deal with these singularities directly,[@smirnov; @balog; @muss; @rmk; @alt] to circumvent this first difficulty, we instead work with chains of large but finite length, $R$. The infinities in the matrix elements are then reduced to terms merely proportional to $R$ which are cleanly cancelled by similar terms in the partition function. As part of this, we will exploit the fact that the matrix elements, even at finite $R$, are computable up to exponentially small corrections. To handle the second difficult, we recognize that the [*infinite*]{} subset of needed matrix elements from the sum in Eqn. (\[eiii\]) are organized according to a Dyson’s equation. This allows us to characterize the subset by resumming a [*finite*]{} number of matrix elements. We now consider how this approach works in practice in two experimentally relevant cases, the transverse field Ising model and the spin-1 chain as represented by the O(3) non-linear sigma model.
[**Transverse field Ising model:**]{} The transverse field Ising model (TFIM) is obtained by taking $S_i\cdot S_i = 3/4$ in Eqn. (1), and setting $J_\perp =0$, ${\bf H} = H\hat x$. In the vicinity of the TFIM’s critical point (i.e. $J_z = H$), this theory has a continuum representation as a free Majorana fermion:[@zamo] $$\label{eiv}
H = \frac{1}{2\pi}\int^R_0 dx \frac{v}{2}({\bar\psi} \partial_x\bar\psi + \psi\partial_x\psi) - i\Delta\psi{\bar\psi}.$$ Here $\psi(x,t)$ and $\bar\psi (x,t)$ are the right and left components of a Majorana Fermi field. The gap, $\Delta$, of the fermions in the disordered regime ($J_z > H$) is given by $\Delta \sim (J_z - H)$. The Hilbert space of the theory on a periodic line of finite length $R$ divides itself into two sectors: Neveu-Schwarz (NS) and Ramond (R). The NS-sector consists of a Fock space built with even numbers of half-integer fermionic modes, i.e. states of the form $|p_1\cdots p_{2N}\rangle_{NS} \equiv a^\dagger_{p_1}\cdots a^\dagger_{p_{2N}}|0\rangle_{NS}$ where a mode’s momentum satisfies, $p_i=2\pi(n_i+1/2)/R$, with $n_i$ an integer, while the R-sector consists of a Fock space composed of odd numbers of even integer fermionic modes, $|k_1\cdots k_{2M+1}\rangle_R \equiv \{ a^\dagger_{k_1}\cdots a^\dagger_{k_{2M+1}}|0\rangle_R\}$, $k_i=2\pi n_i/R$. The energy/momentum, $E(p_i)/P(p_i)$, of a NS state, $|p_1\cdots p_{2N}\rangle_{NS}$, is given simply by $E(p_i)/P(p_i) = \sum_i^{2N} \epsilon(p_i)/\sum_i^{2N}p_i$ where $\epsilon(p) = \sqrt{p^2+\Delta^2}$, with an identical relation holding for states in the R-sector.
To compute the DSF of this model, we need access to the matrix elements, $\langle l | S_z | m \rangle$, of the spin operator $S_z$ at finite $R$. These matrix elements, derived in Refs. (), only are non-zero when $| l \rangle$ and $| m \rangle$ belong to different sectors. For such matrix elements we have $$\begin{aligned}
\label{ev}
_R\langle k_1\cdots k_{2M+1}|S_z(0)|p_1\cdots p_{2N}\rangle_{NS} \!\!&=&\!\! C_R\prod_{i,j}g(\theta_{k_i})g(\theta_{p_j})\cr
&& \hskip -2.3in \times \prod_{i<j}f(\theta_{k_i}-\theta_{k_j})\prod_{i<j}f(\theta_{p_i}-\theta_{p_j})\prod_{i,j} f^{-1}(\theta_{k_i}-\theta_{p_j}),\end{aligned}$$ where $\theta_{p_i}$ parameterizes the momentum $p_i$ via $p_i = \Delta \sinh(\theta_{p_i})$, $C_R = 1 + {\cal O}(e^{-\Delta R})$, $g(\theta) = (\Delta R\cosh(\theta))^{-1/2}$, and $f(\theta) = \tanh (\theta/2)$. Note that up to exponentially small corrections, these matrix elements have the same functional form as at $R=\infty$. The sole difference in the two cases is that at finite $R$, the momenta are quantized. This is a pattern that repeats itself for general integrable models, as emphasized in Ref. , and that we will exploit for our analysis of spin-1 chains.
Crucially all of these matrix elements are finite, a consequence of working at finite $R$. The sole possible divergence comes from the term, $\prod_{i,j} f^{-1}(\theta_{k_i}-\theta_{p_j})$, and occurs as two momenta, $k_i$ and $p_j$, approach one another. But as $k_i$ lies in the R-sector with integer quantization and $p_j$ lies in the NS-sector with half-integer quantization, the two are never exactly equal provided $R$ is finite. In contrast, with $R=\infty$ the distinction between the R- and NS-sectors collapses (via a spontaneous $Z_2$ symmetry breaking). Concomitantly, $S_z$ has matrix elements where $k_i$ and $p_j$ may, in principle, be equal, and so which are infinite. By working at finite $R$, we thus obtain a clean, unambiguous regulation of these infinities.
Even though any given matrix element is finite, we must still sum an infinite number of matrix elements in the Lehmann expansion of Eqn. (\[eiii\]) in order to obtain the DSF, $\chi^I$, at finite T. To do so we employ a Dyson like equation by writing $\chi^{I} (\omega, q)$ in the form, $$\label{evi}
\chi^{I}(\omega, q) = D^{I}(\omega, q)/(1-D^{I}(\omega, q)\Sigma^{I}(\omega, q)).$$ $D^{I}(\omega,q)$ is the DSF in the absence of temperature induced interactions: $D^{I}(\omega,q) = 2\epsilon(q)/((\omega+i\delta)^2+\epsilon^2(q))$. As $\chi^{I}(\omega, q)$ has a well-defined low temperature expansion, so must $\Sigma^{I}(\omega, q)$: $\Sigma^{I}(\omega, q) = \sum_n \Sigma^{I}_n(\omega, q)$, where $\Sigma^{I}_n$ is at best ${\cal O}(e^{-n\beta\Delta})$. We can readily compute $\Sigma^{I}_1$. To do so we expand $\chi^{I}$ to first order in $\Sigma^{I}_1 $: $\chi^{I} = D^{I} + (D^{I})^2\Sigma^{I}_1 + {\cal O}(e^{-2\beta\Delta})$. We then compare this expansion to the expansion of $\chi^{I}$ in terms of the Lehmann expansion of Eqn. (\[eiii\]). To facilitate this we divide $C^{I}(x,\tau)$ into contributions coming from matrix elements with a fixed number of excitations on either side of the operator, $S_z$, i.e. $C^{I}(x,\tau) = \sum_{M,N}C^{I}_{M,N}(x,\tau)$ where $$\begin{aligned}
\label{eix}
C^{I}_{M,N}(x,\tau) \!\!&=&\!\!\!\!\!\! \!\!\sum_{k_1,\cdots,k_{2M+1} \atop p_1,\cdots,p_{2N}}\!\!\!\!\!\!|_R\langle k_1\cdots k_{2M+1}|S_z(0)|p_1\cdots p_{2N}\rangle_{NS}|^2\cr
&& \hskip -.5in \times e^{-\beta E(k_j)}e^{-\tau(E(p_i)-E(k_j))+ix(P(p_i)-P(k_j))}.\end{aligned}$$ Kinematic constraints give that $C^{I}_{M,N}(\omega, q)$ is of order $e^{-\beta{\rm max}(2N\Delta-\theta(\omega)\omega,(2M+1)\Delta+\theta(-\omega)\omega)}$. Keeping terms to at least ${\cal O}(e^{-\beta\Delta})$ and such that ${\rm Im}\chi^{I}(-\omega, q) =-{\rm Im}\chi^{I}(\omega, q)$, we reduce $\chi^{I} (\omega, q)$ to $\chi^{I} = \frac{1}{{\cal Z}}(C^{I}_{01}+C^{I}_{10}
+ C^{I}_{12}+ C^{I}_{21} + e^{-2\beta\epsilon(q)}D^I)$. The final term, $e^{-2\beta\epsilon(q)}D^I$, is a ‘disconnected’ contribution arising from $C_{23}+C_{32}$, exactly cancelling off a similar contribution appearing in $C_{21}+C_{12}$. Comparing these two expansions for $\chi^{I}$ gives us an expression for $\Sigma_1$. First expanding out the partition function, ${\cal Z}^{I} = \sum_{n=0}^\infty Z^{I}_n$ where $Z^{I}_0=1$, $Z^{I}_1 = \sum_{p\in R}e^{-\beta\epsilon(p)}$ and generally $Z^{I}_n$ is ${\cal O}(e^{-n\beta\Delta})$, and then noting that $C^{I}_{01}+C^{I}_{10} = (1-e^{-\beta \epsilon(q)})D^{I}$, we obtain for $\Sigma^{I}_1$, $\Sigma^{I}_1
= (C^{I}_{12}+C^{I}_{21})(D^{I})^{-2}-(Z^{I}_1+e^{-\beta\epsilon(q)})(1-e^{-\beta\epsilon(q)})(D^{I})^{-1}$. To then evaluate $\Sigma^{I}_1$, we compute $C^{I}_{12}$ and $C^{I}_{21}$ numerically. As validation of our use of a finite $R$ regulation of the singularities in the matrix elements, $\Sigma^{I}_1$ remains finite as $R\rightarrow \infty$ even though $C^{I}_{12}$, $C^{I}_{21}$, and $Z^{I}_1$ all diverge.
{height="2.2in" width="2.7in"}
In Fig. 1 we plot the resulting DSF, ${\rm Im}\chi^{I}(\omega, q=0)$, at a variety of temperatures. At $T=0.25\Delta$, the DSF is approximately Lorentzian, but as the temperature is increased to $T=0.6\Delta$, the lineshape develops a marked asymmetry. This asymmetry was also found to be present in a virial-like expansion of the finite T DSF.[@reyes] We quantify the amount of asymmetry in the lineshape by computing the ratio, $A_{\rm LR}$, of the spectral weight to the left and the right of $\omega = \Delta$. In the inset of Fig. 1 we compare our results for $T=0.5\Delta$ (black solid curve) to those arrived at by using a semi-classical approach [@young] (blue curve) and to a Lorentzian approximation thereof (red dashed curve). We see that our computation yields asymmetries in the lineshape far in excess of those found in the semi-classical approach. While the semi-classics has a slight asymmetry at $T=0.5\Delta$, it is close to being Lorentzian.
The origin of this discrepancy between the semi-classics and our treatment lies in two factors. The TFIM, as written in Eqn. (\[eiv\]), is relativistically invariant whereas the semi-classical model used in Ref. only possesses Galilean invariance. It has already been noted that a relativistic dispersion relation, in comparison with a Galilean invariant one, better matches the measured lineshape.[@kenz3] A further difference arises in that the semi-classics for the TFIM is only strictly correct in the $T\rightarrow 0$ limit. At finite $T$ it misses corrections that here are encoded in the form of the matrix elements (Eqn. (\[ev\])).
[**Spin-1 Heisenberg model:**]{} We now apply our approach to the thermal broadening of the coherent mode in a gapped isotropic spin-1 chain (i.e. taking ${\bf S}\cdot{\bf S}=2$, $J_\perp=J_z\equiv J$, and $H=0$ in Eqn. (\[ei\])) . The isotropic spin 1-chain is given in the continuum limit by the O(3) non-linear sigma model:[@haldane] $$\label{exii}
{\cal L} = (2g)^{-1} (\partial_x{\bf n}\cdot\partial_x{\bf n}+\partial_\tau{\bf n}\cdot\partial_\tau{\bf n}).$$ The lattice spin operators, $S_i$, are related to the continuum fields by ${\bf S}_i \simeq (-1)^{i} {\bf n}(ia_0) +
\frac{1}{g}\,{\bf n}\times\partial_t{\bf n}$ (with $a_0$ the lattice spacing).[@o3] In this letter we will focus on the DSF near wavevector $q=\pi$ and so be interested in computing $C(x,\tau) = \langle n^z(x,\tau)n(0)\rangle$. The spectrum and scattering matrix of the O(3) nonlinear sigma model (NLSM) are known exactly. There are three elementary excitations, $A^\dagger_a(\theta)$, $a=x,y,z$, forming a vector representation of O(3). The excitations have a gap behaving as $\Delta \sim J e^{-1/g}$. The excitations’ energy and momentum are parametrized in terms of the rapidity $\theta$ via $\epsilon(\theta)=\Delta\cosh(\theta)$ and $p(\theta)=\Delta\sinh(\theta)$.
Like the TFIM, the eigenstates of the O(3) NLSM can be delineated exhaustively in terms of multi-excitation states, i.e. $|\theta_1,a_1;\cdots,\theta_n,a_n\rangle =
A^\dagger_{a_1}(\theta_1)\cdots A^\dagger_{a_N}(\theta_N)|0\rangle$. However, the matrix elements of these states involving the operator, $n_z$, are considerably more complicated than those of the TFIM. But as we are working at low temperatures, to compute the DSF, $\chi^{O_3}$, we will only need recourse to matrix elements involving a maximum of three excitations (as with the TFIM). In infinite volume they are given by:[@smirnov; @bn] $$\begin{aligned}
\label{exiii}
&&\langle 0|n^a(0)|\theta,b\rangle=\delta_{ab}\ ,\cr
&&\langle \theta_1,a_1|n^a(0)|\theta_3,a_3;\theta_2,a_2\rangle
=-\frac{\pi^{3}}{2}\psi(\hat\theta_{12})\psi(\hat\theta_{13})\psi(\theta_{23})\nn
&&\times\Bigl[
\delta_{aa_1}\delta_{a_2a_3}\theta_{23}
+\delta_{aa_2}\delta_{a_1a_3}\hat\theta_{31}+\delta_{aa_3}\delta_{a_1a_2}\hat\theta_{12}
\Bigr],\end{aligned}$$ where $\psi(\theta)=\frac{\theta+i\pi}{\theta(2\pi i+\theta)}
\frac{\tanh^2(\theta/2)}{\theta}$, $\hat\theta = \theta-i\pi$, and $\theta_{12}=\theta_1-\theta_2$.
As with the TFIM model, we work in finite volume. The sole effect of doing so upon the matrix elements, up to negligible $e^{-\Delta R}$ corrections, is to quantize the momentum (i.e the $\theta$’s) with an attendant effect upon finite volume phase space.[@takacs] Here however the quantization is more complex than that of the TFIM model. We must take into account the non-trivial interactions between the excitations and solve the Bethe ansatz equations. For the calculation at hand, we, at most, most solve the one and two particle Bethe equations for the states, $|\theta_1,a_1\rangle$ and $|\theta_3,a_3;\theta_2,a_2\rangle$. For the one particle state, we have the free quantization condition, $\theta_1 = \sinh^{-1}(2\pi n/R)$ for some integer $n$. For the two particle case, $\theta_2/\theta_3$’s are quantized via $e^{iR\Delta\sinh(\theta_2/\theta_3)}=e^{i\delta_\alpha(\theta_{23}/\theta_{32})}$, where the non-trivial phase, $\delta_\alpha (\theta)$, marks the presence of interactions and depends upon the particular SU(2) representation ($\alpha =$ singlet/triplet/quintet) into which the two particle state falls. Because of the presence of interactions, the finite $R$ matrix elements of the form $\langle \theta_1,a_1|n^a(0)|\theta_3,a_3;\theta_2,a_2\rangle$ are never infinite as the $\theta$’s never coincide. We again see finite $R$ provides a clean regulation of the singularities present at $R=\infty$.
The remainder of the calculation of $\chi^{O_3}$ follows in exact analogy to the TFIM. $\chi^{O_3}(\omega ,q)$ takes the form of Eqn. (\[evi\]) with $D^{O_3} = D^I$. In this case the expansion of $C^{O_3}(x,\tau) \equiv \langle n^z(x,\tau)n^z(0)\rangle$, $C^{O_3}(x,\tau) = \sum_{M,N}C^{O_3}_{M,N}(x,\tau)$, appears as $$\begin{aligned}
\label{exiv}
C^{O_3}_{M,N}(x,\tau) &=& \!\!\!\!\! \sum_{\Theta_1;\cdots;\Theta_M \atop \Theta'_1;\cdots;\Theta'_N}
|\langle \Theta_1,\cdots,\Theta_M|n^z(0)|\Theta'_1;\cdots;\Theta'_N\rangle|^2\cr
&& \hskip -.5in \times e^{-\beta E(\theta_j)}e^{-\tau(E(\theta_i)-E(\theta_j))+ix(P(\theta_i)-P(\theta_j))},\end{aligned}$$ where $\Theta_n \equiv \{\theta_i,a_i\}$. The partition function here admits the expansion, $Z^{O_3} = 1+Z^{O_3}_1 + {\cal O}(e^{-2\beta\Delta})$, and $Z^{O_3}_1 = 3\sum_n e^{\beta\Delta\cosh(\theta_n)}$ With these redefinitions of $C_{M,N}$, $D$ and $Z_n$, $\Sigma^{0_3}_1$ takes the same functional form as $\Sigma^{I}_1$.
{height="2.2in" width="2.8in"}
In Fig. 2 we plot the resulting finite T DSF for the O(3) NLSM. We again find that the lineshape is characterized by an asymmetry that grows with temperature. We observe that this asymmetry is far stronger than that of the semi-classical analysis (see Fig. 10 of Ref. ) whose lineshape is well described by a Lorentzian even up to temperatures of $T = 0.5\Delta$. The origin of this discrepancy between our treatment and the semi-classics is similar to that of the TFIM, i.e. finite energy effects in both the scattering of excitations and in the matrix elements involving the operator, $n_z$.
The asymmetry in the lineshape can be interpreted in terms of a temperature dependent gap, $\Delta (T)$.[@xu] The gap can be extracted as the location of the peak of a Lorentzian fitted to the asymmetric lineshape. We plot $\Delta (T)$ vs $T$ in the left inset to Fig. 2 and compare our computations with the neutron scattering measurements performed on the spin chain, $YBaNiO_5$, in Ref. (). We see good agreement. For other purposes, for example, the analysis of the three magnon scattering continuum in the spin chain compound $CsNiCl_3$ for temperatures $T>0.25\Delta$, we suggest the fitting function, $$\label{exv}
{\rm Im}\chi(\omega,q) = A/((\omega - \epsilon(q) - B)^2+C)^{1-D(x-1)}.$$ In the right inset to Fig. 2 we show that this function provides a good fit of our computed $T=0.4\Delta$ lineshape.
In conclusion we have presented a method by which the lineshape of the coherent mode of gapped spin chains can be determined at finite temperature. This method, employing a continuum integrable representation of the chains, works with finite length, $R$, systems so as to circumvent infinities in matrix elements that appear at $R=\infty$. It further employs a Dyson-like resummation of the matrix elements appearing in a Lehmann expansion of the DSF. The primary conclusion drawn from this analysis is that the lineshape of the mode is asymmetric with an asymmetry increasing with temperature.
We are grateful to A.M. Tsvelik for helpful discussions and communications and G. Xu for access to data from Ref. 15. This work was supported by the EPSRC under grant GR/R83712/01 (FHLE), the DOE under contract DE-AC02-98 CH 10886 (RMK) and the SCCS Theory Institute at BNL (FHLE).
[99]{} F.D.M. Haldane, Phys. Lett. A [**93**]{}, 464 (1983). I. A. Zaliznyak et al. Phys. Rev. Lett. [**87**]{}, 017202 (2001). M. Kenzelmann et al., Phys. Rev. Lett. [**87**]{}, 017201 (2001). M. Kenzelmann et al., Phys. Rev. B [**66**]{}, 024407 (2002). M.D.P. Horton and I. Affleck, Phys. Rev. B[**60**]{}, 9864 (1999); F.H.L. Essler, Phys. Rev. B[**62**]{}, 3264 (2000). M. B. Stone et al., Nature [**440**]{}, 187 (2006). I. Zaliznyak, S. Lee, [*Magnetic Neutron Scattering*]{} in Modern Techniques for Characterizing Magnetic Materials, ed. Y. Zhu, Springer, Heidelberg (2005). F. Smirnov, “Form Factors in Completely Integrable Models of Quantum Field Theory”, World Scientific, Singapore (1992). F. Essler and R. M. Konik in “From Fields to Strings: Circumnavigating Theoretical Physics”, ed. M. Shifman, A. Vainshtein, J. Wheater, World Scientific, Singapore (2005). A. LeClair, G. Mussardo, Nucl. Phys. B [**552**]{}, 624 (1999). R.M. Konik, Phys. Rev. B [**68**]{}, 104435 (2003). B.L. Altshuler, R.M. Konik and A.M. Tsvelik, Nucl. Phys. [**B739**]{}, 311 (2006). S. Sachdev, A. P. Young, Phys. Rev. Lett. 78, 2220 (1997). K. Damle and S. Sachdev, Phys. Rev. B 57, 8307 (1998). G. Xu et al. Science [**317**]{}, 1049 (2007). M. Kenzelmann et al., Phys. Rev. B [**63**]{}, 134417 (2001). M. Kenzelmann et al., Phys. Rev. B [**66**]{}, 174412 (2002). A. Rapp, G. Zarand, Phys. Rev. B [**74**]{}, 014433 (2006). K. Damle, S. Sachdev, Phys. Rev. Lett. 95, 187201 (2005). J. Balog, Nucl. Phys. B [**419**]{}, 480 (1994). P. Fonseca, A. Zamolodchikov, J. Stat. Phys. [**110**]{} 527 (2003). A. Bugrij, Theo. and Math. Phys. [**127**]{} 528 (2001); A. Bugrij, O. Lisovyy, Phys. Lett. A [**319**]{} 390 (2003). B. Pozsgay and G. Takacs, Nucl. Phys. B. [**788**]{} 209 (2008). S.A. Reyes, A. Tsvelik, Phys. Rev. B [**73**]{} 220405(R) (2006). I. Affleck [in]{} [*Fields, Strings and Critical Phenomena*]{}, [eds E. Brézin and J. Zinn-Justin]{}, (Elsevier, Amsterdam, 1989). J. Balog, M. Niedermaier, Nucl. Phys. B [**500**]{}, 421 (1997).
|
---
address: ' $^{1}$ Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany; [dwittor@hs.uni-hamburg.de (D.W.); franco.vazza@hs.uni-hamburg.de (F.V.); mbrueggen@hs.uni-hamburg.de (M.B.)]{} '
---
Introduction
============
The existence of peripheral, elongated and often polarised radio emission in galaxy clusters, so-called radio relics, gives evidence of cosmic-ray electrons being accelerated by shock waves in the intracluster medium (see [@Brunetti_Jones_2014_CR_in_GC] and references therein). Cosmic-ray protons should undergo the same acceleration mechanism, but no evidence of their presence has been found yet. The *Large Area Telescope* on-board of the *Fermi*-satellite [@2009ApJ...697.1071A] is searching for signatures of the cosmic-ray protons, which should produce $\gamma$-ray emission through collisions with the thermal gas. Yet no detection of these $\gamma$-rays has been confirmed and for a variety of clusters the upper flux limits have been estimated to be in the range of $0.5-2.2 \times 10^{-10} \ {\mathrm{ph}}/{\mathrm{s}}/{\mathrm{cm}}^{2}$ above $500 \ {\mathrm{MeV}}$ [@2014ApJ78718A]. Extended searches for the $\gamma$-ray emission in the Coma cluster [@2016ApJ819149A] and the Virgo cluster [@2015ApJ812159A] have set the flux limits above $100 \ {\mathrm{MeV}}$ to $5.2 \times 10^{-9} \ {\mathrm{ph}}/ {\mathrm{s}}/ {\mathrm{cm}}^{2}$ for the former and to $1.2 \times 10^{-8} \ {\mathrm{ph}}/ {\mathrm{s}}/ {\mathrm{cm}}^{2}$ for the latter. Recent results from particle-in-cell simulations [@Caprioli_Spitkovsky_2014_ion_accel_I_eff; @Guo_eta_al_2014_I; @Guo_eta_al_2014_II] suggest that the efficiency of shock acceleration does not only depend on the shock strength but also on the shock obliquity, e.g., the angle between the shock normal and the underlying upstream magnetic field. Cosmic-ray protons should be accelerated more efficiently by diffusive shock acceleration (DSA) in parallel shocks [@Caprioli_Spitkovsky_2014_ion_accel_I_eff]. In contrast, cosmic-ray electrons should prefer a perpendicular configuration as they are first accelerated by shock drift acceleration before they are injected into the DSA cycle [@Guo_eta_al_2014_I; @Guo_eta_al_2014_II].
In our recent work, we have tested if, in galaxy clusters, the additional dependence on the shock obliquity can explain the missing $\gamma$-ray emission and still produce detectable radio relics [@2016arXiv161005305W]. In this contribution we present the most relevant results from that work and include new results.
Methods
=======
Cosmological MHD Simulations
----------------------------
The cosmological magneto-hydrodynamical (MHD) simulation presented in this work has been carried out with the [[*[ENZO]{}*]{}]{}-code [@ENZO_2014]. In our simulation, we solve the MHD equations using a piecewise linear method [@1985JCoPh..59..264C] in combination with hyperbolic Dedner cleaning [@2002JCoPh.175..645D]. We re-simulate a single galaxy cluster with a final mass of $M_{200}(z\approx 0)\approx 9.74 \times 10^{14} \ {\mathrm{M}_{\odot}}$. The cluster shows a major merger at $z \approx 0.27$, which is strong enough to produce detectable radio relics (see [@2016arXiv161005305W] for further information). We simulate a $250^3 \ \sim \ {\mathrm{Mpc}}^3$ comoving volume from $z \approx 30$ to $z \approx 0$, starting from a root grid of $256^3$ cells and $256^3$ dark matter particles. Furthermore, using five levels of Adaptive Mesh Refinement (AMR), we refine $2^5$ times a $\approx$ $25^3$ ${\mathrm{Mpc}}^3$ region centred around our massive cluster, resulting in a final resolution of $31.7 \ {\mathrm{kpc}}$ for a large portion of the cluster volume. For the seeding of the large scale-magnetic fields, we use a primordial magnetic seed field with a comoving value of $B_0 = 10^{-10}\ \mathrm{G}$ along each direction.
Lagrangian Analysis
-------------------
We track the evolution of cosmic rays using Lagrangian tracer particles (see [@2016arXiv161005305W] for more details). The tracer particles follow, both, the advection of the baryonic matter and the enrichment of shock-injected cosmic rays in time. In post-processing, the tracers are advected in a sub-box consisting of $256^3$ cells of the finest grid of the simulation. The sub-box is centred around the mass centre of our galaxy cluster at $z \approx 0$. The tracers are first injected into the box at $z \approx 1$ following the mass distribution of the gas. During run-time, additional tracers are injected from the boundaries following the mass distribution of the entering gas. In total, we generate $N_p \approx 1.33\times10^7$ tracers with a final mass resolution of $m_{\mathrm{tracer}} \approx 10^{8} \ {\mathrm{M}_{\odot}}$.
The tracers are advected linearly in time using the velocities at their location: $\boldsymbol{v}=\boldsymbol{\tilde{v}}+\delta \boldsymbol{v}$. Here, $\boldsymbol{\tilde{v}}$ is the interpolated velocity between the neighbouring cells and $\delta \boldsymbol{v}$ (see Equation (1) in [@2016arXiv161005305W]) is a correction term to cure for a possible underestimate due to mixing in complex flows (see [@Genel_at_al_2014_following_the_flow] for more details).
The local gas values are assigned to every tracer and other properties are computed on the fly. Subsequently, we apply a shock-finding method based on the temperature jump between the positions of a tracer at two consecutive timesteps, similar to the method described in [@Ryu_et_al_2003_shock_waves_large_scale_universe]. Every time a shock is recorded, the Mach number and the corresponding shock obliquity are computed. The latter is calculated using the velocity jump $\Delta \boldsymbol{v} = \boldsymbol{v}_{{\mathrm{post}}} - \boldsymbol{v}_{{\mathrm{pre}}}$ between the pre- and post-shock velocity of the tracer:
$$\theta_i = \arccos \left( \frac{ \Delta \boldsymbol{v} \cdot \boldsymbol{B_i}}{\vert \boldsymbol{v}\vert \vert\boldsymbol{B_i}\vert} \right) \label{eq:theta}.$$
In the equation above, the index $i$ refers to either the pre- or post-shock values. Across each shock, we compute the kinetic energy flux as $F_{\Psi} = 0.5 \cdot \rho_{{\mathrm{pre}}} v_{\mathrm{sh}}^3$, where $\rho_{{\mathrm{pre}}}$ is the pre-shock density and $v_{\mathrm{sh}}$ is the shock velocity. The thermal energy flux, $F_{\mathrm{th}} = \delta(M) F_{\Psi}$, and cosmic-ray energy flux, $F_{\mathrm{CR}} = \eta(M) F_{\Psi}$, are computed using the acceleration efficiencies $\delta\left(M\right)$ and $\eta\left( M\right)$ given in [@2013ApJ...764...95K]. The efficiency, $\eta(M)$, is taken from @2013ApJ...764...95K and it includes the effects of magnetic field amplification at the shocks and thermal leakage of suprathermal particles. We include (as in [@2014MNRAS.439.2662V; @2016arXiv161005305W]) the effect of re-acceleration by computing an effective $\eta_{\mathrm{eff}}(M)$ that is interpolated from the acceleration efficiencies of acceleration and re-acceleration given in @2013ApJ...764...95K.
We use the formula given in [@2007MNRAS.375...77H] to compute the radio emission from the shocked tracers:
$$\begin{split}
\frac{\mathrm{d} P_{{\mathrm{radio}}} \left( \nu_{\mathrm{obs}} \right) }{\mathrm{d} \nu} &= \frac{6.4\cdot10^{34} \ \mathrm{erg}}{{\mathrm{s}}\cdot \mathrm{Hz}} \frac{A}{\mathrm{Mpc}^2} \frac{n_e}{10^{-4} \ {\mathrm{cm}}^{-3}}
\frac{\xi_e}{0.05}\left( \frac{T_d}{7 \ {\mathrm{keV}}} \right)^{\frac{3}{2}} \\
&\times \left( \frac{\nu_{\mathrm{obs}}}{1.4 \ {\mathrm{GHz}}} \right)^{-\frac{s}{2}}
\frac{\left( \frac{B}{\mu {\mathrm{G}}} \right)^{1+\frac{s}{2}}}{\left(\frac{B_{\mathrm{CMB}}}{\mu {\mathrm{G}}} \right)^{2}+ \left(\frac{B}{\mu {\mathrm{G}}} \right)^{2}} \cdot \eta \left( M \right)
\end{split}.\label{eq:HB}$$
The quantities that are taken from the grid are: $A$ the surface area of a tracer, $n_e$ the number density of electrons, $T_d$ the downstream temperature, $B$ the magnetic field strength and the acceleration efficiency $\eta \left(M\right)$ depending on the Mach number $M$ taken from [@2013ApJ...764...95K]. We notice that the application of $\eta (M)$ to Equation (\[eq:HB\]) is limited to spectra flatter than $s \approx -3$, because Equation (\[eq:HB\]) has been derived in energy space while $\eta(M)$ has been derived in momentum space. However, our modelling is accurate enough for the radio frequency, produced by electrons with Lorentz factor of $\gamma > 10^3$, that we are investigating here [@2007MNRAS.375...77H]. The other quantities are the observed frequency band, $\nu_{\mathrm{obs}}=1.4 \ {\mathrm{GHz}}$, the equivalent magnetic field of the cosmic microwave background, $B_{\mathrm{CMB}} = 3.2 \cdot (1+z)^2 \ \mu {\mathrm{G}}$ and electron-to-proton ratio $\xi_e = 0.01$. Following @2007MNRAS.375...77H we assume that the minimum electron energy is $10$ times the thermal gas energy, while the minimum proton energy is fixed to $780 \ {\mathrm{MeV}}$. We use the same approach as in to compute the $\gamma$-ray emission. We refer to Appendix C of our previous publication [@2016arXiv161005305W] for a summary of the method.
Results
=======
In [@2016arXiv161005305W] we studied how linking the shock acceleration efficiency to the shock obliquity can affect the acceleration of cosmic rays by predicting the amount of radio and $\gamma$-ray emission produced by either quasi-perpendicular or quasi-parallel shocks. Following [Figure]{} 3 of @Caprioli_Spitkovsky_2014_ion_accel_I_eff we define quasi-perpendicular shocks as $\theta \in [50^{\circ}, 130^{\circ}]$ and quasi-parallel shocks as $\theta \in [0^{\circ}, 50^{\circ}] \ \& \ [130^{\circ}, 180^{\circ}]$. However, a more detailed analysis in @2016arXiv161005305W showed that the effects on the acceleration of cosmic rays are not very sensitive to the selection of $\theta$.\
We found that the distribution of shock obliquities in a galaxy cluster roughly follows the distribution of random angles in three-dimensional space, $\propto \sin(\alpha)$. Just based on this, one can expect to observe more quasi-perpendicular shocks than quasi-parallel shocks. Hence, the acceleration of cosmic-ray electrons should be more favoured than the acceleration of cosmic-ray protons. For the results on how this affects the radio and $\gamma$-ray emission, we point to our previous publication [@2016arXiv161005305W] as they are similar to the ones presented below.
In this contribution we present a closer analysis of the same cluster at the epoch of the peak of the total radio emission. We show the projection of the gas density overlayed with the radio contours in Figure \[subfig:denspRadio\]. The cluster is still in a very active phase after it experienced a major merger at $z \approx 0.27$, and several smaller gas clumps are still falling onto the cluster.
First, we measure the distribution of pre- and post-shock obliquities at $z \approx 0.2$. The left panel of Figure \[subfig:obliHist\] shows the measured distributions consistent with isotropy. The right panel of Figure \[subfig:obliHist\] shows the distribution of pre-shock obliquities for different selections in the shock Mach numbers. While the obliquity distributions of $M<3$ shocks roughly follow the distribution of random angles, stronger (and rarer) shocks are found to cluster at specific obliquity values, related to single large-scale magnetic structures in the cluster volume.
In the following we perform a similar analysis as in our previous work [@2016arXiv161005305W] to investigate how coupling the shock acceleration efficiencies to the shock obliquity affects the $\gamma$-ray emission \[sss:gamma\] and the radio emission \[sss:radio\].
$\gamma$-ray Emission {#sss:gamma}
---------------------
In Figure \[subfig:gamma\], we show the total integrated $\gamma$-ray emission and radial $\gamma$-ray emission profiles produced in the cluster at this epoch. The total $\gamma$-ray emission received from inside $r_{200}$ is $\sim$1.03$\times 10^{45} \ {\mathrm{ph}}/ {\mathrm{s}}$, which is above the corresponding *Fermi*-limits (see [@2016arXiv161005305W] for the exact computations) of the Coma ($0.035 \times 10^{45} \ {\mathrm{ph}}/ {\mathrm{s}}$) cluster and just below the limits of A2256 ($1.075 \times 10^{45} \ {\mathrm{ph}}/ {\mathrm{s}}$). The $\gamma$-ray emission resulting from cosmic-ray protons that have been accelerated only by quasi-parallel shocks is $\sim$0.31$\times 10^{45}~\ {\mathrm{ph}}/ {\mathrm{s}}$. This is still above the lowest upper limit of the Coma cluster. The observed drop in $\gamma$-ray emission is consistent with the fact that at low Mach numbers only $\sim$one-third of all shocks are quasi-parallel (see Figure \[subfig:obliHist\]).
Consistent with our findings from [@2016arXiv161005305W], we conclude that the missing $\gamma$-ray emission cannot be entirely reproduced by limiting the acceleration of cosmic-ray protons to quasi-parallel shocks.
Radio Emission {#sss:radio}
--------------
We observe two radio relics on the left (hereafter relic one) and right (hereafter relic two) side of the cluster core (see Figure \[subfig:denspRadio\]). Both relics are in the range detectability by modern radio observations. Figure \[subfig:TnvCCpc\] shows the complex geometry of the magnetic field lines in the relic regions. We observe that the morphologies of the relics do not change significantly, if only either quasi-perpendicular (middle panel) or quasi-parallel (right panel) shocks are able to accelerate cosmic-ray electrons.
In the first relic, only $\sim \ 46 \%$ of the radio emission is produced by cosmic-ray electrons that have been accelerated by quasi-perpendicular shocks. This is consistent with the distribution of obliquities (see Figure \[subfig:obliHist\]), as relic one is mostly produced by a higher Mach number shocks with $M \ \sim \ 3-5$ (see Figure \[subfig:mnomask\]), that do not follow the distribution of random angles in a three-dimensional space (see Figure \[subfig:obliHist\]). On the other hand, $\sim$59$\%$ of the shocks producing relic two are quasi-perpendicular, as it is produced by weaker shocks with $M \ \sim \ 2-3$ (see Figure \[subfig:mnomask\]).
However, consistent with [@2016arXiv161005305W], we find that both simulated relics remain visible if the acceleration of electrons is limited to quasi-perpendicular shocks.
Discussion
==========
Combining MHD-simulations and Lagrangian tracers, we continued our study on how restricting the shock acceleration efficiencies to the obliquity affects cosmic rays in galaxy clusters. At the epoch of the highest radio emission, we examined how cosmic rays, that have been accelerated by either quasi-parallel or quasi-perpendicular shocks, contribute to the resulting $\gamma$-ray and radio emission. We chose this epoch for our investigation as the two radio relics are the most prominent.
Our findings agree with our results from [@2016arXiv161005305W]: The distribution of shock obliquities follows the distribution of random angles in a three-dimensional space. Furthermore, we discovered that this only holds for low Mach numbers $M \le 3$. The distribution of shock obliquities for the few high Mach numbers $M \ge 3$ does not show this trend, as they tend to cluster around single magnetic field structures.
Consistent with our findings from [@2016arXiv161005305W], the $\gamma$-ray emission drops by a factor of $\sim$3 if only quasi-parallel shocks are able to accelerate the cosmic rays. Yet, this drop is not large enough the explain the low upper limits set by the *Fermi*-satellite [@2014ApJ78718A], especially in the case of the Coma cluster [@2016ApJ819149A].
On the other hand, the radio emission remains observable if only quasi-perpendicular shocks are able to accelerate cosmic rays. This also holds if the majority of the radio emission is produced by a strong quasi-parallel shock. This supports our conclusion from [@2016arXiv161005305W] that it is possible that the cosmic-ray electrons in observed radio relics have only been accelerated by quasi-perpendicular shocks.
We mention that we do not include any other additional mechanisms, such as cosmic-ray re-acceleration by cluster weather or turbulence (e.g., [@2011MNRAS.412..817B]) which would produce further cosmic-ray protons. On the other hand, we do not allow any spatial diffusion of the cosmic-rays, that would reduce the $\gamma$-ray flux through proton accumulation in the cluster outskirts (e.g., [@2013MNRAS.434.2209W; @2016arXiv160702042L]).
[999]{} \[1\][\#1]{}
, G.; [Jones]{}, T.W. . , [ *23*]{}, 30007.
, W.B.; [Abdo]{}, A.A.; [Ackermann]{}, M.; [Althouse]{}, W.; [Anderson]{}, B.; [Axelsson]{}, M.; [Baldini]{}, L.; [Ballet]{}, J.; [Band]{}, D.L.; [Barbiellini]{}, G.; et al. . , [*697*]{}, 1071–1102.
, M.; [Ajello]{}, M.; [Albert]{}, A.; [Allafort]{}, A.; [Atwood]{}, W.B.; [Baldini]{}, L.; [Ballet]{}, J.; [Barbiellini]{}, G.; [Bastieri]{}, D.; [Bechtol]{}, K.; et al. . , [*787*]{}, 18.
, M.; [Ajello]{}, M.; [Albert]{}, A.; [Atwood]{}, W.B.; [Baldini]{}, L.; [Ballet]{}, J.; [Barbiellini]{}, G.; [Bastieri]{}, D.; [Bechtol]{}, K.; [Bellazzini]{}, R.; et al. . , [*819*]{}, 149.
, M.; [Ajello]{}, M.; [Albert]{}, A.; [Atwood]{}, W.B.; [Baldini]{}, L.; [Barbiellini]{}, G.; [Bastieri]{}, D.; [Bechtol]{}, K.; [Bellazzini]{}, R.; [Bissaldi]{}, E.; et al. . , [*812*]{}, 159.
, D.; [Spitkovsky]{}, A. . , [*783*]{}, 91.
, X.; [Sironi]{}, L.; [Narayan]{}, R. . , [*794*]{}, 153.
, X.; [Sironi]{}, L.; [Narayan]{}, R. . , [*797*]{}, 47.
, D.; [Vazza]{}, F.; [Br[ü]{}ggen]{}, M. . [**2016**]{}, arXiv:1610.05305.
, G.L.; [Norman]{}, M.L.; [O’Shea]{}, B.W.; [Abel]{}, T.; [Wise]{}, J.H.; [Turk]{}, M.J.; [Reynolds]{}, D.R.; [Collins]{}, D.C.; [Wang]{}, P.; [Skillman]{}, S.W.; et al. . , [*211*]{}, 19.
, P.; [Glaz]{}, H.M. . , [ *59*]{}, 264–289.
, A.; [Kemm]{}, F.; [Kr[ö]{}ner]{}, D.; [Munz]{}, C.D.; [Schnitzer]{}, T.; [Wesenberg]{}, M. . , [ *175*]{}, 645–673.
, S.; [Vogelsberger]{}, M.; [Nelson]{}, D.; [Sijacki]{}, D.; [Springel]{}, V.; [Hernquist]{}, L. . , [*435*]{}, 1426–1442.
, D.; [Kang]{}, H.; [Hallman]{}, E.; [Jones]{}, T.W. . , [*593*]{}, 599–610.
, H.; [Ryu]{}, D. . , [*764*]{}, 95.
, F.; [Gheller]{}, C.; [Br[ü]{}ggen]{}, M. . , [*439*]{}, 2662–2677.
, M.; [Br[ü]{}ggen]{}, M. . , [*375*]{}, 77–91.
, F.; [Eckert]{}, D.; [Br[ü]{}ggen]{}, M.; [Huber]{}, B. . , [*451*]{}, 2198–2211.
, J.; [Dolag]{}, K.; [Cassano]{}, R.; [Brunetti]{}, G. . , [*407*]{}, 1565–1580.
, B.; [Tchernin]{}, C.; [Eckert]{}, D.; [Farnier]{}, C.; [Manalaysay]{}, A.; [Straumann]{}, U.; [Walter]{}, R. . , [*560*]{}, A64.
, G.; [Lazarian]{}, A. . , [*412*]{}, 817–824.
, J.; [Oh]{}, S.P.; [Guo]{}, F. . , [*434*]{}, 2209–2228.
, A. . , arXiv:1607.02042.
, E.L. . , [*118*]{}, 1711–1715.
|
---
abstract: 'We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given including a family of quantum walks on the hypercube involving a Clifford Algebra.'
address:
- 'Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin 95302 Cergy Pontoise Cedex, France'
- 'Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, Friedrich-Ludwig-Jahn Str. 15a, 17487 Greifswald, Germany'
author:
- O Lopez Acevedoand T Gobron
title: Quantum walks on Cayley graphs
---
Introduction
============
Recently many effort has been devoted to the construction of new quantum algorithms. In particular a question which has arisen is whether the known algorithms fully exploit the possibilities of quantum mechanics, or if there could exist more efficient ones. A search for new ideas in this direction has been at the origin of a renewed study of quantum walks models [@ambainis_algo_qrw], and a few results have already been obtained, showing that these are definitely relevant in this context.\
The first general characterisation of walks over graphs was presented in [@aakv]. A possible construction for a walk operator is given, based on its classical equivalent, and some quantities relevant in the context of quantum algorithms are defined and computed. One of the principal results states that for bounded degree graphs the mixing time (defined also in the same work) is at most quadratically faster than the mixing time of the simple classical random walk on the same graph. Even if this general result is not so encouraging, some particular graphs have been shown to have properties intrinsically different from their classical equivalents. In particular, a symmetric quantum walk may get across an hypercube in a time linear with the dimension, while its classical counterpart would take an exponentially larger time.\
In algorithmic applications, quantum walks have also shown interesting properties. The first important achievement has been the setting of the quantum search algorithm in the form of a quantum walk over an hypercube [@Shenvi]. Some other similar quantum search algorithms were constructed after this. In one of them [@akr04], the choice of the coin operator was revealed to be of crucial importance, since different operators may achieve different speed-ups (or no speed-up at all) without obvious reasons. A natural question which arises from this problem is whether there exist quantum walks different to than those defined in [@aakv] and if so, to what extent they could be the source of interesting new properties and algorithmic applications. Another problem lies in the dimension of the internal space: it is always possible to enlarge it, and in [@brun_ambainis], it was shown that in an extremal case, the variance of the one dimensional walk recovers the classical behaviour. In a similar direction in [@WD03] and [@WD04] the authors have considered the evolution of a quantum particle governed by a quantum multi-baker map which can be settled as a quantum walk on a line with a multidimensional internal space, the classical limit is also recovereded enlarging the dimension of the internal space. At the opposite, an interesting and still open question is whether there exist quantum walks with local spaces of dimension smaller than that taken in the standard definition. In the context of quantum cellular automata, it is shown in [@meyer2] that for the simple lattice in $d$ dimensions there is no nontrivial walk with an internal space of dimension one, also known as the No-go theorem.\
In this article we make a step in the direction of determining all possible quantum walks for general graphs and characterising their structures. Starting from a general definition of a quantum walk we deduce necessary and sufficient conditions on the coin operators for the evolution to be unitary (section 2). The next section contains a discussion on the solutions of these equations (section 3). In particular, we characterise all possible walks on the Cayley graph of a free group. In the case of abelian groups, the situation is somewhat more complicated, and after a general discussion we present particular solutions. We construct quantum walks over the two dimensional and three dimensional simple lattice with an internal space of dimension smaller than what was previously known and a generalisation to arbitrary dimensions. We also consider the hypercube as a Cayley graph on which we construct a quantum walk where the coin operators are related to elements of the Clifford algebra. Finally, we propose a possible generalisation of a quantum walk where we depart from the image of a particle moving on a lattice and which could be of interest in the context of quantum algorithms (section 4).
Model and unitary relations
===========================
A quantum algorithm is a sequence of transformations on a state of a quantum system. The quantum system is described by a tensor product of two dimensional complex Hilbert spaces. There is a preferred basis of the elementary space where vectors are labelled with the integers zero and one in correspondence to classical bits. Then a basis vector of the entire system is $|x_0 \rangle \otimes \dots \otimes | x_n \rangle$ where $x_i \in \{0,1\}$ and in this way it is possible to associate to each base vector an integer whose binary decomposition coincides with the n-tuple $(x_0, \dots ,x_n)$. The total operator is the product of elementary operators. A presentation of possible sets as well as a demonstration of the universality of these sets may be found in [@barenco].\
A quantum walk is a model for the evolution of a particle over a graph. Many of the choices made in building the model may be explained by the aim of studying them as quantum algorithms. Let $G$ be a directed graph with vertex set $X$ and edge set $E$ such that $G=(X,E)$. Let $\mathcal H$ be the Hilbert space defined by $\mathcal H= \mathcal H_I \otimes \mathcal H_G$. The space $\mathcal H_G=\ell^2(X)$ describes the position of the particle over the graph and the space $\mathcal H_I= \mathbb{C}^d$ describes some internal degrees of the particle.Let $\{|x \rangle\}_{x \in X}$ be a base of $H_G$ and $\{ |1\rangle ,\dots, |d\rangle\}$ a base of $\mathcal H_I$.\
The evolution equation is: $$|\psi_{t+1}\rangle=W |\psi_{t} \rangle$$ where $W$ is a discrete time evolution operator defined as $$\label{eo}
W=\sum_{x\in X}\sum_{z \in E_x} M_{x,z} \otimes T_{x \rightarrow z}$$ where $E_x$ denotes the set of neighbouring sites of $x$ and $T_{x \rightarrow
z}$ translates the particle from $x$ to $z$. $T_{x \rightarrow z}$ is defined by $$\langle x' | T_{x \rightarrow z} \vert \psi\rangle = \langle x'|z \rangle
\langle x | \psi \rangle$$ $M_{x,z}:H_I \to
H_I$ are maps modifying the internal space at the same time as the translation from vertex $x$ to vertex $z$ is applied. Suppose $|\psi_{t}\rangle=\vert i
\rangle \otimes |z \rangle$. Then after one time step the probability of finding the particle in at vertex $y$, a neighbour of $z$, will depend on the previous internal state: $$P(y)= \sum_{j=1}^d | \, \langle j | M_{z,y} |i \rangle |^2$$ One image commonly used to describe the local evolution is that of a coin attached to each vertex and flipped to decide which neighbour the particle will jump to (see for instance [@aakv]) and accordingly the local map $M_{x,y}$ is termed the “coin operator”. Here we follow this usage though our model is more general than the image: in fact, it is important to note that originally the internal state was identified to the set of possible outcomes of the coin flip, or equivalently to the set of neighbours, so that the dimension of the internal space at a given vertex was necessarily equal to the number of outgoing edges. Here we have not considered this identification.\
Unitarity of $W$ is satisfied if and only if: $$\begin{aligned}
W^\dagger W=\mathbbm{1} \Leftrightarrow \sum_{z \in E_x \cap E_{x'}}
M_{x,z}^\dagger M_{x',z}=\delta_{x,x'} \mathbbm{1}_{H_I} \label{co1} \\
W W^\dagger=\mathbbm{1} \Leftrightarrow \sum_{z \in E_x \cap E_{x'}} M_{z,x}
M_{z,x'}^\dagger=\delta_{x,x'} \mathbbm{1}_{H_I} \label{co2}\end{aligned}$$ $\forall x,x'$. When $x \not = x'$, in order to have a non trivial equation, $x$ and $x'$ must be second neighbours and the number of terms in the sum is related to the number of closed paths of length 4 with alternating orientation.\
In the example on the figure 1, one condition equation of the form with three terms is associated with the pair of second neighbours $x$ and $x'$ : $$M_{x,z_1}^\dagger M_{x',z_1}+M_{x,z_2}^\dagger M_{x',z_2}+M_{x,z_3}^\dagger
M_{x',z_3}=0$$
Quantum walks on Cayley Graphs
==============================
We will restrict our study from now on to quantum walks on Cayley graphs. We first recall their definitions and main properties. We follow the presentation given in [@white]. Given a group $\Gamma$ one considers a set $\Delta$ of elements in $\Gamma$ such that $\Delta$ is a generating set for $\Gamma$. The Cayley graph $C_\Delta(\Gamma)=(X,E)$ is defined as the oriented graph with $$\begin{aligned}
X\equiv X(C_\Delta(\Gamma))=\Gamma \label{xcg}\\
E\equiv E(C_\Delta(\Gamma))=\{(x,x\delta)_{\delta} \vert x\in \Gamma ,\delta
\in \Delta \} \label{ecg}\end{aligned}$$ When associating a colour to each element of the generating family, the definition of $C_\Delta(\Gamma)$ makes it a coloured directed graph. In addition a Cayley colour graph is vertex transitive, so that each site is equivalent. Thus we consider internal operators which depend only on the edge colour and direction of the edge $(x,y)$ (i.e. only on the generator $\delta= x^{-1} y$) and not on the starting vertex $x$: $$\begin{aligned}
M_{x,y}= M_{x^{-1} y} \hbox{ for all } (x,y) \in E\end{aligned}$$ Thus the evolution operator $W$ on $\mathcal H$ is $$\begin{aligned}
\label{wcg}
W = \sum_{\delta\in\Delta} M_\delta \otimes T_\delta\end{aligned}$$ where $T_\delta$ is the shift in the direction $\delta$ and is defined for all vertices by the group operation $$T_\delta = \sum_{x \in X} T_{x \rightarrow x \delta}$$ The problem is thus reduced to a local one on $\mathcal H_I$ and the unitarity conditions and now read: $$\begin{aligned}
\label{coc2}
\sum_{\delta_1 \delta_2^{-1} =u } M_{\delta_1}^\dagger M_{\delta_2}
=\delta_{\{u=e\}} \mathbbm 1\\
\label{coc3}
\sum_{\delta_1 \delta_2^{-1} =u } M_{\delta_1} M_{\delta_2}^\dagger
=\delta_{\{u=e\}} \mathbbm 1\end{aligned}$$ where both sums run over all pairs of elements in $\Delta$, $u$ is any element in the set $$\begin{aligned}
\Delta_2=\{ \delta \delta'^{-1}; \delta, \delta' \in \Delta\}\end{aligned}$$ and $e$ is the neutral element in $\Gamma$. The number of equations is twice the cardinality of $|\Delta_2|$ and the number of terms in at least some of these equations will be larger than one as soon as there exists closed paths of length 4 on the graph with an alternating orientation, which in terms of the generators is $$\begin{aligned}
\label{altp4}
\delta_1 \delta_2^{-1} \delta_4 \delta_3^{-1} =e\end{aligned}$$ Because of this relation it will be sometimes useful to define the group $\Gamma$ itself in terms of the “free presentation” $$\begin{aligned}
\Gamma = \langle \Delta'| R\rangle \end{aligned}$$ where $\Delta'$ is a set of generators of a free group and $R$ is the set (which may also be empty) of relations between the elements of $\Delta'$ and their inverses which defines the structure of the group. To define the Cayley graph and in the following we will use the generating set $\Delta$ defined by $$\label{deltadef}
\Delta=\{\gamma:\gamma \in \Delta' \lor \gamma^{-1} \in \Delta'\}$$ where $\Delta'$ is the generating set used in the free presentation of the group. In particular $\Delta$ may contain at the same time a generator and its inverse.\
We now list some generic cases of Cayley groups.
Cayley graphs of free groups
----------------------------
As its name suggests, a free group is a group generated with a (finite) number of generators with no relations between them $$\label{fg}
\Gamma =\langle \Delta' |- \rangle$$ Lets consider the Cayley graph $C_{\Delta}(\Gamma)$ of the precedent group defined by , and . The two sets of equations and can be written as: $$\begin{aligned}
\label{cocf1}
M_{\delta_1}^\dagger M_{\delta_2} &=& M_{\delta_1} M_{\delta_2}^\dagger = 0
\hbox{ for all } \delta_1 \ne \delta_2\\
\label{cocf2}
\sum_{\delta\in \Delta} M_{\delta} M_{\delta}^\dagger &=&
\sum_{\delta\in \Delta} M_{\delta}^\dagger M_{\delta} = \mathbbm 1\end{aligned}$$
On the Cayley graph of the free group , defined by , and , the quantum walk evolution operator is unitary if and only if the internal operators are of the form, $$\label{solcf}
M_{\delta} = U\, P_\delta$$ where $U$ is a unitary matrix of dimension $dim(\mathcal H_I)$ and $\{P_\delta\}_{\delta\in\Delta}$ is a complete family of orthogonal projectors, $$\sum_{\delta\in\Delta} P_\delta = \mathbbm 1$$ The internal space is of dimension larger or equal to $|\Delta|$.\
[**Proof:**]{} First, it is easy to see that is a solution for -. Now suppose -, these equations imply the following relation between the images of the maps $$\begin{aligned}
\label{cocf3}
\mathcal H_I = \oplus_{\delta\in\Delta} \mathcal Im(M_\delta) \\
\label{cocf4}
\mathcal H_I = \oplus_{\delta\in\Delta} \mathcal Im(M_\delta^\dagger)\end{aligned}$$ The fact that a direct sum appears in the right hand sides of - is just a consequence of equations which make all subspaces pairwise orthogonal. The equality (rather than an inclusion) is due to . Define $U\equiv \sum_\delta M_\delta$, an unitary matrix by -, and $P_\delta$ as the orthogonal projector on $\mathcal Im(M_\delta^\dagger)$, follows by considering the elements of a vector basis compatible with the decomposition . The claim that is the general solution is thus proven. $\square$\
One should note however that the right hand side of could be written in many other ways, for instance with its factors written in the opposite order (which makes $P_\delta$ the projector on $Im(M_\delta)$). When the rank of all matrices $M_{\delta}$ is fixed to 1, the dimension on the local Hilbert space takes its minimal value $dim(\mathcal H_I)= |\Delta|$, and if a symmetric presentation for the group is chosen (i.e: $\delta\in\Delta$ implies $\delta^{-1}\in\Delta$), the standard definition of quantum “coin” solution [@aakv] is recovered. Besides these solutions, the only other possibility in the case of free groups consists in taking matrices $M_{\delta}$ of rank different from one and possibly varying with $\delta$.\
The case when the generating set that defines the Cayley graph contains the group identity $e$ and at the same time some generators and their inverses is slightly more involved because the group identity $e$ commutes with all the elements in the group. If both a generator $\delta$ and his inverse $\delta^{-1}$ are in $\Delta$ in addition to equations one has $$\begin{aligned}
\label{identity1}
&&M_{\delta}^\dagger M_{e}+M_{e}^\dagger M_{\delta^{-1}}=0\\
\label{identity2}
&& M_{e} M_{\delta}^\dagger + M_{\delta^{-1}} M_{e}^\dagger =0\end{aligned}$$ for all $\delta\ne e$. Summing all equations in , one gets $M_{e}^\dagger S =- S ^\dagger M_{e}$ where $S=\sum_\delta M_\delta$. Adding again two instances of equations for both a given $\delta$ and its inverse $\delta^{-1}$ gives $$\begin{aligned}
( M_{e}^\dagger S ) (P_\delta + P_{\delta^{-1}}) =
(P_\delta + P_{\delta^{-1}}) ( M_{e}^\dagger S )\end{aligned}$$ for all $\delta \not = e$. Thus $M_e^\dagger S$ is block diagonal in the representation where all the orthogonal projectors $P_\delta$’s are simultaneously diagonal. The problem can essentially be reduced to the one dimensional case which we explore below. This is the first instance of a solution to equations and different to the solution , in the case when there is more than one non-zero term.
### One dimensional walks
The simplest example is a quantum walk in one dimension. Lets consider the group generated by one element $\Gamma=<\delta|->$ and the Cayley graph obtained - using $\Gamma$ and the set $\Delta=\{\delta,\delta^{-1}\}$. The minimal dimension of the internal space is $2$ by the preceeding theorem and the form of the solution follows equation . The evolution operator defined in reads in this case $$\begin{aligned}
\label{wcf1d}
W = (U \otimes Id) (P_\delta \otimes T_\delta
+ P_{\delta^{-1}} \otimes T_{\delta^{-1}} )\end{aligned}$$ where $U$ is a $2\times 2$ unitary matrix. Two quantum walk evolution operators $W$ and $W'$ differing by an unitary transformation $V$ would be equivalent, since this amounts to a change of basis for the initial and final state. We will suppose $V$ of the form of a tensor product $A\otimes
\mathbbm{1}$. Thus equation defines a family of inequivalent quantum walks indexed by 4 real parameters: the 4 parameters associated with the unitary matrix $U$ while the projectors $P_{\delta},P_{\delta^{-1}}$ become the projectors over the spaces spanned by each one of the basis vectors.\
A quantum walk can also be left-right symmetric if it is invariant, up to an unitary transformation $S \otimes \mathbbm 1$, under the transformation $T_\delta \leftrightarrow T_{\delta^{-1}}$. The family of inequivalent and left-right symmetric quantum walks are of the reduced form described before with $U$ $$U=e^{i \delta}\left( \begin{array}{cc} \cos\frac{\theta}{2}& e^{i \alpha}
\sin\frac{\theta}{2}\\-e^{-i\alpha} \sin\frac{\theta}{2}&
\cos\frac{\theta}{2} \end{array} \right)$$ This defines then a 3 parameter family of inequivalent left-right symmetric quantum walks. The unitary $S$ depends also on the 3 parameters. When the identity appears in $\Delta$, two kinds of solutions can be devised depending on whether the two terms appearing in - are separately zero or not. In the first case, one needs to add (at least) one state associated to the identity and the evolution operator becomes $$\begin{aligned}
\label{wcf1de1}
W = (U \otimes Id) (P_\delta \otimes T_\delta
+ P_{\delta^{-1}} \otimes T_{\delta^{-1}} + P_e \otimes Id)\end{aligned}$$ where $U$ is a $3\times 3$ unitary matrix, and appears just as a simple extension of the previous example. However, solutions exist with a two dimensional local Hilbert space, and in such cases the evolution operator is $$\begin{aligned}
\label{wcf1de2}
W = (U \otimes Id) \bigl( \cos(\theta) (P_\delta \otimes T_\delta
+ P_{\delta^{-1}} \otimes T_{\delta^{-1}}) + \sin(\theta) R_{\frac{\pi}{2}}
\otimes Id\bigr)\end{aligned}$$ where $U$ is a $2\times 2$ unitary matrix, and $R_{\frac{\pi}{2}} = \left(
\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$. Equivalent solutions with a two dimensional local Hilbert space were presented first in [@meyer]. In conclusion, we also note that solution remains valid when adding relations between generators. Thus such solutions exist for all groups, in particular for free products of cyclic groups, $$\Gamma=\langle \delta_1, \cdots , \delta_l | \delta_1^{q_1} = \cdots
\delta_l^{q_l} = e \rangle$$ and for free Abelian groups, $$\Gamma=\langle \delta_1, \cdots ,\delta_l | \delta_i \, \delta_j \,
\delta_i^{-1} \delta_j^{-1} = e \;\forall i,j \in \{1,\cdots,l\}\rangle$$ which we consider in the next section.
Cayley graphs of free Abelian groups
------------------------------------
One should note that the commutation relations between elements from the set of generators and their inverses, for instance $$\begin{aligned}
\delta_1 \delta_2^{-1}=\delta_2^{-1}\delta_1\end{aligned}$$ do not necessarily imply the existence of a closed path on the graph with alternate orientation of the edges , except in the case when the inverses of the elements of $\Delta$ are themselves in $\Delta$. The group is defined by: $$\label{sag1}
\Gamma =\langle \delta_1, \dots ,\delta_n |
\delta_i\delta_j\delta_i^{-1}\delta_j^{-1}=e \,\forall i,j \in \{1, \dots ,n\}
\rangle$$ and the set used to construct the Cayley graph is $$\label{sag2}
\Delta=\{ \delta_1, \dots ,\delta_n,\delta_1^{-1}, \dots ,\delta_n^{-1} \}$$ In such a case, equations - read $$\begin{aligned}
\label{cocfc1}
M_{\delta_i}^\dagger M_{\delta_j} + M_{\delta_j^{-1}}^\dagger M_{\delta_i^{-1}}
&=& 0 \hbox{ for all } \delta_i \ne \delta_j\\
\label{cocfc2}
M_{\delta_i} M_{\delta_j}^\dagger +M_{\delta_j^{-1}} M_{\delta_i^{-1}}^\dagger
&=& 0 \hbox{ for all } \delta_i \ne \delta_j\\
\label{cocfc3}
\sum_{\delta\in \Delta} M_{\delta} M_{\delta}^\dagger =
\sum_{\delta\in \Delta} M_{\delta}^\dagger M_{\delta} &=& \mathbbm 1\end{aligned}$$ When $\delta_j= \delta_i^{-1}$, equations - contain a single term and read $$\begin{aligned}
\label{cocfc4}
M_{\delta_i}^\dagger M_{\delta_i^{-1}} = M_{\delta_i^{-1}} M_{\delta_i}^\dagger &=& 0\end{aligned}$$ These are much less restrictive conditions than -, and we lack here the decomposition of $\mathcal H_I$ into orthogonal subspaces which allowed us to give a general answer in the case of free groups. We only notice that equations - are equivalent to the following $$\begin{aligned}
\bigl(\sum_{\delta\in A} \lambda_\delta M_\delta^\dagger) \bigr)
\bigl(\sum_{\delta\in A} \lambda_\delta M_\delta^{-1}) \bigr)=
\bigl(\sum_{\delta\in A} \lambda_\delta M_\delta^{-1}) \bigr)
\bigl(\sum_{\delta\in A} \lambda_\delta M_\delta^\dagger) \bigr) = 0\end{aligned}$$ for all subset $A\in \Delta$ such that $\delta\in A \Rightarrow
\delta^{-1}\not\in A$ and for all families of real parameters $\{\lambda_\delta\}_{\delta\in A}$.\
Equations - imply the following proposition which will help us classify the solutions.
Let $C_{\Delta}(\Gamma)$ be the Cayley graph of the free abelian group with $n$ generators (-) . If a quantum walk operator defined on $G$ is unitary then the image subspaces of any two internal operators $M_{\delta_i}$ and $M_{\delta_j}$ are either orthogonal or contain a common vector subspace. The same implication is valid for the image subspace of their conjugates $M_{\delta_i}^\dagger$ and $M_{\delta_j}^\dagger$: $$\begin{aligned}
\label{ortho}
\bigl(\mathcal Im(M_{\delta_i}) \cap \mathcal Im(M_{\delta_j})
=\{0\}\bigr)\Rightarrow
\mathcal Im(M_{\delta_i}) \bot \,\mathcal Im(M_{\delta_j})\\
\label{orthodag}
\bigl(\mathcal Im(M_{\delta_i}^\dagger) \cap \mathcal Im(M_{\delta_j}^\dagger)
=\{0\}\bigr)\Rightarrow
\mathcal Im(M_{\delta_i}^\dagger) \bot \,\mathcal Im(M_{\delta_j}^\dagger)\end{aligned}$$
[**Proof:**]{} Using for a pair $\delta_i$, $\delta_j^{-1}$, one has $$\begin{aligned}
\label{intersection}
\mathcal Im(M_{\delta_i} M_{\delta_j^{-1}}^\dagger) =
\mathcal Im(M_{\delta_j} M_{\delta_i^{-1}}^\dagger)\end{aligned}$$ and thus $$\begin{aligned}
\mathcal Im(M_{\delta_i} M_{\delta_j^{-1}}^\dagger)
\subset
\bigl(\mathcal Im(M_{\delta_i}) \cap \mathcal Im(M_{\delta_j})\bigr)\end{aligned}$$ Suppose now that $\mathcal Im(M_{\delta_i})$ and $ \mathcal Im(M_{\delta_j})$ have no common vector subspace. Thus $M_{\delta_i} M_{\delta_j^{-1}}^\dagger = 0$, which can be written $$\begin{aligned}
\mathcal Im(M_{\delta_i}^\dagger) \bot
\mathcal Im(M_{\delta_j^{-1}}^\dagger)\end{aligned}$$ and more particularly $$\begin{aligned}
\mathcal Im(M_{\delta_i}^\dagger M_{\delta_j}) \bot
\mathcal Im(M_{\delta_j^{-1}}^\dagger M_{\delta_i^{-1}})\end{aligned}$$ Since the two subspaces are equal (by ) and orthogonal, they are equal to the null vector space and hence we have again $M_{\delta_i}^\dagger M_{\delta_j} =0$, and finally $$\begin{aligned}
\mathcal Im(M_{\delta_i}) \bot \mathcal Im(M_{\delta_j})\end{aligned}$$ The implication is thus proven. The proof of is equivalent, beginning with equation instead of . $\square$\
We can use Proposition (1) to find solutions with an internal space dimension smaller than the number of generators in the following way. First we can write $$\begin{aligned}
\label{dimcc}
{\rm dim}(\mathcal H_I) \ge \sup_{\delta_i,\delta_j}\bigl\{
\sum_{\epsilon_1,\epsilon_2=\pm 1}
{\rm dim}\bigl(\mathcal Im(M_{\delta_i^{\epsilon_1}}) \cap
\mathcal Im(M_{\delta_j^{\epsilon_2}}) \bigr)\bigr\}\end{aligned}$$ where the $\sup$ runs over all pairs $\delta_i, \delta_j$ such that both $\delta_i \ne\delta_j$ and $\delta_i \ne\delta_j^{-1}$. This inequality is true since the four sets appearing in the right hand side are pairwise orthogonal by . A similar equation could be written involving the $M^\dagger$’s. Suppose now that the supremum on the right hand side of is zero, hence giving no direct condition on the dimension of $dim(\mathcal H_I)$. In such a case, all vector subspaces are orthogonal by , which imply $dim(\mathcal H_I) \ge \vert \Delta\vert $. Hence, a necessary condition for the existence of quantum walks with a smaller internal space is that some of the intersections in the sum are non empty. In the following we give some examples.
### A two dimensional walk with a two dimensional internal space.
We consider here the group $\Gamma=\langle \delta_1,\delta_2 | \delta_1
\delta_2 \delta_1^{-1} \delta_2^{-1} = e \rangle$, a symmetric set $\Delta=\{\delta_1,\delta_1^{-1}, \delta_2, \delta_2^{-1}\}$ and define a quantum walk over the associated Cayley graph through the evolution operator which reads here $$\begin{aligned}
W=M_{\delta_1} \otimes T_{\delta_1}+ M_{\delta_1^{-1}} \otimes
T_{\delta_1^{-1}}+M_{\delta_2} \otimes T_{\delta_2} + M_{\delta_2^{-1}} \otimes T_{\delta_2^{-1}}\end{aligned}$$ We suppose that the rank of each matrix $M_{\delta_i}$ is one. In order to impose $dim(\mathcal H_I) =2$, we require that at least two terms in the right hand side of are zero for each possible pair of generators $\delta_i,\delta_j$. We obtain two solutions which transform one derived from the other by changing $\delta_1$ and $\delta_1^{-1}$. Up to an unitary transformation, the solution is $$\begin{aligned}
\fl M_{\delta_1}=UP_1VP_1 \quad M_{\delta_1^{-1}}=UP_2VP_2 \quad M_{\delta_2}=UP_1VP_2 \quad
M_{\delta_2^{-1}} =UP_2VP_1\end{aligned}$$ where $U$ and $V$ are two unitary matrices and $P_1$,$P_2$ two orthogonal projectors. The evolution operator factorises into a product of two one-dimensional operators $$\begin{aligned}
\fl W= ( U\otimes 1) ( P_1 \otimes (T_{\delta_1}T_{\delta_2})^{\frac{1}{2}} +
P_2 \otimes (T_{\delta_1^{-1}}T_{\delta_2^{-1}})^{\frac{1}{2}})\\
(V\otimes 1) (P_1 \otimes (T_{\delta_1}T_{\delta_2^{-1}})^{\frac{1}{2}}
+P_2 \otimes (T_{\delta_1^{-1}}T_{\delta_2})^{\frac{1}{2}})\end{aligned}$$ However a quantum walk with a two dimensional internal space which is symmetric by inversion of only one of the axes or by a rotation of angle $\frac{\pi}{2}$ does not exist.\
This solution generalises in higher dimensions:\
Let $C_{\Delta}(\Gamma)$ be the Cayley graph of the free abelian group with $n$ generators and symmetric presentation and . Then there exists a unitary quantum walk operator on $G$ such that the dimension of the internal space is $n$ if $n$ is even and $n+1$ if $n$ is odd.
[**Proof:**]{} Suppose $n$ even. We consider an internal space of dimension $n$ and decompose it as a direct sum of two dimensional subspaces. We associate to each of these subspaces one different pair of generators. For such a pair $(\delta_i,\delta_j)$, the four operators $M_{\delta_i},M_{\delta_j},M_{\delta_i^{-1}},M_{\delta_j^{-1}}$ act non trivially only on the associated two-dimensional subspace and can be constructed in the same way as the internal operators of the previous example of two dimensional walk. The dimension of the internal space for such a quantum walk is then half the dimension of the free form solution. Suppose now $n$ odd. We can repeat the previous construction for $n-1$ generators, and add a two dimensional space where the internal operators associated to the last generator will have the form of the internal operators of a one dimensional walk. All the internal operators then verify the condition equations -. $\square$\
### Two dimensional walks with a four dimensional internal space.
The impossibility of having a fully symmetric quantum walk does not hold when taking a four dimensional internal space. One possibility is to suppose that all the intersections involved in are of dimension zero, in this case $dim(\mathcal H_I) \ge \vert \Delta \vert = 4$ and the minimal choice of the dimension leads to an evolution operator $W=\sum_\delta P_\delta U
\otimes T_\delta$ where U is a four dimensional unitary matrix. The other possibility is to suppose that all the intersections involved in are of dimension one. In this case the minimal dimension of the internal space is also four. A simple choice of matrices of rank two verifying all the conditions - is: $$\begin{aligned}
M_{\delta_1} &=& \frac{1}{\sqrt{2} }(\,|u_1\rangle \langle v_1 | + |u_2\rangle \langle v_3 |\,)\\
M_{\delta_1^{-1}} &=& \frac{1}{\sqrt{2} }(\,- |u_3\rangle \langle v_4 | + |u_4\rangle \langle v_2 |\,)\\
M_{\delta_2} &=& \frac{1}{\sqrt{2} }(\, |u_1\rangle \langle v_2 | + |u_3\rangle \langle v_3 |\,) \\
M_{\delta_2^{-1}} &=& \frac{1}{\sqrt{2} }(\,- |u_4\rangle \langle v_1 | + |u_2\rangle \langle v_4 |\,)\end{aligned}$$ where $\{ |u_i\rangle\}_{i=1,\cdot,4}$ and $\{|v_i\rangle\}_{i=1,\cdot,4}$ are two orthonormal bases of $\mathcal H_I$. In the following we give the explicit form of the evolution operator supposing that the rank of the matrices $M_\delta$ is one and that the walk is symmetric. A permutation of the vertex set $\Pi$ is associated with a spatial transformation. As in the one dimensional case, the walk is symmetric under this transformation if there exists an unitary $S$ such that $(S \otimes
\Pi)^{\dagger} W (S \otimes \Pi) = W$. In other words, if the initial condition is modified by the transformation $S \otimes \Pi$, the wave function at any time can be deduced from the unmodified wave function by application of the same transformation. We impose invariance under the symmetries of the square lattice by considering the two transformations, $S_i \otimes \Pi_i$ and $S_r \otimes \Pi_r$, being respectively the representation of the inversion along the $x$ axis and the rotation by ${\frac{\pi}{2}}$. The symmetry condition makes $U$ reduce to a product $U=D^{-1} U_0 D$ where $D$ is a diagonal unitary matrix depending on four real parameters and $U_0$ takes the form: $$U_0 = \left( \begin{array}{cccc} a & b & c & c \\ b & a & c & c \\ c &
c & a & b \\ c & c & b & a
\end{array} \right)$$ The matrix $U_0$ depends on $3$ parameters by the unitarity condition. The matrices $S_1$ and $S_2$ depend on the same parameters as the matrix $D$. Then choosing these four parameters equal to one reduces the walk operator to $W=\sum_i P_i U_0 \otimes Ti$ and the matrices $S_1$ and $S_2$ are just the inverse permutation of the generators associated to the spatial transformation.
### A three dimensional walk with a four dimensional internal space.
It has been shown that no nontrivial solution exists in three dimensions with a two dimensional internal space[@Bialynicki]. In the following we give solutions on $\mathbb Z^3$ with a four dimensional internal space. The starting point is again equation . Taking matrices of rank two would not break this condition for $dim(\mathcal H_I)$ provided that each term on the left hand side of is one. Here we thus give the general solution for rank two matrices. Let $\Delta=\{\delta_1,\delta_2,\delta_3,\delta_1^{-1},\delta_2^{-1},\delta_3^{-1}\}$. Defines two orthonormal bases $\{ |u_i\rangle\}_{i=1,\cdot,4}$ and $\{ |v_i\rangle\}_{i=1,\cdot,4}$. Now construct six matrices of rank 2 indexed by the elements of $\Delta$ in the form $$\begin{aligned}
M_{\delta_1} &=& \alpha_1 |u_1\rangle \langle v_2 | + \beta_1 |u_2\rangle \langle v_1 |\\
M_{\delta_1^{-1}} &=& \gamma_1 |u_3\rangle \langle v_4 | + \delta_1 |u_4\rangle \langle v_3 |\\
M_{\delta_2} &=& \alpha_2 |u_1\rangle \langle v_3 | + \gamma_2 |u_3\rangle \langle v_1 | \\
M_{\delta_2^{-1}} &=& \beta_2 |u_2\rangle \langle v_4 | + \delta_2 |u_4\rangle \langle v_2 |\\
M_{\delta_3} &=& \alpha_3 |u_1\rangle \langle v_4 | + \delta_3 |u_4\rangle \langle v_1 |\\
M_{\delta_3^{-1}} &=& \beta_3 |u_2\rangle \langle v_3 | + \gamma_3 |u_3\rangle \langle v_2 |\end{aligned}$$ It is clear that such a choice solves equations . The other equations are solved by taking $$\begin{aligned}
\alpha_2 =\lambda \alpha_1\qquad; \qquad \alpha_3= \mu \alpha_1\\
\beta_2 =\bar\lambda\nu \beta_1\qquad; \qquad \beta_3= -\bar\mu \nu \beta_1\\
\gamma_2 =-\lambda\bar\nu \gamma_1\qquad; \qquad \gamma_3= -\bar\mu \gamma_1\\
\delta_2 =-\bar\lambda \delta_1\qquad; \qquad \delta_3= \mu\bar\nu \delta_1\end{aligned}$$
where $|\nu|^2=1$, $\lambda, \mu \in \mathbb C$ and $$\begin{aligned}
|\alpha_1|=|\beta_1|=|\gamma_1|=|\delta_1|=\frac{1}{\sqrt{1+|\lambda|^2+|\mu|^2}}\end{aligned}$$
Cayley graphs with multiply connected second neighbours
-------------------------------------------------------
In this section we consider Cayley graphs in which any second neighbour is connected by at least two alternating paths. They might be of interest since the condition equations contain at least two terms. Here, we only consider two examples in which each second neighbour is connected by at least two alternate paths. Both are interesting in their own right: the first one admits a scalar solution, while the other admits solutions in terms of a Clifford algebra.
### A simple one dimensional example
Let us consider the commutative group with two generators with one more relation $\delta_1^2 = \delta_2^2$ in the presentation and as defining set for the Cayley graph $\Delta= \{ \delta_1,
\delta_2,\delta_1^{-1},\delta_2^{-1} \}$.\
The four matrices $M_\delta$ have to be taken as solutions of the four equations: $$\begin{aligned}
&&M_{\delta_1}^\dagger M_{\delta_1^{-1}}
+ M_{\delta_2}^\dagger M_{\delta_2^{-1}} =
M_{\delta_1^{-1}} M_{\delta_1}^\dagger +
M_{\delta_2^{-1}} M_{\delta_2}^\dagger = 0\\
&& M_{\delta_1}^\dagger M_{\delta_2^{-1}}
+ M_{\delta_2}^\dagger M_{\delta_1^{-1}} =
M_{\delta_2^{-1}} M_{\delta_1}^\dagger
+ M_{\delta_1^{-1}} M_{\delta_2}^\dagger = 0\\
&&M_{\delta_1}^\dagger M_{\delta_2}
+ M_{\delta_2}^\dagger M_{\delta_1}
+ M_{\delta_1^{-1}}^\dagger M_{\delta_2^{-1}}
+ M_{\delta_2^{-1}}^\dagger M_{\delta_1^{-1}} =0\\
&&M_{\delta_2} M_{\delta_1}^\dagger
+ M_{\delta_1} M_{\delta_2}^\dagger
+ M_{\delta_2^{-1}} M_{\delta_1^{-1}}^\dagger
+ M_{\delta_1^{-1}} M_{\delta_2^{-1}}^\dagger =0\\
&&\sum_\delta M_\delta^\dagger M_\delta =
\sum_\delta M_\delta M_\delta^\dagger=\mathbbm 1\end{aligned}$$ This set of equations admits solutions with a one dimensional internal space, and the evolution operator can be written as $$\begin{aligned}
W= \frac{1}{2} \bigl(e^{i\theta} (\tau_1\pm\tau_2) +
e^{i\varphi} (\tau_1^{-1}\mp\tau_2^{-1}) \bigr)\end{aligned}$$ where $\tau_1$ and $\tau_2$ are the displacements in the directions $\delta_1$ and $\delta_2$. However, as can be seen from the form of the evolution operator, this example is equivalent to a quantum walk on $\mathbb Z$ with a two dimensional internal space by grouping together pairs of second neighbours. What is interesting here is that even on a graph where all sites are equivalent, there may exist scalar solutions provided all second neighbours are multiply connected. The minimal dimension of the internal space would still however have to be questioned since it strongly depends on the choice of the graph and various descriptions appear to be equivalent.
### The hypercube
We consider the group presentation $$\label{hyp}
\Gamma=\langle \delta_1, \cdots \delta_n |\delta_i^2 = e \,\forall i;\,
\delta_i\delta_j\delta_i^{-1}\delta_j^{-1} = e \,\forall i\ne j\rangle$$ whose Cayley graph is the hypercube in $n$ dimensions. The condition equations become: $$\begin{aligned}
M_{\delta_i}^\dagger M_{\delta_j}+M_{\delta_j}^\dagger M_{\delta_i}=0 \label{hyp1}\\
M_{\delta_i} M_{\delta_j}^\dagger+M_{\delta_j} M_{\delta_i}^\dagger=0\label{hyp2}\\
\sum_\delta M_\delta^\dagger M_\delta =\mathbbm{1}\end{aligned}$$ Equations and are valid for all pairs of generators $\delta_i,\delta_j$. Solutions originating from those for a free group of n generators have been studied by various authors ([@kempehyperc]-[@Moore]).
There exists a unitary quantum walk operator on the Cayley graph of the group such that the internal operators are of the form $M_{\delta_i}= \frac{1}{\sqrt n} \sigma_i U$ where $U$ is a unitary matrix of dimension $dim(\mathcal H_I)$ and $\{\sigma_1 \dots \sigma_n\}$ is a set of anticommuting matrices.
[**Proof:**]{} If one requires that all the matrices $M_\delta$ be Hermitian (or anti-hermitian) then the first set of equations - takes the form of an anticommutation relation between all pairs of matrices. Hermitian anticommuting matrices generate a Clifford algebra, it is therefore natural to find solutions among their matrix representations. Let $\{\sigma_1 \dots \sigma_n\}$ such a set of anticommuting matrices and $U$ an unitary matrix. A possible choice for the matrices $M_\delta$ is then $M_{\delta_i}= \frac{1}{\sqrt n} \sigma_i U$. $\square$\
For example, equations for $n=3$ are solved by $M_i=\frac{1}{\sqrt 3} \sigma_i U$ where each $\sigma_i$ is one of the three Pauli matrices and U a unitary matrix in two dimensions. While the dimension of the matrix representation is rather large, (at least $2^{\lbrack\frac{n}{2}\rbrack}$), such solution may nevertheless be useful.
A generalised model of quantum walk
===================================
A quantum walk is a model for the motion of a quantum particle jumping (quantically) over a graph. A particle having a fixed number of internal degrees of freedom, one is naturally led to attach to each point $x$ of the graph a copy of some Hilbert space $\mathcal H_I$ describing them. This is obviously not a necessary hypothesis in the context of a network of quantum processors, and even if we will retain here most of the terminology of quantum walks, we will not base our approach in this section on the interpretation of our quantum object as a physical particle. A second important property is the choice of a discrete time evolution, again motivated by the idea that quantum processors as their classical equivalents would exchange information at discrete times.\
We will continue to consider discrete time evolution but we want to note that quantum walks with continuous time has also been introduced in the context of quantum algorithmics [@FG98] [@CFG01]. As for the discrete time model, the succes of these walks performing particular tasks is dependent on characteristics such as the initial vector state [@BM04], thus indicating that a classification of this model may also be of some interest. Some properties of one dimensional walks have been determined as for example the revival time [@BM05] and a limit theorem demostrated by [@Kon05].\
We consider an oriented graph $G=(X,E)$, $X$ the set of vertices, and $E$ the set of oriented edges. To each vertex $x\in X$, we attach a (finite) Hilbert space $\mathcal H_x$, and define the quantum evolution over $\mathcal H = \oplus_{x \in X}\mathcal H_x$ as follows: For each oriented pair $(x,y)$, we define a linear map $M_{x,y}$ from $\mathcal H_x$ to $\mathcal H_y$, extend it on $\mathcal H$ by setting $M_{x,y}=0$ on $\mathcal H_x^\bot$. We define its conjugate $M_{x,y}^\dagger$ as the map such that $$\begin{aligned}
\langle \Psi' | M_{x,y}\Psi \rangle = \langle M_{x,y}^\dagger \Psi'|\Psi \rangle\end{aligned}$$ for all $|\Psi \rangle$, $|\Psi' \rangle$ in $\mathcal H$. Then we define the evolution of the quantum walk over $\mathcal H$ as: $$\begin{aligned}
|\Psi(t+1)\rangle = W |\Psi(t)\rangle\end{aligned}$$ where $|\Psi(t)\rangle$ is the state of the system at time $t$ and $W$ is the unitary operator $$\begin{aligned}
W =\sum_{(x, y)\in E} M_{x,y}\end{aligned}$$ In order to restrict the sum to the pairs of neighbouring sites and impose $W$ to be unitary we require the following three properties: $$\begin{aligned}
\label{cof1}
M_{x,y} \ne 0 \hbox{ if and only if } (x,y)\in E\end{aligned}$$ $$\begin{aligned}
\label{cof2}
\sum_y M_{x,y}^\dagger M_{z,y} = \sum_y M_{y,x} M_{y,z} ^\dagger = 0 \hbox{ for all } x\ne z\end{aligned}$$ $$\begin{aligned}
\label{cof3}
\sum_y M_{x,y}^\dagger M_{x,y} = \sum_y M_{y,x} M_{y,x} ^\dagger = {\bf 1}_x\end{aligned}$$ where ${\bf 1}_x$ is the projector over $\mathcal H_x$. Conditions and are necessary and sufficient conditions for $W$ to be unitary. Here, it is already interesting to note that even in this more general context quantum “coin” solutions exist provided that on each site the number of incoming edges equals the number of outgoing ones. The construction can be done in the following way: we first set the dimension of all local Hilbert spaces equal to the number of incoming (or equivalently outgoing ) neighbours, $$\begin{aligned}
dim(\mathcal H_x) = |E_x^{in}| = |E_x^{out}|\end{aligned}$$ where we have set $$\begin{aligned}
E_x^{in} = \{y\in X : (y,x)\in E\}\\
E_x^{out} = \{y\in X : (x,y)\in E\}\end{aligned}$$ For all $x\in X$ we fix two orthonormal basis $\mathcal B_x^{in}$ and $\mathcal B_x^{out}$ in $\mathcal H_x$ an label its elements using the list of neighbours, $$\begin{aligned}
\mathcal B_x^{in} =\bigl\{ |\varphi_x^{in}(y)\rangle\bigr\}_{y\in E_x^{in}}\end{aligned}$$ $$\begin{aligned}
\mathcal B_x^{out} =\bigl\{ |\varphi_x^{out}(y)\rangle\bigr\}_{y\in E_x^{out}}\end{aligned}$$ Now setting $$\begin{aligned}
M_{x,y} = |\varphi_y^{in}(x)\rangle \langle \varphi_x^{out}(y)|\end{aligned}$$ just satisfies all conditions , and defines a general quantum “coin” solution even outside the context of a quantum particle on a lattice. In fact we get some more insight on how such solutions works from the point of view of a quantum network: first, each node splits the (partial) wave function along the vectors of a fixed basis $\mathcal B_x^{out}$ and send the resulting complex number to each of its neighbours; then a (partial) wave function is recomposed using the received numbers and the other fixed basis $\mathcal B_x^{in}$. We now want to recover previous definition of quantum walks on a Cayley graph, so we naturally suppose that the properties of the graph are transferred to the walk. In particular all local Hilbert spaces are copies of the same space, $$\begin{aligned}
\mathcal H_x = \mathcal H_0\end{aligned}$$ for all $x$ in $X$ and the complete Hilbert space is equivalent to the direct product of the local space $\mathcal H_0$ with a position space $\mathcal
H_X$. $$\begin{aligned}
\mathcal H \approx \mathcal H_0 \otimes \mathcal H_X\end{aligned}$$ Furthermore, the maps $M_{x,y}$ will depend only on the edge colour and direction of the edge $(x,y)$ (i.e. only on the generator $\delta= x^{-1} y$) and not in the starting vertex $x$: $$\begin{aligned}
M_{x,y}= T_{0,y} M_{x^{-1} y} T_{x,0}\hbox{ for all } (x,y) \in E\end{aligned}$$ where $M_{x^{-1} y}$ is a map on $\mathcal H_0$ and $T_{x,y}$ is the canonical shift map sending $\mathcal H_x$ onto $\mathcal H_y$. Thus the evolution operator $W$ on $\mathcal H$ as a product space reads $$\begin{aligned}
W = \sum_{\delta\in\Delta} M_\delta \otimes T_\delta\end{aligned}$$
Conclusion
==========
We have considered quantum walks on Cayley graphs of groups and addressed the problem of classifying them as a function of the group presentation and the choice of the internal space. A first result is that the smallest possible dimension of the internal space depends strongly on the generating set chosen for constructing the Cayley graph. In the case of free groups, we succeeded in classifying all possible solutions. Standard quantum walk definition is recovered and correspond to an internal space of dimension equal to the number of neighbours (its smallest value) and a free group with a set of generators containing elements of the group different from the identity. When the identity element is present in the generating set used to define the Cayley graph of a free grop, or on other Cayley graphs, we showed that different solutions do exist for which we give a partial characterisation. We presented a few examples of solutions which does not enter in the previously known solutions and which become available as soon as there exist closed paths of length 4 on the graph, with alternating orientation. In particular, we found solutions with a smaller internal dimension that what is usually expected and a new kind of quantum walks on the hypercube based on Clifford algebra representation. We hope that these new possibilities will prove useful in the context of the relationship between quantum walks and quantum algorithms.
Acknowledgements
================
We are grateful to Z. Nagy and F. Millet for helping us in finding Clifford solutions on the hypercube and to J. Avan, J.-P. Kownacki, M. Schürmann and N. Weatherall for useful discussions.\
Research partially supported by European Commission HPRN-CT-2002-00279, RTN QP Applications.
References {#references .unnumbered}
==========
[12]{} Ambainis A 2003 [*Int. J. Quantum Information*]{} [**1**]{} 507 Aharonov D, Ambainis A, Kempe J and Vazirani U 2001 [ *Proc. STOC*]{}\
(Aharonov D, Ambainis A, Kempe J and Vazirani U 2000 [*Preprint*]{} quant-ph/0012090) Shenvi N, Kempe J and BirgittaWhaley K 2003 [ *Phys. Rev. A*]{} [**67**]{} 052307 Ambainis A, Kempe J and Rivosh A 2005 [*Proc. SODA*]{}\
(Ambainis A, Kempe J and Rivosh 2004 [*Preprint*]{} quant-ph/0402107) Brun T A, Carteret H A and Ambainis A 2003 [ *Phys. Rev. A*]{} [**67**]{} 052317 Wójcik D K and Dorfman J R 2003 [*Phys. Rev. Let*]{} [**90**]{} 230602 Wójcik D K and Dorfman J R 2004 [*Physica D*]{} [**187**]{} 223 Meyer D A 1996 [*Phys. Lett. A*]{} [**223**]{} 5-345 Barenco A, Bennett C H, Cleve R, Divicenzo D P, Margolus N, Shor P, Sleator T, Smolin J A, Weinfurter H 1995 [*Phys. Rev. A.*]{} [ **52**]{} 3457 White A T [*Graphs of groups on surfaces*]{} (North-Holland) Meyer D A 1997 [*Phys. Rev. E*]{} [**55**]{} 5-5261 Bialynicki-Birula I 1994 [*Phys. Rev. D*]{} [**49**]{} 12-6920 Kempe J 2003 [*Proc. RANDOM*]{}\
(Kempe J 2002 [*Preprint*]{} quant-ph/0205083) Moore C and Russell A 2002 [*Proc. RANDOM*]{}\
(Moore C and Russell A 2001 [*Preprint*]{} quant-ph/0104137) Farhi E and Gutmann S 1998 [*Phys. Rev. E*]{} [**58**]{} 915 Childs A M, Farhi E and Gutmann S 2002 [ *Quant. Inf. Process*]{} [**1**]{} 35 Mülken O and Blumen A 2005 [*Phys. Rev. E*]{} [**71**]{} 016101 Mülken O and Blumen A 2005 [*Phys. Rev. E*]{} [**71**]{} 036128 Konno N 2005 [*Phys. Rev. E*]{} [**72**]{} 026113
|
---
abstract: 'are widely used in all sorts of applications ranging from industrial automation to search-and-rescue. So far, in these applications they work either isolated with a high mobility or operate in a static networks setup. If mobile work cooperatively, it is in applications with relaxed real-time requirements. To enable such cooperation also in hard real-time applications we present a scheduling approach that is able to adapt real-time schedules to the changes that happen in mobile networks. We present a -model and a heuristic to generate schedules for those networks. One of the key challenges is that running applications must not be interrupted while the schedule is adapted. Therefore, the scheduling has to consider the delay and jitter boundaries, given by the application, while generating the adapted schedule.'
author:
-
bibliography:
- 'ref.bib'
title: 'Adaptive Real-Time Scheduling for Cooperative '
---
Scheduling algorithms, Cyber-Physical Systems, Real-time systems, Wireless networks
Introduction {#sec:intro}
============
In traditional real-time systems a schedule was calculated once and used until the system got another taskset. To switch the taskset of such systems they were stopped completely and started with the new schedule. However, stopping the whole application is not an option in systems that have to perform their tasks continuously. An example for such systems are mobile robots that cooperate to carry a work piece. If a third robots needs to join the group of the carrying robots to weld a second piece to the carried one, the schedules need to be changed without stopping the application on the carrying robots. Stopping the application might lead to dropping the work peace as it is now longer kept in balance by a feedback loop closed between the robots. We call these groups of robots , as they perform a common task cooperatively. To overcome this challenge, mechanisms are needed to generate schedules that introduce the changes, needed to adapt to the new situation, without harming the real-time constraints of running tasks. In this paper we introduce two different mechanisms to overcome these challenges. The first one is a -model that is able the schedule such and also to merge their schedules. The second one is a heuristic algorithm with the same capabilities to be used on embedded devices, as it is less computational complex. This algorithm is based on a hypothesis we introduce and test prior to the discussion of the algorithm. This paper is structured as follows, in we introduce the problem in more detail and discuss our assumptions. In we present the constraints a scheduling algorithm need to fulfill to solve the priorly stated problem. discusses related work from different areas that handles scheduling of problems similar to ours. In we introduce our -model to schedule mobile and to adapt schedules to changes. Afterward we evaluate the computational complexity in . As solving -models is a rather complex task, we introduce and test a hypothesis on which schedules are better adaptable than others in . To calculate schedules on embedded devices in a more predictable time we discuss a heuristic algorithm in . This heuristic algorithm is evaluated in . concludes the paper and its contributions.
Problem Statement {#sec:AllocSchedProblem}
=================
For our scheduling we assume that a distributed real-time system is a wireless network consisting of several nodes. Each node is capable of executing different tasks but only one of them per time-slot. Nodes can only communicate in a half-duplex manner on one channel in one time-slot but may switch channels between two consecutive time-slots. The whole network on the other hand is able to utilize multiple channels at the same time to transfer data in disjoint sets of nodes. The problem of scheduling data transfers over multiple channels can be mapped to the problem of scheduling computation tasks to multiple processors. @coffman1972 [@coffman1972] and @du1989 [@du1989] showed that multiprocessor scheduling of non-preemptive task is NP-Hard. All nodes are able to use the same set of interference free channels. Each channel has the same characteristics for all nodes in the . Further, we assume that all channels have the same characteristics. We also assume all nodes participating in a to be in each others transmission range, thus all communication is single-hop.
Several tasks from a job, the tasks of a job might be executed on several nodes and have dependencies between each other. Each job has a period which is inherited to the tasks. All jobs of a form the set $\omega$. Jobs might share common tasks, these tasks have the shortest period of all jobs they are participating in. Besides the period ($P_i$), each task $T_{i}$ has a maximum jitter ($J_i$) that describes how many time-slots a task might move between two consecutive executions. Additionally, each task $T_{i}$ has a maximum age ($d_{ij}$) that describes how many time-slots the task might be executed before a task ($T_j$) depending on it. The tasks $T_i$ depends on are called its dependencies and are stored in $\Gamma_i$. The matrix $D$, with dimensions $[|\tau| \times |\tau|]$, stores how many time-slots a task $T_i$ is allowed to be scheduled before task $T_j$. All tasks in a form the set $\tau$. The execution time of each task is assumed to be at most one time-slot.
As we assume that the job of a system fails if one task is not executed in time we do not implement priorities in our scheduling.
All tasks of all jobs in a system form a directed graph without circles. This graph might have several entry tasks and leaf tasks. Each job in this graph forms a so-called path that might have several entry tasks but only one leaf task. Entry tasks are tasks without dependencies, they form the set $E$. Leaf tasks are tasks without dependent tasks, all leaf tasks form the set $L$.
The number of time-slots in which the schedule is not repeated is called Hyperperiod($H$). It is defined as the least common multiple of all periods of jobs in $\omega$.
We differentiate between time-slot and slot, a time-slot is a certain portion of time that has a defined start time and end time, all time-slots are of the same length. A time-slot can inhabit multiple slots, as a slot is a time-slot on a certain network resource. Therefore, a time-slot consists of as many slots as interference free network resources are available. An example for interference free network resources are the channels a communication standard defines. In a system with only one communication standard that defines three channels, each time-slot would consist of three slots.
Two tasks are called intersecting if they have common communication partners. This is the case if they have a direct dependency to each other, they are depending on the same task or they have the same depending task. Intersections between tasks are stored in the matrix $I$ with the dimensions $[|\tau| \times |\tau|]$.
depicts an example dependency graph that consists of two jobs. The first job has the entry task with id 5 and the leaf task with id $0$, called job$0$. It consists of the tasks: 5, 4, 3, 2 and $0$. The second job has the entry task with id 5 and the leaf task with id 1, called job1. Consisting of the tasks: 5, 4 and 1. Both of them share the common tasks 5 and 4. The graph is executed on a network with five nodes, the color of each task shows on which node it needs to be executed on. The arrows between tasks describe the dependencies, where the task the arrow is pointing to is depending on the task the arrow is pointing from, e.g. task$0$ is depending on task5.
![Example dependency graph with two jobs consisting in total of six task executed on a network with five nodes. \[fig:AllocSchedGraph\] ](figures/depGraph_explained.pdf){width=".7\linewidth"}
Scheduling Constraints and Objectives {#sec:sched-constr-object}
=====================================
This section discusses the constraints the scheduling has to fulfill in order to generate valid schedules.
\[textcon:1\] It is not allowed that two or more tasks share the same slot.
Sharing slots would lead to transmission interference and loss of data, therefore each task needs its own slot.
\[textcon:2\] Tasks with a common participating node must not be executed in the same time-slot
prohibits the scheduler from scheduling two tasks at the same time that have the same node either receiving or transmitting data. In our example from there are different tasks that can not be scheduled at the same time: task1 and task2 as they are executed at the same node, task2 and task3 as both transmit data that is needed by task$0$. By prohibiting these tasks to be scheduled at the same time, this constraint ensures that the communication can be handled by half-duplex transceiver.
\[textcon:3\] All dependencies of a task must be scheduled before the depending task.
Referring to our example in this constraint ensures that, e.g., task5 is scheduled before task$0$, task3 and task4. To be able to schedule task$0$ it is necessary that task5, task3 and task2 are scheduled before task$0$.
\[textcon:4\] Each dependency of a task must be scheduled no more than its maximal age before the task.
As most data has an age at which it becomes less usable to the requesting task, the providing task or dependency must be scheduled less than this age before the requesting task. E.g., if the maximal age of data provided by task3 is ten time-slots, task3 must not be scheduled more than ten slots before task$0$.
\[textcon:5\] All depending tasks in one job must use the same execution of a common dependency.
To ensure that tasks in one job use the same state of the system, it is necessary that tasks in one job that depend on the same task are scheduled after the same execution of that task. In our example: the job with entry task task5 and leaf task task$0$ is formed by the task ids 5, 4, 3, 2 and $0$. defines that task5 must not be scheduled between the tasks 4, 3, 2 and $0$.
\[textcon:6\] Each leaf task must be scheduled once in its period.
The different jobs might have different periods in which they have to be scheduled in, the leaf task of each jobs might have a different period. Therefore, some of these periods might differ form the hyperperiod, which is the of all periods. To fulfill the requirements of all jobs, the leaf might have be scheduled multiple times in one schedule. Together the ensure that each job is executed the right amount of times per hyperperiod and all tasks in the jobs are executed in the right order.
\[textcon:7\] Two consecutive periods of the same task must not exceed the defined jitter bound for this task.
As described in , each task has a jitter bound that must not be exceeded. Therefore, ensures that the period of a task does not vary more than its jitter bound. E.g., the scheduled period of a task with the defined period of five slots and a maximal jitter of two slots could be decreased to three slots or increased to seven slots. But two consecutive periods of that task may not vary more than two slots, thus, a period change from three slots to seven is not allowed.
A scheduler that enforces all the will generate schedules applicable to a system described in . To be able to adapt to topology changes, the scheduler needs to follow one more constraint:
\[textcon:8\] The difference between the last period in the old schedule and the first period in the new schedule of a task must not exceed the defined jitter bound for this task.
If this constraint is followed the network can switch its schedule to the new one without breaking any real-time constraints.
To make the operation of a network reliable the schedule stability should be maximized. This is especially important when existing schedules need to be adapted to changes in the topology or taskset of a network. Increasing the schedule stability reduces the complexity of switching the schedules and therefore reduces the probability of failures while switching. Therefore, we formulate the general objective for the scheduling as following:
\[obj:general\] Time-slot changes between periods of tasks should be minimized.
Related Work {#sec:AllocSchedRelated}
============
Real-time scheduling is a vast topic, especially if the scope is widened to real-time multiprocessor scheduling. To keep this section of reasonable size we focus on the most applicable related work.
was proposed by @saifullah2010 to schedule networks[@saifullah2010]. It is designed for wireless real-time networks with changing topologies. The priority of a transmission is determined by the time between it is released and its deadline and the number of conflicting transmissions in this time. The highest priority is given to the task with the highest number of conflicting transmissions and the shortest time between release and deadline. need the release time of each transmission prior to scheduling, this is not possible in systems where the release time of a transmission depends on a task that has dependencies. As it is unknown when a dependency is scheduled, the release time of the dependent transmission is also unknown. Therefore, is not applicable for the problem described in .
Another application of wireless real-time networks with changing topologies are networks. @shakkottai2001 introduced an algorithm to schedule a mixture of real-time and non-real-time traffic in networks[@shakkottai2001]. At each time-slot the algorithm calculates which of the packets in the transmission queue has the shortest deadline and schedules this packet into the slot at a certain channel. This done by the eNode-B for each transmission time interval which consists of multiple time-slots. The schedule determined in this manner is only valid for the down-link traffic from the eNode-B to the user equipment. This technique is only possible, as the eNode-B buffers all down-link traffic. In a network such as in there is not one central instance as the eNode-B that buffers all the traffic.
@wang2019 propose a two staged approach to adaptive scheduling in train communication networks[@wang2019]. These networks are often time-triggered and need to handle rapid topology changes in cases where two trains are coupled or decoupled. The first stage is the offline scheduling, it generates schedules for the whole train. The second stage is called online stage, it derives the schedule for the parts of the train during the coupling and decoupling process. A two staged design seems applicable to our system as we also expect rapid topology changes when two need to be merged. On the other hand the characteristics of the network described by @wang2019 are in great contrast to our application. The authors describe train networks as strictly hierarchical multicluster networks with wired connections. Further, the approach does not have concept of dependencies between different data flows.\
A novel approach on how to close feedback control over wireless links is proposed by @baumann2019[@baumann2019]. They propose a system that reduces the communication between different parts of a distributed feedback controller. The reduction is achieved by a co-design called control-guided communication. Control-guided means that the controller tells the communication part of the system its communication demands ahead of time. The communication demand is decreased by a controller that estimates values in between communication. To benefit from the decreased communication demands the schedule needs to reschedule the communication frequently. Therefore, @baumann2019 choose an online scheduling approach. Like the other approaches this approach also lacks concept of dependencies between tasks.
As the approaches discussed above do lack the ability to handle dependencies the scope is widened to scheduling in operations research. In assembly lines dependencies are very common and therefore scheduling approaches in this domain need to handle them from the beginning. The algorithm proposed by @hu1961 forms from a given sets of tasks[@hu1961]. In these each node represents a task and each directed edge represents a dependency. The author assumes an assembly line with a number of equally skilled workers, each of this workers is able to fulfill one task at a time. The task have a predefined order in which they have to be fulfilled, the goal is to find the sequence of tasks for each worker that needs the shortest time to complete all tasks of the given set To clarify why this is applicable to the challenges stated in let the equally skilled worker be equally good channels and the predefined order gives a set of dependencies between tasks. This gives a model of a system which is quite close to the one we depicted above. However, a difference between our use case and assembly line scheduling is, that there is no concept of tasks that can not be executed at the same time because of common child tasks. Further, the tasks in @hu1961’s model have no deadline and the goal is to finish as fast as possible, in contrast our goal is to meet the deadline of all task.
Each of the scheduling concepts mentioned addresses a part of the problem stated in . However, each one is missing some key features needed to fulfill the requirements for scheduling algorithms as they are needed in this work. Some of the solutions used in the following two approaches are inspired by the discussed concepts.
Approach {#sec:AllocSchedMilp}
=========
In this section we introduce an approach to solve the problem described in based on a model. First we introduce the model for a decision problem that generates valid schedules regarding to the assumptions discussed in . Afterwards, we describe the constraints needed by the model to generate valid schedules. Then we introduce the objective that minimizes the jitter.
To ease the modeling we extend the schedule $S$ by a third dimension which is the dimension of tasks in $\tau$, the resulting Matrix is called $A$. Thus, the three dimensions of $A$ are, (i) the considered task in $\tau$, (ii) the channels and (iii) the time-slots. $A$ has the dimensions $|\tau| \times M \times H$ and is defined as followed: $$\forall a_{Tct} \in A : a \in \{0,1\}$$ Where $T$ is the task, $c$ is the channel and $t$ is the time-slot. $$a_{Tct}=
\begin{cases*}
1,& task $T$ is scheduled in channel $c$ \\
& at time-slot $t$ \\
0,& task $T$ is not scheduled in channel $c$\\
& at time-slot $t$
\end{cases*}$$
Following the constraints of the -model are discussed in detail.
\[con:1\] Every slot (timeslot and channel) must at most have one scheduled task $$\sum_{T \in \tau} a_{Tct} \leq 1 \quad$$ $$\forall c \in \mathbb{N}: 1 \leq c \leq M,\;\forall t \in \mathbb{N}: 1 \leq t \leq H$$
guarantees that there are no direct collisions between tasks in the same time-slot and the same channel and therefore implements .
\[con:2\] Tasks with common participating node must not be executed in the same time-slot $$(\iota_{UT} \times \sum_{c=1}^{M} a_{Tct}) + (\iota_{UT} \times \sum_{c=1}^{M} a_{Uct})\leq1$$ $$\quad \forall t \in \mathbb{N}: 1 \leq t \leq H,\; \forall U,T \in \tau$$
As task that have an intersection in their sets of nodes they are communication with must not be scheduled in one time-slot, takes the sum of all scheduled distributions of the interfering tasks $T$ and $U$ in a time-slot $t$ over all channels $1 \leq c \leq M$. This sum has to be less or equal one for all time-slots and all interfering pairs of tasks. Whether tasks $T$ and $U$ are interfering is determined from matrix $I$ at position $UT$. It is the implementation of . The matrix $I$ is defined by and . $$\label{eq:defI}
\forall \iota_{UT} \in I : \iota \in \{0,1\}$$ Where $T$ and $U$ are task in $\tau$ . $$\label{eq:defIota}
\iota_{UT}=
\begin{cases*}
1,& tasks $T$, $U$ have common nodes \\
0,& tasks $T$, $U$ do not have common nodes
\end{cases*}$$
\[con:3\] All dependencies $U$ of task $T$ must be scheduled before $T$ within the minimum of $P_i$ or $t-d_{U}$ time-slots $$\sum_{i=max(1; t-d_{U}; \lfloor t/P_T \rfloor\times P_T)}^{max(1; t)} \sum_{c=1}^{M} a_{Uci} \geq \sum_{c=1}^{M}a_{Tct}$$ $$\forall t \in \mathbb{N}: 1 \leq t \leq H,\; \forall U \in \Gamma_T,\; \forall T \in \tau$$
ensures that all dependencies $\Gamma_T$ to a task $T$ are executed at least as often as the dependent task $T$. It sums up all scheduled executions of a dependency $U$ in the $P_i$ slots and channels before $T$ is scheduled and ensures that this sum is larger than the sum of the scheduled executions of $T$ in time-slot $t$. This guarantees that are enforced.
\[con:4\] Each leaf task must be scheduled once per its period $$\sum_{t=max(1; (p-1)\times P_T}^{p\times P_T} ~~ \sum_{c=1}^{M} a_{Tct} = 1$$ $$\forall p \in \mathbb{N}: 1 \leq p \leq \frac{H}{P_T},\; \forall T \in \tau$$
To ensure that each job is executed once in its period (), sums up all executions of task $T$ during all possible periods and ensures the sum is always one for all tasks.
\[con:5\] Execution of a $T$ must be scheduled in its jitter bounds $$\sum_{i=max(1;t-P_T-J_T)}^{max(1;t-P_T+J_T)} \sum_{c=1}^{M}a_{Tci} \geq \sum_{c=1}^{M}a_{Tct}$$ $$\forall t \in \mathbb{N}: 1 \leq t \leq H,\; \forall T \in \tau$$
As a task $T$ scheduled outside its jitter bound $J_T$ could harm the operation of the system (), prohibits that. Therefore it checks whether there is an execution of $T$ scheduled in the jitter bound $\pm J_T$ one period $P_T$ before the current execution. The following ensure that all tasks of one job use the same execution of a common dependency (). Referring to the graphs in they ensure that task$0$, task3 and task4 all use the same execution of task5. That means, task5 must not be scheduled between task$0$, task3 and task4. This might happen, as jobs can share certain tasks and the periods of some dependencies might be smaller than the period of the job and the depending task. As an example, let job$0$ have a period of ten slots and job1 a period of five slots. With the constraints there is nothing that prohibits to schedule the tasks4 and 5 a second time in between the scheduled executions of task3 and $0$. Doing so could cause task$0$ to operate on different data than task3.
To cope with this we need to introduce a few more variables. Let $\Omega_{T_eT_l}$ be the set of tasks between the entry task $T_e$ of a job and its leaf task $T_l$ with the same period as $T_e$.
For the example above $\Omega_{50}$ would consist of the tasks 3, 2, 0.
On contrast let $\breve\Omega_{T_eT_l}$ be the tasks of the job that have a shorter period than $T_e$. Again, for the example that would mean $\breve\Omega_{50}$ consists of the tasks 5 and 4.
\[con:8\] $\rho_{T_eT_l}$ is the sum of all execution of all Tasks in $\Omega_{T_eT_l}$, $\breve\rho_{T_eT_l}$ is the sum of all execution of all Tasks in $\breve\Omega_{T_eT_l}$ $$\rho_{T_eT_l} = \sum_{T \in \Omega_{T_eT_l} } \sum_{c=1}^{M} \sum_{i=t}^{min(i+P_{T_l}, H)}a_{Tci}$$ $$\forall t \in \mathbb{N}: 1 \leq t \leq H, \forall T_e \in E,\; \forall T_l \in L$$ $$\breve\rho_{T_eT_l} = \sum_{T \in \breve\Omega_{T_eT_l}} \sum_{c=1}^{M} \sum_{i=t}^{min(i+P_{T_l}, H)}a_{Tci}$$ $$\forall t \in \mathbb{N}: 1 \leq t \leq H, \forall T_e \in E,\; \forall T_l \in L$$
is only needed to give $\rho_{T_eT_l}$ and $\breve\rho_{T_eT_l}$ a value that indicates how many tasks of $\Omega_{T_eT_l}$ and $\breve\Omega_{T_eT_l}$ are scheduled in one period of $T_l$.
\[con:10\] If the complete path or non of its tasks is scheduled $\dot{\rho}_{T_eT_l}$ and $\dot{\breve\rho}_{T_eT_l}$ are 1. If only parts are scheduled $\dot{\rho}_{T_eT_l}$ and $\dot{\breve\rho}_{T_eT_l}$ are $0$. $$\rho_{T_eT_l} = |\Omega_{T_eT_l}| \times \dot{\rho}_{T_eT_l} \quad \forall T_e \in E,\; \forall T_l \in L$$ $$\breve\rho_{T_eT_l} = |\breve\Omega_{T_eT_l}| \times \dot{\breve\rho}_{T_eT_l} \quad \forall T_e \in E,\; \forall T_l \in L$$
$\rho_{T_eT_l}$ and $\breve\rho_{T_eT_l}$ are used in to determine whether all task in $\Omega_{T_eT_l}$ or respectively $\breve\Omega_{T_eT_l}$ are scheduled. That is necessary as tasks in $\breve\Omega_{T_eT_l}$ might be scheduled independently but tasks in $\Omega_{T_eT_l}$ must not be scheduled without the tasks in $\breve\Omega_{T_eT_l}$. This in ensured by
\[con:11\] The $\breve\Omega_{T_eT_l}$ might be scheduled alone but $\Omega_{T_eT_l}$ must not $$\dot{\breve\rho}_{T_eT_l} \geq \dot{\rho}_{T_eT_l} \quad \forall T_e \in E,\; \forall T_l \in L$$
Together the define a model for the decision problem. This model is able to generate valid schedules in $A$.
Objectives {#sec:objectives}
----------
The objective of our model is to minimize the jitter between executions of a task. Minimizing the jitter in feedback loops is one of more obvious optimizations for a scheduling. This is due to the fact, that jitter leads to a bigger error in timing of the control task and therefore to larger error in the controlled process.
Minimizing the jitter in the model described above is challenging, as it lacks a concept of how many slot are between different executions of a task. To mitigate this, we minimize the number of tasks that change their relative time-slot in different periods of their job. With this optimization we additionally increase the schedule stability. The schedule stability gives a measure how much a schedule changes between periods.
\[obj:1\] Minimize the number of tasks changing slots between their periods $$Minimize: \frac{1}{N} \times \sum_{T \in \tau}~\sum_{t=1}^{H-P_T}\left(~\left\lvert\sum_{c=1}^{M}a_{Tct} - \sum_{c=1}^{M}a_{Tc(t+P_T)}\right\rvert\right)$$
To evaluate the effectiveness of we scheduled the same set of 130000 tasksets once with and once without any objective. These tasksets were randomly generated under certain constraints, we defined five different hyperperiod lengths (8, 12, 16, 25, 35), four different numbers of dependencies between the tasks (9, 12, 16, 24) and three different numbers of jobs (1 ,3 ,6), the number of tasks were eight or twelve.
![Mean, maximum and minimum jitter over all scheduled tasksets for the two objectives, “None” for no objective and “Jitter” for . \[fig:AllocSchedJitter\] ](figures/jitter.pdf){width=".8\linewidth"}
In the jitter for the two different objectives is shown. “None” refers to the results scheduled without any ojective, “Jitter” refers to the jitter in the schedules optimizes to a minimum jitter by . The jitter shown in is calculated using , it is the mean of the mean jitter of all tasks scheduled in the schedule. $$\label{eq:AllocJitter}
jitter = \frac{ \sum_{T\in \tau}\frac{\sum_{i=1}^{H/P_T} (e_{Ti} - e_{T1}) \: mod \; P_T} {\sum_{t=1}^{H}\sum_{c=1}^{M}a_{tcT}}}{|\tau|}$$ As shows reduces the mean jitter to almost zero. As an objective can not harm the performance, in terms of schedulablility, of our -model, we use the for all further evaluations if not stated otherwise.
Adapting Schedules {#subsec:adaptSchedules}
------------------
This section discusses one of the key contributions of this chapter, the adaptation of existing schedules to changes in the topology of the network or in the taskset. The adaptation needs to be done without harming the real-time requirements of jobs which are present in the existing schedule. formulates this complex goal in a very brief way.
To achieve the goal of adapting schedules, several steps are needed in preparation. First the tasksets of the old schedule and the new tasks need to be merged. The new tasks can either be a second taskset of another or tasks of new job added to the existing . In this work we focus on the first case, where two need to be merged into one. We consider this as the more complex case, as adding a new job is the same despite the fact, that the new job does not have the restrictions of an old schedule. While joining the tasksets ($\tau_1$ and $\tau_2$), task-ids must be kept unique throughout the new taskset $\tau'$. The new hyperperiod $H'$ is the of all periods in $\tau'$.
In the second step the two schedules ($A_1$ and $A_2$) are merged into one schedule $C$ that violates . Thus, tasks of both networks might share one time-slot on the same channel. As $C$ is never to be executed, this does not cause any harm to the networks. $C$ is used in in addition to to generate the new, combined schedule $A'$ that respects the .
\[con:12\] Timeslot of $task_T$ must not differ more than $J_T$ from $C$ to $A'$, changes of channels are ignored $$\sum_{j=t-J_T}^{t+J_T}~\sum_{c=1}^{M}a'_{Tcj} ~\geq ~\sum_{c=1}^{M}c_{Tct} \quad \forall t \leq H',\; \forall T \in \tau'$$
Together with we introduce the new that minimizes the amount of task that are shifted to other time-slots between $C$ and $A'$. Thus, the schedule stability is maximized.
\[obj:reschedule\] Minimize time-slot allocation changes from $C$ to $A'$ $$Maximize: \sum_{T \in \tau'}\sum_{t=1}^{H'} ( \sum_{c=1}^{M}a_{Tct}' \times \sum_{c=1}^{M}c_{TcT})$$
By multiplying the sum of all channel for a certain time-slot and a certain task in the new schedule with the sum of the channels of the same task and time slot in the combined schedule, we get one for each task that was not move and zero for each task that was moved. As each task can at most be scheduled once per time-slot the result of this multiplication can only have the two values, one and zero. By summing this result up over all tasks and time-slots, we get the number of the unchanged time-slot allocations. As we maximize this value, the schedule stability is maximized and the jitter is minimized.
-model and its Complexity {#sec:eval-comp-compl}
=========================
To calculate schedules that respect , we implemented a -model. By solving this -model with its we get a base line of optimal schedules.
As stated in multi-channel real-time scheduling is NP-Hard. Therefore, it is important to evaluate whether the formulated -Model can be solved in a reasonable time according to the use case. Besides that we will also investigate wjhat parameters influence the time it takes to solve the -Model. As the first parameter we evaluated the number of slots in a hyperperiod. We scheduled more than 160,000 different task-sets with five different hyperperiods: 8, 12, 16, 25 and 35 slots on an Intel Xeon W-2195 CPU. These task-sets were randomly generated following guidelines in the hyperperiod length, number of dependencies, nodes, jobs, etc. Form these parameter of the task-sets the hyperperiod length is the one that influences the computation time the most. The time used to solve the model varies from 0.07s to more than 600s. shows the of the five different hyperperiods. The right graph shows only the range from zero to ten seconds of the left graph to show more details. As expected, a longer hyperperiod leads to a longer solve-time, this is due to the larger solution space. The dots in both graphs mark the maximum solve-time. Even though, some task-sets need over 600s the majority is scheduled in less than 3s.
![Impact of different hyperperiod length to the time needed to solve the schedule, shown as \[fig:solvetime\_slots\] ](figures/solveTime_slots.pdf){width=".97\linewidth"}
As a second parameter we evaluated the influence of the number of dependencies in a taskset. To mitigate the influence of the hyperperiod length we only evaluate the solve-time of tasksets with 35 slots. shows the for 9, 12, 16, and 24 dependencies, all together is show 76000 tasksets. As the model gets more complex with more dependencies the solve-time increases as well.
![Impact of different numbers of dependencies to the time needed to solve the schedule, shown as \[fig:solvetime\_dependencies\] ](figures/solveTime_dependencies.pdf){width=".97\linewidth"}
To determine how with multiple jobs would be handled in contrast to with just one job, we evaluate the solve-time of tasksets with a hyperperiod of 35 slots and 16 dependencies. These roughly 20000 task sets have either 1, 3 or 6 jobs. shows that a taskset with more complex jobs, more dependencies, takes longer to be scheduled as an easier one.
![Impact of different numbers of jobs to the time needed to solve the schedule, shown as \[fig:solvetime\_jobs\] ](figures/solveTime_jobs.pdf){width=".97\linewidth"}
Applicability to Embedded Devices
---------------------------------
As most consist of embedded devices with far less computation power than our Intel Xeon W-2195 (fast CPU), we assume the solve-time to be higher. To support this assumption, we used our 10 years old Intel Xeon E5520 (slow CPU) to schedule all tasksets with hyperperiod of 35. The results for the median solve-time are not that different: for the slow CPU and for the fast CPU. However, the mean and maximum solve-times differ a lot: the fast CPU needs a mean solve-time of and a maximum of , the slow CPU needs in mean and at maximum. This huge variety in the solve-times is a problem considering real-time applications. Even if the scheduling has no hard time constraints, waiting up to or even for a schedule is unrealistic in most applications. Therefore, a way to calculate schedules in a more predictable time is necessary.
In we formulate a hypothesis on the adaptability of schedules and validate it to get better insight on how to design an algorithm suitable for the described problem.
Hypothesis on Adaptability of Schedules {#sec:adaptibilityHypothesis}
=======================================
In most scheduling applications it is preferable to schedule all tasks as dense as possible. So the taskset can be scheduled more often in the same amount of time or the executing machines can sleep or take other jobs. In a system where jobs have a defined period there is no need to schedule all task as dense as possible. In contrast, it might have advantages to schedule tasks as sparse as possible. That means, free slots are more uniformly distributed throughout the hyperperiod.
This is especially advantageous if adaptations are taken into consideration. Having free slots throughout the hyperperiod means that tasks in a merged schedule must not be shifted to time-slots as far as in a dense schedule. depicts two densely schedules $S_1$ and $S_2$, $S_1$ has the tasks $A_1$ to $A_3$ and $S_2$ the tasks $B_1$ $B_3$. The merged schedule $S_{12}$ contains all tasks. In this example all tasks are scheduled very densely in the first three of the six time-slots. To merge $S_1$ and $S_2$ the tasks are shifted into other time-slots. $B_3$ has to be shifted three time-slot, therefore $\Delta_{B_3}$ is 3.
![Example of dense schedules. \[fig:denseSchedule\] ](figures/denseSchedule.pdf){width=".8\linewidth"}
on the other hand shows two sparse schedules that also contain three tasks in six time-slots. In contrast to the schedules in , this time the tasks are distributed equally over the six time-slots. To merge the schedules all tasks of $S_2$ have to be shifted only by one time-slot, therefore $\Delta_{B_3}$ is 1.
![Example of sparse schedules. \[fig:sparseSchedule\] ](figures/sparseSchedule.pdf){width=".8\linewidth"}
show the two extreme cases but they illustrate why it might be a good idea to spread the tasks throughout the whole hyperperiod. In the following we introduce our metric for the degree of distribution and investigate under which circumstances the hypothesis, that pairs of schedules which have a higher distribution are more likely to be combinable, is true.
Task Distribution
-----------------
$$\label{metric:distribution}
Distribution = \frac{\sum_{t=1}^{H} x_t}{\sum_{t=1}^{H}\sum_{c=1}^{M}\sum_{T\in \tau}a_{tcT}}$$ gives the distribution as a normalized function of number of used time-slots to unused time-slots divided by the number of all scheduled executions of all tasks. Where $x_t$ indicates an unused time-slot following an used time-slot, as described in . $$\label{eq:x_t}
x_t = \begin{cases*}
1,& if time-slot $t-1$ is used and\\
&time-slot $t$ is unused \\
0,& if time-slot $t-1$ is unused \\
&or time-slot $t$ is used
\end{cases*}
\forall 1 \leq t \leq H$$ A time-slot is called used if there is a task scheduled on at least one channel.
Validity of the Hypothesis
--------------------------
To validate whether the hypothesis is true we generated pairs of two schedules for two different tasksets of similar form, in terms of hyperperiod length, number of jobs, number of dependencies, etc. The distribution of a pair lies between zero and two, as it is the sum of the distribution of both schedules. These pairs were than rescheduled using the -model. shows that pairs with a higher distribution are much more likely to be schedulable than pairs with a lower distribution.
![Reschedulablity over distribution \[fig:reschedulable\_distribution\] ](figures/reschedulable_distribution.pdf){width=".9\linewidth"}
Heuristic Approach {#sec:AllocSchedEuch}
==================
As we have shown in schedules with higher distribution can be merged more easily. Therefore, we propose an algorithm that has three main goals, 1. maximize the distribution, 2. to minimize the jitter of tasks and 3. minimize the number of executions of each task.
In general the algorithm schedules a whole job before it starts to schedule the next job. It starts with the job that has the longest path from the entry to the leaf task and continuous with the next longest job until all jobs are scheduled. We call the job that is currently scheduled active.
Within a job the scheduler starts with the leaf task and places it in the latest possible slot, this way the number of slots to schedule the rest of the jobs is maximized. The slot to schedule the leaf task for the $k$-th subperiod is calculated by utilizing . A subperiod is one period of a job, its length is always a divider of the hyperperiod. $$t_{l,k} = k \times P_l
\label{eq:leafSlot}$$ As slots might be occupied by tasks which were scheduled prior to the current one, we introduce two solutions to choose another slot in which the current task is scheduled. One of them chooses to use other channels before moving to other time-slots. This should lead to a small jitter, as more tasks are scheduled in the calculated time-slot. The other one prefers to use other time-slots first and only uses different channels if no time-slot in the allowed jitter bound is free. This should be beneficial if need to be merged. In this case the can simply use another channel and most conflicts are resolved. These solutions are described in more detail in and . After a free slot is found, this slot in the two dimensional array $S$ is marked occupied with the task. The dimensions of $S$ are given by the number of channels $M$ and length of the hyperperiod $H$.
After the leaf task is scheduled the algorithm uses the so-called backward equation to find the slot for the dependencies of the leaf task, this equation is explained in more detail in . The backward equation maximizes the distance of the execution between the scheduled task and other dependencies but also between the dependency and its dependencies. This process is repeated until all entry tasks of a job are scheduled.
As it might occur that several jobs have common tasks, the backward equation would schedule a common task twice although an already scheduled execution of that task could be used to schedule the rest of the job. Therefore, the algorithm searches the schedule whether there are tasks of the active job scheduled in a certain range of slots. If such a task is found, the algorithm uses this execution and schedules the rest of the active job in the direction from that execution to the leaf task using the so-called forward equations, described in .
While the scheduler is following the dependencies of tasks it might face tasks that have multiple dependencies or multiple tasks that depend on a task. In both cases the scheduler needs to decide in which order these task should be scheduled. We have two different approaches to this issue: the first one orders the task by ascending maximal age. We call this approach *age first*. The idea is to schedule the task first that have a smaller range of time-slots in which they can be scheduled.
The second approach is, to sort the tasks by ascending maximal jitter. This guarantees that the tasks with the hardest jitter constraints are scheduled first. We call this approach *jitter first*. Both approaches will be compared in the evaluation .
Backward Equation {#subsec:backward-equation}
-----------------
The Backward gives a time-slot in which the task should be scheduled, based on the execution time-slot of the tasks that are depending on the task to be scheduled. $k$ gives the subperiod which is currently to be scheduled, $T_c$ is the child (dependent task) of a task and $T_p$ is the parent (dependency) of a task.
needs a special case for the first subperiod $k=1$. This is simply because in this case there is no prior execution of this tasks, so its time-slot can not be taken into the equation. With $|\omega_i|-\delta_{p}$ we calculate how many tasks of the job we have to schedule until the leaf task is reached. The divided $t_{c,k}-1$ or $t_{c,k}-t_{c,k-1}-1$ gives the number of slots left in the subperiod. The division of both gives a time-slot for the parent that has the maximal distance to the child but still leaves enough time-slots to schedule its parents. As the time-slot calculated this way might have a larger distance to the child than the maximal age of the parent the minimum of the division and the maximal age is taken.
If the calculated time-slot is occupied, an alternative is selected applying the solutions from or .
Forward Equation {#subsec:forward-equation}
----------------
Jobs in a taskset might have tasks in common, an example is depicted in . In the example the jobs with the leaf task $T_1$ and $T_2$ have the common task $T_{4}$. This common task is called $T_{com}$. This fact can be used to reduce the total number of task executions in a schedule. To do so we need to find these common tasks in the schedule and decide whether the found execution of such a common task fulfills the timing requirements of the child task that is to be scheduled based on the common task. Therefore, the algorithm defines a range of time-slots in which the execution of a common task has to be located in order to use it. The lower bound of this range $Lower_{com,k}$ for the $k$-th subperiod of $T_{com}$ is defined as the maximum of three values: first, the execution time-slot of its child task in the $k-1$-th subperiod. This guarantees that the order of executions is not altered for prior subperiods. The second value is the time-slot of $T_{com}$s execution in the $k-1$-th subperiod on which the period of the leaf task of the active job $P_l$ is added. To be able to use extra time-slots the maximal jitter $J_{com}$ is subtracted. The last value is the end of the current subperiod $k \times P_l$, to make use of allowed jitter $J_l$ is subtracted as well as the sum of all maximal ages of the tasks from the common task to the leaf task $\sum_{i=com}^{l-1}d_i$. Thus, $Lower_{com,k}$ is defined as follows:
The upper bound of the range is defined as the minimum two values: $t_{l,k} - \delta_{com}$ where $t_{l,k} = k \times P_l + J_L$, which gives the last slot far enough for the end of the subperiod to schedule all tasks between $T_{com}$ and the leaf task, and where $\delta_{com}$ is the distance $T_{com}$ and $T_l$. The second value is the period of the leaf task $P_l$ and maximal jitter of $T_{com}$ added to the time-slot of the last execution of $T_{com}$.
If an execution of $T_{com}$ was found in the range defined by and the scheduler uses to calculate the time-slots the child $T_c$ of $T_{com}$ should be executed. $$\small{
\begin{aligned}
&t_{c,k} = \\
&\begin{cases}
t_{p,k}+min(\lfloor \dfrac{min(k\times P_{l}, t_{p,k+1})-t_{p,k}}{\delta_{p}} \rfloor, d_{p}) &\textit{if}\ T_{p}=T_{com} \\
& and \ k< H/P_{l}\\
t_{p,k}+min(\lfloor \dfrac{k\times P_{l}-t_{p,k}}{\delta_{p}} \rfloor, d_{p}) &\textit{else}\ \\
\end{cases}
\end{aligned}
\label{eq:genForward}
}$$ As describes the $k$-th execution of $T_c$ is scheduled after the $k$-th execution of $T_p$ but before the $k+1$-th execution of $T_p$. Therefore, the execution order defined in the task set is respected. The other limiting factor for the forward equation is the maximal age $d_p$ of $T_p$, the child must be scheduled before $d_p$, otherwise the data produced by $T_p$ is useless. As for the backward equation it is possible that $t_{c,k}$ is already occupied, strategies to handle such situations are described in and .
Time First Shifting {#subsec:timeFristShifting}
-------------------
In the sections above we described how the algorithm determines in which time-slot a task should be scheduled. If this slot is occupied on one channel the algorithm needs to find another slot either on another channel or at another time. This section describes a solution to this challenge, that tries to find another time-slot before it uses other channels. As the algorithm normally schedules the tasks from the leaf to the entry task and thus from right to left in the example, the algorithm first tries to put $T_i$ into $t_{i,k}+1$ on the same channel. This is the time-slot next to $t_{i,k}$ on the right, by going right first the algorithm leaves potentially more space where the yet unscheduled tasks need to be scheduled. If $t_{i,k}+1$ would be occupied as well the algorithm would go to $t_{i,k}-1$ and not $t_{i,k}+2$ to minimize the jitter. After all slots in the jitter bound of $T_i$ are tested and found occupied the algorithm would test $t_{i,k}$ at another channel and repeat the same search pattern if it is occupied as well. Another reason not to take $t_{i,k}$ on the second channel would be that $T_j$ and $T_i$ are interfering tasks, in this case this slot would be handled as occupied.
In this mode the algorithm tries to fit all tasks on one channel. This is done under the hypothesis that two schedules that use one channel primarily are easier to merge, as one of the schedules could be shifted to another channel and most of the conflicts would be resolved.
Channel First Shifting {#subsec:channelFirstShifting}
----------------------
This mode of the algorithm solves the same issue as the one described in but by using all available channels before shifting the execution of $T_i$ in time. After all channels, in the example two, are found occupied for $t_{i,k}$ the algorithm would try then all channels at $t_{i,k}+1$. This mode has the advantages that it minimizes the jitter in the schedule and that it potentially leaves more time-slots entirely empty. Having time-slots empty on all channels might be of advantage when two schedules need to be merged, as tasks can be shifted there to make room for tasks that can not be shifted due to stricter jitter bounds.
In we will evaluate the proposed shifting mechanisms.
Schedule adaption {#subsec:scheduleAdaption}
-----------------
To achieve the goal of adapting schedules, several steps are needed in preparation. First the tasksets of the old schedule and the new tasks need to be merged. The new tasks can either be a second taskset of another or tasks of new job added to the existing . In this work we focus on the first case, where two need to be merged into one. We consider this as the more complex case, as adding a new job is the same despite the fact, that the new job does not have the restrictions of an old schedule. While joining the tasksets ($\tau_1$ and $\tau_2$), task-ids must be kept unique throughout the new taskset $\tau'$. The new hyperperiod $H'$ is the of all periods in $\tau'$. After the tasksets are merged, the same algorithm is used to generate the new schedule.
The result of the backward has no dependencies to tasks outside of the active job, therefore it will not changes unless the job is changed which results in a new job. Thus, the backward equation does not harm the real-time requirements while adapting schedules.
The search range definitions and for $k=1$ sets the upper search bound to $k\times P_{l} + J_{l} - \delta_{com}$, and the lower bound to $|\omega_i| - \delta_{com}$. As the results are not effected by any task outside the job, it will not harm real-time requirements during adaption. For all subperiods where $k>1$, the search box is limited by the previous subperiod, that means that no subperiod with $k>1$ could harm the real-time requirements of a task if the first subperiod did not harm them. The same is true for the forward as it is not effected by tasks outside the job.
As discussed the calculations of the time-slots can not give results outside the jitter bound and the shifting methods described in and do not shift further than the allowed jitter. Thus, no task can be effected by a jitter more than the allowed one during schedule changes. Therefore, the same algorithm can be used to schedule a initially and to adapt its schedule to new topologies or tasksets.
Evaluation {#sec:scheduleAlgoEvaluation}
===========
In this section we evaluate the algorithm discussed in and compare it with the results of the -model. First we show that the presented algorithm has a much more predictable execution time than the -model. Afterwards we compare the different approaches in terms of jitter and the percentage of scheduleable tasksets. In we list the short names used to distinguish the different modes of the algorithm in this section.
**Approach** **Sort tasks on same level by** **Mode**
-------------- --------------------------------- ----------
Age First (0, 0)
Jitter First (0, 1)
Age First (1, 0)
Jitter First (1, 1)
: The four different modes of the scheduling algorithm
\[tb:modes\]
Computational Complexity Comparison {#sec:eval-compl-comp}
-----------------------------------
To compare the computational complexity of the milp-model and our algorithm, we scheduled more than 160,000 different task-sets with five different hyperperiods: 8, 12, 16, 25 and 35 slots on an Intel Xeon W-2195 CPU. These task-sets were randomly generated following guidelines in the hyperperiod length, number of dependencies, nodes, jobs, etc. Form these parameter of the task-sets the hyperperiod length is the one that influences the computation time the most. The time used to solve the model varies from 0.07s to more than 600s. shows the of the five different hyperperiods. The right graph shows only the range from zero to ten seconds of the left graph to show more details. As expected, a longer hyperperiod leads to a longer solve-time, this is due to the larger solution space. The dots in both graphs mark the maximum solve-time. Even though, some task-sets need over 600s the majority is scheduled in less than 3s.
![Impact of different hyperperiod length to the time needed to solve the schedule, shown as \[fig:solvetime\_slots\] ](figures/solveTime_slots.pdf){width=".97\linewidth"}
shows the comparison of the time needed to schedule the same tasksets with the proposed algorithm and the -model. The dotted lines represent the -model and solid lines the heuristic algorithm, the maximum solve time is marked with a diamond for the -model and a dot for the algorithm. The results for the four different modes are very similar, therefore we only discuss the results of the *time first shifting and age first* mode \[0, 0\]. shows that the solve times of the algorithm are much closer to each other and, thus are much less influenced by the length of the hyperperiod. With the vast majority of tasksets scheduled in less than one second, the algorithm is much more likely to finish scheduling in time even on embedded devices. Nevertheless, for short hyperperiods the -model is faster, that suggests that there is room for optimizations, at least in the implementation of the algorithm.
![\[fig:compareSlotsTime\] Influence of hyperperiod length on the schedulability, comparing the four algorithm modes and the -model. For three jobs, nine dependencies and twelve nodes in mode \[0, 0\] ](figures/solveTime_slots_algo.pdf){width="0.9\linewidth"}
Influence of Taskset Parameters to Scheduling Success {#sec:infl-tasks-param}
-----------------------------------------------------
To determine under which conditions the algorithm is able to schedule what percentage of the tasksets, we used the same set of tasksets as in .
The biggest influence is caused by the number of nodes in a , as shown in . In general the results show that a taskset, with all the same parameters except for the number of nodes, is harder to schedule if less nodes are in the . This the due to the fact that in such tasksets there are more intersecting task which makes it more likely that a taskset is unschedulable. Therefore, the algorithm mode should be compared to the -model. This comparison still shows that the algorithms performance deteriorates if fewer nodes are in a . The results of this evaluation confirm the assumption further that *channel first shifting* is the superior mode. They also give rise to the assumption that the algorithm is strongly effected by the number of dependencies.
![\[fig:compareNodes\] Influence of -size on the schedulability, comparing the four algorithm modes and the -model. For three jobs, nine dependencies and 35 slots ](figures/compare_nodes.pdf){width="0.9\linewidth"}
To confirm the assumption we evaluated the performance of the algorithm on tasksets with the same parameters except for the number of dependencies. As for the evaluation above, the algorithms performance needs to be compared with the -model. The results depicted in show a dramatic decline in the percentage of schedulable tasksets between nine and twelve dependencies.
![\[fig:compareDeps\] Influence of the number of dependencies on the schedulability, comparing the four algorithm modes and the -model. For three jobs, and 35 slots and twelve nodes ](figures/compare_deps.pdf){width="0.9\linewidth"}
Due to space limitations we are not able to show the influence the hyperperiod and number of jobs have. In general the hyperperiod length does not have a significant influence, the number of jobs on the other hand has an influence. For a constant number of tasks, a higher number of jobs is harder to schedule than a lower number. This is due to the fact, that the algorithm tends to fill time-slots that are a multiple of the jobs periods first. By having more jobs the chance that this slot in the ones in the jitter range are already occupied is higher.
As the algorithm tries to schedule the leaf task of each job in the last slot of the subperiod, we assume that more jobs lead to a lower performance. To confirm the assumption we evaluated tasksets with the same parameters but the varied number of jobs. show that this assumption is true, it also shows that the performance of *channel first shifting* declines more than the performance of *time first shifting*.
![\[fig:compareJobs\] Influence of the number of jobs on the schedulability, comparing the four algorithm modes and the -model. For nine dependencies, and 35 slots and twelve nodes](figures/compare_jobs.pdf){width="0.9\linewidth"}
All evaluations show that the *channel first shifting* mode was superior to the *time first shifting*, only in some cases the performance of both approaches were close to each other. The evaluation also shows that the order in which the tasks are chosen to be scheduled does not make a noticeable difference in most cases and if there is a difference there is no pattern behind which of them performs better.
Slot Allocation Probability {#sec:slot-alloc-prob}
---------------------------
To investigate further why *channel first shifting* suffers more from an increased job number than *time first shifting* we compare the probability of each time-slot to be allocated for a task. shows these probabilities for both shifting approaches and the -model, it also depicts the value for an equal distributions as a reference. As we discussed above the algorithm always tries to allocate the last time-slot in the subperiod of a task. For the *channel first shifting* this behavior is evident from the results, time-slots that are divisors of 24 have a higher probability to be allocated. The higher the divisor is, the higher is the probability, as there are more jobs that share the end of its subperiod here. For the *time first shifting* this effect is still noticeable but it is mitigated. The -model does not show this effect, as it is not bound to the limitations in the heuristic, but it shows the tendency to allocate the first time-slot.
![\[fig:compareDist\] Comparison of the probability that a certain time-slot is allocated by the different scheduling approaches, for tasksets with hyperperiod length 24 ](figures/compare_distribution.pdf){width=".8\linewidth"}
Allocation Introduced Jitter {#sec:alloc-intr-jitt}
----------------------------
The jitter a scheduling approach introduces is, after its ability to schedule tasksets, one of the most important performance factors of a real-time scheduling approach. To evaluate this factor we compare the jitter introduced by the four modes of the proposed algorithm with the -model in . It shows the minimum, mean and maximum jitter each mode introduces into a set of 56000 tasksets which are all scheduleable by all modes. As in the previous evaluations the *channel first shifting* outperforms the *time first shifting*. Especially with the *ascending age* order the maximum jitter is lower. The mean jitter on the other hand does not differ significantly over all modes. The -model with jitter optimization outperforms all algorithm modes, as it tries to find the minimal possible jitter.
![the mean, minimum and maximal jitter of the four algorithm modes in comparison to the -model \[fig:AllocSchedJitterCompare\]](figures/jitter_compare.pdf){width=".7\linewidth"}
shows that the impact of the ordering is less significant than the impact of the shifting. The impact of the ordering is so small that the lines are actually overlapping in the graph.
The results of this evaluation are very promising especially for an algorithm that is not actively reducing the jitter but only relies on its allocation methods.
Performance of Rescheduling {#sec:perf-resch}
---------------------------
To evaluate how good the different approaches are able to schedule combinations of tasksets we generated over two million combinations of the scheduleable tasksets. From these we randomly chose a set of 250thousand combinations. This set was scheduled by all four modes of the proposed algorithm and the -model as a reference. depicts the percentage of combinations that were scheduleable by each mode. As all evaluations above this shows that the *channel first shifting* is superior to the *time first shifting*.
The *channel first shifting* managed to schedule over 50% of the combinations. This seems to be a quiet low percentage, but taking into consideration that we formed the combinations from tasksets with the same hyperperioid, job number and node number, the effect discussed in is amplified here. As even more tasks need to be scheduled in the same time-slots, these combinations are the worst case combinations. Together with the fact that our tasksets are quite dense, shows that almost all time-slots are used at on least one channel, a success rate of 50% is a promising result.
![\[fig:compareReschedule\] Comparison of the scheduling performance for taskset combinations ](figures/compare_reschedule.pdf){width=".8\linewidth"}
Conclusion {#sec:alloc-concl}
===========
As cooperative are emerging in real-time applications, wireless connections become more and more important. These applications need scheduling algorithms which are able to adapt schedules to new network topologies and application requirements. In we described how a cooperative might be modeled and what the special challenges in these systems are. We discussed the constraints a scheduling algorithm for such systems needs to fulfill in . After a review of existing work in related areas we developed the -model that generates schedules following the constraints discussed earlier. As it can be time consuming to solve the proposed -model, especially on embedded devices, we propose an algorithm to generate schedules which has a lower computation time for almost all cases. To generate schedules that are easy to combine with other schedules we stated the hypothesis that two more sparse schedules are easier to combine than two dense schedules, even with the same hyperperiod and number of tasks. We proof the hypothesis in . In we discuss the proposed algorithm and its four different modes in detail. The evaluations in compares these four modes to each other and to the -model. We show that the *channel first shifting* is the preferable of the two shifting modes. The order in which tasks are scheduled, on the other hand has no distinguishable influence on whether a taskset is scheduleable by the algorithm or not.
|
---
abstract: 'Let $n,p,k$ be three positive integers. We prove that the numbers $\binom{n}{k} \ {}_{3}^{}{F}_{2} (1-k,-p,p-n\ ; \ 1,1-n \ ; \ 1)$ are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application.'
author:
- |
Michel Lassalle\
Centre National de la Recherche Scientifique\
Institut Gaspard Monge, Université de Marne-la-Vallée\
77454 Marne-la-Vallée Cedex, France\
`lassalle @ univ-mlv.fr`\
`http://www-igm.univ-mlv.fr/~lassalle/index.html`
date: '2000 Mathematics Subject Classification : 05A10, 33C20'
title: A new family of positive integers
---
Definition
==========
We use the standard notation for hypergeometric series $${}_pF_q\!\left[\begin{matrix}a_1,a_2,\dots,a_p\\
b_1,b_2,\dots,b_q\end{matrix};z\right]=
\sum _{k=0} ^{\infty}\frac {(a_1)_k\dots(a_p)_k}
{(b_1)_k\dots(b_q)_k}\frac{z^k}{k!},$$ where for an indeterminate $a$ and some positive integer $k$, the raising factorial is defined by ${(a)}_k = a(a+1) \ldots (a+k-1)$.
There are not many families of positive integers which may be defined in terms of hypergeometric functions. Among them stand of course the binomial coefficients $$\binom{n}{p} = {}_2F_1\!\left[\begin{matrix}-p,p-n\\
1\end{matrix};1\right].$$ Actually this expression is obtained by specializing $x=p-n, y=1$ in the celebrated Chu - Vandermonde formula $${\frac{{(y-x)}_p}{{(y)}_p}} = {}_2F_1\!\left[\begin{matrix}-p,x\\
y\end{matrix};1\right].$$
Of course using this relation as a definition of binomial coefficients would be rather tautological. However, quite surprisingly, it is possible to define a new family of positive integers by slightly modifying the Chu - Vandermonde formula.
Indeed for any positive integers $n,p,k$, let us define $${\binom{n}{p}}_{k} = \binom{n}{k} \
{}_3F_2\!\left[\begin{matrix}1-k,-p,p-n\\
1-n,1\end{matrix};1\right].$$ We have obviously $${\binom{n}{p}}_{k} = 0 \quad \textrm{for} \quad k>n \quad , \quad
{\binom{n}{p}}_{k} ={\binom{n}{n-p}}_{k} \quad , \quad
{\binom{n}{p}}_{n} =\binom{n}{p},$$ the last equation following directly from the Chu - Vandermonde formula.
This definition can be rewritten $${\binom{n}{p}}_{k} = \binom{n}{k} \ \sum_{r \ge 0}
\binom{p}{r}\binom{n-p}{r} \frac{\binom{k-1}{r}}{\binom{n-1}{r}}
= \frac{n}{k} \ \sum_{r \ge 0}
\binom{p}{r}\binom{n-p}{r} \binom{n-r-1}{k-r-1}.$$ Thus $\frac{k}{n}\ {\binom{n}{p}}_{k}$ is a positive integer.
One has easily $${\binom{n}{0}}_{k} = \binom{n}{k} \quad, \quad
{\binom{n}{1}}_{k} = k \binom{n}{k}\quad, \quad
{\binom{n}{2}}_{k} = k \binom{n}{k}
+\frac{n(n-3)}{2} \binom{n-2}{k-2},$$ $${\binom{n}{p}}_{0} = 0 \quad \textrm{for} \quad p \neq 0,n \quad ,
\quad {\binom{n}{p}}_{1} = n,
\quad {\binom{n}{p}}_{2} = \frac{n}{2} \left(n-1+p(n-p)\right),$$ and also $$\begin{split}
{\binom{n}{p}}_{n-1} &= n\left[\binom{n-1}{p-1}
+\binom{n-2}{p}\right],\\
{\binom{n}{p}}_{n-2}
&= \binom{n}{2}\left[\binom{n-2}{p}+\binom{n-2}{p-2}\right]
+\frac{n(n-3)}{2}\binom{n-4}{p-2}.
\end{split}$$ These relations suggest that ${\binom{n}{p}}_{k}$ is a positive integer.
Integrality
===========
Using the Chu - Vandermonde formula $$\binom{n-r-1}{n-k} = \sum_{i = 0}^{n-k} {(-1)}^i \binom{r}{i}
\binom{n-i-1}{n-k-i},$$ we get $${\binom{n}{p}}_{k} =\frac{n}{k} \ \sum_{i = 0}^{n-k}
{(-1)}^i \binom{n-i-1}{k-1} \sum_{r \ge 0}
\binom{p}{r} \binom{n-p}{r} \binom {r}{i}.$$ But using again the Chu - Vandermonde formula one has $$\sum_{r \ge 0} \binom {r}{i}
\binom{p}{r} \binom{n-p}{r} = \binom{p}{i}
\sum_{r \ge 0} \binom {p-i}{r-i} \binom {n-p}{r}
= \binom{p}{i} \binom {n-i}{p}.$$ Hence we obtain $$\begin{split}
{\binom{n}{p}}_{k} &=\frac{n}{k} \ \sum_{i = 0}^{n-k}
{(-1)}^i \binom{n-i-1}{k-1} \binom{p}{i} \binom {n-i}{p}\\
&=\frac{n}{k} \ \sum_{i = 0}^{n-k}
{(-1)}^i \binom{n-i-1}{k-1} \binom{n-i}{i} \binom {n-2i}{p-i}\\
&= \sum_{i = 0}^{n-k} {(-1)}^i \binom{n-i}{k} \frac{n}{n-i}
\binom{n-i}{i} \binom {n-2i}{p-i}.
\end{split}$$
Finally we get $${\binom{n}{p}}_{k}=
\sum_{i = 0}^{n-k} {(-1)}^i \binom{n-i}{k} \binom {n-2i}{p-i}
\left[\binom {n-i}{i} + \binom {n-i-1}{i-1}\right].$$ We have thus proved
The positive number ${\binom{n}{p}}_{k}$ is an integer.
Thanks are due to Jiang Zeng for shortening the proof of this result. Note that the previous relations imply immediatly
$$\begin{split}
{\binom{n}{p}}_{k}+{\binom{n}{p-1}}_{k} &=
\sum_{i = 0}^{n-k} {(-1)}^i \binom{n-i}{k} \binom {n-2i+1}{p-i}
\left[\binom {n-i}{i} + \binom {n-i-1}{i-1}\right]\\
&=\frac{k+1}{n+1} \ {\binom{n+1}{p}}_{k+1} -
\sum_{i = 2}^{n-k} {(-1)}^i \binom{n-i}{k} \binom {n-2i+1}{p-i}
\binom{n-i-1}{i-2}.
\end{split}$$
An intriguing problem is to get a combinatorial interpretation for ${\binom{n}{p}}_{k}$.
Generating functions
====================
The following generating function is due to Jiang Zeng.
We have $$\begin{gathered}
\sum_{k,p \geq 0} {\binom{n}{p}}_{k} x^p y^k = {2}^{-n}
\Big[ \left( (1+x)(1+y)+\sqrt{(1+x)^2(1+y)^2-4x(1+y)} \right)^n \\
+\left((1+x)(1+y)-\sqrt{(1+x)^2(1+y)^2-4x(1+y)}\right)^n \Big].\end{gathered}$$
From equation (2) we get $$\begin{split}
\sum_{k, p \geq 0}{\binom{n}{p}}_{k}\ x^p y^k&= \sum_{k, p \geq 0}
x^p y^k \sum_{i \geq 0}(-1)^i \binom{n-i}{k} \frac{n}{n-i}
\binom{n-i}{i} \binom {n-2i}{p-i}\\
&= \sum_{i \geq 0}
(-1)^i (1+y)^{n-i} \frac{n}{n-i} \binom{n-i}{i} x^i(1+x)^{n-2i}\\
&= (1+x)^n(1+y)^n\sum_{i\geq 0}\frac{n}{n-i}\binom{n-i}{i}z^i,
\end{split}$$ with $z=-x/((1+x)^2(1+y))$. But we have the following identity $$\sum_{n > i}\frac{n}{n-i}\binom{n-i}{i}z^i=\left(\frac{1+\sqrt{1+4z}}{2}\right)^n
+\left(\frac{1-\sqrt{1+4z}}{2}\right)^n.$$
We can give another generating function. The following recurrence relation is needed.
We have $$(n-p+1){\binom{n}{p-1}}_k-p{\binom{n}{p}}_k
=\frac{n}{n-1}(n-2p+1){\binom{n-1}{p-1}}_k.$$
This can be easily deduced from equation (1). Indeed up to $n/k$ the left-hand side can be written $$\begin{split}
&\sum_{r \ge 0} \binom{n-r-1}{k-r-1}
\left[ (n-p+1) \binom{p-1}{r}\binom{n-p+1}{r} -p
\binom{p}{r}\binom{n-p}{r} \right]\\ &=
(n-2p+1)\sum_{r \ge 0} \binom{n-r-1}{k-r-1}
\left[\binom{p-1}{r} \binom{n-p}{r} - \binom{p-1}{r-1}
\binom{n-p}{r-1}\right] \\&=
(n-2p+1)\sum_{r \ge 0} \binom{p-1}{r} \binom{n-p}{r}
\left[\binom{n-r-1}{k-r-1}
- \binom{n-r-2}{k-r-2} \right]\\ &=
(n-2p+1)\sum_{r \ge 0}
\binom{p-1}{r} \binom{n-p}{r}\binom{n-r-2}{k-r-1}.
\end{split}$$
We have $$\sum_{k \ge 1} {\binom{n}{p}}_{k} y^k =
n y (y+1)^p \,{}_2F_1\!\left[\begin{matrix}p+1,p-n+1\\
2\end{matrix};-y\right].$$
By recurrence over the integers $n$ and $p$. The property is true for $p=0$, since we have $$\sum_{k \ge 1} {\binom{n}{0}}_{k} y^k =
\sum_{k \ge 1} \binom{n}{k} y^k = (1+y)^n-1 =
n y \,{}_2F_1\!\left[\begin{matrix}1,1-n\\
2\end{matrix};-y\right].$$ From the previous recurrence relation, we deduce that it is enough to prove $$\begin{gathered}
(n-p+1) \ {}_2F_1\!\left[\begin{matrix}p,p-n\\
2\end{matrix};-y\right]
-p (1+y) \ {}_2F_1\!\left[\begin{matrix}p+1,p-n+1\\
2\end{matrix};-y\right]=\\
(n-2p+1) \ {}_2F_1\!\left[\begin{matrix}p,p-n+1\\
2\end{matrix};-y\right].\end{gathered}$$ But this is a classical contiguity relation for ${}_2F_1$ (see for instance [@Ra], Exercice 21.8, page 71), namely $$(c-b-1) \ {}_2F_1\!\left[\begin{matrix}a,b\\
c\end{matrix};y\right]
-a (1-y) \ {}_2F_1\!\left[\begin{matrix}a+1,b+1\\
c\end{matrix};y\right]
=(c-a-b-1) \ {}_2F_1\!\left[\begin{matrix}a,b+1\\
c\end{matrix};y\right].$$
By identification of the coefficients of $y$ we obtain a property which seems difficult to be proved directly, namely $$\begin{split}
{\binom{n}{p}}_{k}&= \frac{n}{p}
\sum_{i=0}^{k-1} \binom{n-p+i}{n-p} \binom{p}{i+1}
\binom{n-p}{k-i-1}\\ &= \frac{n}{p}
\sum_{i=0}^{n-k} \binom{k-1+i}{n-p} \binom{p}{n-k-i} \binom{n-p}{i}.
\end{split}$$ Hence $\frac{p}{n} {\binom{n}{p}}_{k}$ is a positive integer.
This gives a second proof that ${\binom{n}{p}}_{k}$ is a positive integer. Indeed we get $$\begin{split}
{\binom{n}{p}}_{k} &= \frac{p}{n} {\binom{n}{p}}_{k} +\frac{n-p}{n}
{\binom{n}{n-p}}_{k}\\&=
\sum_{i=0}^{n-k} \left[ \binom{k-1+i}{n-p} \binom{p}{n-k-i} \binom{n-p}{i}
+ \binom{k-1+i}{p} \binom{n-p}{n-k-i} \binom{p}{i} \right].
\end{split}$$ Incidentally we have also proved $$\begin{gathered}
{2}^{-n}
\Big[ \left( (1+x)(1+y)+\sqrt{(1+x)^2(1+y)^2-4x(1+y)} \right)^n \\
+\left((1+x)(1+y)-\sqrt{(1+x)^2(1+y)^2-4x(1+y)}\right)^n \Big] =\\
1+ x^n + n y \sum_{p=0}^n x^p (y+1)^p
\,{}_2F_1\!\left[\begin{matrix}p+1,p-n+1\\2\end{matrix};-y\right].\end{gathered}$$
Theory of partitions
====================
Let us now indicate in which situation the integers ${\binom{n}{p}}_{k}$ are naturally encountered.
A partition $\lambda= ( {\lambda }_{1},...,{\lambda }_{n})$ is a finite weakly decreasing sequence of positive integers, called parts. The number $n=l(\lambda)$ of parts is called the length of $\lambda$, and $|\lambda| = \sum_{i = 1}^{n} \lambda_{i}$ the weight of $\lambda$. For any integer $i\geq1$, ${m}_{i} (\lambda) = \textrm{card} \{j: {\lambda }_{j} = i\}$ is the multiplicity of $i$ in $\lambda$. We set $${z}_{\lambda } = \prod\limits_{i \ge 1}^{}
{i}^{{m}_{i}(\lambda)} {m}_{i}(\lambda) ! .$$
Let $X$ be an indeterminate and $n$ a positive integer. We write ${[X]}_{n } = X (X - 1) \cdots (X -n +1)$ for the lowering factorial and $\binom{X}{n} = {[X]}_{n}/n!$. The following result has been proved in [@La] (Theorem 4, p. 275) and in [@Z] : for any positive integers $n,s$ we have $$\sum_{\left|{\mu }\right| = n}
{\frac{{X}^{l(\mu ) - 1}}
{{z}_{\mu }}} \left({ \sum_{i = 1}^{l(\mu )}
{({\mu }_{i})}_{s} }\right)
= (s - 1) ! \sum_{k = 1}^{\min (n,s)}
\binom{s}{k} \binom{X +n - 1}{n - k} .$$ This property generalizes as follows.
Let $X$ be an indeterminate and $n,r,s$ three positive integers. We have $$\sum_{\left|{\mu }\right| = n}
{\frac{{X}^{l(\mu ) - 1}}
{{z}_{\mu }}} \left( \sum_{i = 1}^{l(\mu )}
{({\mu }_{i})}_{r} {({\mu }_{i})}_{s} \right)
= \frac{r! s!}{r+s} \sum_{k = 1}^{\min (n,r+s)}
{\binom{r+s}{s}}_k \binom{X +n - 1}{n - k} .$$
By recurrence over $r$. For $r=0$ the property has been proved since ${\binom{s}{s}}_k=\binom{s}{k}$. Now one has $(i)_{r+1}(i)_{s}=(i)_{r}(i)_{s+1}+(r-s)(i)_r(i)_s$. Thus it is enough to prove $$\frac{(r+1)! s!}{r+s+1}{\binom{r+s+1}{s}}_k=
\frac{r! (s+1)!}{r+s+1}{\binom{r+s+1}{s+1}}_k+
(r-s)\frac{r! s!}{r+s}{\binom{r+s}{s}}_k.$$ But this is the statement of the Lemma.
This result suggests the following conjecture.
Let $X$ be an indeterminate, $n$ a positive integer and $r=(r_1,\ldots,r_m)$ a positive multi-integer with weight $|r| = \sum_{i = 1}^m r_i$. We have $$\sum_{|\mu| = n}
\frac{X^{l(\mu) - 1}}
{z_{\mu }} \left( \sum_{i = 1}^{l(\mu )}
\prod_{k = 1}^m {(\mu_i)}_{r_k} \right)
= \frac{\prod_{j} r_{j}!}{|r|} \sum_{k = 1}^{\min (n,|r|)}
c_k^{(r)} \binom{X +n - 1}{n - k} ,$$ where the coefficients $c_k^{(r)}$ are positive integers, to be computed.
The integers ${\binom{n}{p}}_{k}$ appear also when studying (shifted) Jack polynomials in the spirit of [@La1; @La2] (this application will be given in a forthcoming paper).
[9]{} M. Lassalle, *Une identité en théorie des partitions*, Journal of Combinatorial Theory, Series A, **89** (2000), 270–288. M. Lassalle, *Some combinatorial conjectures for Jack polynomials*, Ann. Combin. **2** (1998), 61–83. M. Lassalle, *Some combinatorial conjectures for shifted Jack polynomials*, Ann. Combin. **2** (1998), 145–163. E. D. Rainville, *Special functions*, Chelsea, New York (1971). Jiang Zeng, *A bijective proof of Lassalle’s partition identity*, Journal of Combinatorial Theory, Series A, **89** (2000), 289–290.
|
---
abstract: |
For any integer $k>2$, the infinite $k$-bonacci word $W^{(k)}$, on the infinite alphabet is defined as the fixed point of the morphism $\varphi_k:\mathbb{N}\rightarrow \mathbb{N}^2 \cup \mathbb{N}$, where $$\varphi_k(ki+j) = \left\{
\begin{array}{ll}
(ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2,\\
(ki+j+1)& \text{if } j =k-1.
\end{array} \right.$$ The finite $k$-bonacci word $W^{(k)}_n$ is then defined as the prefix of $W^{(k)}$ whose length is the $(n+k)$-th $k$-bonacci number. We obtain the structure of all square factors occurring in $W^{(k)}$. Moreover, we prove that the critical exponent of $W^{(k)}$ is $3-\frac{3}{2^k-1}$. Finally, we provide all critical factors of $W^{(k)}$.
author:
- |
[, [P. Sharifani ${}^{ \textrm{b}}$]{} ]{}\
\
\
\
\
\
[Emails: ghareghani@ut.ac.ir, ghareghani@ipm.ir, pouyeh.sharifani@gmail.com.]{}
bibliography:
- 'mybib.bib'
title: '** On square factors and critical factors of $k$-bonacci words on infinite alphabet**'
---
[*Keywords*]{}: k-bonacci words, words on infinite alphabet, square, critical exponent, critical factor.
Introduction
============
The infinite Fibonacci word and finite Fibonacci words are well-studied in the literature and satisfy several extremal properties, see [@cassaigne2008extremal; @de1981combinatorial; @mignosi1998periodicity; @diekert2010weinbaum; @pirillo1997fibonacci]. The infinite Fibonacci word $F^{(2)}$ is the unique fixed point of the binary morphism $0\rightarrow 01$ and $1\rightarrow 0$. The $n$-th finite Fibonacci word $F_n^{(2)}$ is the prefix of of length $f_{n+2}$ of $F^{(2)}$, where $f_n$ is the $n$-th Fibonacci number. A natural generalization of Fibonacci words are $k$-bonacci words which are defined on the $k$-letter alphabet $\{0,1,\ldots, k-1\}$. The infinite $k$-bonacci word $F^{(k)}$ is the unique fixed point of the morphism $\phi_k(0)=01, \phi_k(1)=02,\ldots , \phi_k(k-2)=0(k-1), \phi_k(k-1)=0$ (see [@tan2007some]). The $n$-th finite $k$-bonacci word $F_n^{(k)}$ is defined to be $\phi^n_k(0)$ or equivalently, the prefix of length $f_{n+k}^{(k)}$ of $F^{(k)}$, where $f_{n+k}^{(k)}$ denotes the $(n+k)$-th $k$-bonacci number. While the Fibonacci words are good examples of binary words, $k$-bonacci words are good examples of words over $k$-letter alphabet and they have many interesting properties (see [@tan2007some; @adamczewski2003balances; @bvrinda2014balances; @glen2006sturmian]).
In [@zhang2017some], authors defined the infinite Fibonacci word on infinite alphabet ${\mathbb N}$ as the fixed point of the morphism $\varphi_2: (2i)\rightarrow (2i)(2i+1)$ and $\varphi_2: (2i+1)\rightarrow (2i+2)$. We denote the infinite Fibonacci word on infinite alphabet by $W^{(2)}$. The $n$-th finite Fibonacci word $W_n^{(2)}$ is then defined similar as $F_n^{(2)}$. It is trivial that if digits (letters) of $W^{(2)}$ are computed mod $2$, then the resulting word is the ordinary infinite Fibonacci word $F^{(2)}$. Zhang et al. studied some properties of word $W^{(2)}$. They studied the growth order and digit sum of $W^{(2)}$ and gave several decompositions of $W^{(2)}$ using singular words. Glen et al. considered more properties of $W^{(2)}$ [@glen2019more]. Among other results, they investigated the structure of palindrome factors and square factors of $W^{(2)}$.In [@ghareghani2019arxive], authors introduced the finite (infinite) $k$-bonacci word over infinite alphabet, for $k>2$. The $n$-th finite (res. infinite) $k$-bonacci word over infinite alphabet is denoted by $W^{(k)}$ (resp. $W_n^{(k)}$). They studied some properties of these words and classified all palindrome factors of $W^{(k)}$, for $k\geq 3$.
For a finite word $W$ and a positive integer $n$, $W^n$ is simply obtained by concatenating the word $W$, $n$ times with itself and $W^{\omega}$ is defined as the concatenation of $W$ with itself, infinitely many times; That is $W^{\omega}=W.W.W \ldots$. For a rational number $r$ with $r.|W|\in \mathbb{N}$, the fractional power $W^r$ is defined to be the prefix of length $r.|W|$ of the infinite word $W^{\omega}$. For example if $W= 0102$ then $W^{\frac{5}{2}}= 0102010201$. The index of a factor $U$ of word $W$ is defined as $$\rm{INDEX}(U,W) = \max\{r \in\mathbb{Q}: U^r\prec W \}.$$ Then the [*critical exponent*]{} $E(W)$ of an infinite word $W$ is given by $$E(W)=\sup \{\rm{INDEX}(U,W) : U\in F(W)\setminus \{\epsilon\} \}.$$ A word $U$ is a critical factor of $W$ if $E(W)=\rm{INDEX}(U,W).$ The study of the existence of a factor of the form $U^r$ in a long word and specially computing the critical exponent of a long word is the subject of many papers for example see [@mignosi1989infinite; @vandeth2000sturmian; @berstel1999index; @carpi2000special; @damanik2002index; @justin2001fractional; @blondin2007critical]. Specially, in the case of infinite $k$-bonacci word $F^{(k)}$, it is proved that $E(F^{(k)})=2+\frac{1}{\alpha_k-1}$ (see [@glen2009episturmian]), where $\alpha_k$, the $k$-th generalized golden ratio, is the (unique) positive real root of the $k$-th degree polynomial $x^k-x^{k-1}-\ldots -x-1$. It is proved that $2-\frac{1}{k}< \alpha_k<1$ [@dresden2014simplified; @hare2014three]. Hence, $3<E(F^{(k)})<3+\frac{1}{k-1}$, and $E(F^{(2)})=2+\frac{\sqrt{5}+1}{2}$.
In this work we first investigate some properties of $W_n^{(k)}$. Then, using them, we explore the structure of all square factors of $W_n^{(k)}$. More precisely, we prove that all square factors of $W^{(k)}$ are of the form $ki\oplus C^j(W_n^{(k)})$, for some integers $i>0$ and $j\geq 0$, where $C^j(U)$ denoted the $j$-th conjugate of word $U$. Finally, using the structure of square factors of $W^{(k)}$, we prove that the critical exponent of $W^{(k)}$ is $3-\frac{3}{2^k-1}$.
Preliminaries
=============
In this section we give more definitions and notations that are used in the paper. We denote the alphabet, which is a finite or countable infinite set, by ${\mathcal A}$. When ${\mathcal A}$ is a countable infinite set, we simply take ${\mathcal A}={\mathbb N}$; Then each element of ${\mathcal A}$ is called a digit (instead of a letter). We denote by ${\mathcal A}^*$ the set of finite words over ${\mathcal A}$ and we let ${\mathcal A}^+ ={\mathcal A}^* \setminus \{\epsilon\}$, where $\epsilon$ the empty word. We denote by ${\mathcal A}^{\omega}$ the set of all infinite words over ${\mathcal A}$ and we let ${\mathcal A}^{\infty} ={\mathcal A}^*\cup {\mathcal A}^{\omega}$. If $a\in{\mathcal A}$ and $W \in {\mathcal A}^{\infty}$, then the symbols $|W|$ and ${|W|}_a$ denote the length of $W$, and the number of occurrences of letter $a$ in $W$, respectively.
For a finite word $W=w_1 w_2 \ldots w_{n}$, with $w_i \in {\mathcal A}$ and for $1\leq j\leq j'\leq n$, we denote $W[j,j']=w_j \ldots w_{j'}$, and for simplicity we denote $W[j,j]$ by $W[j]$. Let $U_i\in {\mathcal A}^{*}$, for $1 \leq i \leq n$, then $\prod_{i=n}^{1} U_i$ is defined to be $U_n U_{n-1} \ldots U_1$. For a finite word $W$ and an integer $n$, $n \oplus W$ denotes the word obtained by adding $n$ to each digit of $W$. For example, let $W=01020103$ and $n=5$, then $n \oplus W=56575658$. Similarly, if every digit of $W$ is grater than $n-1$, then $W \ominus n$ denotes the word obtained by subtracting $n$ from each digit of $W$.
A word $V\in {\mathcal A}^+$ is a factor of a word $W \in {\mathcal A}^{\infty}$, if there exist $U\in {\mathcal A}^*$ and $U' \in {\mathcal A}^{\infty}$, such that $W=UVU'$. Similarly, a word $V\in {\mathcal A}^{\infty}$ is a factor of $W \in {\mathcal A}^{\infty}$ if there exists $U\in {\mathcal A}^*$ such that $W=UV$. When $V$ is a factor of $W$ then we denote it as $V\prec W$. A word $V\in {\mathcal A}^+$ (resp. $V\in {\mathcal A}^{\infty}$) is said to be a [*p*refix]{} (resp. [*s*uffix]{}) of a word $W\in {\mathcal A}^{\infty}$, denoted as $V\lhd W$ (resp. $V\rhd W$), if there exists $U\in {\mathcal A}^{\infty}$ (resp. $U\in {\mathcal A}^{*}$) such that $W=VU$ (resp. $W=UV$). If $W\in {\mathcal A}^{*}$ and $W=VU$ (resp. $W=UV$,) we write $V=WU^{-1}$ (resp. $V=U^{-1}W$). The set of all factors of a word $ w $ is denoted by $ F(w) $. If $W=w_1\ldots w_n$ be a finite word and $0\leq j\leq n-1$, then the [*$j$-th conjugatae*]{} of $W$ is defined as $C^j(W)=w_{j+1}\ldots w_nw_1 \ldots w_j$. For example the word $0130102$ is the $4$-th conjugate of $0102013$. A word $ V $ is a [*conjugate*]{} of $ W $ if there exists $0\leq j\leq n-1$ such that $V= C^j(W)$. A factor of the form $UU$ in $W$ is called a square factor or simply a square. For a square factor $UU=W[t,t+2|u|]$ of $W$, the [*center*]{} of the square $UU$ in $W$ is defined to be $c_s(U^2,W)=t+|U|+\frac{1}{2}$.
The $n$-th $k$-bonacci number defined as
$$\label{defkbonum}
f_n^{(k)} = \left\{
\begin{array}{ll}
0\;\; &\text{if } \,\, n = 0,\cdots ,k-2, \\
1\;\; &\text{if } \,\, n = k-1, \\
\sum_{i=n-1}^{n-k}f_i^{(k)}\;\; &\text{if} \,\, n\geq k.
\end{array} \right.$$
The finite (resp. infinite) $k$-bonacci words $W^{(k)}_n$ (resp. $W^{(k)}$) on infinite alphabet $\mathbb{N}$ is defined in [@ghareghani2019arxive], using the morphism $\varphi_k$ given below $$\varphi_k(ki+j) = \left\{
\begin{array}{ll}
(ki)(ki+j+1)\;\; & \text{if } j = 0,\cdots ,k-2 \\
(ki+j+1)& \text{otherwise } .
\end{array} \right.$$ More precisely, $W^{(k)}_n=\varphi_k^n(0)$ and $W^{(k)}=\varphi_k^{\omega}(0)$ (Note that $W^{(k)}_0=F^{(k)}_0=0$). For a fixed value of $k$, the $k$-bonacci words over infinite alphabet are reduced to $k$-bonacci words over finite alphabet when the digits are calculated $\mod\; k$. It is easy to show that for $n \geq 0$, $$\label{size}
|F^{(k)}_n|= |W_n^{(k)}|=f_{n+k}^{(k)}.$$
Some properties of $W_n^{(k)}$
==============================
In this section we provide some basic properties $W_n^{(k)}$, some of which are proved in [@ghareghani2019arxive]. All of these properties are useful for the rest of the work.
[\[Lemma 4 of [@ghareghani2019arxive]\]]{}\[00\] Let $n \geq 0$ and $k>2$. The finite word $W_n^{(k)}$ contains no factor $00$.
Following two lemmas give recursive formulas for computing $W^{(k)}_n$.
[\[Lemma 5 of [@ghareghani2019arxive]\]]{} \[struct1\] For $1 \leq n \leq k-1$, $$\label{struct1eq}
W^{(k)}_n=\prod_{i=n-1}^{0}W^{(k)}_i \, \, n.$$
[\[Lemma 7 of [@ghareghani2019arxive]\]]{}\[struct\] For $n \geq k$, $$\label{structeq}
W^{(k)}_n=\prod_{i=n-1}^{n-k+1}W^{(k)}_i \, (k\oplus W^{(k)}_{n-k}).$$
The following corollary is a direct consequence of Lemmas \[struct1\] and \[struct\] and can be proved using induction on $i$.
\[corki\] Let $i$ and $n$ be two non-negative integers, then $W^{(k)}_n\oplus ki \prec W^{(k)}_{n+ki}$.
Considering the recurrence relations (\[struct1eq\]) and (\[structeq\]) we have the following definitions which are very useful in the next sections.
\[defborder\] Let $j$ be a nonnegative integer, then a factor $A$ of $W^{(k)}_n$ is called a [*bordering factor of type $j$*]{}, for some $n-k+1\leq j\leq n-1$ if $j0 \prec A \prec W^{(k)}_jW^{(k)}_{j-1} \ldots W^{(k)}_{m}$, where $m=\max \{0, n-k+1\}$. Moreover, a [*bordering square factor*]{} of $W^{(k)}_n$ is a bordering factor of $W^{(k)}_n$ which is also a square.
\[defstrad\] Let $n\geq k$, then a factor $A$ of $W^{(k)}_n$ is called a [*straddling factor*]{} of $W^{(k)}_n$ if $A=A_1A_2$, for some nonempty words $A_1$ and $A_2$, with $A_1\rhd W^{(k)}_{n-1} \ldots W^{(k)}_{n-k+1}$ and $A_2\lhd k\oplus W^{(k)}_{n-k}$. Moreover, if a straddling factor of $W^{(k)}_n$ is also a square, it is called an [*straddling square factor*]{}.
[\[Lemma 10 of [@ghareghani2019arxive]\]]{}\[lastdig\] For any $n\geq 1$, the digit $n$ is the largest digit of $W^{(k)}_n$ and appears once at the end of this word.
\[i0\] For every integer $i<n$ we have $i0\prec W^{(k)}_n$.
[ Since $i+1\leq n$, we have $W^{(k)}_{i+1}\prec W^{(k)}_n$. By Lemmas \[struct1\] and \[struct\], $W^{(k)}_i W^{(k)}_{i-1}\lhd W^{(k)}_{i+1}$. Hence, $i0 \prec W^{(k)}_{i+1}\prec W^{(k)}_n$ and the result follows. ]{}
\[i01\] Let $0<i<n$ and $i0= W^{(k)}_n[t,t+1]$, for some $t \in \mathbb{N}$. Then we have $$W^{(k)}_n[t-|W^{(k)}_i|+1,t+1]=W^{(k)}_i0.$$ In other words, if $i0$ appears in $W^{(k)}_n$, then this $i$ appeared as the last digit of a factor $W^{(k)}_i$ of $W^{(k)}_n$.
We prove this by induction on $n$. If $n=2$, then $W^{(k)}_2=0102$ and in this case the only possibility for $t$ is $t=2$ and $W^{(k)}_2[1,3]=W^{(k)}_1 0=010$, as desired. We suppose that the claim is true for all $m\leq n$ we want to prove this for the case $n+1$. If $n+1\geq k$, then by Lemma \[struct\] we have $$W^{(k)}_{n+1}=\prod_{t=n}^{n-k+2}W^{(k)}_t \, (k\oplus W^{(k)}_{n+1-k}).$$ Let $i0$ occurs in $W^{(k)}_{n+1}$, then either $i0\prec W^{(k)}_t$, for some $n-k+2\leq t \leq n$, or $i0$ is a bordering factor of $W^{(k)}_{n+1}$. If $i0\prec W^{(k)}_t$, then by induction hypothesis this $i$ should be the last digit of some factor $W^{(k)}_i$ of $W^{(k)}_{n+1}$. If $i0$ is a bordering factor of $W^{(k)}_{n+1}$, then it is clear that $i$ is the end digit of a factor $W^{(k)}_i$ of $W^{(k)}_{n+1}$.
In the case $n+1<k$, using similar argument as the previous case and Lemma \[struct1\] we obtain the result.
\[lemsuff2w\] Let $2<k<n$ and $B\rhd W_n^{(k)}$ with $|B|=|W_{n-k}^{(k)}|+|W_{n-k-1}^{(k)}|$. Then
- If $n=k+1$, then $|B|_2=1$.
- If $n>k+1$, then $|B|_0>0$.
<!-- -->
- If $n=k+1$, then by (\[struct1eq\]), $B=2.k.(k+1)$, so the result follows.
- If $n>k+1$, then by (\[structeq\]), $$\label{eqsuff2w}
W_{n-k+1}^{(k)}(k\oplus W_{n-k}^{(k)})\rhd W_{n}^{(k)}.$$ Let $D\rhd W_{n-k+1}^{(k)}$ and $|D|=|W_{n-k-1}^{(k)}|$. Then by (\[eqsuff2w\]), to prove the lemma it is suffices to show that $|D|_0>0$. If $k=3$, when $n=5,6$ it is clear that $|D|_0>0$. So, If $n\geq 7$, then by Equation (\[structeq\]), we have $$\begin{aligned}
\nonumber
W_{n-2}^{(n-k+1)}= W_{n-2}^{(k)}=& W_{n-3}^{(k)} W_{n-4}^{(k)} (k\oplus W_{n-5}^{(k)})\\
=& W_{n-3}^{(k)} W_{n-5}^{(k)} \underbrace{W_{n-6}^{(k)} (k\oplus W_{n-7}^{(k)}) (k\oplus W_{n-5}^{(k)})}_{|W_{n-4}^{(k)}|=|D|}\label{eqsuffk=3}
\end{aligned}$$ By (\[eqsuffk=3\]), it is clear that $|D|_0>0$.
If $k>3$ and $n<2k-1$, then by Lemma \[struct1\], we have $0(n-k+1)\rhd W_{n-k+1}^{(k)}$. Since $n>k+1$, we $|W_{n-k-1}^{(k)}|\geq 2$ and hence, $|D|_0>0$.
If $k>3$ and $n\geq 2k-1$, then by Lemma \[struct\], $W_{n-2k+2}^{(k)}(k\oplus W_{n-2k+1}^{(k)})\rhd W_{n-k+1}^{(k)}$. Since $k>3$, $|W_{n-k+1}^{(k)}|>|W_{n-2k+2}^{(k)}|+|W_{n-2k+1}^{(k)}|$ and $|W_{n-2k+2}^{(k)}|_0>0$, we conclude that $|D|_0>0$.
\[lemBnotfactW\] Let $n,k$ and $j$ be nonnegative integers with $3\leq k\leq n$ and $n-k+3\leq j\leq n$. Then $B=W_{n-k+2}^{(k)}k$ is not a factor of $W_j^{(k)}$.
If $j<k$, then $|W_j^{(k)}|_k=0$ and so $B$ is not a factor of $W_j^{(k)}$.
Hence, we shall prove the result for $k\leq j\leq n$. We prove this part by bounded induction on $j$. Let $p=\max \{k, n-k+3\}$. Since, $j\geq k$ and $n-k+3\leq j$, the first step of induction is $j=p$. If $p=k$, then the only occurrence of $k$ in $W_{j}^{(k)}$, is in its last digit, we conclude that if $B\prec W_{j}^{(k)}$, then $B\rhd W_{j}^{(k)}=W_{k}^{(k)}$. Using Lemma \[struct\], we have $W_{j}^{(k)}=W_{j-1}^{(k)}\ldots W_{1}^{(k)}k$. Since $(n-k+2)k \rhd B \rhd W_{k}^{(k)}$, we provide $n-k+2=1$, so $n=k-1<k$, which is a contradiction. If $p=n-k+3$, then by (\[structeq\]), we have $$\label{eq2box}
W_{n-k+3}^{(k)}={ W_{n-k+2}^{(k)}} W_{n-k+1}^{(k)}\ldots W_{n-2k+4}^{(k)} {(k\oplus W_{n-2k+3}^{(k)})}$$ If $B\prec W_{j}^{(k)}=W_{n-k+3}^{(k)}$, then there exist integers $s$ and $t$ such that $B=W_{n-k+3}^{(k)}[s,t+1]$, $W_{n-k+3}^{(k)}[t]=n-k+2$ and $W_{n-k+3}^{(k)}[t+1]=k$. Using Lemma \[lastdig\] and Equation (\[eq2box\]), either $t=|W_{n-k+2}^{(k)}|$ or $t\geq |W_{n-k+3}^{(k)}|-|W_{n-2k+3}^{(k)}|+1$. If $t=|W_{n-k+2}^{(k)}|$, then $W_{n-k+3}^{(k)}[t+1]=0$, which is a contradiction. If $t\geq |W_{n-k+3}^{(k)}|-|W_{n-2k+3}^{(k)}|+1$, then using (\[eq2box\]) and the fact that $|W_{n-k+3}^{(k)}|\leq 2|W_{n-k+2}^{(k)}|$ and $B=|W_{n-k+2}^{(k)}|+1$, we conclude that $s< |W_{n-k+2}^{(k)}|$. Which implies that $|W_{n-k+3}^{(k)}[s,t+1]|_{n-k+2}\geq 2$. But by definition of $B$ and using Lemma \[lastdig\], we have $|B|_{n-k+2}=1$, which is a contradict. Therefore, the first step of induction is true.
We are going to prove that $B$ is not a factor of $W_{j+1}^{(k)}$. For contrary let $B\prec W_{j+1}^{(k)}$. By (\[structeq\]), we have $$\label{eqrecwj+1}
W_{j+1}^{(k)}= W_{j}^{(k)}\ldots W_{j-k+2}^{(k)} (k\oplus W_{j-k+1}^{(k)}).$$
By induction hypothesis $B$ is not a factor of $W_{i}^{(k)}$ for $j-k+1\leq i\leq j$. Since $B$ contains the digit $0$ and $(k\oplus W_{j-k+1}^{(k)})$ does not contain it, there are two following possible cases for $B$:
- [**Case 1.**]{} $B$ is a bordering factor of $W_{j+1}^{(k)}$; Let ${\ell}$ be largest integer such that $B\prec W_{j}^{(k)}\ldots W_{\ell}^{(k)}$. Since for every integer $i$, $W_{i}^{(k)}$ start with $0$, we have $(n-k+2)k\prec W_{\ell}^{(k)}$ which means that ${\ell}> n-k+2$. Therefore, $W_{n-k+3}^{(k)}\lhd W_{\ell}^{(k)}$. Hence, $W_{n-k+2}^{(k)}0\lhd W_{\ell}^{(k)}$. Since $(n-k+2)k\prec W_{\ell}^{(k)}$, there exists integer $\alpha$ such that $W_{\ell}^{(k)}[\alpha,\alpha+1]=(n-k+2)k$. By Lemma \[lastdig\], $\alpha>|W_{n-k+2}^{(k)}|$. Therefore, $B\prec W_{\ell}^{(k)}$, which contradicts to the definition of bordering factor. Hence, $B$ is not a bordering factor of $W_{j+1}^{(k)}$.
- [**Case 2.**]{} $B$ is a straddling factor of $W_{j+1}^{(k)}$; By definition of straddling factor, there exists nonempty word $S$ which is the suffix of $B$ and a prefix of $k\oplus W_{j-k+1}^{(k)}$. Since $j<n$, we have $j-k+2<n-k+2$ and hence using (\[eqrecwj+1\]), we provide that the last two digits of $B$ occur in $k\oplus W_{j-k+1}^{(k)}$. Let $|S|=t+1$, for some $t>0$, this means that $(k\oplus W_{j-k+1}^{(k)})[t,t+1]=(n-k+2)k$. By definition of $S$ we obtain $$\begin{aligned}
\label{eqbs-1}
BS^{-1}\rhd& W_{j+1}^{(k)}[(k\oplus W_{j-k+1}^{(k)})]^{-1}\end{aligned}$$ On the other hand, since $(n-k+2)\prec k\oplus W_{j-k+1}^{(k)}$, we have $n-k+2\geq k$, or $n\geq 2k-2$. If $n=2k-2$, then $W_{j-k+1}^{(k)}[t,t+1]=00$, which contradicts to Lemma \[00\]. Therefore, $n\geq 2k-3$, now, using Equations (\[struct1eq\]) and (\[structeq\]), and the fact that $n-k+3\leq j$, we have $$\label{eqprefWj+1}
W_{n-2k+2}^{(k)} W_{n-2k+1}^{(k)}\lhd W_{n-2k+3}^{(k)}\lhd W_{j-k+1}^{(k)}$$ By Lemma \[lastdig\] and Equation (\[eqprefWj+1\]), we conclude that either $t= |W_{n-2k+2}^{(k)}|$ or $t>|W_{n-2k+2}^{(k)}|+|W_{n-2k+1}^{(k)}| $. First suppose that $t=|W_{n-2k+2}^{(k)}|$. Then by (\[eqprefWj+1\]), we have $S=k\oplus (W_{n-2k+2}^{(k)}0)$. Using (\[eqbs-1\]), we provide $$\begin{aligned}
B[(k\oplus W_{n-2k+2}^{(k)})k]^{-1}\rhd& W_{j+1}^{(k)}[(k\oplus W_{j-k+1}^{(k)})]^{-1}\\
W_{n-k+1}^{(k)}\ldots W_{n-2k+3}^{(k)}\rhd& W_{j}^{(k)}\ldots W_{j-k+2}^{(k)}
\end{aligned}$$ Hence, $n-2k+3=j-k+2$, or $j=n-k+1$, which contradicts to our assumption $n-k+3\leq j$. Now, suppose that $t>|W_{n-2k+2}^{(k)}|+|W_{n-2k+1}^{(k)}| $. Let $D\rhd B$, and $|D|=|W_{n-2k+2}^{(k)}|+|W_{n-2k+1}^{(k)}|$, then $D\prec (k\oplus W_{j-k+1}^{(k)})$. On the other hand using Lemma \[lemsuff2w\] either $|D|_0>0$ or $|D|_2>0$, which is a contradiction.
Squares in $W_n^{(k)}$
======================
In this section we give the structure of all square factors of $W_n^{(k)}$. We first prove that when $n<2k-1$, $W_n^{(k)}$ has no square factor. Then we characterize all square factors of $W^{(k)}$.
\[bordersqu\] For two positive integers $n$ and $k$, there is no bordering square in $W^{(k)}_n$.
[ For contrary suppose that there exists $n-k+2 \leq j \leq n-1$, for which $W^{(k)}_n$ contains a bordering square of type $j$; we denote this word by $A$. By Definition \[defborder\], $j0 \prec A \prec W^{(k)}_jW^{(k)}_{j-1} \ldots W^{(k)}_{n-k+1}$. Since $A$ is a square word, so $|A|_j\geq 2$. but by Lemma \[lastdig\], $$|W^{(k)}_jW^{(k)}_{j-1} \ldots W^{(k)}_{n-k+1}|_j=1.$$ This is a contradiction.]{}
\[sq<k+1\] If $n<k+1$, then $W^{(k)}_{n}$ contains no square factor.
[ We prove this by bounded induction on $n$. By definition $W^{(k)}_0=0$ does not contain any square. Suppose that for any integer $i$, $0\leq i\leq n<k$, $W^{(k)}_n$ does not contain any square. For contrary suppose that $B$ is a square factor of $W^{(k)}_{n+1}$. By (\[struct1eq\]) and (\[structeq\]) we have $$\label{wn+1sq}
W^{(k)}_{n+1} = \left\{
\begin{array}{ll}
W^{(k)}_{n+1}=W^{(k)}_{n}W^{(k)}_{n-1}\ldots W^{(k)}_0 (n+1)\;\; &\text{if } \,\, n+1<k, \\
W^{(k)}_{n+1}=W^{(k)}_{n}W^{(k)}_{n-1}\ldots W^{(k)}_1 (n+1)\;\; &\text{if } \,\, n+1=k.
\end{array} \right.$$ Using induction hypothesis and Lemma \[bordersqu\], we provide that $B$ is a straddling square. By Definition \[defstrad\] and Equation (\[wn+1sq\]), $|B|_{n+1}\geq 2$, which contradicts with Lemma \[lastdig\]. ]{}
\[lemcs>\] let $n> k$ and $A^2$ be a straddling square of $W^{(k)}_{n}$. Then $c_s(A^2,W^{(k)}_{n})> |W^{(k)}_{n}|-|W^{(k)}_{n-k}|$.
By (\[structeq\]), we have $$\label{eqcssq1}
W^{(k)}_n=W^{(k)}_{n-1} \ldots W^{(k)}_{n-k+2} \boxed{ W^{(k)}_{n-k+1}}(k\oplus W^{(k)}_{n-k})$$ For contrary suppose that $c_s(A^2,W^{(k)}_{n})< |W^{(k)}_{n}|-|W^{(k)}_{n-k}|$. Let $A^2=W^{(k)}_{n}[s_1,s_2]$. Since $A^2$ is a straddling square of $W^{(k)}_{n}$, $s_2\geq |W^{(k)}_{n}|-|W^{(k)}_{n-k}|+1$ and $s_1< |W^{(k)}_{n}|-|W^{(k)}_{n-k}|$.
If $c_s(A^2,W^{(k)}_{n})< |W^{(k)}_{n}|-|W^{(k)}_{n-k}|-|W^{(k)}_{n-k+1}|$, then using (\[eqcssq1\]), we conclude that $B=W^{(k)}_{n-k+1}k\prec W^{(k)}_{n-1} \ldots W^{(k)}_{n-k+2}$. By Lemma \[lemBnotfactW\] for any $n-k+2\leq j\leq n-1$, $B$ is not a factor $W^{(k)}_{j}$. Let $s<n$ be largest integer such that $B\prec W_{n-1}^{(k)}\ldots W_{s}^{(k)}$. Since $0 \lhd W_{s}^{(k)}$, we have $(n-k+1)k\prec W_{s}^{(k)}$. Let $W_{s}^{(k)}[\alpha,\alpha+1]=(n-k+1)k$. By Lemma \[lastdig\], $s> n-k+1$. Therefore, $W_{n-k+1}^{(k)}0\lhd W_{n-k+2}^{(k)}\lhd W_{s}^{(k)}$ and by Lemma \[lastdig\], $\alpha>|W_{n-k+1}^{(k)}|$. Therefore, $B\prec W_{s}^{(k)}$, which contradicts to Lemma \[lemBnotfactW\]. Hence, $$\label{eqW=UU}
|W^{(k)}_{n}|-|W^{(k)}_{n-k}|-|W^{(k)}_{n-k+1}|+1<c_s(A^2,W^{(k)}_{n})< |W^{(k)}_{n}|-|W^{(k)}_{n-k}|.$$ This means that the center of $A^2$ happens in $W^{(k)}_{n-k+1}$ which is distinguished by a box in (\[eqcssq1\]). We denote the first occurrences of $A$ in $A^2$ by $A_1$ and the last occurrence of $A$ in $A^2$ by $A_2$. By (\[eqW=UU\]), we conclude that there exist non-empty words $U_1$ and $U_2$, such that $W^{(k)}_{n-k+1}=U_1 U_2$ and $U_2 \lhd A_2$. By Lemma \[lastdig\], $|U_1|_{n-k+1}=0$ and the only occurrence of $n-k+1$ in $U_2$. which is its last digit. We conclude that $|A_2|_{n-k+1}>0$ and all digits of $A_2$ which appear after $n-k+1$ are greater than $k-1$. Hence $|A_1|_{n-k+1}>0$ and all digits of $A_1$ which appear after $n-k+1$ should be also greater than $k-1$. Since $|U_1|_{n-k+1}=0$, we conclude that $U_1 \rhd A_1$ and all occurrence of $n-k+1$ are before the first digit of $U_1$. But this is a contradiction, because $|U_1|_0>0$ and hence there is a digit $0$ which appears after all digit $n-k+1$ in $A_1$.
\[corcentersqcor\] let $A^2$ be a straddling square of $W^{(k)}_{n}$. Then for each $i\leq k-1$, $|A|_i=0$.
[By Lemma \[lemcs>\], $c_s(A^2,W^{(k)}_{n})> |W^{(k)}_{n}|-|W^{(k)}_{n-k}|$. Therefore, using Equation \[structeq\], we conclude that $A\prec (k\oplus W^{(k)}_{n-k})$. This means that all digits of $A$ are greater than $k-1$, as desired. ]{}
\[ww<w\] Let $i<n$ and $n \geq k+1$, then $W^{(k)}_{n}$ contains no factor of the form $W^{(k)}_{i}W^{(k)}_{i}$.
[ We prove this by induction on $n$. If $n=1$, then $W^{(k)}_1=01$ contains no factor $W^{(k)}_0 W^{(k)}_0=00$. By (\[structeq\]), we have $W^{(k)}_n=\prod_{i=n-1}^{n-k+1}W^{(k)}_i \, (k\oplus W^{(k)}_{n-k}).$ For the contrary suppose that $W^{(k)}_{i}W^{(k)}_{i}\prec W^{(k)}_n$ for some $i<n$. By induction hypothesis for any $j<n$, $W^{(k)}_{i}W^{(k)}_{i}$ is not a factor of $W^{(k)}_j$. Now, by Definitions \[defborder\] and \[defstrad\], $W^{(k)}_{i}W^{(k)}_{i}$ should be either a bordering square or a straddling square. By Lemma \[bordersqu\], $W^{(k)}_n$ contains no bordering square. Therefore, $W^{(k)}_{i}W^{(k)}_{i}$ is a straddling square of $W^{(k)}_n$ which can not be occurred by Corollary \[corcentersqcor\]. Hence, there is no factor of the form $W^{(k)}_{i}W^{(k)}_{i}$ in $W^{(k)}_{n}$. ]{}
\[lemsq<2k-1\] If $n<2k-1$, then $W^{(k)}_{n}$ contains no square factor.
[For contrary suppose that there exists a straddling square $A^2$ in $W^{(k)}_{n}$. By definition of straddling factor $A$ contains the digit $n-k+1$. Hence by Corollary \[corcentersqcor\], $n-k+1\geq k$ and $n\geq 2k-1$, which is a contradiction. ]{}
\[lemsq=2k-1\] $kk$ is the only square of $W^{(k)}_{2k-1}$.
[Let $A^2=W^{(k)}_{2k-1}[t,t+|A^2|]$ be a square factor of $W^{(k)}_{2k-1}$. By Lemmas \[lemsq<2k-1\] and \[sq<k+1\], we conclude that for every $j< 2k-1$, $W^{(k)}_{j}$ contains no square factor. By Lemma \[bordersqu\], $W^{(k)}_{2k-1}$ has no bordering square. Hence, $A^2$ is a straddling square. $$\label{eqrecW2k-1}
W^{(k)}_{2k-1}=W^{(k)}_{2k-2} \ldots W^{(k)}_{k}(k\oplus W^{(k)}_{k-1})$$ By Corollary \[corcentersqcor\] and definition of straddling factor of $W^{(k)}_{2k-1}$, $t=|W^{(k)}_{2k-1}|-|W^{(k)}_{k-1}|$. Hence, $kk\lhd A^2$. Hence, either $A^2=kk$ or the number of occurrences of $kk$ in $A^2$ is at least two. If the number of occurrences of $kk$ in $A^2$ is at least two, then $kk\prec (k\oplus W^{(k)}_{k-1})$ which means that $00\prec W^{(k)}_{k-1}$, which contradicts with Lemma \[sq<k+1\]. Hence, the only possibility for $A^2$ is $kk$. ]{}
\[lemsq=strad\] Let $A^2$ be a square of $W_m^{(k)}$. Then there exists integers $i$ and $n\leq m$ such that $A^2\ominus ki$ is a straddling square of $W_n^{(k)}$.
[We prove this using induction on $m$. If $m<2k-1$, then by Lemma \[lemsq<2k-1\], $W_m^{(k)}$ contains no square factor and the result follows. If $m=2k-1$, then by Lemma \[lemsq=2k-1\], $kk$ is the only square of $W_m^{(k)}$ which is a straddling square. If $m>2k-1$, then by (\[structeq\]) and using the induction hypothesis and Lemma \[bordersqu\] the result follows. ]{}
The following corollary is a direct consequence of Lemma \[lemsq=strad\].
\[corsq=strad\] Let $A^2$ be a square of $W^{(k)}$. Then there exists integers $i$ and $n$, such that $A^2\ominus ki$ is a straddling square of $W_n^{(k)}$.
\[centersq2\] let $A^2$ be a straddling square of $W^{(k)}_{n}$. Then $$c_s(A^2,W^{(k)}_{n})\leq |W^{(k)}_{n}|-|W^{(k)}_{n-k}|+|W^{(k)}_{n-2k+1}|+\frac{1}{2}.$$
[ By (\[structeq\]), we have $$\begin{aligned}
W^{(k)}_n&=\prod_{i=n-1}^{n-k+1}W^{(k)}_i \, (k\oplus W^{(k)}_{n-k})\\
&= \prod_{i=n-1}^{n-k+2}W^{(k)}_i (W^{(k)}_{n-k}. \ldots . W^{(k)}_{n-2k+2}) \, (k\oplus W^{(k)}_{n-2k+1})\, (k\oplus W^{(k)}_{n-k}).\label{structsqeq}\end{aligned}$$ We remined that $k\oplus W^{(k)}_{n-2k+1} \prec k\oplus W^{(k)}_{n-k}$. Hence, if $c_s(A^2,W^{(k)}_{n})> |W^{(k)}_{n}|-|W^{(k)}_{n-k}|+|W^{(k)}_{n-2k+1}|$, then $k\oplus ((n-2k+1) W^{(k)}_{n-2k+1})\prec k\oplus W^{(k)}_{n-k}$ it means that $ (n-2k+1) W^{(k)}_{n-2k+1}\prec W^{(k)}_{n-k}$. Using Lemma \[i0\] we conclude that $ W^{(k)}_{n-2k+1} W^{(k)}_{n-2k+1}\prec W^{(k)}_{n-k}$ which contradicts to Lemma \[ww<w\]. ]{}
\[centersq3\] let $A^2=W^{(k)}_{n}[t,t+j]$ be a straddling square of $W^{(k)}_{n}$. Then $t> |W^{(k)}_{n}|-|W^{(k)}_{n-k}|-|W^{(k)}_{n-2k+1}|$.
[ For contrary suppose that $A^2=W^{(k)}_{n}[t,t+j]\prec W^{(k)}_{n}$ for some $t< |W^{(k)}_{n}|-|W^{(k)}_{n-k}|-|W^{(k)}_{n-2k+1}|$. By Lemma \[lemcs>\], $c_s(A^2,W^{(k)}_{n})> |W^{(k)}_{n}|-|W^{(k)}_{n-k}|$. Hence, $$\label{A2<k+w}
W^{(k)}_{n}[t_1,t_2]\prec A \prec k\oplus W^{(k)}_{n-k}.$$ Where $t_1=|W^{(k)}_{n}|-|W^{(k)}_{n-k}|-|W^{(k)}_{n-2k+1}|-1$ and $t_2= |W^{(k)}_{n}|-|W^{(k)}_{n-k}|$. By equation (\[structsqeq\]) an definition of $t_1$ and $t_2$, we have $W^{(k)}_{n}[t_1,t_2]= (n-2k+2)(k\oplus W^{(k)}_{n-2k+1})$. Therefore, by Equation (\[A2<k+w\]) $(n-2k+2)(k\oplus W^{(k)}_{n-2k+1})\prec k\oplus W^{(k)}_{n-k}$, this means that $n-2k+2\geq k$. So, $(n-3k+2) W^{(k)}_{n-2k+1}\prec W^{(k)}_{n-k}$. Now, by Lemma \[i0\], we conclude that $ W^{(k)}_{n-3k+2}W^{(k)}_{n-2k+1}\prec W^{(k)}_{n-k}$. Since, $ W^{(k)}_{n-3k+2} \lhd W^{(k)}_{n-2k+1}$ we conclude that $ W^{(k)}_{n-3k+2}W^{(k)}_{n-3k+2}\prec W^{(k)}_{n-k}$, which is impossible by Lemma \[ww<w\]. ]{}
By Lemma \[centersq2\], $A^2 \prec k\oplus(W^{(k)}_{n-2k+1} W^{(k)}_{n-k})$. Now using Equation(\[structeq\]), we have $$A^2 \prec k\oplus \big(W^{(k)}_{n-2k+1} W^{(k)}_{n-k}\big)
= k\oplus \big(W^{(k)}_{n-2k+1} W^{(k)}_{n-2k+3} [(W^{(k)}_{n-2k+3})^{-1}W^{(k)}_{n-k}]\big)$$
\[defV\] Let $n,k$ be two nonnegative integers with $k\geq 3$ and $n> 2k-1$. We define the word $V^{(k)}_{n}$ as follows:
- If $2k-1< n< 3k-2$, then $V^{(k)}_{n}=W^{(k)}_{n-2k}W^{(k)}_{n-2k-1}\ldots W^{(k)}_{0}$;
- If $n\geq 3k-2$, then $V^{(k)}_{n}=W^{(k)}_{n-2k}W^{(k)}_{n-2k-1}\ldots W^{(k)}_{n-3k+3}$.
\[lemVprefW\] Let $n,k$ be two positive integers with $k\geq 3$ and $n>2k-1$.
- If $n<3k-2$, then $$\begin{aligned}
W^{(k)}_{n-2k+2} &= W^{(k)}_{n-2k+1}V^{(k)}_{n}(n-2k+2),\label{eqwn-2k+2V}\\
W^{(k)}_{n-2k+1} &= V^{(k)}_{n}(n-2k+1). \label{eqwn-2k+1V}
\end{aligned}$$
- If $n=3k-2$, then $$\begin{aligned}
W^{(k)}_{n-2k+2} &= W^{(k)}_{n-2k+1}V^{(k)}_{n}k,\label{eqwn-2k+2V2}\\
W^{(k)}_{n-2k+1} &= V^{(k)}_{n}0k. \label{eqwn-2k+1V2}
\end{aligned}$$
- If $n>3k-2$, then $$\begin{aligned}
W^{(k)}_{n-2k+2} &= W^{(k)}_{n-2k+1}V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2}),\label{eqwn-2k+2V3}\\
W^{(k)}_{n-2k+1} &= V^{(k)}_{n}W^{(k)}_{n-3k+2}(k\oplus W^{(k)}_{n-3k+1}). \label{eqwn-2k+1V3}
\end{aligned}$$
\[lemVunique\] Let $n,k$ be two positive integers with $k\geq 3$ and $n> 2k-1$. Then $V^{(k)}_{n}$ occurs exactly once in $ W^{(k)}_{n-2k+1}$.
[If $2k-1< n< 3k-2$, then using Definition \[defV\] and the fact that $|V^{(k)}_{n}|_{n-2k+1}=0$ we conclude that $V^{(k)}_{n}$ occurs exactly once in $ W^{(k)}_{n-2k+1}$. If $n\geq 3k-2$, then by (\[structeq\]) we have $$W^{(k)}_{n-2k+1}=W^{(k)}_{n-2k}\ldots W^{(k)}_{n-3k+2}(k\oplus W^{(k)}_{n-2k})=V^{(k)}_{n}W^{(k)}_{n-3k+2}(k\oplus W^{(k)}_{n-2k}).$$ By definition of $V^{(k)}_{n}$, $|V^{(k)}_{n}|_{n-2k}=1$ and $(n-2k)0\prec V^{(k)}_{n}$. Using the facts that $|W^{(k)}_{n-3k+2}|_{n-2k}=0$ and $|k\oplus W^{(k)}_{n-2k}|_{0}=0$. Hence, $(n-2k)0$ occurs one time in $W^{(k)}_{n-2k+1}$. So, $V^{(k)}_{n}$ also occurs once in $W^{(k)}_{n-2k+1}$. ]{}
\[corV0\] Let $n,k$ be two positive integers with $k\geq 3$. If $2k-1< n <3k-2$, then $V^{(k)}_{n}$ in $W^{(k)}_{n-2k+1}$ always is followed by digit $n-2k+1$. If $ n \geq 3k-2$, then $V^{(k)}_{n}$ in $W^{(k)}_{n-2k+1}$ always is followed by digit $0$.
[If $2k-1< n <3k-2$, then the result follows using Lemma \[lemVunique\] and Definition \[defV\]. If $n\geq 3k-2$, then by (\[structeq\]), $V^{(k)}_{n}0\prec W^{(k)}_{n-2k+1}$. On the other hand, by Lemma \[lemVunique\], $V^{(k)}_{n}$ occurs exactly once in $W^{(k)}_{n-2k+1}$. Hence, the result follows. ]{}
\[lem3w\] Let $n,k$ be two nonnegative integers with $k\geq 3$ and $n>2k-1$. Then $$\label{eq3wlem}
|W^{(k)}_{n-k}|\geq 3|W^{(k)}_{n-2k+1}|.$$
We can check easily that (\[eq3wlem\]), holds in the cases $k=3$ and $5\leq n\leq 8$. If $k>3$ or $k=3$ and $n\geq 7$, then using (\[structeq\]), we get $$\label{eq3wproof}
|W^{(k)}_{n-k}|\geq |W^{(k)}_{n-k-1}|+|W^{(k)}_{n-k-2}|+|W^{(k)}_{n-k-3}|$$ If $k>3$, then $n-k-3\geq n-2k+1$. Hence, Equation (\[eq3wproof\]) yields the inequality $|W^{(k)}_{n-k}|\geq 3|W^{(k)}_{n-2k+1}|$, as desired. If $k=3$, then $$\begin{aligned}
|W^{(k)}_{n-3}|=& |W^{(k)}_{n-4}|+|W^{(k)}_{n-5}|+|W^{(k)}_{n-6}|\\
=& 2|W^{(k)}_{n-5}|+2|W^{(k)}_{n-6}|+|W^{(k)}_{n-7}| \\
=& 3|W^{(k)}_{n-5}|+|W^{(k)}_{n-6}|-|W^{(k)}_{n-8}|\\
>& 3|W^{(k)}_{n-5}|.\end{aligned}$$
To find all straddling squares of $W^{(k)}_{n}$ we need to give the following definition.
\[defU\] Let $n,k$ be two nonnegative integers with $k\geq 3$ and $n\geq 2k-1$. We define the word $U^{(k)}_{n}$ to be the prefix of $W^{(k)}_{n-2k+1}W^{(k)}_{n-k}$ of size $4|W^{(k)}_{n-2k+1}|$.
We note that Definition \[defU\] is well-defined using Lemma \[lem3w\].
\[lemU=<\] Let $n,k$ be two nonnegative integers with $k\geq 3$ and $2k-1< n< 3k-2$. Then $$U^{(k)}_{n}(k)=W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(n-2k+2)W^{(k)}_{n-2k+1}.$$
Since $W^{(k)}_{n-2k+3}\lhd W^{(k)}_{n-k}$ we have $$\label{eqwn-2kwn-k,2}
W^{(k)}_{n-2k+1} W^{(k)}_{n-k}
=W^{(k)}_{n-2k+1} W^{(k)}_{n-2k+3} [(W^{(k)}_{n-2k+3})^{-1}W^{(k)}_{n-k}]$$
If $k=3$, then applying Equation (\[structeq\]) for $W^{(k)}_{n-2k+3}$ and using (\[eqwn-2k+2V\]) and (\[eqwn-2k+1V\]) we get $$\begin{aligned}
\nonumber
W^{(k)}_{n-2k+3}=& W^{(k)}_{n-2k+2} W^{(k)}_{n-2k+1}(k\oplus W^{(k)}_{n-2k})\\
=& \underbrace{W^{(k)}_{n-2k+1}}_{|W^{(k)}_{n-2k+1}|}\underbrace{V^{(k)}_{n}(n-2k+2)}_{|W^{(k)}_{n-2k+1}|} \underbrace{W^{(k)}_{n-2k+1}}_{|W^{(k)}_{n-2k+1}|} (k\oplus W^{(k)}_{n-2k})\label{eq3W1,k=3}\end{aligned}$$ Using Definition \[defU\] and Equations (\[eqwn-2kwn-k,2\]) and (\[eq3W1,k=3\]) we conclude that $$U^{(k)}_{n}=W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(n-2k+2)W^{(k)}_{n-2k+1}.$$ If $k>3$, then applying Equation (\[structeq\]) for $W^{(k)}_{n-2k+3}$ we get $$W^{(k)}_{n-2k+2} W^{(k)}_{n-2k+1} W^{(k)}_{n-2k}\lhd W^{(k)}_{n-2k+3}\\$$ Hence, using (\[eqwn-2k+2V\]) and (\[eqwn-2k+1V\]) we provide $$\underbrace{W^{(k)}_{n-2k+1}}_{|W^{(k)}_{n-2k+1}|}\underbrace{V^{(k)}_{n}(n-2k+2)}_{|W^{(k)}_{n-2k+1}|} \underbrace{W^{(k)}_{n-2k+1}}_{|W^{(k)}_{n-2k+1}|} W^{(k)}_{n-2k}\lhd W^{(k)}_{n-2k+3}$$ Therefore, by Definition \[defU\] and Equation (\[eqwn-2kwn-k,2\]) we get $$U^{(k)}_{n}(k)=W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(n-2k+2)W^{(k)}_{n-2k+1}.$$
\[lemU=>\] Let $n,k$ be two nonnegative integers with $k\geq 3$ and $n> 3k-2$. Then
- If $k=3$, then $$U^{(k)}_{n}= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2})W^{(k)}_{n-2k+1}(k\oplus W^{(k)}_{n-3k+1}),$$
- If $k>3$, then $$U^{(k)}_{n}= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2})W^{(k)}_{n-2k+1} W^{(k)}_{n-3k+1}.$$
By Equation (\[structeq\]) we have $$\label{eqwn-2kwn-k}
W^{(k)}_{n-2k+1} W^{(k)}_{n-k}
=W^{(k)}_{n-2k+1} W^{(k)}_{n-2k+3} [(W^{(k)}_{n-2k+3})^{-1}W^{(k)}_{n-k}]$$ If $k=3$, then applying Equation (\[structeq\]) for $W^{(k)}_{n-2k+3}$ and using (\[eqwn-2k+2V3\]) and (\[eqwn-2k+1V3\]) we get $$\begin{aligned}
\nonumber
W^{(k)}_{n-2k+3}=& W^{(k)}_{n-2k+2} W^{(k)}_{n-2k+1}(k\oplus W^{(k)}_{n-2k})\\
=& \underbrace{W^{(k)}_{n-2k+1}}_{|W^{(k)}_{n-2k+1}|}\underbrace{V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2}) W^{(k)}_{n-2k+1}(k\oplus W^{(k)}_{n-3k+1})}_{2|W^{(k)}_{n-2k+1}|} [(k\oplus W^{(k)}_{n-3k+1})^{-1}(k\oplus W^{(k)}_{n-2k})]\label{eq3W1,k=3,2}\end{aligned}$$ Using Definition \[defU\] and Equations (\[eqwn-2kwn-k\]) and (\[eq3W1,k=3,2\]) we conclude that $$U^{(k)}_{n}=W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2})W^{(k)}_{n-2k+1}(k\oplus W^{(k)}_{n-3k+1}).$$ If $k>3$, then applying Equation (\[structeq\]) for $W^{(k)}_{n-2k+3}$ we get $$\nonumber
W^{(k)}_{n-2k+2} W^{(k)}_{n-2k+1}W^{(k)}_{n-2k}\lhd W^{(k)}_{n-2k+3}$$ Where, $$\label{eq3W1,k>3,2}
W^{(k)}_{n-2k+2} W^{(k)}_{n-2k+1}W^{(k)}_{n-2k}=
\underbrace{W^{(k)}_{n-2k+1}}_{|W^{(k)}_{n-2k+1}|}\underbrace{V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2}) W^{(k)}_{n-2k+1} W^{(k)}_{n-3k+1}}_{2|W^{(k)}_{n-2k+1}|} [(W^{(k)}_{n-3k+1})^{-1} W^{(k)}_{n-2k}]$$
Using Definition \[defU\] and Equations (\[eqwn-2kwn-k\]) and (\[eq3W1,k>3,2\]) we conclude that $$U^{(k)}_{n}=W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}(k\oplus W^{(k)}_{n-3k+2})W^{(k)}_{n-2k+1} W^{(k)}_{n-3k+1}.$$
In the next lemma we give a formula for $U^{(k)}_{3k-2}$, the proof is similar to the proof of Lemma \[lemU=>\] so it is omitted.
\[lemU==\] Let $n,k$ be two nonnegative integers with $k\geq 3$ and $n= 3k-2$. Then
- If $k=3$, then $$U^{(k)}_{n}= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}kW^{(k)}_{n-2k+1}k,$$
- If $k>3$, then $$U^{(k)}_{n}= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}kW^{(k)}_{n-2k+1}0.$$
\[corU\] If $A^2$ is a straddling square of $W^{(k)}_{n}$, then
- $A^2\prec k\oplus U^{(k)}_{n}$,
- $c_s(A^2\ominus k, U^{(k)}_{n})\leq 2|W^{(k)}_{n-2k+1}|+\frac{1}{2}$.
According to Definition \[defU\] we have
- This is the direct consequence of Lemma \[centersq3\] and equation (\[structsqeq\]).
- This can be deducted easily from Lemma \[centersq2\].
\[lem\*V\*\] Let $n< 3k-1$, then the word $(n-2k+1) V^{(k)}_{n} (n-2k+1)$ occurs exactly once in $U^{(k)}_{n}$.
[Using Lemma \[lemU=<\] and Equation (\[eqwn-2k+1V\]) we have $$\label{eq*V*}
U^{(k)}_{n}(k)=V^{(k)}_{n}(n-2k+1)V^{(k)}_{n}(n-2k+1)V^{(k)}_{n}(n-2k+2)V^{(k)}_{n}(n-2k+1).$$ By Deifinition\[defV\], it is clear that $0\lhd V^{(k)}_{n}$, $|V^{(k)}_{n}|_{n-2k+1}=0$ and $|V^{(k)}_{n}|_{n-2k+2}=0$. Hence, using (\[eq\*V\*\]) we conclude that $ V^{(k)}_{n}$ occurs exactly four times in $U^{(k)}_{n}$. By Equation (\[eq\*V\*\]), the word $(n-2k+1) V^{(k)}_{n} (n-2k+1)$ occurs exactly once in $U^{(k)}_{n}$. ]{}
\[lem\*V0\] Let $n\leq 3k-1$, then the word $(n-2k+1) V^{(k)}_{n} 0$ occurs exactly once in $U^{(k)}_{n}$.
By Deifinition\[defV\], it is clear that $0\lhd V^{(k)}_{n}$, $|V^{(k)}_{n}|_{n-2k+1}=0$ and $|V^{(k)}_{n}|_{n-2k+2}=0$. Hence, using Lemmas \[lemU==\] and \[lemU=>\] and using Lemma \[lemVunique\], we conclude that $ V^{(k)}_{n}$ occurs exactly four times in $U^{(k)}_{n}$. Therefore, by Equations (\[eqwn-2k+1V2\]) and (\[eqwn-2k+1V3\]) it is easy to see that $(n-2k+1) V^{(k)}_{n} 0$ occurs exactly once in $U^{(k)}_{n}$.
\[centersq\] Let $n\geq 2k-1$ and $A^2$ be a straddling square of $W^{(k)}_{n}$ and let $A'^2=A^2\ominus k$. Then $$\label{eqcenterU}
c_s(A'^2, U^{(k)}_{n})\leq |W^{(k)}_{n-2k+1}|+|V^{(k)}_{n}|+\frac{1}{2}.$$
For contrary suppose that $c_s(A'^2, U^{(k)}_{n})>|W^{(k)}_{n-2k+1}|+|V^{(k)}_{n}|+\frac{1}{2}$. We divide the proof in the following cases:
- If $n<3k-2$, then by Lemma \[lemU=<\] and Equation (\[eqwn-2k+1V\]) we conclude that $$\label{eqprefU1}
W^{(k)}_{n-2k+1}V^{(k)}_{n}(n-2k+1)\lhd U^{(k)}_{n}.$$ Using (\[eqprefU1\]) and the fact that $A'^2\oplus k$ is a straddling square of $W^{(k)}_{n}$, we conclude that $(n-2k+1)V^{(k)}_{n}(n-2k+1)$ should occurs at least twice in $A'^2\prec U^{(k)}_{n}$. This is a contradiction with Lemma \[lem\*V\*\].
- If $n<3k-2$, then by Lemma \[lemU=<\] and Equation (\[eqwn-2k+1V\]) we conclude that $$\label{eqprefU2}
W^{(k)}_{n-2k+1}V^{(k)}_{n}0\lhd U^{(k)}_{n}.$$ Using (\[eqprefU2\]) and the fact that $A'^2\oplus k$ is a straddling square of $W^{(k)}_{n}$, we conclude that $(n-2k+1)V^{(k)}_{n}0$ should occurs at least twice in $A'^2\prec U^{(k)}_{n}$. This is a contradiction with Lemma \[lem\*V0\].
The following corollary is the direct consequence of Lemma \[centersq\].
\[centersq5\] let $A^2$ be a straddling square of $W^{(k)}_{n}$. Then $c_s(A^2,W^{(k)}_{n})< |W^{(k)}_{n}|-|W^{(k)}_{n-k}|+|V^{(k)}_{n}|$.
\[lemA2inP\] Let $n\geq 2k-1$, $P^{(k)}_{n}=W^{(k)}_{n-2k+1} W^{(k)}_{n-2k+1}V^{(k)}_{n}$. Then $A^2$ is a straddling square of $W^{(k)}_{n}$ if and only if $A'^2=A^2\ominus k$ is a square of $P^{(k)}_{n}$ satisfying following properties:
- $ |W^{(k)}_{n-2k+1}|\leq c_s(A'^2, P^{(k)}_{n})\leq |W^{(k)}_{n-2k+1}|+|V^{(k)}_{n}|$,
- Let $A'^2=P^{(k)}_{n}[t,t+|A'^2|]$. Then, $t< |W^{(k)}_{n-2k+1}|$.
Let $A^2$ is a straddling square of $W^{(k)}_{n}$. Then by Corollary \[corU\] we have $A^2\ominus k\prec U^{(k)}_{n}$. Using Lemmas \[lemcs>\] and \[centersq\] we conclude that $ |W^{(k)}_{n-2k+1}|\leq c_s(A'^2, P^{(k)}_{n})\leq |W^{(k)}_{n-2k+1}|+|V^{(k)}_{n}|$. Moreover, since $A^2$ is a straddling factor of $W^{(k)}_{n}$ $A'^2$ satisfying (ii).
Now, let $A'^2$ is a square of $P^{(k)}_{n}$ which satisfies (i) and (ii), then we prove that $A^2=A'^2\oplus k$ is a straddling square of $W^{(k)}_{n}$. By Lemma \[lemsq<2k-1\], we conclude that $n\geq 2k-1$. Using Equation (\[structeq\]) for $W^{(k)}_{n}$ and $W^{(k)}_{n-k+1}$ we have $$\begin{aligned}
W^{(k)}_{n} =& W^{(k)}_{n-1}\ldots W^{(k)}_{n-k+1}(k\oplus W^{(k)}_{n-k})\\
W^{(k)}_{n} =& W^{(k)}_{n-1}\ldots W^{(k)}_{n-k+2}
( W^{(k)}_{n-k}\ldots W^{(k)}_{n-2k+2}(k\oplus W^{(k)}_{n-2k+1}))(k\oplus W^{(k)}_{n-k})\end{aligned}$$ Since $W^{(k)}_{n-2k+1}V^{(k)}_{n}\lhd W^{(k)}_{n-k}$, we conclude that [$$\label{eqA+kinP}
W^{(k)}_{n} = W^{(k)}_{n-1}\ldots W^{(k)}_{n-k+2}
W^{(k)}_{n-k}\ldots W^{(k)}_{n-2k+2}\underbrace{(k\oplus W^{(k)}_{n-2k+1} W^{(k)}_{n-2k+1}V^{(k)}_{n})}_{ P^{(k)}_{n}}(k\oplus (W^{(k)}_{n-2k+1}V^{(k)}_{n})^{-1} W^{(k)}_{n-k})$$]{} Using the fact that $A'^2$ satisfies (i) and (ii) and as shown in Equation (\[eqA+kinP\]), we conclude that $A^2=A'^2\oplus k$ is a straddling square of $ W^{(k)}_{n}$.
\[thmsqWn\] Let $0\leq j\leq |V^{(k)}_{n}|$. Then $(C^{(j)} (k\oplus W^{(k)}_{n-2k+1}))^2$ is a straddling square of $W^{(k)}_n$. Moreover, every straddling square $A^2$ of $W^{(k)}_{n}$ is of the form $A^2=(C^{(j)} (k\oplus W^{(k)}_{n-2k+1}))^2$, for some $0\leq j\leq |V^{(k)}_{n}|$.
By Lemma \[lemVprefW\], $V^{(k)}_{n}\lhd W^{(k)}_{n-2k+1}$. Hence, there exists suffix $V'$ of $W^{(k)}_{n-2k+1}$ such that $W^{(k)}_{n-2k+1}=V^{(k)}_{n}V'$. Let $0\leq j\leq |V^{(k)}_{n}|$ and $V_1= V^{(k)}_{n}[1,j]$ and $V_2= V^{(k)}_{n}[j+1,|V^{(k)}_{n}|]$. Then $$\label{eqv1v2v'}
P^{(k)}_{n}= V_1 \overbrace{ V_2 V'V_1 }^{C^{(j)} (W^{(k)}_{n-2k+1})}\underbrace{V_2 V'V_1}_{C^{(j)} (W^{(k)}_{n-2k+1})} V_2$$ Now, using Lemma \[lemA2inP\] and Equation (\[eqv1v2v’\]), $(C^{(j)} (k\oplus W^{(k)}_{n-2k+1}))^2$ is a straddling square of $W^{(k)}_{n}$.
Moreover, let $A^2$ be a straddling square of $W^{(k)}_{n}$. Hence the first $A$ in $A^2$ should contains $n-k+1$. By Lemma \[lemA2inP\] $A'^2=A^2\ominus k$ is a square factor of $P^{(k)}_{n}$ satisfying the conditions of the lemma. Therefore, $|A'|_{n-2k+1}\geq 1$. By Lemma \[lemA2inP\], we can assume that $c_s(A'^2, P^{(k)}_{n})=|W^{(k)}_{n-2k+1}|+j+\frac{1}{2}$ for some $0\leq j\leq |V^{(k)}_{n}|$.
Again using Lemma \[lemVprefW\], $V^{(k)}_{n}\lhd W^{(k)}_{n-2k+1}$. Let $V'\rhd W^{(k)}_{n-2k+1}$ such that $W^{(k)}_{n-2k+1}=V^{(k)}_{n}V'$, $V_1= V^{(k)}_{n}[1,j]$ and $V_2= V^{(k)}_{n}[j+1,|V^{(k)}_{n}|]$. Therefore, for the first $A'$ in $A'^2$ we have $A'\rhd V_1 V_2 V' V_1$ and for the last $A'$ in $A'^2$, $A'\lhd V_2 V' V_1 V_2$. On the other hand $V_1 V_2 V' V_1 \prec W^{(k)}_{n-2k+1}V^{(k)}_{n}$ and by Definition \[defV\] and Lemma \[lastdig\], $|W^{(k)}_{n-2k+1}V^{(k)}_{n}|_{n-2k+1}=1$, hence $|A'|_{n-2k+1}=1$. Since the first place that $n-2k+1$ occurs in $ V_2 V' V_1 V_2$ is $|V_2 V'|$, hence $V_2 V' \lhd A'$. Since the number of occurrences of $V_2 V'$ in $ V_1 V_2 V' V_1$ is once. We conclude that $A'= V_2 V' V_1$.
\[thmallsqW\] Let $k\geq 3$. Then $A^2$ is a square of $W^{(k)}$ if and only if $A\in \{ki\oplus C^j(W^{(k)}_{n-2k+1}): 0\leq j \leq |V^{(k)}_{n}|, i>0, n\geq 0\}$.
If $A= ki\oplus C^j(W^{(k)}_{n-2k+1})$, for some $0\leq j \leq |V^{(k)}_{n}|, i>0, n\geq 0$, then by Theorem \[thmsqWn\] $k\oplus C^j(W^{(k)}_{n-2k+1})=(A^2 \ominus k(i-1))\prec W^{(k)}_{n}$ or equivalently $A^2 \prec W^{(k)}_{n}\oplus k(i-1)$. By Corollary \[corki\], we conclude that $A^2 \prec W^{(k)}_{n+k(i-1)}\prec W^{(k)}$.
On the other hand, if $A^2$ is a square of $W^{(k)}$, then by Corollary \[corsq=strad\], there exist $n>2k-1, i>0$, such that $A^2\ominus k(i-1)$ is a straddling square of $W_n^{(k)}$. By Corollary \[corki\], we conclude that $A^2 \prec W^{(k)}_{n+k(i-1)}$.
We finish this section with the following example.
In this example we provide all square factors of $W^{(3)}_{11}$, which is given bellow. All of these squares are listed in Table \[tableallsq\] according to Theorem \[thmallsqW\]. We note that letters $a$ and $b$ stand for the digits $10$ and $11$. $$\begin{aligned}
W^{(3)}_{11}=
&010201301023401020133435010201301023434353460102013010234010201334353435346343567\\
&010201301023401020133435010201301023434353463435346343567343534667680102013010234\\
&010201334350102013010234343534601020130102340102013343534353463435673435346343567\\
&3435346676834353463435676768679010201301023401020133435010201301023434353460102013\\
&0102340102013343534353463435670102013010234010201334350102013010234343534634353463\\
&4356734353466768343534634356734353466768343534634356767686793435346343567343534667\\
&68676867967689a0102013010234010201334350102013010234343534601020130102340102013343\\
&5343534634356701020130102340102013343501020130102343435346343534634356734353466768\\
&0102013010234010201334350102013010234343534601020130102340102013343534353463435673\\
&4353463435673435346676834353463435676768679343534634356734353466768343534634356767\\
&68679343534634356734353466768676867967689a3435346343567343534667683435346343567676\\
&8679676867967689a67686799a9b
\end{aligned}$$
[ ]{} 1 2 3
-------------------------------- ---- -------------------------- --------- ---
$C^{(j)} (ki\oplus W^{(3)}_0)$ 0 3 6 9
0 34 67 -
1 43 76 -
0 3435 6768 -
1 4353 7686 -
2 3534 6867 -
0 3435346 6768679 -
1 4353463 7686796 -
2 3534634 6867967 -
3 5346343 8679676 -
4 3463435 6796768 -
0 3435346343567 - -
1 4353463435673 - -
2 3534634356734 - -
3 5346343567343 - -
4 3463435673435 - -
5 4634356734353 - -
6 6343567343534 - -
7 3435673435346 - -
0 343534634356734353466768 - -
1 435346343567343534667683 - -
2 353463435673435346676834 - -
3 534634356734353466768343 - -
4 346343567343534667683435 - -
5 463435673435346676834353 - -
6 634356734353466768343534 - -
7 343567343534667683435346 - -
8 435673435346676834353463 - -
9 356734353466768343534634 - -
10 567343534667683435346343 - -
11 673435346676834353463435 - -
12 734353466768343534634356 - -
13 343534667683435346343567 - -
: Square factors of $W^{(3)}_{11}$[]{data-label="tableallsq"}
[ ]{} 1 2 3
------- ---- ---------------------------------------------- --- ---
0 34353463435673435346676834353463435676768679 - -
1 43534634356734353466768343534634356767686793 - -
2 35346343567343534667683435346343567676867934 - -
3 53463435673435346676834353463435676768679343 - -
4 34634356734353466768343534634356767686793435 - -
5 46343567343534667683435346343567676867934353 - -
6 63435673435346676834353463435676768679343534 - -
7 34356734353466768343534634356767686793435346 - -
8 43567343534667683435346343567676867934353463 - -
9 35673435346676834353463435676768679343534634 - -
10 56734353466768343534634356767686793435346343 - -
11 67343534667683435346343567676867934353463435 - -
12 73435346676834353463435676768679343534634356 - -
13 34353466768343534634356767686793435346343567 - -
14 43534667683435346343567676867934353463435673 - -
15 35346676834353463435676768679343534634356734 - -
16 53466768343534634356767686793435346343567343 - -
17 34667683435346343567676867934353463435673435 - -
18 46676834353463435676768679343534634356734353 - -
19 66768343534634356767686793435346343567343534 - -
20 67683435346343567676867934353463435673435346 - -
21 76834353463435676768679343534634356734353466 - -
22 68343534634356767686793435346343567343534667 - -
23 83435346343567676867934353463435673435346676 - -
Critical Exponent and Critical Factors of $W^{(k)}$
===================================================
\[lempower<\] let $A^2$ be a straddling square of $W^{(k)}_{n}$. Then $$\rm{INDEX}(A, W^{(k)}_{n}) = \left\{
\begin{array}{ll}
3-\frac{1}{2^{n-2k+1}}\;\; & \text{if } 2k-1\leq n \leq 3k-3, \\
3-\frac{1}{2^{k-2}}\;\; & \text{if } n =3k-2, \\
3-\frac{|W^{(k)}_{n-3k+2}|+|W^{(k)}_{n-3k+1}|}{|W^{(k)}_{n-2k+1}|}\;\; & \text{if } n> 3k-2.
\end{array} \right.$$
[ By Lemma \[lemA2inP\], $A^2\ominus k \prec P^{(k)}_{n}=W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}$. If $2k-1\leq n\leq 3k-3$, then by Definition \[defV\]. $$\begin{aligned}
P^{(k)}_{n}&= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}V^{(k)}_{n}\\
&= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}W^{(k)}_{n-2k} \ldots W^{(k)}_{0}\\
&= W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1}W^{(k)}_{n-2k+1} (n-2k+1)^{-1}\\
&= (W^{(k)}_{n-2k+1})^{3-\frac{1}{2^{n-2k+1}}}.
\end{aligned}$$ Where the last equality holds since $|W^{(k)}_{n-2k+1}|=2^{n-2k+1}$. If $ n= 3k-2$, then by Definition \[defV\]. $$\begin{aligned}
P^{(k)}_{3k-2}&= W^{(k)}_{k-1}W^{(k)}_{k-1}V^{(k)}_{3k-2}\\
&= W^{(k)}_{k-1}W^{(k)}_{k-1}W^{(k)}_{k-2} \ldots W^{(k)}_{1}\\
&= W^{(k)}_{k-1}W^{(k)}_{k-1}W^{(k)}_{k-1} (0(k-1))^{-1}\\
&= (W^{(k)}_{k-1})^{3-\frac{1}{2^{k-2}}}.
\end{aligned}$$ In the case $n>3k-2$, again by using Definition \[defV\], we conclude that $$P^{(k)}_{n}=(W^{(k)}_{n-2k+1})^{3-\frac{|W^{(k)}_{n-3k+2}|+|W^{(k)}_{n-3k+1}|}{|W^{(k)}_{n-2k+1}|}}.$$ ]{}
In the following example for $k=5$ and $9\leq n\leq 17$, we show that how Lemma \[lempower<\] works.
\[examallp\] In this example we listed all $P^{(5)}_{n}$, when $9\leq n\leq 17$ and for all values of $n$ we present the corresponding power $r$. We note that letters $a$ and $b$ stand for the digits $10$ and $11$.
n $P^{(5)}_{n} \oplus k=(W^{(5)}_{n-9}\oplus k)^r $ r
---- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------
9 55 2
10 56565 $3-\frac{1}{2}$
11 56575657565 $3-\frac{1}{4}$
12 56575658565756585657565 $3-\frac{1}{8}$
13 5657565856575659565756585657565956575658565756 $3-\frac{1}{8}$
14 565756585657565956575658565756a565756585657565956575658565756a 5657565856575659565756585657 $3-\frac{3}{31}$
15 565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a565756585657565956575658 $3-\frac{6}{61}$
16 565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a565756585657565956575658abac565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a565756585657565956575658abac565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a5657565856575659 $3-\frac{12}{120}$
17 565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a565756585657565956575658abac565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a5657565856575659abacabad565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a565756585657565956575658abac565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a5657565856575659abacabad565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a565756585657565956575658abac565756585657565956575658565756a5657565856575659565756585657ab565756585657565956575658565756a $3-\frac{24}{236}$
: Powers of $W^{(5)}_{n-9}$ in $W^{(5)}_{n}$[]{data-label="tableallp"}
As we can see in Table \[tableallp\] in Example \[examallp\], the largest power of $W^{(5)}_{n-9}\oplus k$ in $P^{(5)}_{n}\oplus k$ is $3-\frac{3}{31}$. This power happens when $n=14$, which is the critical exponent of $P^{(5)}_{n}\oplus k$. Moreover, in the following Theorem we show that this $r$ is also the critical exponent of $W^{(5)}$.
\[thmcritical\] Let $k\geq 3$, then the critical exponent of $W^{(k)}$ equals to $3-\frac{3}{2^k-1}$. Moreover, the set of all critical factors of $W^{(k)}$ is $\{P^{(k)}_{3k-1} \oplus ki\}$.
[ By Theorem \[thmsqWn\], for all $n\geq 2k-1$, $W_n^{(k)}$ always contains a square factor. Hence $E(W^{(k)})\geq 2$. Let $A\in F(W^{(k)})$ and $r={\rm INDEX}(A)\geq 2$. We will prove that $r\leq 3-\frac{3}{2^{k-1}}$. Since $r\geq 2$, $A^2$ is a square factor of $W^{(k)}$. By Corollary \[corsq=strad\] there exisit integers $i$ and $n$ such that $A^2\ominus ki$ is a straddling square of $W_n^{(k)}$. Let $m_1=\max\{3-\frac{1}{2^{n-2k+1}}: 2k-1\leq n\leq 3k-3\}$, $m_2=3-\frac{1}{2^{k-2}}$ and $m_3=\max\{3-\frac{|W^{(k)}_{n-3k+2}|+|W^{(k)}_{n-3k+1}|}{|W^{(k)}_{n-2k+1}|}: n\geq 3k-1\}$. Now, using Lemma \[lempower<\] we conclude that $r\leq \max\{m_1, m_2, m_3\}$. It is easy to check that $m_1=m_2=3-\frac{1}{2^{k-2}}$. Since $g(n)=3-\frac{|W^{(k)}_{n-3k+2}|+|W^{(k)}_{n-3k+1}|}{|W^{(k)}_{n-2k+1}|}$ is a decreasing function of $n$, we conclude that $m_3=g(3k-1)=3-\frac{3}{2^k-1}$. Hence $r\leq 3-\frac{3}{2^k-1}$. On the other hand, by Lemma \[lemA2inP\], $P^{(k)}_{3k-1} \oplus k=(W^{(k)}_{k})^{3-\frac{3}{2^k-1}}$. This implies that the set of all critical factors of $W^{(k)}$ equals to $\{P^{(k)}_{3k-1}\oplus ki: i\geq 1\}$. ]{}
In Table \[tablecritical\], we compute the critical exponent and one of the critical factors of $W^{(k)}$, for $3\leq k\leq 8$, according to Theorem \[thmcritical\]. We note that the digits $10,11, \ldots, 16$ are denoted by the letters $a, b, \ldots g$, respectively.
k $P^{(k)}_{3k-1} \oplus k=(W^{(k)}_{k}\oplus k)^r $ $r=3-\frac{3}{2^k-1} $
--- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------
3 343534634353463435 $3-\frac{3}{7}$
4 454645474546458454645474546458454645474546 $3-\frac{3}{15}$
5 565756585657565956575658565756a565756585657565956575658565756a 5657565856575659565756585657 $3-\frac{3}{31}$
6 676867696768676a676867696768676b676867696768676a67686769676867c 676867696768676a676867696768676b676867696768676a67686769676867c 676867696768676a676867696768676b676867696768676a676867696768 $3-\frac{3}{63}$
7 7879787a7879787b7879787a7879787c7879787a7879787b7879787a7879787d 7879787a7879787b7879787a7879787c7879787a7879787b7879787a787978e 7879787a7879787b7879787a7879787c7879787a7879787b7879787a7879787d 7879787a7879787b7879787a7879787c7879787a7879787b7879787a787978e 7879787a7879787b7879787a7879787c7879787a7879787b7879787a7879787d 7879787a7879787b7879787a7879787c7879787a7879787b7879787a7879 $3-\frac{3}{127}$
8 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898e 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898f 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898e 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a89g 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898e 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898f 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898e 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a89g 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898e 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898f 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a898e 898a898b898a898c898a898b898a898d898a898b898a898c898a898b898a $3-\frac{3}{255}$
: the critical exponent and one of the critical factors of $W^{(k)}$[]{data-label="tablecritical"}
|
---
abstract: 'We calculate for a binary mixture of Lennard-Jones particles the time dependence of the solution of the mode-coupling equations in which the full wave vector dependence is taken into account. In addition we also take into account the short time dynamics, which we model with a simple memory kernel. We find that the so obtained solution agrees very well with the time and wave vector dependence of the coherent and incoherent intermediate scattering functions as determined from molecular dynamics computer simulations. Furthermore we calculate the wave vector dependence of the Debye-Waller factor for a realistic model of silica and compare these results with the ones obtained from a simulation of this model. We find that if the contribution of the three point correlation function to the vertices of the memory kernel are taken into account, the agreement between theory and simulation is very good. Hence we conclude that mode coupling theory is able to give a correct quantitative description of the caging phenomena in fragile as well as strong glass-forming liquids.'
address: |
$^{(1)}$ Laboratoire des Verres, Université Montpellier II, F-34095 Montpellier, France\
$^{(2)}$ Institut für Physik, Johannes–Gutenberg–Universität, Staudinger Weg 7, D–55099 Mainz, Germany\
$^{(3)}$ Dipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universitá di Roma [*La Sapienza*]{}, P.le Aldo Moro 2, I-00185 Roma, Italy
author:
- 'Walter Kob$^{(1)}$, Markus Nauroth$^{(2)}$, and Francesco Sciortino$^{(3)}$'
date: '17. September, 2001'
title: 'Quantitative tests of mode-coupling theory for fragile and strong glass-formers'
---
Introduction {#sec1}
============
In the last decade our understanding of the dynamics of supercooled liquids has made significant progress [@proceedings]. In particular it has been shown that for fragile glass-formers the bend one observes if one plots the logarithm of the viscosity as a function of the inverse temperature can be explained very well by means of the so-called mode-coupling theory of the glass transition (MCT) [@gotze99]. In the vicinity of this bend the dynamics of the system changes qualitatively in that the particles start to experience strong caging effects, i.e. they are temporarily trapped by the particles that surround them. MCT gives a self-consistent description of the dynamics of the particles inside this cage as well as how the particles leave this cage, i.e. of the structural relaxation of the supercooled liquid. In the past the predictions of this theory have been checked in many experiments as well as computer simulations and it was found that MCT is indeed able to give a qualitatively correct description of the relaxation dynamics [@gotze99].
However, in principle the theory is supposed to give not only a qualitatively description of the relaxation dynamics of supercooled liquids, but also a quantitative one, [*if*]{} the static properties of the system are known with sufficiently high precision. This attractive feature originates directly from the way the theory is (or can be) derived, namely the Mori-Zwanzig formalism in which one obtains equations of motion for slow variables which involve their [*static*]{} values. Thus, once these static values are known one can, in principle, determine their time dependence. In particular it is possible to calculate from the knowledge of the static structure factor the time dependence of the coherent and incoherent intermediate scattering functions, $F(q,t)$ and $F_s(q,t)$, respectively, where $q$ is the wave vector. Unfortunately, in the past these type of calculations have been done only for very few systems, since one the one hand they are quite involved and on the other hand they require as input structural data with very high quality (better than 1%) [@barrat90; @gotze91; @fuchs92a; @fuchs92b; @nauroth97; @water; @winkler00]. In the present paper we expand these type of calculations in two directions. On the one hand we solve for a simple glass-forming system, a binary Lennard-Jones mixture (BMLJ), the full time and wave vector dependence of the MCT equations, including a realistic short time dynamics, and compare them with results from computer simulations of the same system. On the other hand we calculate the $q-$dependence of the nonergodicity parameters (NEP) for silica (SiO$_2$), a glass-former whose structure is given by an open tetrahedral network and who is the prototype of a strong glass-former, and compare also these results with the ones from simulations of the same system.
Theory {#sec2}
======
In this section we summarize the MCT equations that are needed to calculate the quantities discussed in section \[sec4\]. In order to keep this presentation as simple as possible we will discuss only the equation for the case of a one-component system, although in reality we have used the equations for a two-component system, since the BMLJ as well as SiO$_2$ belong to this class. The full binary equations can be found in Refs. [@barrat90; @nauroth_phd; @sciortino01].
The intermediate scattering function can be defined by $F(q,t)=\langle \delta \rho({\bf q},0) \delta \rho({\bf q},t)\rangle$ where $\delta \rho$ are the density fluctuations. $F(q,t)$ obeys the exact equation of motion
$$\ddot{F}(q,t)+\Omega^2(q)F(q,t)+\int_0^tM(q,\tau)\dot{F}(q,t-\tau)d\tau = 0
\label{eq1}$$
where the frequency $\Omega^2$ is given by $\Omega^2=q^2 k_B T/(m
S(q))$. Here $m$ and $S(q)$ are the mass of the particles and the static structure factor respectively. The function $M(q,t)$ in Eq. (\[eq1\]) is the so-called memory function and it is useful to write it as follows:
$$M(q,t) = M^{REG}(q,t)+
\left\{M^{MCT}[F(k,t)](q)-M^{MCT}[F^B(k,t)](q)\right\}.
\label{eq2}$$
Here $M^{REG}(q,t)$ is that part of the memory function which is responsible for the dynamics of the system at very short times, i.e. after the particles have left the ballistic regime. The functional $M^{MCT}[F(k,t)]$ is the usual memory kernel of MCT which depends on the static structure factor as well as on the three point correlation function $c_3({\bf q}, {\bf k})$ [@gotze99]. Also it contains a short time part. But since we want to describe this time regime by means of $M^{REG}(q,t)$, we have to subtract out this part from $M^{MCT}[F(k,t)]$. This is done in the last term of Eq. (\[eq2\]), where $F^B$ is a function which decays rapidly to zero, but has the correct behavior at short times [@nauroth_phd].
For the regular memory function $M^{REG}(q,t)$ one can make different type of Ansatzes. One which seems to work well at high temperatures is given by [@tankeshwar]
$$M^{REG}(q,t)=\alpha(q) /\cosh(\beta(q) t).
\label{eq3}$$
Here $\alpha(q)$ and $\beta(q)$ are parameters which can be calculated via sum rules from the static structure factor and other static quantities which can be measured in a computer simulation [@nauroth_phd; @boon80]. Hence they are [*not*]{} adjustable fit parameters.
Eqs. (\[eq1\])-(\[eq3\]) are a self-contained set of equations of motion from which one thus can calculate the full time and wave vector dependence of $F(q,t)$, and similar equations exist for the incoherent intermediate scattering function $F_s(q,t)$. We have solved these equations by discretizing $q$-space into 100 points that covered the $q-$range up to 3-4 times the location of the main peak in $S(q)$.
Models and details of the simulations {#sec3}
=====================================
The first model investigated is a 80:20 mixture of Lennard-Jones particles. In the following we will call the majority and minority species A and B particles, respectively. Both of them have the same mass $m$ and they interact via a potential $V_{\alpha\beta}=
4\epsilon_{\alpha\beta}[(\sigma_{\alpha\beta}/r)^{12}-
(\sigma_{\alpha\beta}/r)^6]$, $\alpha,\beta\in \{\rm A,B\}$. The parameters $\epsilon_{\alpha\beta}$ and $\sigma_{\alpha\beta}$ are given by $\epsilon_{\rm AA}=1.0$, $\sigma_{\rm AA}=1.0$, $\epsilon_{\rm AB}=1.5$, $\sigma_{\rm AB}=0.8$, $\epsilon_{\rm BB}=0.5$, and $\sigma_{\rm BB}=0.88$. This potential is truncated and shifted at a distance $\sigma_{\alpha\beta}$. In the following we will use $\sigma_{\rm AA}$ and $\epsilon_{\rm AA}$ as the unit of length and energy, respectively (setting the Boltzmann constant $k_{\rm B}=1.0$). Time will be measured in units of $\sqrt{m\sigma_{\rm AA}^2/48\epsilon_{\rm AA}}$. In the past the structural and dynamical properties of this system have been studied in great detail [@kob_lj; @gleim98]. More detail on this can be found in Ref. [@kob_99].
For the present work we only needed to determine the three point correlation function $c_3$ since the time and temperature dependence of $F(q,t)$ and $F_s(q,t)$, as well as the one of $S(q)$, can be found in the mentioned literature. For this we simulated a system of 800 A particles and 200 B particles in a box with volume $(9.4)^3$. The total time of this simulation was about $10^8$ time steps from which we obtained roughly 12,000 independent configurations. This large number was necessary to determine $c_3$ with sufficient precision. Due to this large computational effort we did this calculation only for one temperature, $T=1.0$. Thus in the following we will assume that the temperature dependence of $c_3$ is weak.
The second model we study is amorphous silica, SiO$_2$. For this we use the potential proposed by van Beest [*et al.*]{} which has the functional form [@beest90]
$$\phi_{\alpha \beta}(r)=
\frac{q_{\alpha} q_{\beta} e^2}{r} +
A_{\alpha \beta} \exp\left(-B_{\alpha \beta}r\right) -
\frac{C_{\alpha \beta}}{r^6}\quad \alpha, \beta \in
[{\rm Si}, {\rm O}].
\label{eq4}$$
The values of the constants $q_{\alpha}, q_{\beta}, A_{\alpha
\beta}, B_{\alpha \beta}$, and $C_{\alpha \beta}$ can be found in Ref. [@beest90]. The potential has been truncated and shifted at 5.5 Å. In the past it has been shown that this potential is able to give a reliable description of silica in its molten phase as well as in the glass (see [@horbach99; @horbach01; @saika01] and references therein). For the present calculations to determine $c_3$ we used 600 ions in a box with volume (20.4Å)$^3$. The total length of the simulation was $2\times
10^7$ time steps, from which we obtained at 4000 K around 2000 independent configurations.
Results {#sec4}
=======
We start by considering first the dynamics of the BMLJ at intermediate and high temperatures. In this $T-$range it can be expected that the effect of the memory kernel of MCT is not relevant and thus we will set it to zero. In Fig. \[fig1\] we show the time dependence of $F_{\rm AA}(q,t)$ for various temperatures. The wave vector is 7.25, the location of the main peak in $S_{\rm AA}(q)$. The dashed lines with symbols are the result of the simulation whereas the full lines are the prediction of the theory. As can be seen, the theory works very well at high temperatures but starts to break down at intermediate temperatures in that it underestimated the correlation function at intermediate times. Thus we see that even at the intermediate temperature $T=1.0$, which is more than twice the MCT temperature $T_c=0.435$, cage effects become important.
In order to see whether the memory kernel $M^{MCT}$ is able to take into account these effects we have solved Eqs. (\[eq1\])-(\[eq3\]) by taking now into account also this contribution to the memory function $M(q,t)$. In doing this we had to face a problem which we had encountered already some time ago [@nauroth97], namely that MCT is not able to predict reliably the value of the critical temperature $T_c$. For the BMLJ the simulations show that $T_c\approx 0.435$ [@kob_lj], whereas the theory predicts a value around 0.92 [@nauroth97]. This means that the theory is not able to predict correctly the [*absolute*]{} value of the time scale for the $\alpha-$relaxation, although it is able to predict the shape of the correlation functions (see below). Therefore we had to use [*one*]{} adjustable parameter, a temperature which we will denote by $T_f$, which is the temperature at which the vertices in the MCT-functional $M^{MCT}$ are evaluated. The value of $T_f$ was adjusted such that the time scale for $F_{\rm AA}(q,t)$ for $q=7.25$ was reproduced correctly.
In Fig. \[fig2\] we show the time dependence of the coherent as well as the incoherent intermediate scattering function for $q=7.25$ and $q=9.98$, the location of the first peak and the first minimum in $S_{\rm AA}(q)$, respectively. The temperature is 2.0, i.e. a value for which we find that the [*regular*]{} memory kernel is no longer able to give a good description of the relaxation dynamics (see Fig. \[fig1\]). From this figure we see that in general the agreement between the simulation and the theory is very good in that the shape of the curves as well as their position is correctly predicted. (The discrepancy found for $F_{\rm AA}(q=9.98,t)$, where the theory predicts a pronounced shoulder at around $t=2$ whereas the simulation shows only a weak shoulder in that time regime, is probably related to the fact that the description of the short time dynamics is not yet completely adequate [@nauroth_phd]). Thus we conclude that MCT is indeed able to give a correct description of the relaxation dynamic of the system at intermediate temperatures, i.e. at temperatures where the cage effect starts to become noticeable.
We now check whether this conclusion is also correct if the temperature is so low that the cage effect becomes very important. For this we have solved the MCT equations for $T=0.466$, i.e. a temperature for which the relaxation dynamics is about $10^4$ times slower than the one at high $T$. The time dependence of $F_{\rm AA}(q,t)$ and of $F_{\rm A}^s(q,t)$ as predicted from the theory is shown in Fig. \[fig3\]. Also included are the results from the computer simulations from Ref. [@kob_lj]. As in the case of intermediate temperatures, Fig. \[fig2\], we find that also for this $T$ the agreement between theory and simulation is very good. The main discrepancy is again seen for $F_{\rm AA}(q,t)$ at $q=9.98$, and the reason for it is the same as the one given above. All in all we thus conclude that for this system the theory is indeed able to predict the full time and wave vector dependence of the coherent and incoherent scattering function.
The temperature dependence of the relaxation time of the BMLJ system shows significant deviations from an Arrhenius law [@kob_lj]. As mentioned above, these deviations are believed to be related to a change in the transport mechanism of the particles which show a hopping type of motion at low temperatures whereas at high $T$ they show a more collective/flow-like behavior. It is of interest that recently it has been suggested that even silica shows such a crossover in the transport mechanism, although this crossover occurs at relatively high temperatures (around 3300K) [@horbach99]. Therefore one might ask whether MCT is able to give a reliable description of the relaxation dynamics of this important glass-forming system also. Note that from a structural point of view the BMLJ system and amorphous silica are very different, since the former one resembles to the random close packing of hard spheres whereas the latter is given by an open network structure similar to the continuous random tetrahedral network proposed long time ago by Zachariasen [@zachariasen32]. Since, as pointed out in the Introduction, MCT uses only structural information to predict the dynamics, it is of great interest to see whether the theory is also able to give a correct quantitative description if the structure is very different from the one of closed packed hard spheres.
To check this we have calculated for amorphous silica the wave vector dependence of the nonergodicity parameters, i.e. the height of the plateau in the intermediate scattering function at intermediate times (see, e.g., Fig. \[fig3\]). Before we discuss the results, we have to mention a technical point which makes the calculation of the NEP for the case of silica much harder than for the case of the BMLJ. In Sec. \[sec2\] we mentioned that the memory kernel of the MCT contains only static quantities, namely the static structure factor $S(q)$ and the three point correlation function $c_3({\bf q}, {\bf k})$. From a simulation it is quite easy to determine $S(q)$ with high precision. For the function $c_3$ this is, however, not the case, since due to the two vectorial arguments the statistics for this quantity is very bad. Therefore we had to make very long simulations in order to determine $c_3$ with sufficient accuracy. More details on this can be found in Ref. [@sciortino01].
In the following we will discuss the results for the NEP for the BMLJ as well as for the case of silica. Since all of the results presented in Figs. \[fig1\]-\[fig3\] were obtained with the approximation that $c_3\equiv 0$, one has of course to check whether or not they do not change if this assumption is not made. We mention, however, already here, that some time ago Barrat [*et al.*]{} showed that this approximation is very good for the case of a soft-sphere system, i.e. a system which is relatively similar to the BMLJ considered here [@barrat89]. Whether this result holds also for the case of a system with an open network structure has, however, so far not been investigated. We also mention that in order to calculate the NEPs it is not necessary to introduce any fit parameter of any kind. The only input to the data are the partial structure factors [@gotze99]. Thus for this type of calculation the above discussed problem with the $T_f$ does not exist.
In Fig. \[fig4\] we show the wave vector dependence of the NEP for the coherent functions. (Note that since this is a binary system, there are three of them. Furthermore we mention that for reasons of convenience we show the NEP multiplied by the corresponding partial structure factors.) In each panel we show three curves: The circles are the result from the simulation published in Ref. [@gleim98]. The dashed and full line is the theoretical result for the cases that $c_3$ is set equal to zero and $c_3\neq 0 $, respectively. First of all this figure shows that the theory is able to reproduce with excellent accuracy the data of the simulation without any adjustable parameter. Furthermore we recognize from Fig. \[fig4\] also that the theoretical prediction hardly depends on whether or not $c_3$ is taken into account, in agreement with the finding of Barrat [*et al.*]{} [@barrat89].
For silica the situation is quite different as can be inferred from Fig. \[fig5\] where we show the wave vector dependence of the NEP for this system. We see that in this case the theoretical prediction for $c_3=0$ differs strongly from the one if this function is taken into account. Thus we find that for the case of a network structure the contribution of the three point correlation function to the memory function is very important. It is remarkable that if the contributions of $c_3$ are taken into account, the theoretical prediction agrees very well with the result of the simulation, which were presented in Ref. [@horbach01]. Thus we conclude that the theory is also able to give a quantitative correct prediction for this type of glass-former.
Summary {#sec5}
=======
The goal of this work was to check to what extend the mode-coupling theory of the glass transition is able to give a correct [*quantitative*]{} description of the dynamics of glass-forming liquids. This was done for two very different types of systems: A binary mixture of Lennard-Jones particles, whose structure is similar to the one of a close packing of hard spheres and whose temperature dependence of relaxation times makes it a glass-forming liquid of intermediate fragility. On the other hand we have studied silica, who has a open network structure and which, in the temperature region where experiments are feasible, is considered to be the prototype of a strong glass-former.
For the BMLJ system we have solved the MCT equations to obtain the full time and wave vector dependence of the relaxation dynamics. In particular we have also included a realistic Ansatz for the dynamics at short times so that the theoretical curves should be able to give also a good description in this time regime. By comparing the theoretical curves for $F(q,t)$ and $F_s(q,t)$ with the ones obtained from computer simulations of the same system, we find that cage effects become noticeable already at relatively high temperatures. The theory is able to give a very reliable quantitative description of the relaxation dynamics for all temperatures considered. The only discrepancy found is probably related to the fact that our understanding of the dynamics at [*short*]{} times is still incomplete.
For the case of silica we have calculated the $q-$dependence of the nonergodicity parameters. We have found that for this system it is important to include in the evaluation of the memory function also those contributions that stem from three point correlation functions. Most probably this finding is related to the open network structure of the system. We find that once $c_3$ is taken into account the prediction of the theory for the NEP are in very good agreement with the results from computer simulations.
In summary we have shown that MCT is able to give a very good [*quantitative*]{} description of the relaxation dynamics of fragile as well as strong liquids, at least at high and intermediate temperatures. Thus we conclude that this theory is able to rationalize [*at least*]{} the first few decades of the slowing down of a very large class of glass-forming liquids on a quantitative level.
Acknowledgments: We thank L. Fabbian for contributing to the early development of this research, and W. Götze for useful discussions on this work. Part of this work was supported by the DFG through SFB 262. F.S. acknowledges support from INFM-PRA-HOP and MURST-PRIN2000. W. K. thanks the Universitá La Sapienza for a visiting professorship during which part of this work was carried out.
See, e.g., proceedings of [*Third International Discussion Meeting*]{}, J. Non-Cryst. Solids [**235-237**]{} and proceedings of [*Unifying Concepts in Glass Physics*]{} Trieste Sep. 1999; J. Phys.: Condens. Matter [**12**]{} (2000).
W. Götze, J. Phys.: Condens. Matter [**10**]{}, A1 (1999).
J.-L. Barrat and A. Latz, J. Phys.: Condens. Matter [**2**]{}, 4289 (1990).
W. Götze and L. Sjögren, Phys. Rev. A [**43**]{}, 5442 (1991). M. Fuchs, I. Hofacker and A. Latz, Phys. Rev. A [**45**]{}, 898 (1992). M. Fuchs, W. Götze, S. Hildebrand, and A. Latz, Z. Phys. B [**87**]{}, 43 (1992).
M. Nauroth and W. Kob, Phys. Rev. E [**55**]{}, 657 (1997).
L. Fabbian, A. Latz, R. Schilling, F. Sciortino, P. Tartaglia, and C. Theis Phys. Rev. E. [**60**]{} 5768 (1999); C. Theis, F. Sciortino, A. Latz, R. Schilling, and P. Tartaglia, Phys. Rev. E [**62**]{}, 1856 (2000).
A. Winkler, A. Latz, R. Schilling, and C. Theis, Phys. Rev. E [**62**]{}, 8004 (2000).
M. H. Nauroth, Ph.D. Thesis (Technical University of Munich, 1999).
F. Sciortino and W. Kob, Phys. Rev. Lett. [**86**]{}, 648 (2001).
K. Tankeshwar, K. N. Pathak, and S. Ranganathan, J. Phys. C [**20**]{}, 5749 (1987); J. Phys. Condens. Matter [**7**]{}, 5729 (1995).
J. P. Boon and S. Yip [*Molecular Hydrodynamics*]{} (Dover, New York, 1980).
W. Kob and H. C. Andersen, Phys. Rev. E [**51**]{}, 4626 (1995); Phys. Rev. E [**52**]{}, 4134 (1995).
T. Gleim, W. Kob, and K. Binder, Phys. Rev. Lett. [**81**]{}, 4404 (1998); T. Gleim, Ph.D. Thesis (Mainz University, 1999).
W. Kob, J. Phys.: Condens. Matter [**11** ]{}(1999) R85.
B. W. H. van Beest, G. J. Kramer and R. A. van Santen, Phys. Rev. Lett. [**64**]{}, 1955 (1990).
J. Horbach and W. Kob, Phys. Rev. B [**60**]{}, 3169 (1999).
J. Horbach and W. Kob, Phys. Rev. E in press.
I. Saika-Voivod, P. H. Poole and F. Sciortino, Nature [**412**]{}, 514 (2001)
W.H. Zachariasen, J. Am. Chem. Soc. [**54**]{}, 3841 (1932).
J.-L. Barrat, W. Götze, and A. Latz, J. Phys.: Condens. Matter [**1**]{}, 7163 (1989).
Figures {#figures .unnumbered}
=======
|
---
abstract: 'Topological phases of matter are protected from local perturbations and therefore have been thought to be robust against decoherence. However, it has not been systematically explored whether and how topological states are dynamically robust against the environment-induced decoherence. In this Letter, we develop a theory for topological systems that incorporate dissipations, noises and thermal effects. We derive novelly the exact master equation and the transient quantum transport for the study of dissipative topological systems, mainly focusing on noninteracting topological insulators and topological superconductors. The resulting exact master equation and the transient transport current are also applicable for the systems initially entangled with environments. We apply the theory to the topological Haldane model (Chern insulator) and the quantized Majorana conductance to explore topological phases of matter that incorporate dissipations, noises and thermal effects, and demonstrate the dissipative dynamics of topological states.'
author:
- 'Yu-Wei Huang'
- 'Pei-Yun Yang'
- 'I-Chi Chen'
- 'Wei-Min Zhang'
title: Dissipative topological systems
---
Topological phases of matter are the most active research fields in modern condensed matter physics today [@KT1972; @TKNN1982; @Haldane1988; @Wen1991; @Kitaev2001; @Kane2005; @Bernevig2006; @Bernevig2013; @Wen2017; @Chiu2016]. They comprise several exotic quantum phases such as topological insulators and superconductors [@Hasan2010; @Qi2011], Weyl semimetals [@Xu2015], fractional quantum Hall effect [@Tsui1982] and Majorana zero modes [@HZhang2018], etc. These quantum phases of matter have been largely explored during the last decade [@Bernevig2013; @Wen2017; @Chiu2016; @Hasan2010; @Qi2011; @Ando2015; @Lutchyn2018]. However, realistic systems in nature have inevitable interactions with the surrounding environments. When system-environment interactions are not negligible, the dynamics of the systems are strongly influenced by dissipations and noises, which has become the main obstacle in practical realizations of quantum computing. Although topological phases of matter have been thought to be robust against the environment-induced decoherence, a theory that incorporates dissipations, noises and thermal effects for demonstrating such robustness has been barely established. In this Letter, we attempt to develop a dissipative quantum theory for topological phases of matter.
In contrast to an isolated quantum system, whose states are governed by Schrödinger equation, the quantum state evolution of an open quantum system (the system interacting with environment) is determined by the master equation [@Breuer08; @Weiss08]. Exact master equation for arbitrary open systems has only been formally formulated through the operator-projection method by Nakajima [@Nakajima58] and Zwanzig [@Zwanzig60]. However, in practice, very few systems can be solved from Nakajima-Zwanzig master equation [@Breuer08; @Weiss08]. Therefore, most of investigations for open quantum system dynamics are often based on the Born-Markov approximation with Lindblad-type master equation [@Lindblad76; @GKS76], including some recent applications to topological systems [@Viyuela2012; @Cheng2012; @Rivas2013]. These investigations are valid only in the weak system-environment coupling regime. There are some exceptions that one can derive the exact master equation for open quantum systems, using Feynman-Vernon influence functional approach [@Feynman63]. A prototype example is the quantum Brownian motion (QBM), its exact master equation has been derived [@Leggett83; @Haake85; @HPZ1992; @Grabert88] in 1980’s-1990’s. In the last decade, the exact master equation has also been derived for a large class of open systems described by particle-particle exchanges between the system and environments for both boson and fermion open systems [@Tu2008; @Jin2010; @Lei2012; @Zhang2012; @Zhang2018], from which we also obtain the transient quantum transport theory that can reproduce explicitly the Schwinger-Keldysh’s non-equilibrium Green function technique [@Haug2008; @Yang2017]. Very recently, we have extended the exact master equation for Majorano zero modes influenced by the gate-induced charge fluctuations [@Lai2018; @Schmidt2012].
In this Letter, we will derive novelly the exact master equation and the transient quantum transport for noninteracting topological insulators incorporating with initial system-reservoir entanglement. Then we generalize the theory to topological superconductors with Bogoliubov-de Gennes Hamiltonian that has potential applications in topological quantum computing. As a result, a dissipative quantum theory for topological phases of matter is established. We take the topological Haldane model (Chen insulator) [@Haldane1988; @Jotzu2014] and the quantized Majorana conductance in superconductor-semiconductor hybrid systems [@HZhang2018; @Lutchyn2018] as two type applications, to clarify the role of dissipations, noises and thermal effects in topological phases of matter.
*1. Open quantum systems with initial system-environment entanglement for noninteracting topological insulators.* We begin with the open systems (either bosons or fermions) coupled to their environments that are described by the following Hamiltonian, $$\begin{aligned}
H_{\mathrm{tot}}(t)
& = H_{\textsc{s}}(t) + H_{\textsc{e}}(t) + H_{{\textsc{s}}{\textsc{e}}}(t) \notag\\
& = {\begin{pmatrix} \bm{a}^{\dagger}& \bm{c}^{\dagger}\end{pmatrix}} {{\cdot}}{\begin{pmatrix} {\bm{\epsilon}}_{\textsc{s}}(t) & \bm{V}_{{\textsc{s}}{\textsc{e}}}(t) \\ \bm{V}^{\dagger}_{{\textsc{s}}{\textsc{e}}}(t) & {\bm{\epsilon}}_{\textsc{e}}(t) \end{pmatrix}} {{\cdot}}{\begin{pmatrix} \bm{a} \\ \bm{c} \end{pmatrix}}, \label{FAH}\end{aligned}$$ where $H_{\textsc{s}}(t)$ and $H_{\textsc{e}}(t)$ are the Hamiltonian of the system and the environment, respectively, and $H_{{\textsc{s}}{\textsc{e}}}(t)$ is the interaction between them. The notation $\bm{a} \equiv (a_1, a_2, a_3, \dotsc)^{\intercal}$ is a one-column matrix and $a_i$ is the annihilation operator of the $i$-th energy level of the system. Similarly, $\bm{c} \equiv (c_{k}, c_{k'}, c_{k''}, \dotsc)^{\intercal}$ and $c_k$ is the annihilation operator of the continuous spectrum mode $k$ of the environment, while ${\bm{\epsilon}}_{\textsc{s}}(t)$ and ${\bm{\epsilon}}_{\textsc{e}}(t)$ are the spectra of the system and the environment, respectively, and $\bm{V}_{{\textsc{s}}{\textsc{e}}}(t)$ is the coupling strength matrix between them.
Equation (\[FAH\]) is applicable to both topological and non-topological open quantum systems. Topological structures can be manifested through energy eigen-wavefunctions. For open systems, states of the system are described by the reduced density matrix which is determined from the total density matrix (a highly entangled state) of the system and environment: $\rho_{\textsc{s}}(t) \equiv {\operatorname{Tr}}_{\textsc{e}}[\rho_{\mathrm{tot}}(t)]$. The total density matrix is governed by the von Neumann equation: $\frac{{\mathrm{d}}}{{\mathrm{d}}t} \rho_{\mathrm{tot}}(t) = \frac{1}{i\hbar} [H_{\mathrm{tot}}(t), \rho_{\mathrm{tot}}(t)]$. Taking a partial trace over the environment states from the von Neumann equation, we have $$\begin{aligned}
\frac{{\mathrm{d}}}{{\mathrm{d}}t} \rho_{\textsc{s}}(t)
= \frac{1}{i\hbar} & [H_{\textsc{s}}(t), \rho_{\textsc{s}}(t)]
+ \bm{a}^{\dagger}{{\cdot}}\bm{A}(t) + \bm{A}^{\dagger}(t) {{\cdot}}\bm{a} \notag\\
& - ( \bm{a} {{\cdot}}\bm{A}^{\dagger}(t) + \bm{A}(t) {{\cdot}}\bm{a}^{\dagger})
, \label{fsms}\end{aligned}$$ where the collective operator $\bm{A}(t) \equiv \frac{1}{i\hbar} {\operatorname{Tr}}_{\textsc{e}}[\bm{V}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}\bm{c} \rho_{\mathrm{tot}}(t)]$ which contains all the influence of the environment on the system dynamics. Here we have also used the fact that ${\operatorname{Tr}}_{\textsc{e}}[H_{\textsc{e}}(t), \rho_{\mathrm{tot}}(t)] = 0$. Our aim is to carry out explicitly the partial trace in the collective operator $\bm{A}(t)$, from which the master equation can be novelly and straightforwardly obtained, and also the noise, thermal effects and dissipations in topological phases of matter can be explicitly explored.
For the initial system-environment decoupled or partitioned states [@Leggett1987]: $\rho_{\mathrm{tot}}(t_0) =\rho_{\textsc{s}}(t_0)
\otimes \rho_{\textsc{e}}(t_0)$, where $\rho_{\textsc{e}}(t_0) =\frac{1}{Z} e^{-\beta H_{\textsc{e}}(t_0)}$ is the thermal state of the environment, the exact master equation of Eq. (\[fsms\]) has been derived [@Tu2008; @Jin2010; @Lei2012; @Zhang2012], and the partial trace in the operator $\bm{A}(t)$ has also been explicitly computed [@Jin2010] using the Feynman-Vernon influence functional [@Feynman63]. Now we consider the system and the environment in a partition-free initial state, $\rho_{\mathrm{tot}}(t_0) = \frac{1}{Z}
e^{-\beta H_{\mathrm{tot}}(t_0)}$. In this situation, the system is highly entangled with the environment from the beginning so that the Feynman-Vernon influence functional [@Feynman63] is no longer applicable. In experiments, most of realistic open quantum systems start with a partition-free initial state. Typical examples are various quantum devices which are usually equilibrated to the environment before one starts to manipulate them. One often uses different quench methods to drive the system away from the equilibrium state to control the states of the system or to study its nonequilibrium dynamics. This can be practically realized by the time-dependent parameters in Eq. (\[FAH\]).
Because of the quadratic nature of Eq. (\[FAH\]), with the explicit time-dependent Hamiltonian $H_{\mathrm{tot}}(t)$, the total density matrix is drived away from the initially entangled equilibrium state $\rho_{\mathrm{tot}}(t_0) = \frac{1}{Z}
e^{-\beta H_{\mathrm{tot}}(t_0)}$, but it always lives in a Gaussian-type state. Therefore, in coherent state representation [@zhang90], we have $
{\langle \bm{\xi} |} \rho_{\mathrm{tot}}(t) {| \bm{\xi}' \rangle} = \frac{1}{Z(t)} \exp(\bm{\xi}^{\dagger}{{\cdot}}\bm{\Omega}(t) {{\cdot}}\bm{\xi}')
$, where ${| \bm{\xi} \rangle} \equiv \exp(\bm{a}^{\dagger}{{\cdot}}\bm{\xi}_{\textsc{s}}+ \bm{c}^{\dagger}{{\cdot}}\bm{\xi}_{\textsc{e}}) {| 0 \rangle}$ is the unnormalized coherent eigenstates of the particle annihilation operators $(\bm{a}, \bm{c})$ with eigenvalue $\bm{\xi} = (\bm{\xi}_{\textsc{s}},
\bm{\xi}_{\textsc{e}})$ which are complex numbers for bosons and Grassmann numbers for fermions, and $
\bm{\Omega}(t) = {\left(\begin{smallmatrix} \bm{\Omega}_{{\textsc{s}}{\textsc{s}}}(t) & \bm{\Omega}_{{\textsc{s}}{\textsc{e}}}(t) \\ \bm{\Omega}_{{\textsc{e}}{\textsc{s}}}(t) & \bm{\Omega}_{{\textsc{e}}{\textsc{e}}}(t) \end{smallmatrix}\right)}
$ is the Gaussian kernel of the total density matrix. By partially tracing over all the environment states, it is not difficult to find that $
{\langle \bm{\xi}_{\textsc{s}}|} \bm{A}(t) {| \bm{\xi}_{\textsc{s}}' \rangle}
= \frac{1}{i\hbar} \bm{V}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}(\bm{1} \mp \bm{\Omega}_{{\textsc{e}}{\textsc{e}}}(t))^{-1}
{{\cdot}}\bm{\Omega}_{{\textsc{e}}{\textsc{s}}}(t) {{\cdot}}\bm{\xi}_{\textsc{s}}' {\langle \bm{\xi}_{\textsc{s}}|} \rho_{\textsc{s}}(t) {| \bm{\xi}_{\textsc{s}}' \rangle}
$, from which we obtain: $$\bm{A}(t)
= \frac{1}{i\hbar} \bm{V}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}(\bm{1} \mp \bm {\Omega}_{{\textsc{e}}{\textsc{e}}}(t))^{-1}
{{\cdot}}\bm{\Omega}_{{\textsc{e}}{\textsc{s}}}(t) {{\cdot}}\rho_{\textsc{s}}(t) \bm{a}
, \label{cas1}$$ where the upper (lower) sign of $\mp$ correspond to boson (fermion) systems. Substituting this result into Eq. (\[fsms\]), we novelly obtain the exact master equation for the reduced density matrix of the system.
However, the key ingredient in the derivation of the master equation is to characterize explicitly the dissipation and noises induced by the environment, which are embedded in the time-dependent Gaussian kernel $\bm{\Omega}(t)$. Our aim is to find the relation between $\bm{\Omega}(t)$ and the physical measurable quantities such that dissipation and noise dynamics can be observed. Note that under the Gaussian state, the Wick’s theorem is always valid, and higher-order correlation functions can always be decomposed in terms of the single-particle correlations. A direct calculation shows that
$$\begin{aligned}
& \bm{n}_{\textsc{s}}(t)
= \bm{\Omega}_{\textsc{s}}(t) {{\cdot}}(\bm{1}\mp\bm{\Omega}_{\textsc{s}}(t))^{-1}, \\
& \bm{n}_{{\textsc{e}}{\textsc{s}}}(t)
= (\bm{1} \mp \bm{\Omega}_{{\textsc{e}}{\textsc{e}}}(t))^{-1} {{\cdot}}\bm{\Omega}_{{\textsc{e}}{\textsc{s}}}(t) {{\cdot}}(\bm{1} \mp \bm{\Omega}_{\textsc{s}}(t))^{-1}.\end{aligned}$$
where $n_{{\textsc{s}},ij}(t) \equiv {\langle a_j^{\dagger}(t) a_i(t) \rangle}$ and $n_{{\textsc{e}}{\textsc{s}},ki}(t) \equiv {\langle a_i^{\dagger}(t) c_k(t) \rangle}$ are the single particle correlations, and $
\bm{\Omega}_{\textsc{s}}(t)
= \bm{\Omega}_{{\textsc{s}}{\textsc{s}}}(t) \pm \bm{\Omega}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}(\bm{1} \mp \bm{\Omega}_{{\textsc{e}}{\textsc{e}}}(t))^{-1} {{\cdot}}\bm{\Omega}_{{\textsc{e}}{\textsc{s}}}(t)
$ which is given by $
{\langle \bm{\xi}_{\textsc{s}}|} \rho_{\textsc{s}}(t) {| \bm{\xi}_{\textsc{s}}' \rangle}
= \frac{1}{Z_{\textsc{s}}(t)} \exp( \bm{\xi}_{\textsc{s}}^{\dagger}{{\cdot}}\bm{\Omega}_{\textsc{s}}(t) {{\cdot}}\bm{\xi}_{\textsc{s}}' )$, and from which one can also prove that $ \bm{a} \rho_{\textsc{s}}(t) = \rho_{\textsc{s}}(t) \bm{\Omega}_{\textsc{s}}(t) {{\cdot}}\bm{a}$.
Furthermore, the time evolution of the system operators $a_i(t)$ can be directly solved from Eq. (\[FAH\]) with the Heisenberg equation of motion. The solution can be written as $\bm{a}(t) = \bm{u}(t,t_0) {{\cdot}}\bm{a}(t_0) + \bm{F}(t)$, where $u_{ij}(t,t_0) \equiv {\langle [a_i(t), a_j^\dag(t_0)]_{\mp} \rangle}$ is the retarded Green function that describes the dissipation, and $F_i(t)$ linearly depends on $c_k(t_0)$ that characterizes noises, see the explicit solution given in supplemental materials [@SM]. Then $$\bm{n}_{\textsc{s}}(t)
= \bm{u}(t,t_0) {{\cdot}}\bm{n}_{\textsc{s}}(t_0) {{\cdot}}\bm{u}^{\dagger}(t,t_0) + \bm{v}(t,t),$$ where $\bm{v}(t,t)$ generalizes the Keldysh’s correlation Green function that also includes initial system-environment entanglement [@Yang2015]. Also, the electron transient current flowing from the environment into the system is $$\begin{aligned}
I(t)
& \equiv -e \frac{{\mathrm{d}}}{{\mathrm{d}}t} {\langle \bm{c}^{\dagger}(t) {{\cdot}}\bm{c}(t) \rangle}
= \frac{e}{i\hbar} \bm{V}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}\bm{n}_{{\textsc{e}}{\textsc{s}}}(t) + \text{H.c.} \notag\\
& = -e [ \bm{\kappa}(t,t_0) {{\cdot}}\bm{n}_{\textsc{s}}(t) + \bm{\lambda}(t,t_0) + \text{H.c.} ]
, \label{tc}\end{aligned}$$ where the dissipation and noise coefficients $\bm{\kappa}(t,t_0) = \frac{1}{i\hbar} {\bm{\epsilon}}_{\textsc{s}}(t) -
\dot{\bm{u}}(t,t_0) {{\cdot}}\bm{u}^{-1}(t,t_0)$ and $\bm{\lambda}(t,t_0) = \dot{\bm{u}}(t,t_0) {{\cdot}}\bm{u}^{-1}(t,t_0)
{{\cdot}}\bm{v}(t,t) - \dot{\bm{v}}(t,t)$ are also determined explicitly by Green functions $\bm{u}(t,t_0)$ and $\bm{v}(t,t)$. Combining all the above results together, Eq (\[cas1\]) becomes $$\bm{A}(t)
= \! - [\bm{\kappa}(t,t_0) {{\cdot}}\bm{a} \rho_{\textsc{s}}(t) \!+\! \bm{\lambda}(t,t_0) {{\cdot}}(\rho_{\textsc{s}}(t) \bm{a} \mp \bm{a} \rho_{\textsc{s}}(t))]
, \label{as}$$ which captures explicitly all the dissipation and noises induced by the environment. The master equation (\[fsms\]) and the transient current (\[tc\]) simply become
\[current\_Mas\] $$\begin{aligned}
& \frac{{\mathrm{d}}\rho_{\textsc{s}}(t)}{{\mathrm{d}}t}
= \frac{1}{i\hbar} \big[ H_{\textsc{s}}(t), \rho_{\textsc{s}}(t) \big]
+ \big[ {\mathcal{L}}^{+}(t) + {\mathcal{L}}^{-}(t) \big] \rho_{\textsc{s}}(t)
, \label{eme} \\
& I(t)
= e {\operatorname{Tr}}_{\textsc{s}}\big[ {\mathcal{L}}^{+}(t) \rho_{\textsc{s}}(t) \big]
= -e {\operatorname{Tr}}_{\textsc{s}}\big[ {\mathcal{L}}^{-}(t) \rho_{\textsc{s}}(t) \big],\end{aligned}$$
where the current superoperators ${\mathcal{L}}^{+}(t) \rho_{\textsc{s}}(t) = \bm{a}^{\dagger}{{\cdot}}\bm{A}(t) + \bm{A}^{\dagger}(t) {{\cdot}}\bm{a}$ and ${\mathcal{L}}^{-}(t) \rho_{\textsc{s}}(t)
= - \bm{a} {{\cdot}}\bm{A}^{\dagger}(t) - \bm{A}(t) {{\cdot}}\bm{a}^{\dagger}$ carry the information current flowing into and out of the system, respectively.
It is easy to check that for fermionic systems, Eq. (\[current\_Mas\]) reproduces the exact master equation and the transient transport current incorporating with the initial system-environment correlations given in Ref. [@Yang2015]; For noninteracting bosonic systems, except for a special case [@Tan2011], this gives a general dissipative theory incorporating initial system-environment entanglement. The master equation and the transient current also have the same universal form derived from the Feynman-Vernon influence functional for the case of no initial system-environment entanglement [@Tu2008; @Jin2010; @Lei2012; @Zhang2012], whereas the initial system-environment entanglement is fully embedded into the correlation Green function $\bm{v}(t,t)$, as shown in [@Tan2011; @Yang2015].
*2. Open systems for topological superconductors.* Now, we generalize the exact master equation to the open systems containing paring couplings to the environment, such as the superconductor-semiconductor hybrid systems in the study of topological quantum computing. Through a Bogoliubov transformation, the paring terms in the Hamiltonian of the system or the environment can always be switched into the coupling Hamiltonian between the system and the environment. Then the general Hamiltonian can be expressed as $$\begin{aligned}
H_{\mathrm{tot}}(t)
= & \bm{a}^{\dagger}{{\cdot}}{\bm{\epsilon}}_{\textsc{s}}(t) {{\cdot}}\bm{a} + \bm{c}^{\dagger}{{\cdot}}{\bm{\epsilon}}_{\textsc{e}}(t) {{\cdot}}\bm{c} \notag \\
& + {\begin{pmatrix} \bm{a}^{\dagger}& \bm{a} \end{pmatrix}} {\begin{pmatrix} \bm{V}_{{\textsc{s}}{\textsc{e}}}(t) & \bm{\Delta}_{{\textsc{s}}{\textsc{e}}}(t) \\ \pm \bm{\Delta}_{{\textsc{s}}{\textsc{e}}}^*(t) & \pm \bm{V}_{{\textsc{s}}{\textsc{e}}}^*(t) \end{pmatrix}}{\begin{pmatrix} \bm{c} \\ \bm{c}^{\dagger}\end{pmatrix}},\end{aligned}$$ where the last term is the Bogoliubov-de Gennes Hamiltonian matrix describing effectively the pairing processes between the system and environment.
Following the same procedure, taking a partial trace over the environmental states from the von Neumann equation, one obtains the same master equation (\[fsms\]) for the reduced density matrix $\rho_{\textsc{s}}(t)$. The only difference is the collective operator $\bm{A}(t)$ which is now given by $$\begin{aligned}
\bm{A}(t)
\equiv \frac{1}{i\hbar} {\operatorname{Tr}}_{\textsc{e}}[(\bm{V}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}\bm{c} + \bm{\Delta}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}\bm{c}^{\dagger}) \rho_{\mathrm{tot}}(t)].\end{aligned}$$ Similarly, if one can carry out explicitly the partial trace over the environmental states for the above operator $\bm{A}(t)$, the exact master equation involving pairing couplings can also be novelly and straightforwardly obtained. Indeed, using the same procedure, we obtain $$\begin{aligned}
{\begin{pmatrix} \bm{A}(t) \\ -\bm{A}^{\dagger}(t) \end{pmatrix}}
= & - {\mathcal{K}}(t,t_0) {\begin{pmatrix} \bm{a} \rho_{\textsc{s}}(t) \\ \bm{a}^{\dagger}\rho_{\textsc{s}}(t) \end{pmatrix}} \notag\\
& - {\mathcal{Z}} \Lambda(t,t_0) {\begin{pmatrix}
\rho_{\textsc{s}}(t) \bm{a} \mp \bm{a} \rho_{\textsc{s}}(t) \\
\bm{a}^{\dagger}\rho_{\textsc{s}}(t) \mp \rho_{\textsc{s}}(t) \bm{a}^{\dagger}\end{pmatrix}}
, \label{tsa}\end{aligned}$$ where ${\mathcal{Z}} = {\left(\begin{smallmatrix} \bm{1} & \bm{0} \\ \bm{0} & \mp\bm{1} \end{smallmatrix}\right)}$, and ${\mathcal{K}}(t,t_0)$ and $\Lambda(t,t_0)$ are given later, see Eq. (\[sopc\]). Thus, the master equation for topological superconductor open systems with arbitrary pairing couplings has exactly the same form as Eq. (\[eme\]) but the collective operator $\bm{A}(t)$ is modified by Eq. (\[tsa\]).
Because the topological superconductor open systems involving pairing interactions, the explicit form of the master equation is more complicated. Substituting the solution of Eq. (\[tsa\]) into Eq. (\[fsms\]), we get $$\begin{aligned}
\frac{{\mathrm{d}}}{{\mathrm{d}}t} \rho_{\textsc{s}}(t)
= & \frac{1}{i\hbar} [H'_{{\textsc{s}}}(t,t_0), \rho_{\textsc{s}}(t)] \notag\\
& + \sum_{ij} \gamma_{ij}(t,t_0) \begin{aligned}[t] \big[
& 2 a_j \rho_{\textsc{s}}(t) a_i^{\dagger}\\
& - \rho_{\textsc{s}}(t) a_i^{\dagger}a_j - a_i^{\dagger}a_j \rho_{\textsc{s}}(t)
\big] \end{aligned} \notag\\
& + \sum_{ij} \widetilde{\gamma}_{ij}(t,t_0) \begin{aligned}[t] \big[
& a_i^{\dagger}\rho_{\textsc{s}}(t) a_j \pm a_j \rho_{\textsc{s}}(t) a_i^{\dagger}\\
& - \rho_{\textsc{s}}(t) a_j a_i^{\dagger}\mp a_i^{\dagger}a_j \rho_{\textsc{s}}(t)
\big] \end{aligned} \notag\\
& \mp \frac{1}{2} \sum_{ij} \big\{ \lambda_{ji}(t, \begin{aligned}[t]
&t_0) \big[ 2 a_j^{\dagger}\rho_{\textsc{s}}(t) a_i^{\dagger}- a_i^{\dagger}a_j^{\dagger}\rho_{\textsc{s}}(t) \\
& - \rho_{\textsc{s}}(t) a_i^{\dagger}a_j^{\dagger}\big] + \text{H.c.} \big\} \end{aligned}
\label{pairMEM}\end{aligned}$$ The first term is the renormalized Bogoliubov-de Gennes Hamiltonian of the system $
H_{\textsc{s}}'(t,t_0)
= \frac{1}{2} {\left(\begin{smallmatrix} \bm{a}^{\dagger}& \bm{a} \end{smallmatrix}\right)}
{\left(\begin{smallmatrix} \bm{E}_{\textsc{s}}'(t,t_0) & \bm{\Delta}_{\textsc{s}}'(t,t_0) \\ \bm{\Delta}_{\textsc{s}}'^{\dagger}(t,t_0) & \pm \bm{E}_{\textsc{s}}'^{\intercal}(t,t_0) \end{smallmatrix}\right)}
{\left(\begin{smallmatrix} \bm{a} \\ \bm{a}^{\dagger}\end{smallmatrix}\right)}
$. The second and the third terms describe the dissipation and noise dynamics which are very similar to the cases without including pairings [@Tu2008; @Jin2010; @Lei2012; @Yang2015]. The last term comes from pairing-process induced dissipation. Explicitly, those time non-local dissipation and noise coefficients
\[sopc\] $$\begin{aligned}
& {\begin{pmatrix} \bm{E}_{\textsc{s}}'(t,t_0) & \bm{\Delta}_{\textsc{s}}'(t,t_0) \\ \bm{\Delta}_{\textsc{s}}'^{\dagger}(t,t_0) & \pm \bm{E}_{\textsc{s}}'^{\intercal}(t,t_0) \end{pmatrix}}
= {\mathcal{E}}_{\textsc{s}}(t) + \frac{\hbar}{2i} ( {\mathcal{K}}(t) - {\mathcal{K}}^{\dagger}(t) )\notag\\
& \quad\quad
= - \frac{\hbar}{2i} ( {\mathcal{Z}} \dot{{\mathcal{U}}}(t,t_0) {\mathcal{U}}(t,t_0)^{-1} - \text{H.c.} ), \\
& {\begin{pmatrix} \bm{\gamma}(t,t_0) & \bm{\gamma}'(t,t_0) \\ \bm{\gamma}'^{\dagger}(t,t_0) & \mp \bm{\gamma}^{\intercal}(t,t_0) \end{pmatrix}}
= \frac{1}{2} ( {\mathcal{K}}(t,t_0) + {\mathcal{K}}^{\dagger}(t,t_0) ) \notag\\
& \quad\quad
= - \frac{1}{2} ( {\mathcal{Z}} \dot{{\mathcal{U}}}(t,t_0) {\mathcal{U}}(t,t_0)^{-1}+ \text{H.c.} ), \\
& {\begin{pmatrix}
\widetilde{\bm{\gamma}}(t,t_0) & \bm{\lambda}(t,t_0) \\
\bm{\lambda}^{\dagger}(t,t_0) & 2\bm{\gamma}^{\intercal}(t,t_0) {\pm} \widetilde{\bm{\gamma}}^{\intercal}(t,t_0)
\end{pmatrix}}
= - \Lambda(t,t_0) - \Lambda^{\dagger}(t,t_0) \notag\\
& \quad\quad
= \tfrac{{\mathrm{d}}}{{\mathrm{d}}t} {\mathcal{V}}(t,t) - ( \dot{{\mathcal{U}}}(t,t_0) {\mathcal{U}}(t,t_0)^{-1} {\mathcal{V}}(t,t) + \text{H.c.} ),\end{aligned}$$
are all determined by the retarded and correlation Green functions ${\mathcal{U}}(t,t_0)$ and ${\mathcal{V}}(t,t)$ incorporating pairing interactions [@SM]. Those matrices are Hermitian, so we have $\bm{E}_{\textsc{s}}'^{\dagger}(t,t_0) = \bm{E}_{\textsc{s}}'(t,t_0)$ and $\bm{\Delta}_{\textsc{s}}'^{\intercal}(t,t_0)
= \pm \bm{\Delta}_{\textsc{s}}'(t,t_0)$, $\bm{\gamma}^{\dagger}(t,t_0) = \bm{\gamma}(t,t_0)$ and $\bm{\gamma}'^{\intercal}(t,t_0) = \mp \bm{\gamma}'(t,t_0)$, $\widetilde{\bm{\gamma}}^{\dagger}(t,t_0) = \widetilde{\bm{\gamma}}(t,t_0)$ and $\bm{\lambda}^{\intercal}(t,t_0) = 2\bm{\gamma}'(t,t_0) \pm \bm{\lambda}(t,t_0)$. The experimentally measured transport current flowing from the environment into the system is given by $$\begin{aligned}
I(t) = \frac{e}{i\hbar} \bm{V}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}\bm{n}_{{\textsc{e}}{\textsc{s}}}(t) - \frac{e}{i\hbar} \bm{\Delta}_{{\textsc{s}}{\textsc{e}}}(t) {{\cdot}}\bm{q}_{{\textsc{e}}{\textsc{s}}}(t) + \text{H.c.}
, \label{pairingcurrent}\end{aligned}$$ where $q_{{\textsc{e}}{\textsc{s}},ij}(t) \equiv {\langle a_j^{\dagger}(t) c_i^{\dagger}(t) \rangle}$. From this transient current one can study Majorana quantum transport dynamics that we will discuss latter.
*3. Applications.* The first application is the topological Haldane model which describes quantum Hall effect in honeycomb lattice without magnetic field [@Haldane1988] and has been experimentally realized with ultracold fermionic atoms [@Jotzu2014], and its Hamiltonian can be written as $$\begin{aligned}
H
= & M \sum_{i} \left( a_{i}^{\dagger}a_{i} - b_{i}^{\dagger}b_{i} \right)
- J_1 \sum_{\langle i,j \rangle} \left( a_{i}^{\dagger}b_{j} + \text{H.c.} \right) \notag\\
& + J_2 \sum_{\langle\!\langle i,j \rangle\!\rangle} \left(
e^{i\phi} a_{i}^{\dagger}a_{j} + e^{-i\phi} b_{i}^{\dagger}b_{j} + \text{H.c.}
\right)
, \label{hmh}\end{aligned}$$ where $a_i$ ($b_i$) is the annihilation operator of A (B) site electrons, and $J_1$ and $J_2$ are the nearest neighbor and the next-nearest neighbor coupling strengths, respectively, see Fig. \[pdhm\](a). The energy difference $M$ between A and B sites breaks inversion symmetry, and the phase $\phi$ in the next-nearest neighbor couplings breaks time-reversal symmetry that topologically leads to quantum Hall effect. The non-trivial topological phase is located in the region of $|M| < 3\sqrt{3}|J_2\sin\phi|$ (see Fig. \[pdhm\](b)), in which the band gap is closed at the edge of lattice. Here we attempt to dynamically probe this topological structure in Haldane model from the open quantum system point of view by coupling an adatom to the honeycomb lattices.
Putting an adatom ($H_a = \epsilon_0 c_d^{\dagger}c_d$) on the edges or bulk of lattices, described by the coupling Hamiltonian $H_I = V c_d^{\dagger}a_j+ \text{H.c.}$, where $j$ is the coupled site, we can study the dissipative dynamics of the adatom under the influence of the topological structure of the Haldane model. We treat the honeycomb lattice with the Haldane Hamiltonian (\[hmh\]) as the environment of the adatom. Then the solution of the reduced density matrix of the adatom can be determined effectively by the occupation number $n_a(t) = {\operatorname{Tr}}_{\textsc{s}}[c_d^{\dagger}c_d \rho_{\textsc{s}}(t)] $. By dynamically solving the occupation number of the adatom (initially occupied), we find that its steady-state solution manifests the whole topological structure of the Haldane model, as shown in Fig. \[pdhm\](c), as a result of dissipation. In Fig. \[pdhm\](c) the dark color corresponds to the complete dissipation (zero occupation in the adatom in the steady-state limit but initially it is fully occupied) in the topological phase. Such dissipation is built up only when the lattice energy gap closes, which occurs at the edge of non-trivial topological phase, see the right plot in Fig. \[pdhm\](c). This provides indeed a very useful method of probing topological structures for more complicated topological systems through the study of dissipative dynamics of adatoms (impurities).
Another application is the quantized Majorana conductance in superconductor-semiconductor hybrid system that has been very recently observed [@HZhang2018]. The Hamiltonian of the total system is modeled as a tight-binding $N$-site p-wave superconductor, with its left/right ends of superconductor coupled respectively with the left/right leads. One can solve the large number chain of superconductor with zero chemical potential [@Schmidt2012], in which two Majorana zero modes are localized at the ends of the chain with exponentially decaying wave function along the chain. Focusing only on the zero modes, we have the interaction Hamiltonian of the zero modes coupled with the two leads, $$\begin{aligned}
H_I
& = \sum_k
{\begin{pmatrix} c_{L,k}^{\dagger}& c_{L,k} \end{pmatrix}}
{\begin{pmatrix} V'_{L,k} & \Delta'_{L,k} \\ -\Delta'_{L,k} & - V'_{L,k} \end{pmatrix}}
{\begin{pmatrix} b_0 \\ b_0^{\dagger}\end{pmatrix}} \\
& + \sum_k
{\begin{pmatrix} c_{R,k}^{\dagger}& c_{R,k} \end{pmatrix}}
{\begin{pmatrix} V'_{R,k} & -\Delta'_{R,k} \\ \Delta'_{R,k} & - V'_{R,k} \end{pmatrix}}
{\begin{pmatrix} b_0 \\ b_0^{\dagger}\end{pmatrix}}.
\end{aligned}$$ where $b_0$ is zero mode annihilation operator, and $V'_{\alpha,k} = \frac{\sqrt{1-\delta^2}}{2} (1 + \delta^{(N-1)/2})
V_{\alpha,k}$, $\Delta'_{\alpha,k} = \frac{\sqrt{1-\delta^2}}{2} (1 - \delta^{(N-1)/2}) V_{\alpha,k}$. It shows that the tunneling strength $V'_{\alpha,k}$ and pairing parameter $\Delta'_{\alpha,k}$ only depend on dimensionless parameter $\delta = (\Delta - w)/(\Delta + w)$. And for large $N$, the tunneling strength is almost equal to the pairing parameter, which makes the superconducting system evolve into a half-filled state.
Applying bias to the leads, one can measure the current through the superconductor. From Eq. (\[pairingcurrent\]), we study the transient current and transient differential conductance of superconductor, and find a relation between the spectral density and the conductance in the steady state limit. Especially when two spectral densities $J_{\alpha}(\epsilon)$ are same and symmetric, we have, $$\begin{aligned}
G(\mu,t\to\infty) = {\textstyle\int {\mathrm{d}}\epsilon} \, G_0(\epsilon) \tfrac{\partial}{\partial\mu} f(\mu,\epsilon),\end{aligned}$$ where $G_0(\mu) = 1/\big[\big(\tfrac{\mu - \Delta'(\mu)}{J'(\mu)/2}\big)^2 + 1\big]$ is the conductance at zero temperature in the unit of $2e^2/h$, and $f(\mu,\epsilon)$ is the Fermi-Dirac distribution, $J'(\epsilon) = \sum_{\alpha,k} 2\pi
(|V_{\alpha,k}'|^2 + |\Delta_{\alpha,k}'|^2) \delta(\epsilon_{\alpha,k} - \epsilon)$ is the spectral density, and $\delta \mu'(\epsilon)
= {\mathcal{P}}\int \frac{{\mathrm{d}}\epsilon'}{2\pi} \frac{J'(\epsilon')}{\epsilon - \epsilon'}$ is the corresponding energy shift, which is anti-symmetric because of the symmetric spectral density. It shows that at zero temperature, the conductance at zero bias is precisely the quantized Majorana conductance, $2e^2/\hbar$ [@Law2009], recently measured in experiment [@HZhang2018]. It has a Lorentzian function shape deformed by the energy shift and the decay rate, and thermal fluctuations broaden and lower down the zero-bias peak by convolution. The buildups of zero-bias peak are shown in Fig. \[zbp\]. The transient behavior of current in different bias involves different frequencies, which induces the oscillation of differential conductance. It shows that Majorana conductance is indeed the observation of the dissipation and thermal fluctuations of the Majorana zero mode.
![\[zbp\]Buildups of zero-bias conductance peak in the time domain at (a) zero temperature, (b) low temperature ($k_BT = 0.005W$), and (c) high temperature ($k_BT = 0.05W$). Here we take $J'(\epsilon) = \Gamma \sqrt{1 - (\frac{\epsilon}{W/2})^2}$, which is solved from the tight-binding model, and set height-width ratio as $\Gamma/W = 0.095$. ]({conductance-Gamma=0.19}.pdf){width="\linewidth"}
In conclusion, we novelly derive a dissipation theory for noninteracting topological systems, which allows one to investigate the dynamics of topological states incorporating dissipations, noises and thermal effects. By applying the theory to the Haldane model and the quantized Majorana conductance in a superconductor-semiconductor hybrid system, we demonstrate how dissipation and noises make topological structures observed in experiments. On the other hand, dissipation and noises are the sources of decoherence. Therefore, topological states cannot be immune from decoherence.
J. M. Kosterlitz and D. J. Thouless, *Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)*, J. of Phys. C: Solid State Phys. **5**, L124 (1972).
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, *Quantized Hall conductance in a two-dimensional periodic potential*, Phys. Rev. Lett. **49**, 405 (1982).
F. Haldane, *Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly*, Phys. Rev. Lett. **61**, 18 (1988).
X. G. Wen, *Topological Orders in Rigid States*, Int. J. Mod. Phys. B. **4**, 239 (1990); *Non-Abelian Statistics in the FQH states*, Phys. Rev. Lett. **66**, 802 (1991).
A. Y. Kitaev, *Unpaired Majorana fermions in quantum wires*, Physics Uspekhi **44**, 131 (2001); *Fault-tolerant quantum computation by anyons*, Ann. Phys. **303**, 2 (2003).
C. Kane and E. Mele, *$Z_2$ topological rrder and the quantum spin Hall effect*, Phys. Rev. Lett. **95**, 226801 (2005).
B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, *Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells*, Science **314**, 1757 (2006).
B. A. Bernevig, and T. L. Hughes, *Topological Insulators and Topological Superconductors* (Princeton Univ. Press, New Jersey, 2013).
C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, *Classification of topological quantum matter with symmetries*, Rev. Mod. Phys. **88**, 035005 (2016).
X. G. Wen, *Colloquium: Zoo of quantum-topological phases of matter*, Rev. Mod. Phys. **89**, 041004 (2017).
M. Z. Hasan and C. L. Kane, *Colloquium: Topological insulators*, Rev. Mod. Phys. **82**, 3045 (2010).
X. L. Qi and S. C. Zhang, *Topological insulators and superconductors*, Rev. Mod. Phys. **83**, 1057 (2011).
S. Y. Xu, *et al*, *Discovery of a Weyl fermion semimetal and topological Fermi arcs*, Science **349**, 613 (2015).
D. C. Tsui, H. L. Stormer, and A. C. Gossard, *Two-Dimensional Magnetotransport in the Extreme Quantum Limit*, Phys. Rev. Lett. **48**, 1559 (1982).
H. Zhang, *et al.* *Quantized Majorana conductance*, Nature **556**, 74 (2018).
Y. Ando and L. Fu, *Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials*, Ann. Rev. of Conden. Matter Phys. **6**, 361 (2015).
R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, *Majorana zero modes in superconductor-semiconductor heterostructures*, Nature Review Materials, **3**, 52 (2018).
P. Breuer and F. Petruccione, *The Theory of Open Quantum Systems*, see particularly Chapters 10 $\sim$ 12 (Oxford Univ. Press, New York, 2002).
U. Weiss, *Quantum Dissipative Systems*, 3rd Ed. (World Scientific, Singapore, 2008).
S. Nakajima, *On quantum theory of transport phenomena: steady diffusion*, Prog. Theo. Phys. **20**, 948 (1958).
R. Zwanzig, *Ensemble method in the theory of irreversibility*, J. Chem. Phys. **33**, 1338 (1960).
G. Lindblad, *On the generators of quantum dynamical semigroups*, Comm. Math. Phys. **48**, 119 (1976).
V. Gorini, A. Kossakowski, E.C.G. Sudarshan, *Completely positive dynamical semigroups of N-level systems*, J. Math. Phys. **17**, 821 (1976).
O. Viyuela, A. Rivas, M.A. Martin-Delgado, *Thermal instability of protected end states in a 1-D topological insulator*, Phys. Rev. B **86**, 155140 (2012).
M. Cheng, R. M. Lutchyn, and S. Das Sarma, *Topological protection of Majorana qubits*, Phys. Rev. B **85**, 165124 (2012).
A. Rivas, O. Viyuela, M.A. Martin-Delgado, *Density Matrix Topological Insulators*, Phys. Rev. B **88**, 155141 (2013).
R. P. Feynman and F. L. Vernon, *The theory of a general quantum system interacting with a linear dissipative system*, Ann. Phys. **24**, 118 (1963).
A. O. Caldeira, and A. J. Leggett, *Path Integral Approach To Quantum Brownian Motion*, Physica A **121**, 587 (1983).
F. Haake and R. Reibold, *Strong damping and low-temperature anomalies for the harmonic oscillator*, Phys. Rev. A **32**, 2462 (1985).
B. L. Hu, J. P. Paz, Y. H. Zhang, *Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise*, Phys. Rev. D **45**, 2843 (1992).
H. Grabert, P. Schramm, and G.-L. Ingold, *Quantum Brownian motion: The functional integral approach*, Phys. Rep. **168**, 115 (1988).
M. W. Y. Tu and W. M. Zhang, *Non-Markovian decoherence theory for a double-dot charge qubit*, Phys. Rev. B **78**, 235311 (2008).
J. S. Jin, M. W. Y. Tu, W. M. Zhang, and Y. J. Yan, *Non-equilibrium quantum theory for nanodevices based on the Feynman-Vernon influence functional*, New J. Phys. **12**, 083013 (2010).
C. U. Lei, and W. M. Zhang, *A quantum photonic dissipative transport theory*, Ann. Phys. **327**, 1408 (2012).
W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W. Y. Tu, and F. Nori, *General non-Markovian dynamics of open quantum systems*, Phys. Rev. Lett. **109**, 170402 (2012).
W. M. Zhang, *Exact master equation and general non-Markovian dynamics in open quantum systems*, arXiv:1807.01965 (2018).
H. Haug and A. P. Jauho, *Quantum Kinetics in Transport and Optics of Semiconductors* (Springer Series in Solid-State Sciences Vol. 123, 2008).
P. Y. Yang and W. M. Zhang, *Master equation approach to transient quantum transport in nanostructures*, Frontiers of Physics, **12**, 127204 (2017).
H. L. Lai, P. Y. Yang, Y. W. Huang, and W. M. Zhang, *Exact master equation and non-Markovian decoherence dynamics of Majorana zero modes under gate-induced charge fluctuations*, Phys. Rev. B **97**, 054508 (2018).
M. J. Schmidt, D. Rainis, and D. Loss, *Decoherence of Majorana qubits by noisy gates*, Phys. Rev. B **86**, 085414 (2012).
G Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif and T. Esslinger, *Experimental realization of the topological Haldane model with ultracold fermions*, Nature **515**, 237 (2014).
A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, *Dynamics of the dissipative two-state system*, Rev. Mod. Phys. **59**, 1 (1987).
W. M. Zhang, D. H. Feng and R. Gilmore, *Coherent states: theory and some applications*, Rev. Mod. Phys. **62**, 867 (1990).
See Supplemental Materials
P. Y. Yang, C. Y. Lin and W. M. Zhang, *Master equation approach to transient quantum transport in nanostructures incorporating initial correlations*, Phys. Rev. B **92**, 165403 (2015).
H. T. Tan and W. M. Zhang, *Non-Markovian dynamics of an open quantum system with initial system-reservoir correlations: A nanocavity coupled to a coupled-resonator optical waveguide*, Phys. Rev. A **83**, 032102 (2011).
K. T. Law, P. A. Lee, and T. K. Ng, *Majorana fermion induced resonant Andreev reflection*, Phys. Rev. Lett. **103**, 237001 (2009).
|
---
abstract: 'Several families of irregular moons orbit the giant planets. These moons are thought to have been captured into planetocentric orbits after straying into a region in which the planet’s gravitation dominates solar perturbations (the Hill sphere). This mechanism requires a source of dissipation, such as gas-drag, in order to make capture permanent. However, capture by gas-drag requires that particles remain inside the Hill sphere long enough for dissipation to be effective. Recently we have proposed that in the circular restricted three-body problem particles may become caught up in ‘sticky’ chaotic layers which tends to prolong their sojourn within the planet’s Hill sphere thereby assisting capture. Here we show that this mechanism survives perturbations due to the ellipticity of the planet’s orbit. However, Monte Carlo simulations indicate that the planet’s ability to capture moons decreases with increasing orbital eccentricity. At the actual Jupiter’s orbital eccentricity, this effects in approximately an order of magnitude lower capture probability than estimated in the circular model. Eccentricities of planetary orbits in the Solar System are moderate but this is not necessarily the case for extrasolar planets which typically have rather eccentric orbits. Therefore, our findings suggest that these extrasolar planets are unlikely to have substantial populations of irregular moons.'
author:
- |
Sergey A. Astakhov$^{1,2}$[^1] and David Farrelly$^{2}$[^2]\
$^{1}$John von Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany\
$^{2}$Department of Chemistry and Biochemistry, Utah State University, UT 84322-0300, USA
date: 14 August 2004
title: 'Capture and escape in the elliptic restricted three-body problem'
---
celestial mechanics – methods: $N$-body simulations – planets and satellites: formation – planetary systems: formation
Introduction
============
While the large regular moons of the giant planets follow almost circular, low inclination, prograde orbits, the aptly named irregular moons tend to do the opposite: i.e., they often have highly eccentric, high inclination orbits which may be retrograde [*or*]{} prograde. These moons, therefore, provide actual observational examples of the complexities of three-body dynamics. Irregular moons are thought to have been captured during the early stages of the Solar System but the detailed mechanism has not been fully elucidated. Based on a study of the circular restricted three-body problem we have recently proposed that chaos played a key role in the initial stages of capture [@cac2003]; in this mechanism – called chaos assisted capture (CAC) – particles may initially become entangled in chaotic layers which separate directly scattering from regular (bound) regions of phase space. This temporary trapping serves to extend the lifetimes of particles within the Hill sphere thereby providing the breathing space necessary for relatively weak dissipative forces (e.g., gas-drag) to effect permanent capture. While this basic scenario may be responsible for the [*initial*]{} formation of families of irregular moons, other factors may affect their subsequent long term evolution including collisions; gravitational perturbations from other planets; and deviations from circularity of the planet’s orbit. Here we investigate the effect of orbital ellipticity of the planet on the capture mechanism using Monte Carlo simulations and phase space visualisations based on the Fast Lyapunov Indicator. Two representative star-planet binaries are considered: the actual Sun-Jupiter system and a hypothetical extrasolar cousin of higher orbital eccentricity. We find that the chaos-assisted capture mechanism is robust to moderate ellipticities.
Recent discoveries of numerous irregular moons at giant planets [@gladman2001; @jewitt2003; @kavel] have spurred the development of theories of their origin, which is generally thought to involve capture (@carruba2002 [@yokoyama2003; @cac2003; @burns2004]; @nesvorny2004; @neto2004 [@kavel]). These moons are considered to be primordial pieces of the Solar System captured in the amber of time. While there is, as yet, no consensus on the detailed capture mechanism of these minor bodies a number of recent studies have, nevertheless, attemped to simulate their post-capture evolution in order to explain salient aspects of their orbits. For example, several attempts have recently been made to explain the observed clustering of irregular satellites as being the result either of the breakup of a larger parent body or from catastrophic collisions between planetesimals [@nesvorny2003; @burns2004; @nesvorny2004; @kavel]. These hypotheses are supported empirically by photometric surveys [@grav2004] which indicate that members of the clusters have similar surface colours suggesting that they may have had common progenitors. However, whatever post-capture orbital evolution occurs [*within*]{} the Hill sphere there has to have been an initial capture mechanism efficient enough to populate the Hill sphere with moonlets in the first place. We have recently proposed one such mechanism – chaos-assisted capture [@cac2003] – in which long-term, but temporary, trapping occurs in the ‘sticky’ chaotic layers lying close to Kolmogorov-Arnold-Moser [KAM, @ll] tori embedded within the planet’s Hill sphere. If a particle gets entangled in one of these layers then even a moderate level of dissipation (or, alternatively, slow planetary growth, @neto2004) may be sufficient to make capture permanent. It is likely that Comet-Shoemaker-Levy 9 was a recent example of an object trapped in such a chaotic zone for the best part of the last century [@chodas]. However, the absence of a large gas cloud at Jupiter makes contemporary permanent capture of such objects by gas-drag unfeasible.
Qualitatively, in the CAC model the capture probability of a planet is largely determined by the volume of phase space (at a given energy) taken up by chaotic zones within the Hill sphere. Simulations in the circular restricted three-body problem (CRTBP) show that capture becomes less probable at higher energies because most of the phase space is directly scattering and relatively few KAM tori survive; those that do are surrounded by relatively thin chaotic layers which are less effective at trapping intruders [@cac2003]. Factors that are not included in the CRTBP may also affect the capture mechanism, e.g., the eccentricity of the parent planet’s orbit. A more realistic extension of CAC beyond the CRTBP is, therefore, needed. As a first step we extend the model to the [*elliptic restricted three-body problem*]{} (ERTBP, @sze [@llibre; @benest]; @dvorak; @scheeres; @mako) which allows us to model perturbations induced by deviations of the planet’s orbit from circularity. While orbital eccentricities are relatively small, although not negligible, for the giant planets of the Solar System they may be appreciable for extrasolar planets (@schneider [@marcy]; @fischer; @marzari [@goldreich; @beauge; @michtchenko]). For example, compare Jupiter’s mean orbital eccentricity [^3] $e=0.04839$ to typical extrasolar planet eccentricities which lie in the range $e \sim 0.2-0.6$ [@tremaine]. Up to now there have been more than 100 extrasolar planetary systems detected [^4] and an intriguing question is whether these massive planets might harbour moons [@barnes; @death; @williams] which might be habitable [@habitable]. As an exemplary extrasolar captor, we study a system similar in mass ratio to the idealized Sun-Jupiter binary but which follows a highly elliptic orbit. This provides a comparative estimate for the probability of capture and also suggests the ranges of energy and orbital inclination over which extrasolar irregular moons might be expected to exist.
The paper is organized as follows: Section 2 introduces the Hamiltonian for the ERTBP which, in the limit of zero ellipticity, reduces to the CRTBP. We work in a coordinate system whose origin coincides with the planet and, in actual integrations, regularise the dynamics to deal with two-body collisions [@Aarseth]. Because the ERTBP is explicitly time dependent it is not possible to compute conventional Poincaré surfaces of section [SOS, @ll] even in the planar limit so as to visualise the structure of phase space. Therefore we use the notion of a Fast Lyapunov Indicator [FLI, @fro2000] to visualise the structure of phase space. Section 3 briefly discusses the CAC mechanism as applied to the CRTBP in order to facilitate comparison with the ERTBP. In particular, FLI on the surfaces of section are computed which can be compared directly with SOS in the CRTBP. This is done in Sec. 4 where Monte Carlo simulations of capture are performed. Unlike in @cac2003 in these simulations we do not include dissipation and, instead, focus on the distributions of particles that can be temporarily trapped in chaotic zones for very long time periods. This avoids complications associated with the best choice of dissipative force [see, e.g., @burns2004]. Finally, conclusions are contained in Sec. 5.
Hamiltonian and methods
=======================
The CRTBP describes the dynamics of a test particle having infinitesimal mass and moving in the gravitational field of two massive bodies (the ‘primaries’ – e.g., a planet and a star) which revolve around their center of mass on a circular orbit. The equations of motion are, therefore, most naturally presented in a non-inertial coordinate system that rotates with the mean motion of the primaries [@murray]. In the rotating coordinate system the positions of the primaries are fixed. When the planet’s orbit is elliptic rather than circular a nonuniformly rotating-pulsating coordinate system is commonly used. These new coordinates have the felicitous property that, again, the positions of the primaries are fixed; however the Hamiltonian is explicitly time-dependent [@sze].
Hamiltonian
-----------
For our purposes it is most convenient to locate the origin at the planet (Fig. \[zvs\]) because angular momentum will be measured with respect to the planet. Then, following @sze and @llibre, we obtain the planetocentric ERTBP Hamiltonian $H_e$ after introducing an isotropically pulsating length scale
$$H_e=E_e=\frac{1}{2}((p_x+y)^2 + (p_y-x)^2 + {p_z}^2 + z^2)-$$ $$\biggl(\frac{1-\mu }{\sqrt{(1+x)^2+y^2+z^2}} + \frac{\mu }{\sqrt{x^2+y^2+z^2}} +$$ $$\frac{1}{2}(x^2 + y^2 + z^2) +
(1-\mu)x +\frac{1}{2}(1-\mu) \biggr)\bigg/(1+e\,\cos\,f). \label{ertbp}$$
The semimajor axis of the orbit of the primaries $a_p$ has been scaled to unity and $e$ is the eccentricity of the planet’s orbit; $\mu = m_{1}/(m_{1}+m_{2})$, where $\mu, m_{1}$ and $m_{2}$ are the reduced mass and masses of the planet and star, respectively ($\mu = 9.5359 \times 10^{-4}$ for Sun-Jupiter). The true anomaly $f$, i.e. the planet’s angular position measured from the pericenter, is related to the physical time $t$ through [@sze] $$\frac{df}{dt} = \frac{(1+e\,\cos\,f)^{2}}{(1-e^{2})^{3/2}}. \label{true}$$
The (planet-centered) CRTBP Hamiltonian [@cac2003]
$$H_c=E_c=\frac{1}{2}({p_x}^2 + {p_y}^2 + {p_z}^2)-(x\,p_{y}-y\,p_{x})-$$ $$\frac{1-\mu }{\sqrt{(1+x)^{2}+y^{2}+z^{2}}}-\frac{\mu }{\sqrt{x^{2}+y^{2}+z^{2}}}-$$ $$(1-\mu)x - \frac{1}{2}(1-\mu), \label{crtbp}$$
is recovered when $e=0$. In both cases (\[ertbp\]) and (\[crtbp\]), $h_z = x\,p_{y}-y\,p_{x}$ is the $z$-component of angular momentum ${\bf h}=(h_x,h_y,h_z)$ with respect to the planet. Note that the orbital inclination $i = \arccos
{h_z}/{|{\bf h}|}$ is invariant under isotropic pulsating rescaling from CRTBP to ERTBP. The orbit is said prograde if $i < \pi/2$ ($h_z >0$) and retrograde otherwise.
The Hamiltonian of the elliptic problem (\[ertbp\]) is a periodic function of the true anomaly $f$ (which plays the role of time) and, hence, generates a non-autonomous dynamical system. Unlike the circular problem, the ERTBP does not possess an energy integral and, evidently, no such useful guiding concept as a static zero-velocity surface [@murray] can be introduced. Furthermore, due to the extra dimension associated with the explicit time dependence, construction of the SOS and, indeed, any visual analysis of phase space seems impossible even in the planar limit.
![\[zvs\] Level curves of the zero velocity surface at high (a) and low (b) energies in the planetocentric planar CRTBP along with the five Lagrange equillibrium points (labelled $L_1$ – $L_5$). The circle shaded grey is the Hill sphere centered on the planet (P). The star (S) is at $(-1,0)$.](fig1.eps){width="6cm"}
Numerical methods
-----------------
We have performed numerical integrations in which fluxes of particles are simulated as having passed from heliocentric orbits into the region surrounding the planet as defined by the Hill sphere. This region roughly corresponds to the region between the points labelled $L_1$ and $L_2$ in Fig. \[zvs\]. In practice it is convenient to go to extended phase space by introducing an additional pair of conjugate variables – ‘coordinate’ $f$ and ‘momentum’ $p_f = -H$ [@ll] for which the new Hamiltonian becomes conservative. In our numerical simulations we have used this method in combination with Levi-Civita (in 2D) and Kustaanheimo-Stiefel (in 3D) regularising techniques [@KS; @Aarseth] to avoid problems associated with two-body collisions [@nagler]. Numerical integrations were done using a Bulirsch-Stoer adaptive integrator [@NR].
{width="170mm"}
One approach to visualising phase space structures in systems with greater than 2 degrees-of-freedom relies on computations of short-time Lyapunov exponents (LE) over sets of initial conditions of interest (for recent developments and relevant applications see, e.g., @fro2000; @sandor [@dvorak]; @simo2003; @sandor2004 and references therein, although the idea of using finite-time LE itself dates back at least to @lorenz). As was demonstrated by @fro2000 various dynamical regimes (including resonances) can be distinguished by monitoring time profiles of a quantity called the Fast Lyapunov Indicator. Given an $n$-dimensional continuous-time dynamical system, $${d{\bf x}}/{dt} = {\bf F}({\bf x}, t), {\bf x} = (x_1, x_2,..., x_n), \label{EOM}$$ the Fast Lyapunov Indicator is defined as [@fro2000] $$FLI({\bf x}(0), {\bf v}(0), t) = \ln |{\bf v}(t)|, \label{fli}$$ where [**v**]{}(t) is a solution of the system of variational equations [@tancredi] $$\frac{d{\bf v}}{dt} = \biggl(\frac{\partial{\bf F}}{\partial{\bf x}} \biggr) {\bf v}. \label{variation}$$
The complementary system (\[variation\]) contains spatial derivatives which only aggravate the singularities in the restricted three body problem (RTBP) and so the use of regularisation becomes even more important. For this reason, all calculations of FLI reported herein were made using regularised versions of (\[EOM\]) and (\[variation\]).
As distinct from the largest Lyapunov characteristic exponent [@ll] $$\lambda = \lim_{t\to\infty} \frac{1}{t} \ln \frac{|{\bf v}(t)|}{|{\bf v}(0)|}, \label{lyap}$$ which just tends to zero for any regular orbit, the more sensitive FLI can help discriminate between resonant and non-resonant regular orbits [@fro2000]. Although, in practice, detection of resonances may be tricky, since real differences in FLI for quasiperiodic and periodic orbits of the same island are not always huge.
Chaos assisted capture
======================
In this section we briefly summarize the CAC mechanism described in @cac2003 as applied to the CRTBP. Even though the basic mechanism applies in three-dimensions, it is simpler here to outline the general scheme in terms of the planar version of the CRTBP. Fig. \[zvs\] shows the relevant zero velocity surface [ZVS, @murray] together with the five Lagrange equilibrium points.
![\[escape\] A paradigmatic escaping ‘sticky’ trajectory (a) in the planar Sun-Jupiter CRTBP; its multiple returns (b) to the Hill sphere (shaded grey, $R_H \simeq 0.068$) after heliocentric excursions; its radial distance from the planet $r=\sqrt{x^2+y^2}$ (c) and time profile of the Fast Lyapunov Indicator (d). The dotted vertical line indicates the first crossing of the Hill sphere (escape).](fig3.eps){width="7cm"}
Level curves of the ZVS, similar to a potential energy surface (PES), serve to limit the motion in the rotating frame and so define an energetically accessible region that may intersect the Hill sphere. However, unlike a PES, because the ZVS is defined in a rotating frame, it is possible for energy maxima to be stable as is, in fact, the case for $L_4$ and $L_5$ at which points Jupiter’s Trojan asteroids are situated [@murray]. The Hill sphere (see Fig. \[zvs\]) roughly occupies the region between the saddle points $L_1$ and $L_2$ which we term the ‘capture zone’ with its radius being given by $R_H=a_p/(\mu/3)^{1/3}$ where $a_p$ is the planet’s semimajor axis [@murray]. In the case of ERTBP the ZVS pulsates, which defines periodically time-dependent capture regions [@mako]. The two Lagrange saddle points $L_1$ and $L_2$ act as gateways between the Hill sphere and heliocentric orbits. A key finding of @cac2003 is that at energies close to (but above) the Lagrange points only prograde orbits can enter (or exit) the capture zone. At higher capture energies the distribution shifts to include both senses of $h_z$ until, finally, retrograde capture becomes more likely. The statistics of inclination distributions will, therefore, be expected to depend strongly on energy, i.e., how the curves of zero-velocity intersect the Hill sphere. Fig. \[sos\] portrays the structure of phase space in the planar limit ($z = p_z = 0$) in a series of Poincaré surfaces of section at four energies. At the lowest energy shown in Fig. \[sos\](a) many of the prograde orbits are chaotic whereas all the retrograde orbits are regular. Because incoming orbits cannot penetrate the regular KAM tori, prograde orbits must remain prograde while retrograde orbits cannot be captured nor can already bound retrograde orbits escape. After $L_2$ has opened in Fig. \[sos\](b) the chaotic ‘sea‘ of prograde orbits quickly ‘evaporates’ except for a ‘sticky’ layer of chaos which clings to the KAM tori. With increasing energy this front evolves from prograde to retrograde motion. Chaotic orbits close to the remaining tori can become trapped in almost regular orbits for very long times. In the presence of dissipation these chaotic orbits can be smoothly switched into the nearby KAM region and almost always preserve the sign of angular momentum. Thus, at low (high) energy permanent capture is almost always into prograde (retrograde) orbits.
Results and discussion
======================
In this section we describe the results of our simulations in the planar ERTBP using the FLI and our Monte Carlo simulations in the spatial ERTBP.
Planar ERTBP and FLI
--------------------
We are primarily interested here in obtaining a qualitative picture of the volume and structure of phase space occupied by the chaotic layer in the ERTBP as compared to the CRTBP. For this we find a good diagnostic to be $FLI(t)$ whose increase for chaotic orbits can usually be detected before, or no later than, a test particle finally escapes from the capture zone (Fig. \[escape\]). These measurements, when made over the capture region (which contains permanently bound regular trajectories [*even at energies well above $L_1$ and $L_2$*]{}), provide an estimate of the number of orbits that could be captured as illustrated in Fig. \[fract\]. The number of chaotic orbits inside the capture zone, detected by computing FLI, decreases with increasing ellipticity for moderate eccentricities (Fig. \[fract\](a)) signifying that the chaotic layers get weaker and, therefore, capture is expected to become less probable than it is at $e=0$.
Ideally the problem of separating the fractions of temporarily trapped and almost immediately escaping trajectories could be better quantified by partitioning the phase space into disjoint (e.g. ‘inner’, Hill sphere, and ‘outer’, heliocentric space) regions and computing the fluxes across the barriers between them. But, given the current state of phase space transport theory, this has not yet been shown possible in practice for essentially 3D problems such as the spatial CRTBP and ERTBP Hamiltonians. Interestingly enough, despite strong theoretical grounds [@wiggins], and exhausting attempts, efforts to describe quantitatively spatial three-body dynamics by constructing global invariant manifolds [@belbruno], that would presumably contain all possible incoming and outgoing chaotic trajectories through $L_1$ and $L_2$ saddle points, have been not quite successful in approaching the 3D problem so far. Part of the reason is that multiple escapes and recurrences of high energy trajectories to the Hill sphere (see example on Fig. \[escape\]a,b) make local manifolds near multidimensional saddles extremely difficult to use as rigorous surfaces of no return. This is even more pronounced for chaotic ionization of atomic Rydberg electrons (@atom; @lee) which is closely related to RTBP dynamics. Recent progress in pursuit of manifolds for the spatial three-body problem is reported by @marsden [@scheeres]; @waalkens.
![\[fract\] Fraction of chaotic orbits within the Hill sphere that have $FLI > 8$ at the cut-off time $T_{cut}=200$ years in the planar Sun-Jupiter ERTBP as a function of eccentricity. The phase space was sampled randomly with initial energies, true anomalies and eccentricities taken also at random. For reference, shown in parentheses are the mean eccentricities of Jupiter, Saturn, Neptune, Uranus, Pluto, some extrasolar planetary systems (HD 209452, HD 169830, HD 73526, HD 108147, HD 74156b), and binary trans-Neptunian objects on elliptic orbits ($2001~QC_{297}$,$1998~WW_{31}$, $2001~QW_{322}$, $2001~QC_{298}$, $1999~TC_{36}$, $1998~SM_{165}$).](fig4.eps){width="7cm"}
To analyse the structure of phase space in ERTBP, we first computed FLI in the planar (2D) circular case ($e = 0, z = p_z = 0$), where direct comparison with surfaces of section (in the Hill limit, $\mu
\ll 1$, @simo [@cac2003]) can be made. Figs. \[FLI\](a–c) confirm that phase space visualisation via computing short-time FLI (\[fli\]) works well in the planar CRTBP, reproducing correctly all the relevant features visible in the SOS shown in Fig. \[sos\]. In particular, the chaotic layer, and its evolution with increasing energy, can be easily identified by high values of FLI (shown in yellow). Note also that FLI measurements make sense even for relatively short time integrations, and are more economical than constructing the corresponding SOS (compare cut-off times given in the Fig. \[FLI\] and Fig. \[sos\] captions). As for the sensitivity, it is not clear how reliable the numerical distinctions (red-purple-blue) between the FLI for quasiperiodic and resonant orbits are, but this is irrelevant for our current purposes.
In the planar limit, initial conditions were generated randomly within the Hill radius on the surface of section (see Fig. \[FLI\] caption) and assuming that, initially, $f = \pi/2$ in which case the ERTBP initial conditions reduce to those of CRTBP. This guaranties that all ERTBP initial conditions are generated with identical initial energies $p_f(0)=-E_e$, and these are, in fact, true CRTBP energies $E_c$. The setting, thereby, allows for a direct comparison between the SOS of Fig. \[sos\] and results obtained using the FLI. Due to the dimensionality of the ERTBP, relaxation of the above constraint, e.g. choosing the initial true anomalies at random, produced FLI maps with no distinct structure. On the contrary, consistency in initial conditions at fixed $f = \pi/2$ helps track the smooth changes from CRTBP to ERTBP.
As the planet’s orbital eccentricity is increased to reach its actual value for Jupiter, FLI maps reveal a reduction of the density of orbits within the chaotic layers (yellow paterns) visible on Fig. \[FLI\](d-f) at each of the corresponding energies shown. However, at this moderate eccentricity, the phase space structures responsible for CAC (‘sticky’ KAM tori surrounded by the chaotic layers) generally survive any deviations introduced by ellipticity. This suggests robustness of the CAC mechanism with respect to actual ellipticities of the giant planets’ orbits which lie well below $e \simeq 0.4$ (see Fig. \[fract\]). We further confirm this by direct Monte Carlo simulations of capture probability (Sec. 4.2) which reveal that CAC indeed survives weak ellipticity in the RTBP. An important implication for future studies can be drawn by noticing that the heliocentric orbits of Pluto (with Charon) as well as of some recently discovered binary trans-Neptunian objects [^5] are quite eccentric (see examples on Fig. \[fract\]), but not to the extent characteristic of extrasolars (although the relative orbit of the binary partners as distinct from the heliocentric orbit of the combine can be very eccentric). These binary objects can be viewed in the framework of the Hill approximation too, so the weakly elliptic chaos-assisted mutual capture, probably stabilized by the fast exchange of energy and angular momentum with a fourth body, may have been involved in early stages of their formation (@gold; @cac2003; @hut).
The picture at much higher eccentricities evolves towards significant distorsions of the phase space stuctures as visualised by FLI. In Fig. \[FLI\](g–i) the KAM tori (red–blue islands) shrink, giving way to regions of scattering and chaotic orbits with short (200 years in this example) residence times inside the Hill sphere. These short-lived trajectories are the main contributions to the rapidly increasing number of chaotic orbits observed in Fig. \[fract\] for $e > 0.4$. This does not mean, however, that capture will necessarily be enhanced, because a high density of strongly chaotic orbits does not correlate with the number of very long-lived trajectories trapped close to KAM tori.
On the other hand, Lyapunov exponents computed for individual trajectories cannot serve as a predictor of global stability, i.e. as an indicator of whether an orbit will stay long enough in a bound region, or if it escapes quickly. This aspect in computing Lyapunov exponents for escaping (captured) trajectories concerns the notion of ‘stable chaos’ [@stable]. This term was coined to refer to chaotic motions that are locally hyperbolic, but demonstrate macroscopic long-term stability, so that the Lyapunov time $T_L$ (the inverse of the largest Lyapunov exponent (\[lyap\])) is substantially less than the ’event’ time $T_E$, e.g. the time for a trajectory to leave a certain region. However, there are indications of a possible simple linear correlation between $T_E$ and $T_L$ as shown in @murison, which suggests that knowledge of the Lyapunov time could allow one to predict an ’event’ time. Concerning the residence (escape) time near the planet, simulations in the planar CRTBP, however, show that such a correlation with the Lyapunov time does not exist (Fig. \[L\_times\], see also discussion by @morbidelli and @varvoglis). Among the randomly chosen escaping chaotic trajectories originating in the Hill sphere, there are many very long-lived examples that have diverse, even quite small, Lyapunov times. This means that local (microscopic) instabilities in the chaotic layer alone cannot explain the distributions of survival probability. Rather, as was pointed out by @tsiganis, long-term trapping with short $T_L$ near ‘sticky’ KAM structures can be attributed to the existence of phase space quasi-barriers and approximate integrals of motion. In the case of RTBP, these are the phase space structures around the saddle points at $L_1$ and $L_2$ and the $z$-component of angular momentum with respect to the planet, respectively. In 3D, the latter is approximately conserved [@conto] for long-lived chaotic orbits which explains the unexpected strong correlation between initial and final inclinations of captured particles [‘inclination memory’, @cac2003].
Monte Carlo simulations in the spatial ERTBP
--------------------------------------------
We simulated capture statistics in the spatial (three-dimensional) ERTBP by integrating isotropic fluxes of test particles that bombard the capture zone producing equal probabilities of initial conditions. Initial conditions were generated as follows: the particle’s position vector was chosen uniformly and randomly on the surface of the Hill sphere. Velocities were also chosen uniformly and randomly in accordance with the value of the CRTBP energy $E_c$. In turn, the starting energy (Jacobi constant $C_J=-2E_c$) was chosen uniformly random between its minimum possible value (as defined by the energy of $L_1$ when $f = \pi/2$) and its highest value (determined empirically such that above it the capture probability was essentially zero.) Then, by randomizing the initial true anomaly $f$ we model equal chances for a test particle to have any phase with respect to the mutual revolution of the primaries. The trajectories were integrated until one of the following occurred: the particle exited the Hill sphere; it penetrated a sphere, centred on the planet, of a given radius (see Fig. \[prob\] caption); it survived for a predetermined cut-off time.
![\[L\_times\] Escape time [*versus*]{} Lyapunov time for 2D CRTBP chaotic trajectories with initial conditions chosen randomly inside the Hill sphere. Escape was defined as the first crossing of the Hill sphere.](fig6.eps){width="8cm"}
Choosing particles on the Hill sphere, where the motion is chaotic or scattering, minimizes (although does not eliminate) the risk of accidentally starting orbits inside impenetrable KAM regions [@neto2004]. Although permanently bound, these orbits could never actually have been captured because KAM regions cannot be penetrated at all in 2D and only exponentially slowly in 3D. It is only in the chaotic layer between scattering and stability that capture can happen. An initial swarm of incoming particles produces broad distribution of survivors with different residence times inside the Hill sphere. We monitored its dynamics up to certain cut-off times (of the order of several thousand years) to find those long-lived chaotic orbits that may have been vulnerable to capture by a relatively weak dissipative force such as gas drag. Only these orbits, in the CAC model, could be the precursors to the currently observed distributions of irregular satellites. In a statistical sense, any short-time flybys are unlikely to have contributed to the primordial population of potential moons. Also, since the Sun-Jupiter system was chosen as an example, we eliminated test particles that penetrated the inner region of the Hill sphere occupied by Jupiter’s most influencial regular moon Callisto. The massive regular moons may have acted as a source of strong perturbation removing some temporarily captured moonlets following prograde orbits. This provides a possible reason for the observed prograde-poor distribution of jovian irregulars [@cac2003].
Fig. \[prob\] shows the results of these simulations. The capture probability density is plotted on the plane of initial and final inclinations. Unlike other properties, e.g., energy, inclination can be defined consistently in both the CRTBP and the ERTBP ($e=0$ and $e > 0$ in isotropically pulsating coordinates). As the distributions in Fig.\[prob\] confirm, inclination is approximately conserved during temporary capture in the ERTBP, so that the RTBP energy/inclination dependent CAC mechanism survives additional perturbations caused by deviations of the planet’s orbit from circularity. This is clearly seen upon comparing Fig.\[prob\] (a) and (b). The latter shows the capture probability for the Sun-Jupiter system with its actual eccentricity being used. We also verified by similar Monte Carlo runs that the basic energy–inclination correlation discussed in Sec. 3 remains valid in the 3D ERTBP. We note, however, that since the energy is not a conserved quantity in the elliptic problem, a better representation of results is given in terms of inclinations.
The most prominent effect observed in the ERTPB compared to the circular case is the overall decrease of capture probability as eccentricity increases. Simulations indicate that, given the same time scale, the actual ellipticity of Jupiter’s orbit accounts for approximately an order of magnitude lower capture probability than predicted by the circular model, while the relative number of progrades [*versus*]{} retrogrades remains unaffected.
Capture by typical extrasolar planets will be expected to become even more suppressed due to the wildly eccentric nature of these, essentially ERTBP, star–planet systems. To test this we used an eccentricity $e=0.4839$ which is ten times greater than that of Jupiter’s orbit but quite similar to what has been estimated, for instance, for several already detected extrasolars (the values of $e$ are given in parentheses): HD 108147 (0.498), HD 168443b (0.53), HD 82943c (0.54), HD 142415 (0.5), HD 4203 (0.46), HD 210277 (0.46), HD 147513 (0.52), HD 190228 (0.5), HD 50554 (0.5), HD 33636 (0.53). For consistency, we left unchanged all of the other parameters and environmental factors (including Callisto whose hypothetical extrasolar analogs may well exist around exo-Jupiters) exactly as they were used in the simulations for Sun-Jupiter. Fig.\[prob\](c) indicates that, due to the ellipticity of the orbits and assuming similar time scales and other conditions, capture by extrasolar planets may be as much as aproximately ten times less efficient than it could be by the giant planets of our system.
The relatively low capture probability, predicted in ERTBP, adds to other destructive mechanisms possibly operating on as yet undiscovered irregular satellites of extrasolar planets. The loss of satellites through Yarkovsky [@death] or tidal [@barnes] effects may also diminish the possibility for highly elliptic extrasolar captors to marshal large populations of irregular moons. On the other hand, exo-planets with masses from one-half to four times Jupiter’s mass on low eccentricity orbits ($e < 0.02$) may be considered candidates for having irregular satellites. These may include, e.g., already known extrasolars such as HD 179949, 55 Cnc b, HD 169830c, HD 187123, Tau Boo, HD 75289, 51 Peg, Ups And b, HD 195019.
![\[prob\] Capture probability (the density of survivors normalized by the number of trajectories in the incoming fluxes) as a function of initial and final inclinations in spatial circular (a), elliptic Sun-Jupiter (b) and exemplary elliptic extrasolar (c) RTBP systems. The values of eccentricity $e$ are shown. Chaotic trajectories with random energies ($C_J \in [2.995,{C_J}^{(L_1)}]$), true anomalies ($f \in [0,2\pi]$), velocities and coordinates (on the Hill sphere with the radius $R_H \simeq 0.068$) were integrated until they either escaped from the sphere or crossed Callisto’s orbit at $r=2.42 \times 10^{-3}$ (Jupiter’s orbital semimajor axis $a_p=1$). Each of the distrubutions was drawn from orbital elements of $N \simeq 20000$ long-lived chaotic orbits that entered and survived within the Hill sphere for $T_{cut}=20000$ years. Incoming fluxes consisted of $3 \times 10^7$ (a), $1.6 \times 10^8$ (b), $4.6 \times 10^9$ (c) particles.](fig7.eps){width="8cm"}
Conclusions
===========
The complexity of many-body systems evidently goes far beyond that of the simplest non-trivial case, the three-body problem. However, when considered on a hierarchical level, interactions between gravitional centers can often be decomposed to relatively low degree-of-freedom subsystems for which analysis from a dynamical systems point of view becomes possible. In the problem described here, the existence of ’sticky’ volumes of phase space where regular regions are surrounded by chaotic layers explains why several bodies may become temporarily (but for rather long time periods) trapped by mutual chaos-assisted capture. These quasi-stable configurations can subsequently be stabilised (or destroyed) on much longer time scales by various forces (dissipative or not). Here we have considered the effect introduced by a slow (compared to time-scales of motion in the chaotic layer) periodic parametric time dependence in the three-body problem. Our simulations show that chaos-assisted capture applies and is, in fact, only slightly perturbed in cases when the primaries move on an elliptic orbit. It is expected to be of significant interest (e.g. in applications to essentially many-particle systems such as star clusters and the asteroid belt) to consider chaos-assisted capture at the next hierarchical level, i.e. when the primary two-body configuration is not restricted to a regular orbit, which may be the case in three-body encounters of comparably sized objects.
acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by grants from the US National Science Foundation through grant 0202185 and the Petroleum Research Fund administered by the American Chemical Society. All opinions expressed in this article are those of the authors and do not necessarily reflect those of the National Science Foundation. S. A. A. also acknowledges support from Forschungszentrum Jülich, where part of this work was done.
[7]{}
Aarseth S. J., 2003, Gravitational N-body Simulations: Tools and Algorithms. Cambridge Univ. Press, Cambridge
Astakhov S. A., Burbanks A. D., Wiggins S., Farrelly D., 2003, Nature, 423, 264
Barnes J. W., O’Brien D. P., 2002, ApJ, 575, 1087
Beaugé C., Michtchenko T. A., 2003, MNRAS, 341, 760.
Benest D., 2003, A&A, 400, 1103
Belbruno E., 2004, Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton Univ. Press, Princeton and Oxford
Brunello A. F., Uzer T., Farrelly D., 1997, Phys. Rev. A, 55, 3730
Burns J. A., Ćuk M., 2002, BAAS, 34, DPS 34th Meeting, abstr. No. 42.01
Carruba V., Burns J. A., Nicholson P. D., Gladman B. J., 2002, Icarus, 158, 434
Chiang E. I., Fischer D., Thommes E., 2002, ApJ, 564, L105
Chodas P. W., Yeomans D. K., 1996, in Knoll K. S., Weaver H. A., Feldman P. D., ed., The collision of Comet Shoemaker-Levy 9 and Jupiter. Cambridge Univ. Press, Cambridge
Cincotta P. M., Giordano C. M., Simó C., 2003, Physica D, 182, 157
Contopoulos G., 1965, ApJ, 142, 802
Ćuk M., Burns J. A., 2004, Icarus, 167, 369
Froeschlé C., Guzzo M., Lega E., 2000, Science, 289, 2108
Funato Y., Makino J., Hut P., Kokubo E., Kinoshita D., 2004, Nature, 427, 518
Gladman B. et al., 2001, Nature, 412, 163
Goldreich P., Sari R., 2003, ApJ, 585, 1024
Goldreich P., Lithwick Y., Sari R., 2002, Nature, 420, 643
Gómez G., Koon W. S., Lo M. W., Marsden J. E., Masdemont J., Ross S. D., 2004, Nonlinearity, 17, 1571
Grav T., Holman M. J., 2004, ApJ, 605, L141
Kavelaars J. J. et al., 2004, Icarus, 169, 474
Lecar M., Franklin F., Murison M., 1992, AJ, 104, 1230
Lee E., Brunello A. F., Cerjan C., Uzer T., Farrelly D., 2000, in Yeazell J., Uzer T., ed., The Physics and Chemistry of Wave Packets, Wiley, NY, p. 95
Lichtenberg A. J., Lieberman M. A., 1992, Regular and Chaotic Dynamics, 2nd edn. Springer-Verlag, NY
Llibre J., Piñol J., 1990, Celest. Mech. Dynam. Astronom., 48, 319
Lorenz E. N., 1965, Tellus, 17, 321
Makó Z., Szenkovits F., 2004, Celest. Mech. Dynam. Astronom., in press
Marcy G. W., Butler R. P., 2000, PASP, 112, 137
Marzari F., Weidenschilling S. J., 2002, Icarus, 156, 570
Michtchenko T. A., Malhotra R., 2004, Icarus, 168, 237
Milani A., Nobili A. M., 1992, Nature, 357, 569
Morbidelli A., Froeschlé C., 1996, Celest. Mech. Dynam. Astronom., 63, 227
Murray C. D., Dermott S. F., 1999, Solar System Dynamics. Cambridge Univ. Press, Cambridge
Nagler J., 2004, Phys. Rev. E, 69, 066218
Nesvorný D., Alvarellos J. L. A., Dones L., Levison H. F., 2003, AJ, 126, 398
Nesvorný D., Beaugé C., Dones L., 2004, AJ, 127, 1768
Neto E. V., Winter O. C., Yokoyama T., 2004, A&A, 414, 727
Pilat-Lohinger E., Funk B., Dvorak R., 2003, A&A, 400, 1085
Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., 1999, Numerical Recipes in C, 2nd edn. Cambridge Univ. Press, Cambridge
Sándor Z., Balla R., Téger F., Érdi B., 2001, Celest. Mech. Dynam. Astronom., 79, 29
Sándor Z., Érdi B., Széll A., Funk B., 2004, Celest. Mech. Dynam. Astronom., in press
Schneider J., 1999, CR Acad. Sci. II B, 327, 621
Sheppard S., Jewitt D., 2003, Nature, 423, 261
Simó C., Stuchi T. J., 2000, Physica D, 140, 1
Stiefel E. L., Scheifele G., 1971, Linear and Regular Celestial Mechanics: Perturbed Two-Body Motion, Numerical Methods, Canonical Theory. Springer-Verlag, New York
Szebehely V., 1967, Theory of Orbits: the Restricted Problem of Three Bodies. Acad. Press, NY and London
Tancredi G., Sánchez A., Roig F., 2001, AJ, 121, 1171
Tsiganis K., Anastasiadis A., Varvoglis H., 2000, Chaos, solitons and fractals, 11, 2281
Tremaine S., Zakamska N. L., 2003, preprint (astro-ph/0312045)
Varvoglis H., Anastasiadis A., 1996, AJ, 111, 1718
Villac B. F., Scheeres D. J., 2004, A simple algorithm to compute hyperbolic invariant manifolds near $L_1$ and $L_2$, 14th AAS/AIAA Space Flight Mechanics Meeting, February 2004, Maui, Hawaii
Waalkens H., Burbanks A., Wiggins S., 2004, J. Phys. A: Math. Gen, 37, L257
Wiggins S., Haller G., Mezic I., 1994, Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Applied Mathematical Sciences, Vol 105). Springer-Verlag, NY
Williams D. M., 2003, BAAS, 35, DPS 35th Meeting, abstr. No. 27.10
Williams D. M., Kasting J. F., Wade R. A., 1997, Nature, 385, 234
Yokoyama T., Santos M. T., Cardin G., Winter O. C., 2003, A&A, 401, 763
[^1]: E-mail: s.astakhov@fz-juelich.de; www.astakhov.newmail.ru
[^2]: E-mail: david@habanero.chem.usu.edu
[^3]: http://nssdc.gsfc.nasa.gov/planetary/planetfact.html
[^4]: http://www.obspm.fr/planets\
http://exoplanets.org
[^5]: http://www.johnstonsarchive.net/astro/asteroidmoons.html
|
---
abstract: 'Electronic wave functions of planar molecules can be reconstructed via inverse Fourier transform of angle-resolved photoelectron spectroscopy (ARPES) data, provided the phase of the electron wave in the detector plane is known. Since the recorded intensity is proportional to the absolute square of the Fourier transform of the initial state wave function, information about the phase distribution is lost in the measurement. It was shown that the phase can be retrieved in some cases by iterative algorithms using *a priori* information about the object such as its size and symmetry. We suggest a more generalized and robust approach for the reconstruction of molecular orbitals based on state-of-the-art phase-retrieval algorithms currently used in coherent diffraction imaging. We draw an analogy between the phase problem in molecular orbital imaging by ARPES and of that in optical coherent diffraction imaging by performing an optical analogue experiment on micrometer-sized structures. We successfully reconstruct amplitude and phase of both the micrometer-sized objects and a molecular orbital from the optical and photoelectron far-field intensity distributions, respectively, without any prior information about the shape of the objects.'
address: 'Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland'
author:
- 'P Kliuiev, T Latychevskaia, J Osterwalder, M Hengsberger and L Castiglioni'
bibliography:
- 'iopart-num.bib'
title: 'Application of iterative phase-retrieval algorithms to ARPES orbital tomography'
---
June 2016
[*Keywords*]{}: phase retrieval, ARPES, orbital tomography, molecular orbital
Introduction
============
Organic semiconductors play a key role in modern devices such as organic light-emitting diodes and photovoltaic cells [@oregan1991; @mathew2014]. More recently, organic molecules have been used as catalysts in photolytic water splitting, a promising route towards production of hydrogen as renewable energy source [@li2015]. Tailoring the physical properties of molecular optoelectronic devices [@browne2009; @pan2015; @sharifzadeh2015] crucially depends on a deep understanding of the charge transfer mechanisms at metal-organic interfaces. The time-resolved spatial visualization of such processes would hence be highly desirable.
The frontier orbitals, i. e. the highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals, largely determine the chemical reactivity and electronic properties of molecular systems. Detailed information about the electronic structure of molecular systems can be inferred from angle-resolved photoelectron spectroscopy (ARPES) of well-ordered molecular layers on single-crystalline substrates [@puschnig2009; @puschnig2013; @dauth2014; @lueftner2014; @wiessner2014; @weiss2015]. The photoemission intensity
$I(\textit{\textbf{k}}_{\mathrm{f}\parallel}, E_{\mathrm{kin}})$ is derived from Fermi’s golden rule as $$\begin{aligned}
I(\textit{\textbf{k}}_{\mathrm{f}\parallel}, E_{\mathrm{kin}})&\propto&\sum\limits_{i} \left| \right\langle\psi_\mathrm{f}(\textit{\textbf{k}}_{\mathrm{f}\parallel}, E_{\mathrm{kin}}, \textit{\textbf{r}}) \left| \textit{\textbf{A}}\cdot\textit{\textbf{p}}\left| \psi_i(\textit{\textbf{k}}_{i\parallel}, \textit{\textbf{r}})\right\rangle\right|^2
\nonumber \\
~&~&\times \delta(E_{\mathrm{kin}}+\Phi+E_{i}-\hbar\omega)\times\delta(\textit{\textbf{k}}_{\mathrm{f}\parallel}-\textit{\textbf{k}}_{i\parallel}-\textbf{G}_{\parallel}),\end{aligned}$$ where $\psi_{i}$ and $\psi_{\mathrm{f}}$ denote initial and final state wave functions with corresponding momentum components $\textit{\textbf{k}}_{i\parallel}$ and $\textit{\textbf{k}}_{\mathrm{f}\parallel}$ parallel to the surface, respectively. The delta functions in the second line comprising photon energy $\hbar\omega$, sample work function $\Phi$ and reciprocal lattice vector $\textit{\textbf{G}}_{\parallel}$ ensure energy and momentum conservation in the photoemission process. The transition matrix element is given in the dipole approximation, where $\textit{\textbf{p}}$ and $\textit{\textbf{A}}$ denote the momentum operator and the vector potential of the exciting light. The photocurrent $I(\textit{\textbf{k}}_{\mathrm{f}\parallel}, E_{\mathrm{kin}})$ is obtained by summation over all transitions from occupied initial states $\psi_i$ to the final state $\psi_{\mathrm{f}}$ characterised by the kinetic energy $E_{\mathrm{kin}}$ and the parallel component of the final state momentum $\textit{\textbf{k}}_{\mathrm{f}\parallel}$ of the photoelectron. The photoemission final state $\psi_{\mathrm{f}}$ can be approximated by a plane wave $\propto \rme^{i\textit{\textbf{k}}_{\mathrm{f}}\textit{\textbf{r}}} $ provided the following conditions are fulfilled [@puschnig2009; @puschnig2013; @goldberg1978]: (i) photoelectrons are emitted from $\pi$-orbitals of large planar molecules, for which all the contributing orbitals are of the same $p_z$ character; (ii) the molecules consist of mainly light atoms (H, C, N, O) and final state scattering effects can thus be neglected. Under these assumptions, the measured ARPES intensity becomes proportional to the squared modulus of the Fourier transform of the initial state wave function $\psi_i$ weakly modulated by a slowly varying angle-dependent envelope function [@puschnig2009; @puschnig2013]: $$I(\textit{\textbf{k}}_{\mathrm{f}\parallel}, E_{\mathrm{kin}})\propto|\textit{\textbf{A}}\cdot\textit{\textbf{p}}|^2|\mathcal{F}\{\psi_i (\textit{\textbf{k}}_{i\parallel}, \textit{\textbf{r}})\}|^2$$
The recorded intensity pattern, however, does not contain any information about the phase of the complex-valued electron wave distribution in the detector plane, which inhibits the direct reconstruction of the molecular wave function via computation of an inverse Fourier transform. In certain cases, phase information can be inferred from the parity of the wave function [@puschnig2009] or from dichroism measurements [@wiessner2014] and be imposed onto the measured data. However, the reconstruction of the molecular wave functions in such a way is not applicable to the most general type of problems when the phase distribution cannot be deduced from symmetry considerations. This issue was addressed by Lüftner et al. [@lueftner2014] by suggesting an iterative phase retrieval procedure similar to the Fienup algorithm [@fienup1978]. In the suggested procedure, one iterates back and forth between real and reciprocal spaces by computing Fourier transforms and satisfying the constraints in both domains. In real space, the wave function is confined to a rectangular box which roughly corresponds to the van der Waals size of the molecule and thus represents the support of the object. The absolute value of the wave function is reduced to 10$\%$ outside this confinement box at each iteration step. In reciprocal space, the computed value of the amplitude is replaced by the measured one and the phase is kept.
In this work, we suggest that the phase problem in ARPES-based molecular orbital imaging can be solved in a more robust manner by utilizing the analogy to the phase problem in coherent diffraction imaging (CDI) [@miao1999]. Both in CDI and orbital imaging, the far field pattern in the detector plane is proportional to the squared modulus of the Fourier transform of the object distribution. Provided the far-field intensity pattern is measured at the oversampling condition [@miao1998; @miao2003], both the amplitude and the phase of the object can be reconstructed from the experimentally available modulus of its Fourier transform using the phase retrieval algorithms [@fienup1978] as it is done in CDI [@miao1999]. Therefore, we suggest to directly apply state-of-the-art phase-retrieval algorithms, currently used in CDI, for the reconstruction of molecular orbitals. These algorithms were specifically optimized for objects described by a complex-valued transmission function [@harder2010], which makes them ideal for the reconstruction of electron wave functions. Moreover, recent advancements in CDI allowed for the solution of the phase problem without need for the precise knowledge of the shape of the object, which is instead found in the course of the reconstruction using the shrinkwrap algorithm [@marchesini2003]. To facilitate a better understanding of the CDI phase-retrieval algorithms in view of their applicability to reconstruction of molecular wave functions, we designed an optical analogue experiment and performed CDI on micrometer-sized structures produced by means of photolithography. Available CDI phase-retrieval algorithms [@fienup1978; @harder2010; @marchesini2003] were employed for the reconstruction of the micrometer-sized object. Eventually, the same algorithms were applied to a set of ARPES data and the lowest unoccupied molecular orbital (LUMO) of pentacene was reconstructed.
Methods
=======
Optical CDI of a microstructure
-------------------------------
The microstructures for the optical CDI experiments were patterned in a 105 nm-thick Cr film deposited on a 1.7 mm-thick fused silica substrate, thus providing transparent objects in a non-transparent medium. The individual microstructures had an identical shape but different sizes and were separated from one another by several millimeters to avoid interference between the neighbouring objects. The size of the microstructures was selected in such way that the ratio between microstructure length (e.g., $\SI{15}{\micro m}$) and employed laser wavelength ($\SI{0.532}{\micro m}$) was comparable to the ratio between length of pentacene molecule ($\approx \SI{1.5}{nm}$) and de Broglie wavelength of the electrons ($\approx \SI{0.17}{nm}$) at the used photon energy ($\SI{50}{eV}$). The experimental setup for optical CDI is shown in Fig. 1. The laser beam profile had a Gaussian distribution as shown in the inset. For CDI experiments, the laser beam is usually spatially filtered and then expanded using two lenses, which ensures that the intensity profile in the object plane is constant [@thibault2007]. In our experiment, we employed the laser beam without expansion because the light intensity variations on the length-scale of the microstructure were negligible. The microstructure was illuminated from the side of the Cr film. The far-field distribution of the scattered wave was imaged onto a semitransparent screen and the diffraction patterns were recorded with a 10-bit CCD camera (Hamamatsu C4742-95) placed behind the screen as shown in Fig. 1. In order to increase the dynamic range, we recorded several diffraction patterns at different exposures by using a rotatable neutral density filter with optical densities ranging from 0 to 4.0. The recorded images were then combined into one high-dynamic-range (HDR) image by a procedure proposed by Debevec [@devebec1997].
![\[fig1\] Experimental setup of optical CDI. The distance between sample and screen was set to 22.5 cm. The size of the imaged screen area comprised $40\times40$ cm$^2$ sampled with $1000\times1000$ pixels. Inset: Intensity distribution of the laser profile.](figure1.eps)
ARPES of pentacene/Ag(110)
--------------------------
A well-ordered sub-monolayer of pentacene molecules adsorbed on Ag(110) served as model system for orbital tomography. Pentacene ARPES data has been acquired during a beamtime of A. Schöll and coworkers (University of Würzburg) at the NanoESCA beamline at Elettra synchrotron (Trieste, Italy) and has been provided to us for validation of our phase retrieval algorithm [@grimm]. The crystal was prepared according to standard procedures [@feyer2014] and pentacene molecules [@puschnig2009] were deposited from a home-built Knudsen cell [@wiessner2014]. ARPES constant binding energy (CBE) momentum maps of the pentacene LUMO were recorded with the p-polarized light at a photon energy of $\SI{50}{eV}$ using the photoemission electron microscope (PEEM) [@schneider2012; @patt2014]. The setup of the PEEM and the experimental geometry are shown in Fig. 2(a) and Fig. 2(b), respectively. The microscope was operated in the momentum mode and allowed for detection of electrons with the acceptance angle of $\alpha=\pm90^0$ corresponding to slightly less than $\pm \SI{3}{\AA^{-1}}$ at $\SI{50}{eV}$ photon energy without any sample rotation. The CBE map was integrated over a $\SI{200}{meV}$ energy window, which is of the order of the electron analyzer resolution and of the full-width at half-maximum of the pentacene LUMO at the binding energy of $\SI{0.1}{eV}$.
![\[fig2\] (a) Schematic of the PEEM setup. (b) Experimental geometry. The photon energy was 50 eV and the light was p-polarized with an incidence angle of $65^0$. The photoemitted electrons were collected by the PEEM objective lens with an acceptance angle of $\alpha=\pm90^0$. $\textit{\textbf{k}}_{\mathrm{f}\parallel}$ and $\textit{\textbf{k}}_{\mathrm{f}\bot}$ denote parallel and normal components of the final state momentum of the photoelectrons.](figure2a_2b.eps)
Algorithms
----------
Prior to reconstruction of the pentacene LUMO, we tested the performance of the algorithms on the optical CDI data set, taking advantage of the high dynamic range of these data. We employed a combination of the phase-constrained [@harder2010] hybrid input-output [@fienup1978] (PC-HIO) and error reduction [@fienup1978] (ER) algorithms. The usage of both algorithms in an alternating scheme has been shown to eliminate stagnation problems and to provide faster convergence [@fienup1978; @harder2010; @williams2006].
![\[fig3\] Iterative phase retrieval scheme.](figure3.eps)
The support of the object was found using the shrinkwrap algorithm [@marchesini2003]. Following the conventional procedure of this algorithm, the initial estimate of the object support was obtained from the autocorrelation of the object by computing the inverse Fourier transform of the experimental diffraction pattern $I(X,Y)$, convolving it with a Gaussian function (width $\sigma=$ 5 pixels) and applying a threshold at $10\%$ of its maximum. The pixel values below the threshold were zeroed. The reconstruction began with 40 iterations of the PC-HIO algorithm followed by 2 iterations of the ER algorithm. We found that this number of iterations is sufficient to yield a resonable estimate of the object shape and thus to perform the first update of the object support by using the shrinkwrap procedure [@marchesini2003] described in detail below. The scheme of the iterative phase retrieval procedure is shown in Fig. 3, which included the following steps:
1. In the first iteration $k=1$, the experimental amplitude $|F(X,Y)|=\sqrt{I(X,Y)}$ was combined with a random phase and the inverse Fourier transform supplied an initial input object distribution $g_k(x,y)$, where $(X,Y)$ and $(x,y)$ denote the coordinates in the detector and object planes, respectively. We assume the most general case of a complex-valued object distribution and keep both its real and imaginary parts.
2. By computing the Fourier transform of $g_k(x,y)$, we obtain the complex-valued distribution $G_k(X,Y)=\mathcal{F} \left\{g_k(x,y)\right\}$.
3. By replacing the calculated amplitude $|G_k(X,Y)|$ with the experimental amplitude $|F(X,Y)|$, while keeping the calculated phase distribution, we obtain an updated complex-valued field distribution in the detector plane $G'_k(X,Y)$.
4. Inverse Fourier transform of $G'_k(X,Y)$ provides the output object distribution $g'_k(x,y)$.
5. In the PC-HIO algorithm [@fienup1978; @harder2010], the input object for the next iteration $g_{k+1}(x,y)$ is obtained as
$$g_{k+1}(x,y)=\left\{
\begin{array}{@{}ll@{}}
g'_k(x,y), & \text{if}\ (x,y)\in \gamma, \\
g_k(x,y)-\beta g'_k(x,y), & \text{if}\ (x,y)\notin \gamma,
\end{array}\right.$$
where $\beta=0.9$ is a feedback parameter and $\gamma$ corresponds to a set of points which comply with the object domain constraints (belong to the support region and have their phases within an expected range). In the ER algorithm [@fienup1978], the object distribution $g_{k+1}(x,y)$ is calculated as
$$g_{k+1}(x,y)=\left\{
\begin{array}{@{}ll@{}}
g'_k(x,y), & \text{if}\ (x,y)\in \gamma, \\
0, & \text{if}\ (x,y)\notin \gamma,
\end{array}\right.$$
where $\gamma$ fulfills the same criteria as in the PC-HIO algorithm.
The output object distribution $g'_k(x,y)$ obtained in the last iteration of the ER cycle was used to update the object support. This was done by convolving $g'_k(x,y)$ with a Gaussian function and setting a threshold at $12\%$ of its maximum, as it is typically done in the shrinkwrap algorithm [@marchesini2003]. The width of the Gaussian was initially set to 2.5 pixels. Upon the first update of the support, the algorithm continued with alternating cycles of 20 iterations of the PC-HIO algorithm followed by 2 iterations of the ER algorithm [@fienup1978; @harder2010; @williams2006]. The end of each cycle was finalized by computing a new distribution of the object support. The threshold value and the Gaussian width were chosen empirically so that no part of the reconstructed pattern was truncated, but instead the support converged smoothly towards the shape of the object. The latter requirement was ensured by reducing the width of the Gaussian at every support update by $1\%$ as it is conventionally done in the shrinkwrap algorithm [@marchesini2003]. The quality of the reconstructions was estimated by computing the mismatch between the iterated and the experimental amplitudes [@fienup1978; @fienup1982; @fienup1986]:
$$\label{eq:error}
E=\sqrt{\frac{\sum_{X,Y=0}^{N-1}||F(X,Y)|-|G_{\mathrm{it}}(X,Y)||^2}{\sum_{X,Y=0}^{N-1}|F(X,Y)|^2}},$$
where $|F(X,Y)|$ is the experimental amplitude, $|G_{\mathrm{it}}(X,Y)|$ is the iteratively obtained amplitude.
Oversampling requirements
-------------------------
The solution of the phase problem requires the fulfillment of the oversampling condition [@miao1998; @miao2003]. Given an $N\times N$ pixel sampled amplitude $|F(X,Y)|=|\sum_{X,Y=0}^{N-1} f(x,y) e^{-2\pi i(xX+yY)/N}|$ in reciprocal space, we obtain a set of $N^2$ equations, which have to be solved in order to find both the amplitude and phase of $f(x,y)$. Miao et al. [@miao1998] defined the oversampling ratio as
$$\sigma=\frac{N_{\mathrm{total}}}{N_{\mathrm{unknown}}},$$
where $N_{\mathrm{total}}$ is the total number of pixels and $N_{\mathrm{unknown}}$ is the number of pixels with unknown values. The set of equations is solved by dense sampling of the diffraction pattern so that the object distribution is surrounded by a zero-padded region with $\sigma>2$ [@miao1998]. In each dimension of a 2D data set, we can define a linear oversampling ratio
$$\label{eq:los}
\O=\frac{N\Delta r}{a},$$
where $N$ is the linear number of pixels, $\Delta r$ is the size of the pixel in the object domain and $a$ is the largest extent of the object. The oversampling requirement then corresponds to $\O>\sqrt{2}$ [@miao1998].
Results and discussion
======================
Optical CDI: Reconstruction of the micrometer-sized structures
--------------------------------------------------------------
Fig. 4 shows the results of the reconstruction of the micrometer-sized structures. In optical CDI, we employed micrometer-sized structures of $30\times12$ $\SI{}{\micro m^2}$ (sample 1) and $14.8\times6$ $\SI{}{\micro m^2}$ (sample 2). The scanning electron microscope (SEM) images of samples 1 and 2 are shown in Fig. 4(a,b) next to the experimental diffraction patterns (Fig. 4(c,d)). The size of the diffraction patterns sampled with $1000\times1000$ pixel was $40\times40$ cm$^2$ in each case, thus giving the size of the pixel in the detector plane $\Delta p=400$ $\SI{}{\micro m}$. The size of the pixel in the object plane $\Delta r$ can be related to the distance $z=22.5$ cm from the object to the detector plane and to the employed laser wavelength $\lambda=532$ nm [@zuerch2015]. The linear oversampling ratio defined by Eq. \[eq:los\] can be rewritten as [@miao1998]:
$$\O=\frac{z\lambda}{a\Delta p}$$
For the samples 1 and 2 with the lengths $a_1=\SI{30}{\micro m}$ and $a_2=\SI{15}{\micro m}$, the linear oversampling ratios fulfilled the oversampling condition and were $\O_1\approx10$ and $\O_2\approx20.2$, respectively.
![\[fig4\] Reconstruction of the micrometer-sized objects. (a,b) SEM images. (c,d) Experimental diffraction pattern intensities shown on logarithmic scale. (e,g) Reconstructed amplitudes. (f,h) Reconstructed phases.](figure4a_4h.eps)
Prior to application of the phase retrieval algorithms, the experimental diffraction patterns were pre-processed: First, each of the recorded $1000\times1000$ pixel images was centered. Centering of the experimental diffraction pattern was shown to have a strong effect on the quality of the reconstruction in CDI [@zuerch2013]. The noise of the CCD camera (average count rate of 50 counts) was subtracted from each pixel and the images were truncated to $500\times500$ pixels around their centers because of the low signal-to-noise ratio at the peripheral parts. The central part of each diffraction pattern was dominated by an intense laser spot due to the partial transparency of the chromium film to the laser beam. Pixel values exceeding the thresholds of $1.5\cdot10^5$ counts (sample 1) and $6\cdot10^3$ counts (sample 2) were defined as missing and their values were updated in the course of the reconstruction by using the corresponding pixel values of the calculated amplitudes in the detector plane [@marchesini2003; @latychevskaia2015]. In each case, the square root of the resulting diffraction pattern was fed into the algorithm. We found that 10 alternating cycles of the PC-HIO and ER algorithms, each followed by an update of the support, were enough to achieve a stable reconstruction. Further increase in the number of the reconstruction cycles was not necessary since it did not improve the quality of the reconstructed object distribution. At the end of 10 cycles, each reconstruction was stabilized by 100 iterations of the ER algorithm [@latychevskaia2015]. In total, we performed 1000 independent reconstructions by employing a random phase distribution for each reconstruction run. Eventually, the 50 reconstructions with the smallest error $E$ as defined by Eq. \[eq:error\] were selected and averaged [@latychevskaia2015] and are shown in Fig. 4(e-h). The reconstructed amplitudes correctly reproduce the shape and dimension of the microstructures. Furthermore, as it was expected for a purely transmitting object illuminated by a Gaussian beam with an almost planar wavefront at the object site, the phase distributions turned out to be almost constant. The lower quality of the reconstructed amplitude of sample 2 (Fig. 4 (g)) can be attributed to the low signal-to-noise ratio in the respective diffraction pattern.
ARPES orbital tomography: Reconstruction of the pentacene LUMO
--------------------------------------------------------------
We then applied the same algorithm to the ARPES data. Fig. 5 shows the results of the reconstruction of the pentacene LUMO. The experimental CBE map is shown in Fig. 5(a). Given the resolution in reciprocal space of $\Delta k\approx$ $\SI{0.01}{\AA^{-1}}$ and the length of the pentacene molecule $a\approx\SI{15}{\AA}$, the linear oversampling ratio in the ARPES experiment can be calculated using Eq. \[eq:los\]. Taking the relation $\Delta r \Delta k = \frac{2\pi}{N}$ between the pixel size in object space, $\Delta r$, and reciprocal space, $\Delta k$, into account, the linear oversampling ratio can be expressed as
$$\O=\frac{2\pi}{a\Delta k}.$$
The linear oversampling ratio was $\O\approx42$ and thus fulfilled the oversampling condition [@miao1998]. The experimental CBE map was pre-processed following similar steps as those applied to the reconstruction of the micrometer-sized objects: First, the image was centered and the quasi-constant noise of the CCD camera (average count rate of 50 counts) was subtracted from each pixel. To ensure a sufficient number of pixels allocated per unit length of the molecule, we zero-padded the experimental CBE map to $2000\times2000$ pixels around its center. The square root of the processed CBE map was fed into the algorithm with the same parameters as used for the reconstruction of the micrometer-sized structures. Varying these parameters did not lead to any substantial improvements in the quality of the reconstruction. In total, we performed 1000 reconstructions of the pentacene LUMO. About $56\%$ of the reconstructed objects $g(x,y)$ were reconstructed together with their conjugate $g^*(-x,-y)$ or twin images [@fienup1986]. The identification of the twin images could be automated by a procedure proposed by Fienup [@fienup1986], but here they were easily identified by visual inspection and discarded. From the remaining reconstructions, 50 with the smallest error $E$ as defined by Eq. \[eq:error\] were selected and averaged. The reconstructed amplitude and phase of the pentacene LUMO are shown in Fig. 5 (b-c) together with the overlayed carbon frame of the molecule for comparison.
![\[fig5\] Reconstruction of the pentacene LUMO. (a) CBE map recorded with PEEM from a sub-monolayer of pentacene on Ag(110) at 50 eV photon energy. (b) Reconstructed amplitude of the LUMO. (c) Reconstructed phase. Image transparency is weighted with the corresponding amplitude values for illustration purposes. (d) The same CBE map as in (a), but symmetrized with respect to the center. Reconstructions of (e) amplitude and (f) phase obtained from (d).](figure5a_5f.eps)
It should be noted that we did not perform a normalization of the ARPES intensity by the angle-dependent factor $|\textbf{A}\cdot\textbf{p}|^2$ nor did we enforce any symmetry constraints in the course of the reconstruction onto the amplitude and phase shown in Fig. 5 (b-c). The object distribution was let to freely evolve until the stable solution was reached, which makes the utilized algorithm independent of any symmetry properties imposed onto the object under reconstruction. Furthermore, we note that the recorded CBE map shown in Fig. 5(a) contains features coming from the Ag(110) substrate (mostly at high momenta), but they do not seem to have a profound effect on the results of the reconstruction. By comparing our results with the literature, we find that the phase distribution weighed with the correspondent amplitude values as well as the shape of the orbital correctly reproduce the DFT calculations [@lueftner2014; @ules2014] as well as the data reconstructed by Lüftner et. al [@lueftner2014].
Finally, in order to assess the robustness of the algorithm in terms of the quality of reconstruction from the unsymmetrized CBE map, we made use of the symmetry properties of the pentacene LUMO amplitude and phase and symmetrized the CBE map shown in Fig. 5 (a) around its center. The symmetrical version is shown in Fig. 5 (d). In optics, the far field diffraction pattern is symmetric only in two cases: either due to the real-valued nature of an object or in case of an even complex-valued object distribution with an even amplitude and an even phase. In the latter case, the Fourier transform of the even complex-valued function is an even function as well and the far field intensity distribution is therefore symmetric. In the case of the complex-valued wave function of the pentacene LUMO, the symmetrization is justified purely due to the symmetry of the LUMO amplitude and the phase as it is known from the DFT calculations [@lueftner2014; @ules2014]. The symmetrized CBE map was pre-processed following the same procedure as described above and the results of the reconstruction are shown in Fig. 5(d-f). Qualitatively, the reconstructions from the unsymmetrized CBE map are as good as the reconstructions from the symmetrized data set, except for some minor differences in the shapes of the lobes due to the intrinsic asymmetry of the CBE in Fig. 5(a). This agreement further proves the robustness of the employed algorithm for the reconstruction of molecular orbitals with arbitrary symmetry properties.
Summary and Conclusion
======================
In this work, we show that the state-of-the-art phase retrieval algorithms currently employed in CDI can be successfully used for the reconstruction of complex-valued wave functions of molecules adsorbed on single-crystalline substrates. We tested and applied these algorithms in an optical analogue experiment and then successfully applied them to the reconstruction of the LUMO of pentacene adsorbed on Ag(110). The advantage of using modern CDI algorithms and in particular the shrinkwrap algorithm for the reconstruction of molecular orbitals is that they do not require any *a priori* information about the shape of the object. Instead, they smoothly converge to the correct shape of the object in the course of the reconstruction. In case of molecular wave functions, this is highly important, since precise estimation of the object support is difficult and cannot be guaranteed in every case. This applies, for instance, if the orbital tomography technique aims at visualizing chemical reactions or following the dynamics of excited states, where effective electronic wave functions are unknown. The availability of a general and robust reconstruction algorithm is thus an important step for further advancement of orbital tomography.
Financial support by the Swiss National Science Foundation through NCCR MUST is greatefully acknowledged. We thank Achim Schöll and co-workers (University of Würzburg) for making the pentacene ARPES data available to us. The Center for Micro- and Nanoscience (ETH Zurich) is acknowledged for design and production of the photomask. SEM imaging was performed with equipment maintained by the Center for Microscopy and Image Analysis (University of Zurich).
References {#references .unnumbered}
==========
|
---
abstract: 'We prove that all zeros of the polynomials orthogonal with respect to a measure $d \mu(x;a) = d \mu(x) + M \delta(x-a)$, where $d\mu$ is a nonatomic positive Borel measure and $M>0$, are increasing functions of the mass point $a$. Thus we solve partially an open problem posed by Mourad Ismail.'
address: |
Departamento de Matemática Aplicada\
IBILCE, Universidade Estadual Paulista\
15054-000 Saõ José do Rio Preto, SP, Brazil.
author:
- 'Dimitar K. Dimitrov'
title: Monotonicity of zeros of polynomials orthogonal with respect to a discrete measure
---
Introduction
============
Let $d\alpha(x;\tau)$ be a family of positive Borel measures depending on the parameter $\tau \in (c,d)$ and $\{ p_n(x;\tau)\}$ be the sequence of polynomials orthogonal with respect to $d\alpha(x;\tau)$. For a given $n\in \mathbb{N}$, denote by $x_{nk}(\tau)$, $k=1,\ldots,n$, the zeros of $p_n(x;\tau)$. Markov [@Markov] and Stieljes [@Sti] were the first who studied $x_{nk}(\tau)$ as functions of the the parameter $\tau$ in two fundamental papers, both published in 1886. Markov’s beautiful result [@Markov] (see also [@Ismail Theorem 7.1.1] and [@Szego Theorem 6.12.1]) concerns the case when the measure is absolutely continuous on $(a,b)$, that is $d\alpha(x;\tau) = \omega(x;\tau) dx$, where, for any $\tau\in (c,d)$, $\omega(x;\tau)$ is a weight function supported on $(a,b)$.
\[A\] Let $\{ p_n(x;\tau) \}$ be orthogonal with respect to $d \alpha(x;\tau)$, $$d \alpha(x;\tau) = \omega(x;\tau) d \alpha(x),$$ on an interval $I=(a,b)$ and assume that $\omega(x;\tau)$ is positive and has continuous first derivative with respect to $\tau$ for $x\in (a,b)$, $\tau \in T=(\tau_1,\tau_2)$. Furthermore, assume that $$\int_{a}^{b} x^j \omega_\tau(x;\tau)\, d\alpha(x),\ \ j=0, 1, \ldots, 2n-1,$$ converge uniformly for $\tau$ in every compact subinterval of $T$. Then the zeros of $p_n(x;\tau)$ are increasing (decreasing) functions of $\tau$, $\tau \in T$, if $\partial \{ \ln \omega(x;\tau)\} /\partial \tau$ is an increasing (decreasing) function of $x$, $x\in I$.
The lack of similar results in the case when the measure contains discrete masses motivated Ismail to formulate the following problem (see [@Ismail Problem\[24.9.1]):
Extend Theorem A to the case when $$d \alpha(x;\tau) = \omega(x;\tau) d x + d \beta(x;\tau),$$ where $\beta(x;\tau)$ is a jump function or a step function.
Then Ismail emphasises the importance of a result when only the discrete part of the measure depends on the parameter: “The case of purely discrete measures is of particular interest so we pose the problem of finding sufficient conditions on $d \beta(x;\tau)$ to guarantee the monotonicity of the zeros of the corresponding orthogonal polynomials when the mass points depend on the parameter $\tau$.”
The above problem, as stated, is challenging because of its generality. However, the absolutely continuous part and the discrete part of the measure may force the zeros to move either in the same or in the opposite direction. In the latter situation the influence of each part of of the measure may be rather complex so that one could hardly expect reasonable sufficient conditions. The above comment of Ismail might have been motivated by the latter argument as well as by the fact that the influence of the absolutely continuous part is described in Theorem \[A\]. Therefore, we concentrate on the discrete part only. In other words, from now on we consider the case $$d \alpha(x;\tau) = \omega(x)\, dx + d \beta(x;\tau).
\label{dal}$$ In general, $$d \beta(x;\tau) = \sum_{j=0}^{\kappa} M_j(\tau)\, \delta(x-a_j(\tau)).
\label{dbeta}$$ where $M_j(\tau)>0$, $a_j(\tau) \in \mathbb{R}$ and $\delta$ stands for the Dirac delta. Each term depends on both $M_j(\tau)$ and $a_j(\tau)$. When all positions $a_j(\tau)$ are fixed, that is $a_j(\tau) = a_j$, and only $M_j(\tau)$ depend on $\tau$, one may simply apply Theorem A with an obvious modification: extend $M_j(\tau)$ to a positive function $M_j(x;\tau)$ on $(a,b)$ and look at $\alpha(x)$ as a Stietjes distribution with jumps at $a_j$. The corresponding results about the monotonic behaviour of the zeros of the classical discrete orthogonal polynomials can be found in [@MCOMP Theorem A and Theorem 2.1].
Observe also that the influence of the distinct discrete parts in (\[dbeta\]) could be pretty complex, so that we concentrate on a discrete measure with a single mass, that is we reduce the problem to the case $$d \beta(x;\tau) = \sum_{j=0}^{\kappa} M_j(\tau)\, \delta(x-a_j(\tau))$$
Therefore, we arrive at the question about the monotonicity of zeros of polynomials orthogonal with respect to the location of a single mass point. Thus we let $$\label{dmat}
d \mu(x;\tau) = d \mu(x) + M \delta(x-a(\tau)),$$ where $d\mu$ is a nonatomic positive Borel measure, and prove:
If $d \mu(x;\tau)$ is defined by (\[dmat\]) and $a(\tau)$ is an increasing function of $\tau$, then all zeros $x_{nk}(\tau)$ of the polynomials $p_n(x;\tau)$, orthogonal with respect to $d \mu(x;\tau)$, are increasing functions of $\tau$.
A simpler equivalent statement is
\[corr\] If $d \mu(x;a) = d \mu(x) + M \delta(x-a)$, then all zeros $x_{nk}(a)$ of the polynomials $p_n(x;a)$, orthogonal with respect to $d \mu(x;a)$, are increasing functions of $\tau$.
Succinctly, our main result states that a moving mass point of a measure “pulls” all the zeros of the corresponding orthogonal polynomials towards the same direction it moves.
Proof
=====
We prove Corollary \[corr\].
The idea of the proof is rather simple and natural. One needs to approximate Dirac’s delta by the normal distribution centred at $a$, consider $a$ as a parameter and apply the classical Markov’s theorem, that is, Theorem A to the normal distribution. Thus, the proof is a formalisation of this straightforward idea.
First we prove the statement in the case when $d \mu(x)$ is an absolutely continuous measure, $d \mu(x)=\omega(x) dx$.
Let $$\mathcal{N}(x;a,\gamma) = \frac{1}{\sqrt{\pi} \gamma} e^{-((x-a)/\gamma)^2}.$$ Then $$\int_{-\infty}^{\infty} \mathcal{N}(x;a,\gamma)\, dx = 1\ \ \mathrm{for\ every}\ a\in \mathbb{R}\ \ \mathrm{and}\ \ \gamma>0.$$ It is well known that $\mathcal{N}(x;a,\gamma)$ converges to $\delta(x-a)$ in the sense of distributions when $\gamma$ goes to zero with positive values. Let $p_n(x;a,\gamma)$ be the polynomials orthogonal with respect to $d \mu(x;a,\gamma) = \left\{ \omega(x) + M\, \mathcal{N}(x;a,\gamma) \right\} dx$. Rather straightforward calculations show that for every $a\in \mathbb{R}$ and $\gamma>0$ the moments $m_k(a,\gamma)$ of $\mathcal{N}(x;a,\gamma) dx$ are explicitly given by $$\begin{aligned}
m_k(a,\gamma) & = & \int_{-\infty}^{\infty} x^k\, \mathcal{N}(x;a,\gamma)\, dx \\
\ & = & \frac{1}{\sqrt{\pi}} \sum_{j=0}^{k} {k \choose j} \frac{1+(-1)^j}{2}\, \Gamma \left( \frac{j+1}{2} \right)\, a^{k-j}\, \gamma^j.\end{aligned}$$ Then, for any fixed $a\in \mathbb{R}$ and $k\in \mathbb{N}$, $m_k(a,\gamma) \rightarrow a^k$ as $\gamma \rightarrow +0$. On the other hand, the $k$-th moment of $\delta(x-a)$ is exactly $a^k$. Therefore the moments of $d \mu(x;a,\gamma)$ converge to the moments of $d \mu(x;a)$ as $\gamma \rightarrow + 0$. It follows from the representation of the orthogonal polynomials as quotients of determinants involving the moments of the measure (see [@Szego (2.26)] or [@Ismail (2.1.6)]) that, for any $n\in \mathbb{N}$ and $a \in \mathbb{R}$, the Taylor coefficients of the polynomial $p_n(x;a,\gamma)$ converge to those of $p_n(x;a)$ when $\gamma$ converges to zero with positive values. This immediately yields that $p_n(x;a,\gamma)$ converges locally uniformly to $p_n(x;a)$. Then a well-known theorem of Hurwitz (see [@Hille Theorem 14.3.2]) implies that the zeros $x_{nk}(a,\gamma)$ of $p_n(x;a,\gamma)$ converge to the zeros $x_{nk}(a)$ of $p_n(x;a)$ when $\gamma \rightarrow +0$. Moreover, by the implicit function theorem $x_{nk}(a,\gamma)$ are smooth functions of $\gamma$, for $\gamma\in (0,\infty)$.
Let us fix $\gamma>0$ and consider the behaviour of $x_{nk}(a,\gamma)$ as functions of $a$. It is clear that $$\begin{aligned}
\mathcal{N}_a(x;a,\gamma) & = & \frac{\partial}{\partial a} \mathcal{N}(x;a,\gamma) \\
\ & = & \frac{2}{\sqrt{\pi} \gamma^3}\ (x-a)\ e^{-((x-a)/\gamma)^2}\end{aligned}$$ is a continuous function of $a \in \mathbb{R}$ as well as that the moments of $\mathcal{N}_a(x;a,\gamma) $ are integrals that converge uniformly for $a$ in any subinterval of the real line. Since $$\frac{\partial}{\partial a} \ln \mathcal{N}(x;a,\gamma) = \frac{2 (x-a)}{\gamma^2}$$ is an increasing function of $x$, then, by Theorem A, all $x_{nk}(a,\gamma)$ are increasing functions of $a$. Taking a limit, as $\gamma \rightarrow +0$, we conclude that $x_{nk}(a)$ are increasing functions of $a$ too. This completes the first proof in the case when $d \mu(x)$ is absolutely continuous.
Since a nonatomic Borel measure measure contains an absolutely continuous part and an enumerable quantity of jumps, we may approximate the corresponding delta functions either by normal distributions or even by $C^\infty$ compactly supported test functions (see [@Gel]) and then we proceed as above.
The main result may be applied to deduce the monotonicity of zeros of sequences of polynomials orthogonal with respect to a mesure of the form (\[dal\]), where $\omega$ is associated with the classical sequences of orthogonal polynomials (see [@CMC; @CMVQ]).
[99]{}
I. Area, D. K. Dimitrov, E. Godoy and V. G. Paschoa, Zeros of classical orthogonal polynomials of a discrete variable, Math. Comp. 82 (2013), 1069–1095.
E. J. H. Cejudo, F. Marcellán, H. P. Cabrera, An electrostatic model for zeros of perturbed Laguerre polynomials, Proc. Amer. Math. Soc. 142 (2014), 1733–1747.
E. J. H. Cejudo, F. Marcellán, M. F. P. Valero and Y. Quintana, A Cohen type inequality for Laguerre-Sobolev expansions with a mass point outside their oscillatory regime, Turkish J. Math. 38 (2014), 994–1006.
I. M. Gel’fand and G. E. Shilov, Generalized Functions: Volume 1 Properties and Operations, Academic Press, New York, 1964.
E. Hille, Analytic Function Theory, Vol. 2, Second Edition, AMS Chelsea Publishing, Providence, RI, 2002.
M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.
A. Markov, Sur les racines de certaines équations (second note), Math. Ann. 27 (1886), 177–182.
T. J. Stieltjes, Sur les racines de certaines l’équation $X_n=0$, Acta Math. 9 (1886), 385–400.
G. Szegő, Orthogonal Polynomials, 4th edition, Amer. Math. Soc., Providence, RI. 1975.
|
---
author:
- 'S. Demers ,'
- 'P. Battinelli'
date: 'Received; accepted'
title: 'The C star outer disk population of M31 seen with the SLOAN filters [^1]'
---
Introduction
============
Photometric identification of intermediate-age carbon stars in nearby galaxies has recently been done using a combination of broad and narrow-band photometry. The (CN – TiO) technique has been applied by, for example, Battinelli et al. (2003), Nowotny et al. (2003) and Harbeck et al. (2004) while the near infrared approach has been adopted by, for example, Demers et al. (2002) and Cioni & Habing (2005). The SLOAN Digital Sky Survey (SDSS) photometric system, described by Fukugita et al. (1996), has been used to identify carbon stars in the Galactic halo. Indeed, Krisciunas et al. (1998) were first to show that carbon stars can be differentiated from M stars in the (r$'$ – i$'$) — (g$'$ – r$'$) diagram. Faint halo carbon stars were discovered, using this technique, from the SDSS database by Margon et al. (2002). The first catalogue of these halo C stars has recently been published by Downes et al. (2004) where 251 stars were identified in a 3000 deg$^2$ area. The authors evaluate that over 50% of the sample constitute nearby dwarf C stars. Obviously only a global survey like the SDSS can tackle the Galactic halo. Surveys of nearby galaxies can, however, be done with a more conventional approach.
Battinelli et al. (2003) have identified nearly one thousand C stars in the southern outer disk of M31. Approximately 0.6 deg$^2$ were surveyed using the CFH12K mosaic. C stars are identified from their position on the (CN – TiO) vs (R – I) colour-colour diagram. The reliability of this technique is quite secure, as it was demonstrated by Brewer et al. (1995) and Albert et al. (2000). The M31 sample offers the opportunity to compare the (CN – TiO) technique with the SDSS colour-colour approach. We therefore describe here our new observations which include part of the fields already observed by us and extend the M31 disk survey to slightly larger radii.
Observations and data reduction
===============================
The observations presented in this paper consist of one Megacam field, centered on $\alpha$ = 00:37:02.3, $\delta$ = +39:40:50 (J2000.0), obtained in Service Queue Observing mode in August 2003. The Megacam camera is installed at the prime focus of the 3.66 m Canada-France-Hawaii Telescope. The camera consists of a mosaic of 36 2048 $\times$ 4612 pixels CCDs providing a field of view of nearly one deg$^2$, with a resolution of 0.187 arcsecond per pixel. Images were obtained through g$'$, r$'$ and i$'$ SDSS filters. The observations were secured under non photometric and partly cloudy conditions. For this reason, three short exposures were taken under excellent conditions and have been used to calibrate the long exposures. It turns out, however, that our adopted exposure times are somewhat too short to fully survey the very red stars of M31. Table 1 presents the journal of the observations. The g$'$, r$'$ and i$'$ magnitudes, are calibrated with bright first generation SDSS standards (Smith et al. 2002), as explained in the CFHT/Megacam website.
{width="10cm"}
The data distributed by the CFHT have been detrended. This means that the images have already been corrected with the master darks, biases, and flats. This pre-analysis produces 36 CCD images, of a given mosaic, with the same zero point and magnitude scale.
$$% \begin{array}{p{0.5\linewidth}clr}
\begin{array}{lllcc}
\hline
\noalign{\smallskip}
{\rm Date}&{\rm Filter}&{\rm exposure}&{\rm seeing}\ ('')& {\rm airmass} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
2003/08/24&r'& 430\ s&1.34&1.12\\
2003/08/24&r'& 430\ s&0.93&1.12\\
2003/08/24&i'& 200\ s&0.78&1.12\\
2003/08/24&g'& 550\ s&0.97&1.11\\
2003/08/24&g'&550\ s&0.91&1.12\\
2003/08/24&g'& 550\ s&0.91&1.11\\
2003/08/24&r'& 430\ s&0.78&1.09\\
2003/08/24&g'& 550\ s&0.95&1.08\\
2003/08/24&g'& 550\ s&0.90&1.07\\
\\
2003/09/18&r'& 43\ s&0.84&1.06\\
2003/09/18&i'& 20\ s&0.84&1.06\\
2003/09/18&g'& 55\ s&0.94&1.06\\
\noalign{\smallskip}
\hline
\end{array}$$
The photometric reductions were done by Terapix, the data reduction center dedicated to the processing of extremely large data flow. The Terapix team, located at the Institut d’Astrophysique de Paris, matches and stacks all images taken with the same filter and, using SExtractor (Bertin & Arnouts 1996), provides magnitude calibrated catalogues of objects in each of the combined images. SExtrator classifies objects into star or galaxy but the classification scheme breaks down for faint magnitudes. It is essentially useless for the M31 stars. A flag is attached to each object, flag = 0 corresponds to isolated object not affected by neighbours. As can be seen from Figure 1, the northern part of the Megacam field is closer to the center of M31. For this reason a substantial stellar density gradient is observed across the field thus numerous stars with flag $\ne$ 0 are present in the northern half. Since the astrometric calibration of the images has been done by the CFHT Service Observing team, we have equatorial coordinates as well as calibrated colours and magnitude for each object in the field.
Results
=======
The data
--------
Each calibrated $i'$, r$'$ and g$'$ file contains about 300,000 objects, $\sim$230,000 of them with flag = 0. We shall employ and analyse this subset. The number of stars, having a photometric error smaller than a given value, varies from one file to the next. For example, the $i'$, r$'$ and g$'$ files contain respectively 83,000, 121,000 and 110,000 stars with err $<$ 0.10 mag. However, when $i'$ and r$'$ are combined, the number of stars with colour error, $\sigma_{ri}$ $<$ 0.10, is 50,500 while for g$'$ and r$'$, the number of stars with $\sigma_{gr}$ $<$ 0.10 is 56,700. When the three files are combined and, following our standard criterion only stars with $\sigma_{irg}$ = $(\sigma_{ri}^2 + \sigma_{gr}^2)^{1/2}$ $<$ 0.125 are retained, the number of stars drops to 37,000. We shall see later that this small number is due to the presence of numerous faint red stars not well observed in the g$'$ and r$'$ filters. Finally the remaining $\sim$65,000 stars with flags = 1,2, or 3 and corresponding to objects affected by close neighbours or/and originally blended will not be used for the magnitude and colour comparisons but will be needed later to cross-identify known C stars.
The colour-magnitude diagrams
-----------------------------
A one square degree field in the direction of M31 must obviously include stars of different population. The major axis of its disk runs roughly diagonally across the Megacam field, from 18 kpc to 33 kpc. At such distances the bulge population is completely negligible (Windrow et al. 2003), thus we see disk and halo stars of M31, our Galactic contributions (halo and disk stars) and numerous unresolved galaxies. We present, in Figure 2, the two colour-magnitude diagrams (CMD) corresponding to the whole Megacam field for stars with flag = 0 and colour error $<$ 0.10. The broken lines represent the limiting magnitudes which correspond to the magnitudes where the luminosity function drops to 50% of its peak value. These are found to be: i$'$ = 22.5, r$'$ = 22.9 and g$'$ = 24.0.
The top panel shows the i$'$ magnitudes versus the (r$'$ – i$'$) colours. Two features are conspicuous: the bright end of the red giant branch of M31, starting at i$'$ $\sim$ 21 and extending far to the red; and the vertical ridge at (r$'$ – i$'$) $\sim$ 0.10. This ridge corresponds to Galactic G dwarfs, at the MS turnoff, seen along the line of sight. The colour location of this ridge is indicative of the reddening. According to Schlegel et al. (1998) the Galactic contribution to the reddening in this direction amounts to E(B–V) = 0.06, which translates to E(r$'$–i$'$) = 0.04.
The second CMD (lower panel of Fig. 2) has more interesting features. Blue main sequence stars are well separated from the bulk of M31’s stars, differential reddening within the star forming regions must be responsible for the diffuse appearance of the main sequence. The narrow vertical plume at (g$'$ – r$'$) $\sim$ 1.4 corresponds to stars of spectral type M, as the synthetic colours from Fukugita et al. (1996) demonstrate. This plume is also seen in the simulation of Galactic stellar objects by Fan (1999). The vertical ridge corresponding to Galactic G dwarf is also seen. Finally we see, in this CMD, a small population of stars extending at (g$'$ – r$'$) $>$ 1.7. We identify these stars with extreme (g$'$ – r$'$) colours to C stars observed for the first time in the SLOAN colours by Krisciunas et al. (1998). This C star tail is seen to curve [*below the the limiting magnitude of the data*]{}, suggesting that some C stars must have been missed.
{width="15cm"}
The colour-colour diagram
-------------------------
The colour-colour diagram of the Megacam field is displayed in Figure 3. Again, only stars with flag = 0 are included, we plot only stars fainter than $i'$ = 17.5 to exclude some ($\sim$ 2000) of the foreground Galactic stars. C stars are seen to the right of the diagram, with colours reaching (g$'$ – r$'$) = 3.
We note, however, a number of points with extreme negative (r$'$ – i$'$) colours, seen at the bottom of the diagram. Investigation of their spatial distribution reveals that they are nearly all on the West side of the North-South CCD borders. The spurious $r'$ magnitudes close to the borders are due to a $\sim$80 pixel shift of one of the exposures relative to the two other ones. Thus, the instrumental magnitudes of about 200 stars near the CCD borders are wrong because they are not based on the right number of exposures. It is rather difficult to delete these stars from the database because we have lost their original x,y CCD coordinates since we are now using equatorial coordinates.
{width="10cm"}
Cross identification of M31 stars
---------------------------------
The goal of our study is to obtain the SDSS magnitude and colours of the known C stars in the disk of M31 which were identified from their position in the (R – I) vs (CN – TiO) plane. The (CN – TiO) technique is designed to identify cool N-type C stars. In the (R – I) vs (CN – TiO) plane, the warmer and fainter C stars are mixed with the late K or early M stars. For this reason the differentiation of C stars and M stars is limited to stars with (R – I)$_0$ $>$ 0.90. Therefore, our approach will be somewhat different from the one employed by Margon et al. (2002) in their identification of faint high-latitude carbon stars (FHLCs) because we intend to deal exclusively with cool red C stars.
The first step in the cross identification is to match all stars with identical equatorial coordinates. To do so, we select two datasets: the CFH12K SW1 field consisting of 62,007 stars for which we have I, (R – I) and (CN – TiO) and a second set consisting of the 37,000 stars in the Megacam field for which we have i$'$, (r$'$ – i$'$) and (g$'$ – r$'$). After a few iterations to minimize the $\Delta\alpha$ and $\Delta\delta$ we retain some 8000 pair of stars matched within 0.8 arcsec. The reason why such relatively small number of matches is obtained can be seen from Fig. 1, only 63% of SW1 overlaps with 20% of the Megacam field. In the region common to both fields there are $\sim$ 10,000 stars from Megacam and $\sim$ 35,000 in the SW1 field.
We compare, in Figure 4, the magnitudes and colours of these matched stars. The i$'$ magnitudes are fainter than the I magnitudes by $\sim$ 0.25 mag. The downward bulge, seen at the faint magnitude end, is simply due to the natural increase of the scatter for the faintest magnitudes. The asymmetry of the dispersion is explained by the missing stars with faint i$'$ magnitudes. These stars are present in the i$'$ file but, since they don’t have matches in the r$'$ or g$'$ files, disappear from the i$'$r$'$g$'$ Megacam dataset.
The relationship between the magnitudes can be expressed in the following ways, obtained from linear regressions: $$i' = I + (0.289\ \pm 0.010) + (0.1404\ \pm 0.0078)\times(R-I)\eqno(1)$$ or $$I = i' - (0.291\ \pm 0.008) - (0.1717\ \pm 0.0081)\times(r'-i').\eqno(2)$$ The comparison of the colours shows a significant scatter at the red end. This is simply a consequence of the rather bright r$'$ limiting magnitude of our observations. The faint red stars have a lower photometry quality. The colour relationship is obtained by using only 900 stars with i$'$ $<$ 20.5. These stars have (r$'$ – i$'$) $<$ 2.0. $$(r'-i') = -(0.129 \pm 0.017) + (0.857\ \pm 0.013)\times(R-I),\eqno(3)$$ this is to be compared to the synthetic colours relation calculated by Fukugita et al. (1996). They quote (r$'$ – i$'$) = 0.98(R – I) – 0.23; for (R – I) $<$ 1.15 and (r$'$ – i$'$) = 1.40(R – I) – 0.72; for (R – I) $>$ 1.15. We do not see in our data a break in the colour relation up to (r$'$ – i$'$) $\approx$ 2.0, where the brighter stars can be seen. For comparison, the Fukugita et al. relation is drawn on Fig. 4.
![Comparison of the magnitudes and colours of the 8000 stars matched between the two datasets. The top panel shows that the i$'$ magnitudes are fainter than the I magnitudes, the identity line is drawn. For the bottom panel, the lines represent the Fukugita et al. (1996) colour relation. []{data-label="FigComp"}](fig4comp.ps){width="10cm"}
Cross identification of C stars
-------------------------------
Of the nearly one thousand M31 C stars identified by Battinelli et al. (2003) in fields SW1 and SW2, only 644 are in the Megacam field. Cross identification, with the same criterion described above, yields barely 129 matches, just 20% of the C stars in the field. There are two major reasons for this low success rate. The disk of M31 shows a gradient of stellar surface density. The northern part of the Megacam field being more densely populated contains more C stars that are in a somewhat crowded environment. Many northern C stars were not matched. If we do not take into account the flag assigned by SExtrator to each star, and accept all stars irrespective of their flag, the number of carbon star matched increases to 245. This is done without relaxing the photometric error criterion. The second explanation for the low success rate comes from the fact that the limiting r$'$ and g$'$ magnitudes are not as faint as the CFH12K data, thus the redder C stars are missed. Figure 5 compares the (R – I)$_0$ colour distribution of the known C stars with those cross identified in the i$'$ and g$'$ files. Obviously the red stars are missing in the g$'$ file and also, to a lesser extent in the r$'$ file, not shown here.
![Colour distributions of the C stars and those cross-identified with the Megacam data. []{data-label="colour distributions"}](fig5his.ps){width="6cm"}
The magnitude and colour distributions of the known C stars and those identified in our Megacam data are shown in Figure 6. The top panel indicates the number of C stars cross identified depends moderately on the apparent i$'$ magnitude. This implies that the crowding is the most important factor. The lower panel confirms that red stars are missing from our Megacam sample.
![Magnitude and colour distributions of the 644 known C stars are compared to the distribution of the 245 C stars cross-identified with our three magnitude Megacam file (shaded histograms). []{data-label="Mag and colour distributions"}](fig6his2.ps){width="10cm"}
Figure 7 presents the SDSS colour-colour diagram of the 245 C stars recovered in our Megacam data, when no flag restriction is applied. The three solid lines outline the acceptance limits adopted by Margon et al. (2002) for the FHLCs. As expected, our C stars are redder than these limits in (r$'$ – i$'$) as well as in (g$'$ – r$'$). Since our earlier adopted limit of (R – I)$_0$ = 0.90 corresponds to (r$'$ – i$'$) = 0.64, very few if any C stars should have bluer (r$'$ – i$'$) colours. These Megacam observations, sampling the disk of M31 where numerous K giants are present, are certainly not ideal to identify bluer C stars which can easily be confused with the bulk of the K giants. We draw, somewhat arbitrarily, the dashed lines corresponding to our adopted colour limits for the M31 cool C star population. The extreme upper and lower points correspond to stars that have calculated magnitude and colour (given by Eqs 2 and 3) which differ substantially from the CFH12K values. The variability of C stars could be responsible or, more likely, spurious matches are always possible in a crowded field. The 129 C stars matched to the flag = 0 data are indistinguishable in the colour-colour plane.
![Colour-colour diagram of the 245 C stars, identified from their (CN – TiO) index. The three solid lines define the C star acceptance limits adopted by Margon et al. (2002). The dashed lines trace our adopted upper and lower limits. []{data-label="Colour-colour plot"}](fig7ccM.ps){width="8cm"}
Selection criteria for cool C stars
-----------------------------------
A close-up of Fig. 3, along with our adopt boundaries for C stars is displayed in Figure 8. It is obvious that the blue limit adopted by Margon et al. (2002) is of little use to us. Indeed, the numerous K stars seen in the M31 disk, overwhelm the few C stars with (g$'$ – r$'$) $\approx$ 1.5. This, to a so large extent, that to exclude as much as possible K and M stars it would seem necessary to adopt a conservative blue limit around (g$'$ – r$'$) $\approx$ 1.7. We describe how we can better determine this colour limit.
![Close-up of the colour-colour diagram for stars with flag = 0. The dashed lines represent the adopted boundaries. []{data-label="CC plot"}](closecc.ps){width="8cm"}
For the 8000 stars cross-identified we have their (R – I), (CN – TiO), (g$'$ – r$'$) and (r$'$ – i$'$). We can then identify the C and M stars by applying our criteria based on (R – I) and (CN – TiO) colours. Thus we can calculate the number of C or M stars having a (g$'$ – r$'$) larger than a certain limit. The numbers of C stars (N$_C$) and M stars (N$_M$) selected for different (g$'$ – r$'$) lower limits are displayed in Figure 9. As we shift the blue limit to redder colours the number of M stars drops appreciably while the number of C stars decreases slightly. Figure 10 presents the variation of the ratio of the number of C stars to the sum (N$_C +$ N$_M$) for various (g$'$ – r$'$) limits. The dashed curve is the ratio of N$_C$ relative to the original 245 C stars. We conclude, from this figure, that a reasonable colour limit is (g$'$ – r$'$) = 1.55. At this limit 86% of the stars are C stars and we lose 20% of the C stars which have bluer colours. Furthermore, as we have previously explained, the C stars detected by this technique represent only a fraction of the total cool C star population because our Megacam observations do not have sufficiently deep exposures.
![ The numbers of C and M stars retrieved from samples having different colour limits show that the selection of M stars is very sensitive to the colour limit. []{data-label="C and M stars"}](fig9CM.ps){width="8cm"}
![ The change of the ratio (N$_C$/(N$_C$ + N$_M$) for different colour limits. The dashed curve is the percentage of the 245 C star retained. []{data-label="ratios of C/M stars"}](fig10ratio.ps){width="8cm"}
Discussion
==========
Properties of C stars
---------------------
Applying the above colour limits to our Megacam data file, for stars with flag = 0 and colour $\sigma_{irg}$ $<$ 0.125, yields a sample of 480 C stars candidates. According to Fig. 9, there should be some 70 K or M stars polluting this sample. Cross-identification with the known C stars in M31 results in 102 matches. However, a cross-identification with the whole CFH12K SW1 database reveals that 17 stars are in fact M stars. These few stars are thus deleted from our sample. The equatorial coordinates, given in degrees, the magnitude, the colours and their attached errors as determined by SExtractor of these remaining 463 stars are listed in Table 2. The $\sigma_{gr}$ being twice as large than the $\sigma_{ri}$ is explained by the faintness of C stars in the g$'$ band. 361 of them are newly identified C stars located almost all at larger radial distances than the ones previously known.
The i$'$ luminosity function of the 463 C stars candidates is displayed in Figure 11. A Gaussian is fitted by eyes over the distribution. Their mean apparent magnitude $\langle i' \rangle$ = 20.64, with a variance of 0.31, their mean colours are: $\langle(r' - i')\rangle$ = 0.86 and $\langle(g' - r')\rangle$ = 1.86. Using eq. 2, this mean magnitude corresponds to $\langle I \rangle$ = 20.20, a value to be compared with $\langle I_0 \rangle$ = 19.94, obtained for the M31 C star population by Battinelli et al. (2003). Taking into account a mean extinction, of the order of A$_I$ = 0.12, implies that we have acquired essentially the same stellar population even though using different colour criteria.
$$% \begin{array}{p{0.5\linewidth}clr}
\begin{array}{lcccccccc}
\hline
\noalign{\smallskip}
id&RA&Dec&i'&\sigma_{r}&(r'-i')&\sigma_{r-i}&(g'-r')&\sigma_{g-r} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
1& 8.6219311 &39.6322746& 20.877& 0.030& 0.994& 0.043& 2.415& 0.113\\
2& 8.6224728 &39.7282753& 20.107& 0.018& 0.850& 0.025& 1.791& 0.042\\
3& 8.6282034 &39.8302460& 20.585& 0.026& 1.134& 0.040& 2.209& 0.097\\
4& 8.6309576 &39.3995628& 20.726& 0.028& 0.471& 0.034& 1.571& 0.042\\
5& 8.6383266 &39.3832932& 20.548& 0.036& 0.899& 0.048& 1.721& 0.073\\
6& 8.6383543 &39.8645439& 21.190& 0.036& 0.950& 0.051& 2.027& 0.107\\
7& 8.6440039 &39.7481689& 21.465& 0.043& 0.610& 0.054& 2.024& 0.093\\
8& 8.6491823 &39.6935310& 20.527& 0.022& 0.974& 0.032& 2.636& 0.097\\
9& 8.6501064 &40.1469650& 19.649& 0.018& 0.793& 0.023& 1.691& 0.035\\
10& 8.6624346 &39.3817787& 20.508& 0.021& 1.049& 0.033& 2.192& 0.075\\
11& 8.6695499 &39.4705505& 20.596& 0.023& 0.917& 0.032& 2.112& 0.067\\
12& 8.6699352 &39.5931396& 20.262& 0.020& 0.844& 0.028& 1.918& 0.051\\
\noalign{\smallskip}
\hline
\end{array}$$
Complete Table 2 is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5). A portion is shown here for guidance regarding its form and content. Units of right ascensions and declination (J2000) are in degrees.
![ Luminosity function of the 463 C stars identified by applying our colour criterion. []{data-label="C LF"}](fig11LF.ps){width="8cm"}
The M31 disk and its intermediate-age population
------------------------------------------------
C stars can be used to map the spatial distribution of the intermediate-age population. To do so, we calculate the surface density per arcmin$^2$ in elliptical annuli of 10$'$ width having the shape of the M31 apparent disk. Such ellipses are sketched in Fig. 1. We adopt for the disk of M31 a position angle of the major axis of 37.7$^\circ$ and an ellipticity $\epsilon$ = 0.787. Because of the huge size of M31, even compared to our one deg$^2$ field, we are observing only sectors of annuli of quite different angular length. Even though the radial distance along the major axis reaches only $\sim$ 150$'$, we can sample larger radial distances on the edge of the field away from the major axis. The top panel of Figure 12 presents the C star surface density as a function of the radial angular distances. The bottom panel shows the surface density of the all stars detected by SExtractor fainter than i$'$ = 17.5 to exclude the brighter Galactic stars. These stars are mostly M31’s red giants. Their surface density reaches a plateau at $\sim$ 170$'$ ($\sim$ 35 kpc) this position should correspond to the edge of the stellar disk. If we adopt, by averaging the last 7 points, 36.54 stars/arcmin$^2$ for the density outside of the disk, the declining slope of the surface density corresponds to a scale length of 4.9 $\pm$ 0.4 kpc. This is in excellent agreement with the scale length of C stars determined by Battinelli et al. (2003) and the one derived by Walterbos & Kennicutt (1988) from multicolour integrated surface photometry.
The density profile of C stars is much more irregular. The flattening of the profile at short distances is most probably due to the fact that SExtractor has difficulties dealing with crowded fields. For C stars only stars with $\sigma_{irg}$ $<$ 0.125 are selected while stars of all errors are included in the bottom panel. Contrary to the bulk of stars, C stars are seen up to $\sim$ 190$'$, ($\sim$40 kpc) where a sharp drop is observed. The last three points represent only five C stars farther than 200$'$. This confirms the identification of C stars along the major axis of the disk by Battinelli & Demers (2005) who found one C star at 40 kpc, well outside of the Megacam field.
The fact that C stars are seen beyond “the edge of the disk” suggests that a tenuous thick disk, containing intermediate-age stars, must be present beyond the detectable edge. It density contrast, relative to the halo population, may be too low for easy detection. Since C stars are seen behind a zero foreground they can be seen even in extremely low density environment.
![ The surface densities, stars per arcmin$^2$ for C stars (top) and all stars with i$'$ $>$ 17.5 (bottom). []{data-label="Density profiles"}](fig12prof.ps){width="8cm"}
The C/M ratio
-------------
We have confirmed that the SDSS colours are useful to identify cool C stars. The next step is to adopt a colour criterion applied to the M stars, thus allowing the determination of the C/M ratio. According the Hawley et al. (2002), M0 dwarfs have $\langle(r' - i')\rangle$ = 0.91 $\pm 0.24$. Unfortunately, because of the width and upper colour limit of the C star zone in the colour-colour diagram it appears quite difficult to discriminate M stars from C stars near such (r$'$ – i$'$) colour. However, M3 stars have $\langle(r' - i')\rangle$ = 1.29 $\pm$ 0.32 a colour that cannot be confused with C stars. Therefore, we adopt for colour limits of the M3+ stars: (r$'$ – i$'$) = 1.3 and (g$'$ – r$'$) = 1.5, to separate them slightly from C stars. We have $\sim$ 6000 such stars (with $\sigma_{ir}$ $<$ 0.10) in our database. The limiting magnitude of this sample, because of the redness of the stars, reaches i$'$ $\approx$ 22, which translates into I = 21.5.
M3+ counts, in the same 10$'$ wide elliptical annuli previously adopted, show a plateau for distances larger than 170$'$. We average these outer counts to obtain an estimate of the foreground/background density. In our case it corresponds to 1.06 $\pm$ 0.03 M3+ stars per arcmin$^2$. A value we substract from the observed M3+ density. This density is to be compared to the M3+ density of 0.8 stars/arcmin$^2$ we calculate from Durrell’s et al. (2001) observations of a remote field near M31.
Figure 13 presents the C/M3+ ratio as a function of galactocentric distances. As expected, for a decrease of metallicity with radial distances, the number of C stars increases relative to the number of M3+ stars. Such behavior, for example, has been observed in M33 by Rowe et al. (2004).
![ The C/M3+ ratio determined for elliptical annuli of the shape of M31’s disk. []{data-label="C/M profile"}](fig13cmratio.ps){width="8cm"}
Colour and magnitude trend with the galactocentric distance
-----------------------------------------------------------
Our previous two investigations of the outer disk of M 31 have revealed the existence of a mild decrease of the I luminosity of C stars with increasing galactocentric distances (see Battinelli & Demers, 2005). It is therefore worth to inspect the behaviour of the newly identified C stars which extend the previous surveys up to nearly 240$'$. Figure 14 shows the magnitude and colour as a function of the galactocentric distance for the newly identified C stars along with the nearly 1000 previously know (Battinelli et al., 2003; Battinelli & Demers, 2005). Megacam $i'$ were converted into Kron-Cousins I using eq. (2).
![ Colour and Magnitude of C stars as a function of the galactocentric distances. []{data-label="gradients"}](fig14trend.ps){width="8cm"}
A certain radial fading of the luminosity and a gradual disappearance of the reddest C stars is evident. In principle, the luminosity fading may be explained by an increase of the metallicity at large galactocentric distances. Indeed, it is well know that the higher the metallicity the lower is the C star luminosity. This explanation is however unsatisfactory since both the observed colour gradient, shown in the top panel of Fig. 14, and the C/M behaviour in Fig. 13 point definitely towards a metallicity decrease in the outer part of the Andromeda disk. Age could, however, be responsible for the observed luminosity fading. Indeed, theoretical models for simple stellar populations of intermediate age (see e.g. fig. 11 in Marigo et al., 1999) suggest that the brightest C stars disappear when the age of the population increases. A radial increase of the age of the youngest C stars (or equivalently a decrease of the mass of the most massive C stars) can overwhelm the metallicity effect and explain the luminosity trend. It is clear that, beyond this qualitative considerations, an answer to the question is possible only through a full modeling – which could also account for the observed metallicity and density radial trends – of the composite stellar population in the outer disk.
Conclusion
==========
The SDSS filters offer an alternative method to identify C stars and also late M stars. Contrary to the (CN – TiO) technique, it is difficult with the SDSS filters to isolate C stars from M0 stars. Furthermore, because numerous C stars have quite large (g$'$ – r$'$) colour the exposure time to reach the desired g$'$ magnitude can be very long relative to the i$'$ exposure. In term of telescope time both techniques require approximately the same total exposures since the CN and TiO exposures must be at least three to four times the I exposures.
The SDSS approach provides, however, uncontested advantages for two aspects of the extragalactic C stars survey. These filters are available on the new generation of large mosaic detectors, such as Megacam. They allow the survey of an entire nearby galaxy in a relatively short time. We have recently followed this approach to survey 4 deg$^2$ around NGC 6822 where nearly 900 C stars are already known in its extended halo (Letarte et al. 2002). Optical imagers, albeit of small field size, available on some large telescopes, such as Gemini, offer - sometimes exclusively - SDSS filters. Because of the lack of the general availability of the CN and TiO filter, the SDSS approach must be adopted for any attempt to survey C stars among the neighbours of the Local Group.
This research is funded in parts (S. D.) by the Natural Science and Engineering Council of Canada. We are grateful to Yannick Mellier and the Terapix team to have so promptly accepted to measure our Megacam data.
Albert, L., Demers, S., & Kunkel, W. E. 2000, AJ, 119, 2780
Battinelli, P., & Demers, S. 2005, A&A, 430, 905
Battinelli, P., Demers, S., & Letarte, B. 2003, AJ, 125, 1298
Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393
Brewer, J. P., Richer, H. B., & Crabtree, D. R. 1995, AJ, 109, 2480
Cioni, M.-R. L., & Habing, H. J. 2005, A&A, 429, 837
Demers, S., Dallaire, M., & Battinelli, P. 2002, AJ, 123, 3428
Demers, S., Battinelli, P., & Letarte, B. 2003, A&A, 410, 795
Downes, R. A., Margon, B., Anderson, S. et al. 2004, AJ, 127, 2838
Durrell, P.R., Harris, W. E., & Pritchet, C. J. 2001, AJ, 121, 2557
Fan, X. 1999, AJ, 117, 2528
Fukugita, M., Ichikawa, T., Gunn, J. E., Doi, M., Shimasaku, K., & Schneider, D. P. 1996, AJ, 111, 1748
Harbeck, D., Gallagher, J. S., & Grebel, E. K. 2004, AJ, 127, 2711
Hawley, S. L., Covey, K. R., & Knapp, G. R. 2002, AJ, 123, 3409
Krisciunas, K., Margon, B., & Szkody, P. 1998, PASP, 110, 1342
Letarte, B., Demers, S., Battinelli, P., & Kunkel, W. E. 2002, AJ, 123, 832
Marigo, P., Girardi, L., & Bressan, A., 1999, A&A, 344, 123
Margon, B., Anderson, S. F., Harris, H. C. et al. 2002, AJ, 124, 1651
Nowotny, W., Kerschbaum, F., Olofsson, H., & Schwarz, H. E., 2003, A&A, 403, 93
Rowe, J. F., Richer, H. B., Brewer, J. P., & Crabtree, D. R. 2004, AJ, in press
Schlegel, D., Finkbeiner, D., & Davis, M. 1998, ApJ, 500, 525
Smith, J. A., Tucker, D. L, Kent, S., et al. 2002, AJ, 123, 2121
Walterbos, R. A. M., & Kennicutt, R. C. 1988, A&A, 198, 61
Windrow, L. M., Perrett, K. M., & Suyu, S. H. 2003, ApJ, 588, 311
[^1]: Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii.
|
---
abstract: 'Based on a suite of Monte Carlo simulations, I show that a stellar-mass-dependent lifetime of the gas disks from which planets form can explain the lack of hot Jupiters/close-in giant planets around high-mass stars and other key features of the observed semimajor axis distribution of radial velocity-detected giant planets. Using reasonable parameters for the Type II migration rate, regions of planet formation, and timescales for gas giant core formation, I construct synthetic distributions of Jovian planets. A planet formation/migration model assuming a stellar mass-dependent gas disk lifetime reproduces key features in the observed distribution by preferentially stranding planets around high-mass stars at large semimajor axes.'
author:
- Thayne Currie
title: 'On the Semimajor Axis Distribution of Extrasolar Gas Giant Planets: Why Hot Jupiters Are Rare Around High-Mass Stars'
---
Introduction
============
The semimajor axis distribution of Jovian-mass planets discovered in radial velocity surveys reveals striking trends (Figure \[fg1\], top panels). Many Jovian planets around solar-mass stars have semimajor axes a$_{p}$ $\gtrsim$ 0.5 AU. Many also have a$_{p}$ $\lesssim$ 0.1–0.2 AU (’hot Jupiters’), and few have intermediate values (’the period valley’, @Cumming08). While hot Jupiters comprise $\approx$ 20% of planets around $<$ 1.5 M$_{\odot}$ stars, surveys have yet to detect hot Jupiters orbiting $>$ 1.5 M$_{\odot}$ stars. All radial velocity-detected planets around $>$ 1.5 M$_{\odot}$ stars have semimajor axes $\gtrsim$ 0.5 AU [@Jj07a; @Jj07b; @Jj08; @Sa08; @Wright09]. Planetary migration may explain aspects of the semimajor axis distribution of Jovian planets, including the origin of hot Jupiters (which cannot form in situ ) and the period valley for solar-mass stars [e.g. @Il04; @Bi07]. However, the cause for the lack of hot Jupiters and other close-in giant planets around high-mass stars is less clear. One possibility is that hot Jupiters surrounding high-mass stars are engulfed as the stars evolved off the main sequence [@Sa08]. However, most high-mass stars with planets (subgiants) are physically too small to engulf hot Jupiters [@Jj07b].
In this paper, I show that models of planet formation/migration with a stellar-mass-dependent lifetime of the gaseous circumstellar disks from which planets form can explain the dearth of hot Jupiters around high-mass stars as suggested by radial velocity surveys. If gas disks disappear much faster around high-mass stars than around solar and subsolar-mass stars, then inward migration is halted and the planets are stranded at large semimajor axes. The arguments described here build on work by @Bi07 who studied how a stellar-mass-dependent gas disk lifetime may explain why the period valley is more pronounced for planets orbiting F stars than G and K stars. In Section 2, I make simple analytical arguments to show that the observed semimajor axis distribution of giant planets may emerge from a stellar-mass-dependent disk lifetime. In Section 3, I perform numerical modeling similar to recent work [e.g. @Il04; @Bi07] to test my hypothesis. I construct synthetic distributions of giant planets using a suite of Monte Carlo simulations with a range of stellar mass-dependent disk lifetimes and as well as a mass-independent lifetime. A mass-independent lifetime poorly reproduces the observed semimajor axis distribution of planets, while a mass-dependent lifetime reproduces observed trends.
Analytical Motivation
=====================
Planets form in disks around young stars. Once planets grow to Jovian masses, they can open a gap in the disk and undergo ’Type II migration’ [@LinPap85; @Ward97] with a migration rate regulated by the local viscous diffusion time. For regions interior to $\approx$ 20 AU, migration is inward. The migration rate can be parameterized assuming a Minimum Mass Solar Nebula (MMSN) model as prescribed in @Il04 scaled to the star’s mass where the initial gas column density is $\Sigma_{g, 1 M_{\odot}, MMSN}$ = 2400 g cm$^{-2}$ at 1 AU: $$\frac{da}{dt}\sim1.3\times10^{-5} AU/yr\times(\frac{a_{p}}{1 AU})^{0.5}(\frac{M_{J}}{M_{p}})(\frac{\alpha}{10^{-3}})(\frac{M_{\star}}{M_{\odot}})^{1.5}e^{-4t/\tau_{g}}.$$ In this equation, M$_{p}$/M$_{J}$ is the planet mass in Jovian masses, $\alpha$ is the viscosity parameter, M$_{\star}$ is the stellar mass, t is time, and $\tau_{g}$ is the timescale for the local disk surface density to drop to $\lesssim$ 1–2% of its original value[^1].
The maximum Type II migration rate is the radial velocity of the gas, da$_{p}$/dt $\sim$ -1.5$\alpha(\frac{H}{a_{p}})^{2}a_{p}\Omega$, where H is the disk scale height at a$_{p}$ and $\Omega$ is the Keplerian frequency. For a MMSN surface density profile this rate is $$\frac{da_{p}}{dt}\sim 2.3\times10^{-5}AU yr^{-1}(\frac{M_{\star}}{M_{\odot}})^{0.5}(\frac{\alpha}{10^{-3}}).$$ This equation governs the migration rate for disk masses that are much larger than the planet’s mass. The nominal migration timescale from Equation 1, a$_{p}$/(da$_{p}$/dt), is then $\tau_{m,II}$ $\approx$ 0.16 Myr for a Jovian mass planet at 1 AU with $\alpha$ = 5$\times$10$^{-4}$ if $\tau$$_{m,II}$ $<<$ $\tau_{g}$. The timescale from the maximum drift rate (Equation 2) is about half this value. If the local surface density in a gaseous disk is drained to $\lesssim$ 1% of its initial value (t $\approx$ $\tau_{g}$), Equation 1 implies a migration rate of $\lesssim$ 1 AU/20Myr. Because nearly all gaseous circumstellar disks disappear by $\approx$ 10 Myr [@Cu07a; @Cu07c], migration is essentially halted. Thus, migration is halted if $\tau_{g}$ $\lesssim$ $\tau_{m,II}$.
Simple arguments show that the gas disk lifetime may depend on stellar mass ($\tau_{g}$ $\propto$ M$_{\star}$$^{-\beta}$) and that the ratio of $\tau_{g}$ to $\tau_{m,II}$ may *strongly* depend on stellar mass [see also @Bi07]. Optical spectroscopic studies of 1–10 Myr-old star-forming regions reveals a strong trend between accretion rate and stellar mass, dM$_{\star}$/dt $\propto$ M$_{\star}$$^{2}$, for 0.03–3 M$_{\odot}$-mass stars [@Calvet04; @Muzerolle05]. The timescale for a star to accrete some fraction of its mass, x, is then $\tau_{g}$ $\approx$ 3Myr$\times$$(\frac{x}{0.1})(\frac{M_{\star}}{M_{\odot}})^{-1}$ or $\tau_{g}$ $\propto$ M$_{\star}$$^{-1}$. Thus, gas disks may dissipate faster around high-mass stars than around low-mass stars.
A shorter gas dissipation timescale eventually leads to a slower migration rate. According to Equation 1, at t/$\tau_{g, 1M_{\odot}}$=0.2, a planet orbiting a 3 M$_{\odot}$ star migrates inward at a slower rate than one orbiting a 1 M$_{\odot}$ star if $\beta$ $\approx$ 1. At slightly later times (e.g., t/$\tau_{g, 1 M_{\odot}}$ = 0.3), the migration rate for a planet orbiting 3 M$_{\odot}$ star is half that for one orbiting a 1 M$_{\odot}$ star. Therefore, migration around higher-mass stars decelerates earlier; $\tau_{g}$/$\tau_{m,II}$ will be smaller for higher-mass stars. Because smaller timescale ratios will strand more gas giants at larger semimajor axes (e.g. $\sim$ 1 AU), the relative frequency of hot Jupiters and other close-in planets should be lower for higher-mass stars.
Numerical Model
===============
To test whether a stellar-mass-dependent gas disk lifetime can explain the observed distribution of extrasolar gas giant planets, I produce synthetic populations of exoplanets in semimajor axis vs. stellar mass space using a suite of Monte Carlo simulations. My model generally follows the approach of @Bi07 who show simulations for a 1 M$_{\odot}$ star and vary the range in disk lifetimes (1–10 Myr, 3–30 Myr, 10-100 Myr) to show how a stellar mass-dependent disk lifetime regulates the distribution of exoplanets. Here, I perform simulations for a range of stellar masses and include an explicit power law dependence for the gas disk lifetime.
Stellar masses for the synthetic population are randomized between 0.3 and 3 M$_{\odot}$ with a probability distribution weighted towards solar/subsolar-mass stars (P(M$_{\star}$) $\propto$ M$_{\star}$$^{-2.5}$: a Salpeter-like IMF). To model the regions of planet formation, the planets’ initial semimajor axes are chosen between 1 and 25 AU. For my fiducial model, I assume a gas disk lifetime that scales inversely with stellar mass ($\tau_{g}$ = 4 Myr$\times$(M$_{\star}$/M$_{\odot}$)$^{-1}$).
For Jovian-mass planets to form, I require that they reach an isolation core mass of M$_{iso}$=5 M$_{\oplus}$ [e.g. @Alibert2005]. I assume a 2.5$\times$ scaled MMSN model from @Il04 to account for the median disk mass needed to form cores [@Kb08] and set the metallicity comparable to the median metallicity of stars with detected Jovian-mass planets (\[Fe/H\] $\sim$ 0.15, @Wright09). I set the ice line location equal to values from Figure 1 of @Kk08 at 0.3 Myr interpolating between values for stars of different masses[^2]. Finally, I require that enough gas is left when the core mass is 5 M$_{\oplus}$ to form a Jovian-mass planet. This condition is equivalent to $$M_{g, iso} = 1.34 M_{J}\times(\frac{a_{p}}{1AU})^{0.75}(\frac{\Sigma_{1AU, 1M_{\odot}}}{2400 g cm^{-2}})^{1.5}(\frac{M_{\star}}{M_{\odot}}),$$ where the gas feeding zone size is 10 Hill radii. The Jovian planet-forming regions are $>$ 2 AU for 0.5 M$_{\odot}$ stars, $>$ 3 AU for 1 M$_{\odot}$ stars, and $>$ 5 AU for 2.5 M$_{\odot}$ stars.
From their birthplaces, I track the semimajor axis evolution of planets from Type II migration according to Equations 1 and 2, assuming a viscosity parameter of $\alpha$=5$\times$10$^{-4}$. Planets reaching within $\approx$ 2–5 R$_{\star}$ ($\sim$ 2–5 R$_{\odot}$(M$_{\star}$/M$_{\odot}$)$^{0.75}$ may be affected by magnetospheric disk truncation [@Lin96], which is not treated in this simple model. Any planet that reaches this small separation is given a final semimajor axis that is randomized between 2 and 5 R$_{\star}$. I make two simplifying assumptions in the fiducial model that will be removed later. First, I assume that all planets are the mass of Jupiter. Second, I assume that all planets form at 1 Myr: a characteristic time for the formation of 5 M$_{\oplus}$ cores at 5 AU around solar-mass stars [@Kb08].
Table 1 lists simulation results for a total of 20,000 planets (20 simulations of 1000 planets; N$_{total}$). The table shows the total number of planets with final semimajor axes $\le$ 3 AU and in three semimajor axis bins ($<$ 0.2 AU, 0.2–0.5 AU, and 0.5–3 AU) for each of the stellar mass bins (0.3–0.5 M$_{\odot}$, 0.8–1.5 M$_{\odot}$, and 1.5–3 M$_{\odot}$). The lower left panel of Figure \[fg1\] shows the final semimajor axis versus stellar mass distribution of these planets (black circles). I overplot the distribution of radial velocity-detected planets with well-constrained stellar masses (258; red stars[^3]). The lower right panel shows a histogram plot of the semimajor axes for the synthetic population in three mass bins: 0.3–0.5 M$_{\odot}$ (dashed line), 0.8–1.5 M$_{\odot}$ (dotted line) and 1.5–3 M$_{\odot}$ (solid line).
Despite its simplicity, the model yields good agreement with observed semimajor axis distributions. In the synthetic population, stars with masses between 0.3 M$_{\odot}$ and $\sim$ 1 M$_{\odot}$ have many hot Jupiters. The right panel of Figure \[fg1\] shows that few 0.8–1.5 M$_{\odot}$ stars have planets with a$_{p}$ $\sim$ 0.1 and 0.5 AU, consistent with the ’period valley’ in the observed population. Most strikingly, the synthetic population shows a sharp drop in the number of hot Jupiters from $\sim$ 1.3 M$_{\odot}$ to 1.5 M$_{\odot}$. For my model assumptions, the synthetic population lacks any hot Jupiters for stars with M$_{\star}$ $\gtrsim$ 1.6 M$_{\odot}$ and lacks any planets with intermediate (0.2–0.5 AU) distances for M$_{\star}$ $\gtrsim$ 1.8 M$_{\odot}$. This trend is consistent with the clear lack of detected hot Jupiters around 1.5–3 M$_{\odot}$ stars. The $\chi^{2}$ values comparing the relative frequencies of hot Jupiters, planets at 0.2–0.5 AU, and those at 0.5–3 AU with the observed distribution confirm that agreement is good, especially for $\gtrsim$ 0.8 M$_{\odot}$ stars.
The semimajor axis distribution shows far poorer agreement if the gas disk lifetime is independent of or weakly dependent on stellar mass (Figure \[fg2\]). Assuming $\tau_{g}$=4 Myr for all masses turns all planets around $>$ 1.5 M$_{\odot}$ stars into hot Jupiters, weakens the period valley for solar-mass stars, and confines all planets around low-mass stars to semimajor axes with a$_{p}$ $>$ 10 AU (top-left panel). These properties are inconsistent with the observed distribution. The model with $\tau_{g}$ = 2 Myr (top-right panel) for all stars yields the correct distribution for high-mass stars but eliminates all hot Jupiters around solar/subsolar-mass stars, and strands all planets around subsolar-mass stars at a$_{p}$ $>$ 10 AU. Models with $\tau_{g}$ $\propto$ M$_{\star}$$^{-0.5}$ fare marginally better (bottom panels). The model with $\tau_{g, 1 M_{\odot}}$ = 4 Myr predicts a pileup of hot Jupiters, many planets at $\sim$ 3–10 AU, and a dearth of planets at $\sim$ 0.2–0.5 AU for all stars with masses $>$ 0.8 M$_{\odot}$. It also confines planets around $<$ 0.5 M$_{\odot}$ stars to $>$ 0.3 AU with a peak at $\sim$ 10 AU. The model with $\tau_{g, 1 M_{\odot}}$ = 2 Myr confines all planets to a$_{p}$ $>$ 3 AU. These features are clearly not present in the observed population. The $\chi^{2}$ values for all of these simulation runs exceed 100 for low-mass stars and (sometimes) high-mass stars.
Motivated by the success of the fiducial model with $\tau_{g}$ $\propto$ M$_{\star}$$^{-1}$, I remove assumptions regarding the planet’s mass and core formation timescale and run a second set of simulations. First, I set the planet’s mass equal to the minimum gap-opening mass for Type II migration, requiring that the planet’s Hill radius is larger than the local disk scale height [e.g. @LinPap85]: $$M_{II} \sim 0.4 M_{J}\times(\frac{M_{\star}}{M_{\odot}})(\frac{a_{p}}{1 AU})^{0.75}.$$ Second, I estimate the formation timescale for each core explicitly, require that the core can form before gas is dissipated, and require that enough gas is left to form a migrating planet after core formation. From the @Kb08 results, I extrapolate the formation timescale of 1 Myr at 5 AU around a solar-mass star with $\Sigma_{d}$ = 2.5 g cm$^{-2}$ to different distances, stellar masses, and dust column densities. I assume that $\tau_{core}$ $\propto$ ($\Sigma_{d}$$\Omega$)$^{-1}$, which is a reasonable approximation of the numerical results [@Kb08b].
Results from the second set of simulations (the bottom half of Table 1, Figure \[fg3\]) show good agreement between the synthetic distribution and the observed distribution if $\tau_{g}$ $\propto$ M$_{\star}$$^{-\beta}$, where $\beta$ = 0.75-1.5. The semimajor axis versus stellar mass distribution (shown for $\tau_{g}$=2 Myr$\times$ M$_{\star}$$^{-1}$, top-left panel) features the same sharp drop in the frequency of hot Jupiters and planets with a$_{p}$ = 0.2–0.5 AU (log(a$_{p}$)= -0.7– -0.3) for M$_{\star}$ $\ge$ 1.5 M$_{\odot}$ exhibited by the fiducial model. The histogram plots for $\beta$ = 1 (top right panel) and $\beta$ = 0.75 (second row, left panel) confirm that there is a lack of hot Jupiters and other planets with a$_{p}$ $<$ 0.5 AU for high-mass stars. The models successfully reproduce the two other features of the observed distribution: the period valley for solar-mass stars and the lack of long-period planets around low-mass stars. The agreement is confirmed quantitatively as models with $\beta$ = 0.75–1.5 typically have $\chi^{2}$ values less than $\sim$ 20 for all stellar mass bins.
I also calculate the percentage of planets that achieve a gap-opening mass and undergo Type II migration (N$_{gap}$) for each of the stellar mass bins (100$\times$N$_{gap}$/N$_{total}$). For $\beta$ $<$ 1.5, the frequency is lowest for M stars. This percentage probes the relative sizes of planet-forming regions for a given scaled disk mass (2.5 $\times$ a scaled MMSN) and is different from but related to the frequency of planets [@Jj07b]. For calculations with lower scaled disk masses (e.g., 1.25$\times$ MMSN scaled) the planet-forming regions shrink, and only solar-to-high mass stars form migrating planets. For calculations with an even lower scaled disk mass, only high-mass stars form migrating planets. The percentage of migrating gas giants for M stars is lowest because their disks have lower masses: M$_{disk}$ $\propto$ f$_{g}$ $\propto$ M$_{\star}$ [@Kk08].
Models with $\tau_{o}$ = 2 Myr and $\beta$ $\le$ 0.5 (third row) begin to show disagreement with the observed distribution as they predict a pileup of hot Jupiters and period valley for high-mass stars and a lack of hot Jupiters for low-mass stars. The $\chi^{2}$ values for low-mass stars and high-mass stars for $\beta$ $\le$ 0.5 are significantly higher than for $\beta$ = 0.75–1.5. Therefore, according to my simulations, the observed semimajor axis distribution of gas giant planets results from a stellar-mass-dependent gas disk lifetime.
Discussion
==========
Through a series of Monte Carlo simulations, I have shown that a stellar-mass-dependent gas disk lifetime can explain the observed semimajor axis distribution of extrasolar gas giant planets, including the lack of hot Jupiters and other planets with a$_{p}$ $<$ 0.5 AU around high-mass stars. Synthetic distributions of planets produced assuming that the gas disk lifetime, $\tau_{g}$, scales as M$_{\star}$$^{-\beta}$ with $\beta$ = 0.75–1.5 reproduce key features in the observed semimajor axis distribution. Distributions from models lacking a stellar-mass-dependent disk lifetime quantitatively provide a poorer match to observations.
This work extends and complements the investigation of @Bi07 who use Monte Carlo simulations and semi-analytical prescriptions for planet growth to explain exoplanet trends for solar-mass stars. While high-mass stars lack a pronounced period valley, predicted by @Bi07, the dearth of planets at 0.2–0.5 AU agrees with their predictions. Moreover, this work shows that a stellar mass-dependent disk lifetime, invoked by @Bi07 to explain exoplanet trends for solar-mass stars, may explain trends for planets around stars with a wide range of masses. Future modeling work is necessary to test this hypothesis more conclusively. Future modifications include incorporating a more sophisticated treatment of circumstellar gas accretion, modeling Type I migration, determining the sensitivity of the planets’ synthetic distributions to the migration rate (i.e., value of $\alpha$), and tracking the migration of planets *while* they are accreting gas [e.g. @Alibert2005].
Recent studies of young stars in clusters support a stellar mass-dependent gas disk lifetime [@Kk09]. Based on optical spectroscopy, the frequency of gas accretion in 2–15 Myr-old clusters is significantly higher for stars with M$_{\star}$ $<$ 1 M$_{\odot}$ than for higher-mass stars [e.g., IC 348, Tr37, and h and $\chi$ Persei @Dahm08; @Ck08; @Si06; @Cu07c]. Secondary characteristics of gas-rich disks (optically thick thermal infrared emission) are also rarer for high-mass stars [e.g., @Ca06; @Cl09]. Combining cluster data to empirically constrain $\tau_{g}$(M$_{\star}$) may be possible. However, uncertainties in stellar ages for stars in the youngest ($<$ 3 Myr) clusters present a strong challenge to constructing an empirically based gas disk lifetime. I will address these issues in a future paper.
If the gas disk lifetime strongly depends on stellar mass, my model simulations suggest that future radial velocity surveys will find few gas giant planets orbiting at small separations from high-mass stars. If $\approx$ 10% of high-mass stars have gas giant planets with a$_{p}$ $<$ 3 AU [@Jj07b], the simulations with $\beta$ = 0.75–1.5 imply that out of 1,000 high-mass stars targeted for radial velocity surveys, fewer than $\approx$ 15 will have planets at a$_{p}$ $\le$ 0.5 AU while more than $\approx$ 75 will have planets at a$_{p}$ $\ge$ 0.5 AU. Ongoing surveys will provide a larger sample from which to compare observed and predicted frequencies of hot Jupiters and other gas giants at small separations (J. Johnson, in preparation). I thank John A. Johnson and Jason Wright for discussions on exoplanet data and for encouraging the writing of this paper; Geoff Marcy, Scott Kenyon, Dan Fabrycky, Ruth Murray-Clay, Brad Hansen, and Jonathan Irwin for valuable comments; and the referee for a rapid and insightful report. This work is supported by NASA Astrophysics Theory Grant NAG5-13278 and NASA Grant NNG06GH25G.
Alibert, Y., et al., 2005, , 626, 57L Burkert, A., Ida, S., 2007, , 660, 845 Calvet, N., et al., 2004, , 128, 1294 Carpenter, J., et al., 2006, , 651, 49L Cumming, A., et al., 2008, , 120, 531 Currie, T., Kenyon, S. J., 2009, , submitted (arXiv:0801.1116) Currie, T., Lada, C. J., Plavchan, P., Irwin, J., Kenyon, S. J., 2009, , in press (arXiv:0903.2666) Currie, T., et al., 2007(a),, 659, 599 Currie, T., et al., 2007(b), , 669, 33L Dahm, S., 2008, , 136, 521 Ida, S., Lin, D. N. C., , 2004, 604, 338 Johnson, J. A., et al., 2007a, , 665, 785 Johnson, J. A., et al., 2007b, , 670, 833 Johnson, J. A., et al., 2008, , 675, 784 Kennedy, G., Kenyon, S. J., 2008, , 673, 502 Kennedy, G., Kenyon, S. J., 2009, , in press (arXiv:0901.2608) Kenyon, S. J., Bromley, B., 2008, , 179, 451 Kenyon, S. J., Bromley, B., 2009, , 690, 140L Lin, D. N. C., Bodenheimer, P., Richardson, D., 1996, Nature, 380, 606 Lin, D. N. C., Papaloizou, J., 1985, in Protostars and Planets II, ed. D. C. Black and M. S. Matthews (Tucson, AZ: Univ. of Arizona Press), 981 Muzerolle, J., et al., 2005, , 625, 906 Sato, B., et al., 2008, , 60, 539 Sicilia-Aguilar, A., et al., 2006, , 132, 2135 Ward, W., 1997, Icarus, 126, 261 Wright, J. T., Upadhyay, S., Marcy, G. W., Fischer, D. A., Ford, E. B., Johnson, J. A., 2009, , in press (arXiv:0812.1582)
[llllllllllllllllllllll]{}
Obs.&-&-&11&176&20&0.73& 0.27& 0.0& 0.18& 0.13& 0.0 &0.09& 0.60& 1.00 & –&–&– &1.8$^{a}$& 4.2$^{a}$& 8.9$^{a}$\
M1&4&1&5524&2198&925&0.50&0.42&0.05&0.09&0.10&0.05&0.41&0.49&0.89&28.8&6.3&3.0\
“ &4&0.5&1957&2469&3499& 0.10&0.48& 0.52& 0.11& 0.09& 0.09& 0.79& 0.43& 0.39&157.0&13.1&114.5\
” &4&0&602&2787&4782& 0&0.52&0.94& 0& 0.09& 0.03& 0.99&0.39&0.04&245.2&19.0&317.9\
“ &2&0.5& 329& 255&20& 0&0&0&0&0&0&1.00&1.00&1.00&245.2&44.0&0\
” &2&0& 92& 365&1290&0&0&0.14&0&0&0.09&1.00&1.00&0.77&245.2&44.0&14.2\
M2&2&1&948& 893&1146 & 0.64& 0.46& 0.06& 0.05& 0.10& 0.10& 0.32& 0.44& 0.84&13.7&11.0&6.9&13.4&20.8&20.9\
“&2&1.5&1337& 629& 234 & 0.76& 0.48& 0.0& 0.04& 0.10& 0.03 & 0.2& 0.41& 0.97 &5.7&14.3&0.3&19.6&21.4&14.5\
”&2&0.75& 783& 911&1222&0.53& 0.52& 0.15& 0.07& 0.10& 0.10& 0.40& 0.38& 0.75&25.1&19.7&16.7&11.1&20.7&22.4\
“&2&0.5& 546& 584& 780& 0.40& 0.55& 0.24& 0.08& 0.09& 0.13& 0.52& 0.36& 0.63&53.5&24.2&37.2&9.2&20.4&26.1\
”&2&0.1& 319& 671& 975 & 0.31& 0.53& 0.51& 0.09& 0.08& 0.12 & 0.60& 0.40& 0.36&78.3&19.4&120.4&5.8&22.0&30.3\
“&3&1&1173& 810& 751& 0.74& 0.65& 0.26&0.05& 0.08& 0.12&0.21& 0.27& 0.62&5.5&45.0&39.9&18.1&26.6&25.8\
”&4&1&1499&961&998 & 0.78& 0.78& 0.46& 0.04& 0.04& 0.10&0.18& 0.18& 0.45&5.3&78.3&92.2&21.8&29.2&30.6 \[table\]
[^1]: I set the parameter f$_{g}$ in @Il04 equal to M$_{\star}$/M$_{\odot}$. Equation 65 in @Il04 is missing a factor of $\sqrt{M_{\odot}/M_{\star}}$.
[^2]: @Il04 and @Bi07 assume an optically-thin disk in determining the ice line position. Like @Kk08, I assume that the disk is not optically thin when planetesimals grow into gas giant cores.
[^3]: Downloaded from http://exoplanet.eu.
|
---
address: |
Laboratoire Astroparticule et Cosmologie,\
10 rue Alice Domon et Léonie Duquet,\
75205 Paris, France\
[E-mail: jdawson@in2p3.fr]{}
author:
- 'J. V. DAWSON on behalf of the Double Chooz collaboration'
title: STATUS OF THE DOUBLE CHOOZ EXPERIMENT
---
Introduction {#intro}
============
Neutrino oscillation has been clearly established via the study of solar, astmospheric, reactor and beam neutrinos. Combination of these results requires the existence of (at least) three-neutrino mixing. In the current view, the PMNS mixing matrix relates the three neutrino mass eigenstates to the three neutrino flavour eigenstates parameterized by three mixing angles ($\theta_{12}$, $\theta_{13}$ and $\theta_{23}$) and one CP violating phase $\delta_{cp}$ (for Dirac neutrinos). Great progress has been made in measuring the mixing angles and the two squared mass differences $\Delta m^2_{ij} = m^2_i - m^2_j$ (for a good review of neutrino oscillation experiments see for example [@Dore2008]). However, the mixing angle $\theta_{13}$, the mass hierarchy and the $\delta_{cp}$ phase are still currently unknown. Indeed only upper limits to the value of $\theta_{13}$ have been found, indicating that this angle is very small with respect to the other two. Whilst a measurement of $\theta_{13}$ would complete the knowledge of the mixing angles, even a more stringent upper limit would be useful since the size of $\theta_{13}$ has a great bearing on the possibility to observe CP violation in the leptonic sector with upcoming neutrino experiments (see for example [@Schwetz2007] for a discussion of $\theta_{13}$ and CP violation discovery in forthcoming experiments).
A three-flavour global analysis on existing data gives an upper bound of sin$^2\theta_{13} < 0.035$ at 90% C.L [@Schwetz2008]. This value is dominated by the bound given by the reactor experiment, CHOOZ [@apollonio2003], in which no oscillation was observed $R = 1.01
\pm 2.8\%(stat) \pm 2.7\%(sys)$.
Reactor experiments search for the disappearance of electron anti-neutrinos emitted from the cores of the nuclear reactors. Equation \[eqn:survival\] gives the survival probability of a $\bar{\nu_e}$ from a reactor, where $E$ is the neutrino energy and $L$ is the distance from the source to the detector.
$$\begin{split}
P(\bar{\nu_e}\rightarrow \bar{\nu_e}) = 1 - sin^2(2\theta_{13}) sin^2\frac{\Delta m_{31}^2 L}{4E} - cos^4 \theta_{13} sin^2(2\theta_{12})sin^2\frac{\Delta m_{21}^2 L}{4E}\\ + 2sin^2\theta_{13} cos^2\theta_{13} sin^2\theta_{12}\left(cos \frac{(\Delta m_{31}^2 - \Delta m_{21}^2)L}{2E} - cos \frac{\Delta m_{31}^2 L}{2E} \right)
\end{split}
\label{eqn:survival}$$
For short baselines only the first two terms are relevant. With a well positioned detector (such that L/E is $\sim$0.3 km/MeV), a detector might observe less neutrinos than anticipated indicating a non-zero value of $\theta_{13}$ and therefore these experiments are termed ’disappearance’ experiments. ’Appearance’ experiments i.e. long baseline accelerator experiments aim to measure the appearance of $\nu_e$s in a $\nu_\mu$ beam.
Reactor based $\theta_{13}$ experiments have some advantages over long baseline accelerator experiments. They suffer less from parameter degeneracies, being independent of $\delta_{cp}$ and the sign of $\Delta m_{31}$ and having only a weak dependence of $\Delta m^2_{21}$. Since the neutrino energies are low, $\sim$1 to 10 MeV, and the detectors are positioned at short distances, there are no matter effects. The major drawback to this type of experiment is that there is limited knowledge on the neutrino production processes inside the reactors.
Double Chooz
============
The Double Chooz experiment [@LOI2006] is located at Chooz, the same site as the original Chooz experiment, in the Champagne-Ardennes region in France. The site contains two closely neighbouring nuclear reactors each with a thermal power of 4.27 GW. The Double Chooz concept is to use two identical detectors; one near, to effectively measure the neutrino spectrum and flux from the reactor, and one far, to observe any neutrino disappearance.
The far detector is located in the same underground laboratory as the original Chooz experiment (1 km from the two cores). This site is perfect for three reasons; an ideal L/E of 0.3 MeV/km, the cost is significantly reduced due to the existing laboratory, and the experimental background rate i.e. from muons, neutrons and rock radioactivity etc are already well measured with reactor-off data. The near detector underground laboratory will be 400m from the two reactors and must be constructed.
The target is a Gadolinium loaded scintillator, with an interacting anti-neutrino of energy greater than 1.8 MeV causing an inverse-beta decay of a proton. $$\bar\nu_e + p \rightarrow n + e^+$$ The positron slows depositing its kinetic energy in the scintillator. It quickly annihilates; releasing two 511 keV gammas. The total prompt visible energy seen is some 1 to $\sim$8 MeV and is directly related to the energy of the neutrino $E_\nu = E_{visible} + 0.8$MeV. After a characteristic delay, the neutron slows and is captured; on Gadolinium (absorption time of 30 $\mu$s) or on Hydrogen. Gamma cascades from the captures give energy deposits of $\sim$8 MeV (from Gadolinium) and 2.2 MeV from Hydrogen.
As the interaction cross-section rises (with the square of the energy) and the reactor neutrino spectrum falls in a similar fashion, the convolution of these two, the observed spectrum is roughly Gaussian in shape with a peak visible energy of $\sim$4 MeV.
Detector Design
===============
Figure \[fig:detector\] shows the detector and laboratory design. Both detectors are identical from the buffer tank (inner-most stainless steel vessel) inwards which is a physics requirement. Shielding against the radioactivity of the rock is provided by 15 cm of demagnetised steel for the far detector but less stringent shielding is required for the near detector.
Each detector is formed from a series of nested cylinders with each volume filled with different liquids; insensitive buffer oil for shielding, Gd-doped scintillator as the target and undoped scintillators for gamma rays, fast neutrons and muons.
The two inner vessels are acrylic and transparent to photons above 400 nm. The inner-most vessel is the Target, with a diameter of 2.3 m, which contains 10 m$^{3}$ of Gadolinium-doped scintillator; such that the scintillator contains 1 g/l of Gadolinium. In this volume neutron-capture on Gadolinium can occur releasing cascade gammas with an energy of $\sim$8 MeV. More than 80% of neutron captures are on Gadolinium rather than Hydrogen. The definition of a neutrino candidate event is one in which neutron capture on Gadolinium occurs.
Enclosing the target is the Gamma-Catcher volume, with a diameter of 3.4m, which contains 22 m$^{3}$ of undoped scintillator. The purpose of this volume is to detect the gammas emitted in both the neutron-capture process and positron annihilation in the target, such that gammas emitted from neutrino events occurring in the outer volume of the target are detected. This results in a well-defined target volume.
Since the photomultipliers are the most radioactive component of the detector, the inner volumes are shielded by a buffer volume, with a diameter of 5.5m, filled with non-scintillating paraffin oil. Events occurring in the acrylic volumes are detected by 390 10 inch low background photomultiplier tubes (Hamamatsu R7081 [@hamamatsu]) fixed to the inside of the steel buffer tank. Uniquely the photomultiplier tubes are angled to improve the uniformity of light collection efficiency in the inner-most volumes. We anticipate to achieve 7% energy resolution at 1 MeV.
The outer detector volume is steel walled, with a diameter of 6.6m, and filled with scintillator. 78 8 inch photomultipliers (Hamamatsu R1408[@hamamatsu]) line the outermost wall which is painted with a reflective white coating. This volume is the Inner Veto with the purpose of detecting and tracking muons and fast neutrons.
On top of the detector sits the Outer Veto. This comprises strips of plastic scintillator and wavelength-shifting fibres. The veto extends further than the detector diameter with the purpose of detecting and tracking muons. The precision of the entry point of a muon, X-Y position, will be far more precise than that achieved by the Inner Veto and detector. One of the main objectives is to tag near-miss muons, which interact in the surrounding rock (and not in the detector) but produce fast neutrons. Another important goal is to determine whether a muon entered the inner detector. Muons that do so can produce cosmogenic isotopes (i.e. via a photonuclear interaction on $^{12}$C), some of which will produce backgrounds for the experiment.
Backgrounds
===========
As each neutrino produces two time-correlated signals; that of the positron and a delayed capture of a neutron (with characteristic decay time of 30 $\mu$s), backgrounds can come from two sources; accidental and time-correlated.
The accidental component comes from the random chance that two events of appropriate energy interact within this characteristic time. Since these two events are unrelated this rate can easily be measured, based on the singles rate. The main source of events come from radioactive contamination with the dominant source being the photomultiplier tubes. For the accidental component to be well constrained, strict radioactive contamination limits have been placed on all parts.
The most difficult backgrounds to study are those that are, like our signal, time-correlated. From the experience of Chooz it is anticipated that Double Chooz will observe some $\sim$ 1.5 events/day of false neutrino-like events. The Chooz experiment had a period of data-taking before operation of the nuclear reactors began and so the background could be very thoroughly investigated. The sources of the neutrino-like events observed were attributed to fast neutrons (muon-induced neutrons) and cosmogenically produced isotopes (also muon produced).
Fast neutrons can mimic neutrino signals by producing a proton-recoil (positron-like signal) and a delayed neutron capture. If the muon is seen by the experiment then these events can be tagged. More dangerous, however, are near-miss muons which interact in the rock releasing fast neutrons which interact in the detector. The primary purpose of the Outer Veto is to identify these events by covering an area wider than the detector itself.
Those cosmogenically produced isotopes that are dangerous for the experiment are those that result in electron emission followed by neutron emission, as these mimic well our neutrino signal. Two isotopes, $^{8}$He and $^{9}$Li, have long decay times (119 ms and 174 ms respectively)[@Hagner2000] rendering a hardware veto impractical. Coupling information from the Outer Veto (with precise muon entry points), Inner Veto and inner detector will allow reconstruction of muon tracks to identify muons that cross the inner detector.
Improvements on Chooz {#improvements}
=====================
Improvements on the original Chooz experiment have been made in two ways; the detector design and the two-detector concept. The new detector target is more than twice as large as the original Chooz detector. The scintillator technology has improved, and Gadolinium-loaded scintillator now is very stable (on the timescale of years) allowing a longer run time $\sim$5 years. The number of neutrinos detected in the far detector assuming 3 years of running will be $\sim$60,000 compared to 2,700 in the Chooz experiment, reducing the statistical error, 2.8% in Chooz, to 0.4%.
The aim is to reduce the systematic error, 2.7% in Chooz, to less than 0.6%. There are three sources of systematic error; the reactor, the detector and the analysis. With two detectors, each reactor component systematic; flux and cross-section, reactor power and energy per fission, reduce to below 0.1%. Making a relative measurement, between the two detectors, reduces many of detector systematics to similar orders.
The scintillators will be produced for both detectors in one batch, reducing the systematic on the number of H and Gd atoms in each detector. With a well performing scintillator the number of observed photons should be high enough such that all of the positron signal is observed so there is no systematic introduced by cutting on the positron spectrum. The improved detector design, with target and gamma-catcher vessels, provides a fixed fiducial volume such that positional cuts are not needed in the analysis eliminating another important source of systematic error.
In general, controlling the relative systematics between the two detectors is far easier than the absolute. Two detectors, however, introduces one new systematic - the live time, as both detectors must operate simultaneously.
Construction Progress {#progress}
=====================
The near detector site has been chosen, some $\sim$400 m from the two reactors, and the civil engineering study made. The excavation and construction of the new laboratory is foreseen to be completed by the end of 2010. The new laboratory will be slightly deeper than the site originally proposed giving a shielding of 115 m.w.e (metres water equivalent). At this new site we anticipate detecting $\sim$ 500 neutrinos per day.
Good progress has been made on the far detector and related infra-structure. The far detector shielding, buffer vessel and inner veto photomultipliers have been installed. The laboratory has been thoroughly cleaned, all surfaces painted and clean tents and a protective wall installed. During March 2009, the buffer vessel was simultaneously welded and lowered in to place. Scaffolding was erected inside this vessel so that the inner detector photomultipliers can be installed. After this is complete, the inner acrylic vessels can be installed. The liquid handling systems will be completed in parallel. Already the scintillator (and oil) tanks and associated filling systems have been installed in the liquid storage facility close to the entrance of the tunnel. An aggressive schedule is proposed such that the detector is anticipated to commence operation by the end of 2009.
Conclusion
==========
Double Chooz will be the first next generation reactor experiment to commence operation. The construction of the far detector of the Double Chooz experiment will be completed in 2009 with detector commissioning occurring at the end of the year. The first phase of data-taking will occur with the far detector only. Figure \[fig:sensitivity\] shows the improvement in sensitivity as a function of time; the far detector only phase and the two detector phase. Whilst the experiment is less sensitive without the near detector, it will still be more sensitive than the original Chooz detector and should reach a sensitivity to sin$^2(2\theta_{13})$ of 0.06 with one year of data. With two detectors, Double Chooz will be able to measure sin$^2(2\theta_{13})$ to 3$\sigma$ if sin$^2(2\theta_{13})>0.05$ or exclude sin$^2(2\theta_{13})$ down to 0.03 at 90% for $\Delta
m_{31}^2 = 2.5 \times 10^{-3} eV^2$ with three years of data with both near and far detectors.
[10]{} U. Dore and D. Orestano, [*Rep. Prog. Phys.*]{} [**71**]{} (2008) 106201. T. Schwetz, [*Phys.Lett.B*]{}, [**648**]{} (2007) 54-59. T. Schwetz et al, [*New J. Phys.*]{} [**10**]{} (2008) 113011 (arXiv:0808.2016). M. Apollonio et al., Chooz Collaboration, [*Eur. Phys. J. C*]{} [**27**]{} (2003) 331.
F. Ardellier et al, Double Chooz Collaboration, Double Chooz: A search for the neutrino mixing angle $\theta_{13}$ (2006) preprint arXiv:hep-ex/0606025v4.
Hamamatsu Corporation, http://sales.hamamatsu.com/. T. Hagner et al,[*Astroparticle Physics*]{} [ **14**]{} (2000) 33-47.
|
---
abstract: 'It is well-known to be impossible to trisect an arbitrary angle and duplicate an arbitrary cube by a ruler and a compass. On the other hand, it is known from the ancient times that these constructions can be performed when it is allowed to use several conic curves. In this paper, we prove that any point constructible from conics can be constructed using a ruler and a compass, together with a single fixed non-degenerate conic different from a circle.'
author:
- 'Seungjin Baek, Insong Choe, Yoonho Jung, Dongwook Lee and Junggyo Seo'
title: |
Constructions by ruler and compass,\
together with a fixed conic
---
introduction
============
Trisecting an arbitrary angle and doubling the cube by ruler and compass are two of the famous problems posed by Greeks which are known to be unsolvable. The first proof of the impossibility for the geometric constructions is attributed to Wantzel (1837). For history on this subject, we refer the reader to [@S pp. 25-26].
It is also well known that the trisection and duplication are possible if it is allowed to use one or more conic sections in addition to ruler and compass([@V]). The old constructions used hyperbolas and parabolas as conic sections. Recently, Hummel [@H] and Gibbins–Smolinsky [@GS] independently found the constructions using ellipses.
Gibbins and Smolinsky, at the end of their paper, asked if one can reduce the number of types of ellipses for the involved constructions. More specifically Hummel, in the final remark of his paper, asked if all points constructible from ellipses are constructible using the same ellipse in addition to a ruler and a compass. The goal of this paper is to show that the answer is Yes. More generally, we prove that every conic-constructible point can be obtained using a ruler and a compass, together with a single fixed non-degenerate conic different from a circle (Theorem \[main\]). It can be pointed out that our result resembles the Poncelet-Steiner’s theorem which states that any construction with a ruler and a compass can be accomplished by a ruler together with a fixed circle and its center(see [@E §3.6]).
The precise meaning of the conic-constructibility will be reviewed in the next section. In §3, we will state the main result and give a proof.
Conic-constructible points
==========================
The point $(x,y)$ in the plane $\mathbb{R}^2$ is identified with a complex number $x + i y \in \mathbb{C}$. Starting from the initial set $P=P_0$ of points, perform the drawings:
\(i) Given two points in $P$, draw a line through the two points using a ruler.
\(ii) Given two points $z_1$ and $z_2$ in $P$, draw a circle centered at $z_1$ with radius $|z_1-z_2|$ using a compass.
Let $P_1$ be the set of points by adjoining all the intersections of lines and circles from the above drawings. Now, replacing $P_0$ by $P_1$ and running the above process, we get the set $P_2$. Repeating this inductively, we get a sequence of sets $$P= P_0 \subset P_1 \subset P_2 \subset \cdots.$$ A point is called *constructible from $P$* if it is inside $P_\infty := \bigcup_{i=0}^\infty P_i$. A point is called *constructible* if it is constructible from the initial set $\{ 0,1 \}$ in ${\mathbb{C}}$. It is well known that a point is constructible if and only if it lies in a subfield $F$ of ${\mathbb{C}}$ which has a finite sequence of subfields starting from rational numbers: $$\mathbb{Q} = F_0 \subset F_1 \subset F_2 \subset \ldots \subset F_n = F$$ such that $[F_{i+1}: F_i] = 2$ for each $i$.
Now in addition to the above drawings (i) and (ii), consider another one:
\(iii) Draw all parabolas, ellipses, and hyperbolas having foci in $P$, directrix lines which are obtained from (i), and eccentricities equal to the length of a segment $\overline{z_1 z_2}$ for some $ z_1, z_2 \in P$. Let $P=Q_0$ and let $Q_1$ be the set of points by adjoining all the intersections of lines and circles and conics from the drawings (i), (ii), and (iii). Replacing $Q_0$ by $Q_1$ and running the same kind of process, we get $Q_2$. Again by induction, we get a sequence of sets $$P= Q_0 \subset Q_1 \subset Q_2 \subset \cdots.$$ A point is called *conic-constructible from $P$* if it is inside $Q_\infty := \bigcup_{i=0}^\infty Q_i$. A point is called *conic-constructible* if it is conic constructible from the initial set $\{ 0,1 \}$ in ${\mathbb{C}}$. Note that for any $P$, both the set of constructible points derived from $P$ and the set of conic-constructible points derived from $P$ are subfields of ${\mathbb{C}}$, because the process (i) and (ii) are already enough to produce the complex numbers given by the operations $+, -, \times, \div$. The $x$-coordinates of the constructible (or conic-constructible) points derived from $P$ again form a subfield of $\mathbb{R}$, which are called *constructible (or conic-constructible) numbers derived from $P$*. Videla found a useful criterion:
\[videla\] [([@V Theorem 1, 2])]{}\
(1) The set of conic-constructible points forms the smallest subfield of ${\mathbb{C}}$ containing 0, 1, and $i$ which is closed under conjugation, square roots and cube roots.\
(2) A point in $ {\mathbb{C}}$ is conic-constructible if and only if it is contained in a subfield of ${\mathbb{C}}$ of the form $\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_n)$, where $\alpha_1^{k_1} \in \mathbb{Q}$ and $\alpha_i^{k_i} \in \mathbb{Q}(\alpha_1, \ldots, \alpha_{i-1})$ for $2 \le i \le n$, where $k_i \in \{2,3\}$ for each $i = 1, 2, \ldots, n$.
The point of Proposition \[videla\] (2) is that taking square and cube roots of a given complex number are precisely the constructions required to obtain all the conic-constructible points. In other words, providing the constructions for real cube roots and trisections of angles are sufficient to get all conic-constructible points.
Let $K$ be a subfield of $\mathbb{R}$ such that every positive number $x \in K$ has a square root in $K$. A line passing through two points of $K^2$ is called a *line in $K$*. A circle is called a *circle in $K$* if its center is in $K^2$ and it passes through a point in $K^2$. Similarly, a conic (parabola, ellipse, or hyperbola) is called a *conic in $K$* if its foci are in $K^2$, its directrix line is in $K$, and its eccentricity is in $K$. Recall the following useful fact.
\[useful\] Let $E$ be a non-degenerate conic (different from a circle), defined by the equation $ax^2 + bxy +cy^2 +dx+ ey+f = 0$. Then $E$ is in $K$ if and only if the coefficients $a,b, c, d, e, f$ are in $K$ (after suitable rescaling). \[fact\]
The proof for the case of ellipse is given in [@GS Proposition 3] and the argument can be modified to work also for parabolas and hyperbolas.
result
======
We introduce two more notions: $C$-constructible points and $e$-constructible points. Let $C$ be a non-degenerate conic in the field of constructible numbers. For the initial set $P \subset {\mathbb{C}}$, a point is *$C$-constructible from $P$* if it is obtained by the same process as conic-constructible points except that the conics in step (iii) are confined to the fixed one: $C$.
We call an ellipse or a hyperbola of eccentricity $e>0$ to be “regular” if it is given by the equation $$(1-e^2) (x-a)^2 + (y-b)^2 = \lambda^2$$ for some $a, b, \lambda \in \mathbb{R}$. Also, we call a parabola to be regular if it is of the form $$x = \lambda(y-a)^2 +b$$ for some $a, b, \lambda \in \mathbb{R}$.
Let $e>0$ be a constructible number. A point is *$e$-constructible from $P$* if it is obtained by the same process as conic-constructible points except that
\(1) the conics in step (iii) are confined to the regular ones of eccentricity $e$, and
\(2) the intersections of two conics, neither of which is a circle, are not adjoined.\
In other words, (1) a conic is drawn only when it is regular with eccentricity $e$, and (2) for any drawn conic different from a circle, only the intersections with lines and circles are counted for the next stage.\
As before when $P = \{0,1 \}$, we say $C$-constructible and $e$-constructible respectively, for short. By definition, if $C$ is a regular conic with eccentricity $e$, then
$C$-constructible $\Rightarrow$ $e$-constructible $\Rightarrow$ conic-constructible.
\[e\] Let $e>0$ be any constructible number. Every conic-constructible point is $e$-constructible.
Let $z \in {\mathbb{C}}$ be a conic-constructible point. By Proposition \[videla\], $z \in \mathbb{Q} (\alpha_1, \alpha_2, \ldots, \alpha_n)$, where either $\alpha_i^2$ or $\alpha_i^3$ is contained in $ \mathbb{Q} (\alpha_1, \ldots, \alpha_{i-1})$ for each $i$. We will show that $z$ is $e$-constructible by induction on $n$. Suppose that all the points in $H:= \mathbb{Q} (\alpha_1, \alpha_2, \ldots, \alpha_{n-1})$ are $e$-constructible. If $\alpha_n^2 \in H$, then $\alpha_n$ is constructible from $H$.
Now suppose $\alpha_n^3 = re^{i \theta} \in H$. Let $q = \cos \theta$ and let $K$ be the field of constructible numbers derived from $0,1$ and $r, q$.
First consider the intersection of the following circle and conic: $$x^2 + y^2 - rx - y = 0 \ \ \text{and} \ \ (1-e^2)x^2 + y^2 - rx - (1-e^2)y = 0 .$$ By Proposition \[fact\], these are a circle and a conic in $K$. From these two equations, we get $x^4 - rx = 0$. Hence the $x$-coordinate of the intersection points other than the origin corresponds to the cube root of $r$.
Next for $q = \cos \theta$, by triple-angle formula, $\cos \left( {\theta}/{3} \right)$ is a real solution to the equation $$4x^3 - 3x - q = 0.$$ To get this equation, we intersect the following circle and conic: $$x^2 + y^2 - \frac{q}{4}x - \frac{7}{4}y = 0 \ \ \text{and} \ \ (1-e^2)x^2 + y^2- \frac{q}{4}x - \left(\frac{7}{4} - e^2 \right)y = 0 .$$ Again, these are a circle and a conic in $K$. Note that the conics in the above are regular of eccentricity $e$. From these intersections, we constructed the numbers $\sqrt[3]{r}$ and $ \cos (\theta/3)$, and eventually the point $\alpha_n$. Thus so far we have shown that $\alpha_n$ is an $e$-constructible point derived from 0, 1, and $r, q$. Since we assumed that all the points in $H$ are $e$-constructible and $\alpha_n^3 = re^{i \theta} \in H$, $r$ and $q= \cos \theta$ are also $e$-constructible. Therefore, $\alpha_n$ is $e$-constructible and thus all the points in $\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_{n})$ are $e$-constructible.
Now comes our main result: Every conic-constructible point can be constructed by using a single fixed conic in addition to a ruler and a compass.
Let $C$ be any non-degenerate conic, different from a circle, in the field of constructible numbers. Then every conic-constructible point is $C$-constructible. \[main\]
Let $e>0$ be any constructible number. By Lemma \[e\], it suffices to prove that every $e$-constructible point is $C$-constructible. We first introduce a hierarchy on the $e$-constructible points starting from the field of constructible points ${\mathbb{F}}_0$: $${\mathbb{F}}_0 \subset {\mathbb{F}}_1 \subset {\mathbb{F}}_2 \subset \cdots .$$ Draw all the regular conics in ${\mathbb{F}}_0$ of eccentricity $e$. Let $Q_{1}^e$ be the set of points by adjoining to ${\mathbb{F}}_0$ all the intersections of lines and circles with any of the drawn regular conics. Let ${\mathbb{F}}_1$ be the field of constructible points derived from $Q_1^e$.
Inductively, for each $k \ge 0$, let $Q_{k+1}^e$ be the set of points by adjoining to ${\mathbb{F}}_k$ all the intersections of lines and circles with any of the drawn regular conics in ${\mathbb{F}}_k$ of eccentricity $e$. And define ${\mathbb{F}}_{k+1}$ as the field of constructible points derived from $Q_{k+1}^e$. Then the set of $e$-constructible points coincides with $\bigcup_{k=0}^\infty {\mathbb{F}}_k$.
Now we prove that every $e$-constructible point is $C$-constructible by induction on $k$. Certainly the constructible points ${\mathbb{F}}_0$ are $C$-constructible. Now assume that all the points in ${\mathbb{F}}_k$ are $C$-constructible. Take any point $z \in {\mathbb{F}}_{k+1} \setminus {\mathbb{F}}_k$ which is obtained as an intersection of a circle $R_k$ in ${\mathbb{F}}_k$ and a regular conic $C_k$ in ${\mathbb{F}}_k$ of eccentricity $e$.
We may assume that $C$ is regular. Indeed, we may rotate $C$ by an angle $\theta$ where $\cos \theta$ and $\sin \theta$ are constructible, to send it to a regular conic, because $C$ is in the field of constructible numbers. From now on, we assume that $C$ is regular.
Since both $C$ and $C_k$ are in ${\mathbb{F}}_k$ of the same eccentricity, $C_k$ can be obtained from $C$ by a magnification by a factor $\lambda >0$ and a translation in $a+bi$, where $a,b$ and $\lambda$ are in ${\mathbb{F}}_k$. Therefore, the intersection point $z \in R_k \cap C_k$ can be obtained as follows:
\(a) Translate the circle $R_k$ by a point in ${\mathbb{F}}_k$ and magnify by a factor in ${\mathbb{F}}_k$ to get $R_k'$.
\(b) Intersect $R_k'$ with $C$ to get the corresponding intersection point $z'$.
\(c) Reverse the process (a) to get $z$ from $z'$.
Since we assumed that all of the points in ${\mathbb{F}}_k$ are $C$-constructible, the intersection $z' \in R_k' \cap C$ is a $C$-constructible point, because $R_k'$ is a circle in ${\mathbb{F}}_k$. We conclude that $z$ is a $C$-constructible point.
Since every point in ${\mathbb{F}}_{k+1}$ is constructible from ${\mathbb{F}}_k$ and the intersection points $z \in R_k \cap C_k$ obtained as in the above, it is $C$-constructible. This completes the proof.
*Acknowledgment.* This paper grew out from an R&E project “A study on the numbers constructible from conic sections”, which was supported by Seoul Science High School during March-December 2012.
[99]{}
Eves, H.: *College Geometry*, Johns and Bartlett Publisher, Boston, 1995.
Gibbins, A.; Smolinsky, L.: *Geometric constructions with ellipses*, Math. Intelligencer **31**, no. 1 (2009), 57–62.
Hummel, P.: *Solid constructions using conics*, The PME Journal, **11**, no. 8 (2003), 429–435.
Stillwell, J.: *Mathematics and its history*, 2nd edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2002.
Videla, Carlos, R.: *On points constructible from conics*, Math. Intelligencer **19**, no. 2 (1997), 53–57.
Insong Choe\
Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-Gu, Seoul 143-701, Korea.\
Email: `ischoe@konkuk.ac.kr`\
\
Seungjin Baek, Yoonho Jung, Dongwook Lee and Junggyo Seo\
Seoul Science High School, Uamgil 63 (Hyewha-dong 1-1), Jongro-ku, Seoul 110-530, Korea.\
Emails: `bjh7790@naver.com`, `yoonhoim@nate.com`, `dwleeid@naver.com`, `sjgceo@naver.com`
|
---
abstract: 'We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.'
address:
- 'Department of Ecology and Biology (DEB) Tuscia University Largo dell’Università, 01100 Viterbo, Italy'
- 'Department of Mathematics and Computer Science University of FlorenceViale Morgagni 67/A, 50134 Firenze - Italy'
author:
- Dimitri Mugnai
- Edoardo Proietti Lippi
title: 'Linking over cones for the Neumann Fractional $p-$Laplacian'
---
Keywords: fractional $p-$Laplacian, Neumann boundary conditions, linking over cones, lack of Ambrosetti-Rabinowitz condition.
2010AMS Subject Classification: 35A15, 47J30, 35S15, 47G10, 45G05.
Introduction
============
In this paper we are concerned with the problem $$\label{plink}
\begin{cases}
(-\Delta)^s_p u =\lambda |u|^{p-2}u + g(x,u) & $ in $ \Omega,
\\
\mathscr{N}_{s,p}u=0 & $ in $ {\mathbb{R}}^N \setminus \overline{\Omega}.
\end{cases}$$ Here $p\in(1,\infty)$, $\Omega$ is a bounded domain with Lipschitz boundary, $\lambda \geq 0$ and $g:\Omega\times {\mathbb{R}}\to {\mathbb{R}}$ is a Carathéodory function. The novelty of our investigation relies on the fact that we study a quasilinear fractional problem in presence of nonlocal [*Neumann*]{} boundary conditions, namely we require that $$\mathscr{N}_{s,p}u(x):=\int_\Omega \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy=0$$ for every $x\in {\mathbb{R}}^N \setminus \overline{\Omega}$. As a matter of fact, such a condition is the natural $p-$Neumann boundary condition associated to the operator $$(-\Delta)^s_pu(x):=P.V.\int_{{\mathbb{R}}^N} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy$$ for $x\in \Omega$, $P.V.$ being the Cauchy Principal value, see [@BMPS; @DPROV; @mupli] (see also [@mazon] for a related case and [@warma] for the restricted or regional fractional $p-$Laplacian. See also [@mbrs] for a general overlook on nonlocal operators).
Under suitable assumptions on $g$, we will show that problem admits solutions. As usual, we shall deal with weak solutions, belonging to a suitable function space. In our case, solutions will be sought in the space $$X:=\left\{u:{\mathbb{R}}^N\to {\mathbb{R}}\quad \text{measurable such that }\|u\|<\infty\right\},$$ where $$\|u\|:=\left(\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy +\|u\|_{L^p(\Omega)}^p\right)^\frac{1}{p},$$ and ${\mathcal{Q}}={\mathbb{R}}^{2N}\setminus (C\Omega)^2$, $C\Omega={\mathbb{R}}^N\setminus \Omega$.
It is clear that, when $\Omega$ is sufficiently regular, as in our case, in the integral above we can equally consider ${\mathbb{R}}^N\setminus \Omega$ or ${\mathbb{R}}^N\setminus \overline\Omega$.
We will deal with the following standard
Let $u\in X$. We say that $u$ is a weak solution of if $$\frac{1}{2}\int\int_{\mathcal{Q}}\frac{J_p(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\, dxdy= \lambda\int_\Omega |u|^{p-2}uv\,dx+\int_\Omega g(x,u)v\,dx$$ for every $v\in X$, where $J_p(u(x)-u(y))=|u(x)-u(y)|^{p-2}(u(x)-u(y)) $, provided that the last integral makes sense.
Of course, below we will give conditions which ensure that the definition above makes sense.
We observe that we shall consider only the case $\lambda\geq0$. Indeed, the case $\lambda<0$ makes the situation different, since one can apply the Mountain Pass Theorem with the Cerami or with the Palais-Smale condition (see [@mupli]). In our case the natural geometric structure for the associated functional is the one of [*linking over cones*]{}, as introduced in [@dela], for which some suitable topological notions are needed. As usual when dealing with linking structures, it is natural to consider the eigenvalues of the underlying operator; in this case we will employ the sequence of eigenvalues found in [@mupli] by using the Fadell-Rabinowitz index. All these preliminary tools will be recalled in Section \[secback\] below. We also recall that the use of linking theorems for fractional operators with [*Dirichlet boundary conditions*]{} has already appeared in related situations (see [@pp] and [@sv]).
As for the nonlinear source, in Section \[seclinking\] we assume that $g$ has $p-$superlinear growth and satisfies different sets of assumptions: in the first case, we will assume that $g$ satisfies the usual Ambrosetti-Rabinowitz condition, while in the second case we will exploit a different general assumption, introduced in [@MP]. We remark that in both cases we encounter the difficulty of determining the topological structure of the associated functional, while in the second case we have the additional complication related to the proof of the Cerami condition. Finally, in Section \[secsaddle\] we consider the case in which $g$ has $p-$linear growth.
As a matter of fact, there are two examples with $p=2$ that are covered by our results and which explain the nature of our results better: $$\begin{cases}
(-\Delta)^s u = \lambda u +|u|^{q-2}u\quad $ in $ \Omega,
\\
\mathscr{N}_{s,2}u=0 \quad \quad $ in $ {\mathbb{R}}^N \setminus \overline{\Omega},
\end{cases}$$ with $q>2$ and $q<\frac{2N}{N-2s}$ if $N>2s$, and $$\label{es2}
\begin{cases}
(-\Delta)^s u = \lambda u +f(x)\quad $ in $ \Omega,
\\
\mathscr{N}_{s,2}u=0 \quad \quad $ in $ {\mathbb{R}}^N \setminus \overline{\Omega},
\end{cases}$$ with $\lambda<0$ and $f\in L^2(\Omega)$. For the first problem the idea is to apply a standard Linking Theorem, while in the second case the variational structure is the one of the classical Weierstrass Theorem. In our results the first situation is widened to cover the quasilinear form of the fractional $p-$Laplacian, which doesn’t let us apply the classical Linking theorem directly, since the nonlinear operator $(-\Delta)^s_p$ does not have linear eigenspaces; thus, the use of Linking over cones provides an original opportunity, see [@dela], [@liu], [@28], [@29] for related cases in the local situation.
. Moreover, the possibility of treating nonlinear terms non verifying the classical Abrosetti-Rabinowitz condition, makes our results new also in the easier case $p=2$. On the other hand, the easy situation described in problem is enlarged to cover quasilinear problems where a nonlinear term is allowed to be not far from 0, as $\lambda$ is in (see Theorem \[thsad\]).
Background {#secback}
==========
First we recall some notions regarding the eigenvalues of fractional $p-$Laplacian, see [@BMPS] and [@mupli]. Consider the nonlinear eigenvalue problem $$\label{probla}
\begin{cases}
(-\Delta)^s_p u = \lambda |u|^{p-2}u \quad $ in $ \Omega,
\\
\mathscr{N}_{s,p}u=0 \quad \quad $ in $ {\mathbb{R}}^N \setminus \overline{\Omega},
\end{cases}$$ with $\lambda \in {\mathbb{R}}$. As usual, if admits a weak solution we say that $\lambda$ is an eigenvalue of $(-\Delta)^s_p$ with $p-$Neumann boundary conditions. So, there exists a sequence $\lambda_m$ of eigenvalues defined as $$\label{lm}
\begin{aligned}
\lambda_m:=\inf \left\lbrace \sup_{u\in A}\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy
\right.&: \, A\subseteq M , A \text{ is symmetric,}\\
&\left. \text{ compact and } i(A) \geq m \right\rbrace,
\end{aligned}$$ where $i$ is the $\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz (see [@fara]) and $$M:=\left\lbrace u\in X :\, \int_\Omega |u|^p\,dx=1 \right\rbrace.$$ Notice that $\lambda_1=0$ is the first (simple) eigenvalue with associated eigenspace made of constant functions (see [@mupli]).
For each $\lambda_m$, we can define the cones $$\label{c-}
C_m^-:=\left\lbrace u\in X :\, \int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy \leq \lambda_m \int_\Omega |u|^p\,dx \right\rbrace$$ $$\label{c+}
C^+_m:=\left\lbrace u\in X :\, \int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy \geq \lambda_{m+1} \int_\Omega |u|^p\,dx \right\rbrace.$$ For further use, we also introduce the notation $$[u]=\left(\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy\right)^{1/p},$$ which is closely related to the fractional Gagliardo seminorm.
Now we recall some notions on linking sets and Alexander-Spanier cohomology, referring to [@dela].
Let $D,S,A,B$ be four subsets of a metric space $X$ with $S \subseteq D$ and $B\subseteq A$. We say that $(D,S)$ *links* $(A,B)$, if $S\cap A= B\cap D =\emptyset$ and, for every deformation $\eta:D\times[0,1] \to X\setminus B$ with $\eta(S\times[0,1])\cap A=\emptyset$, we have that $\eta(D\times \{1\})\cap A\neq \emptyset$.
To prove the existence of critical points we will use a particular case of [@fri Theorem 3.1]. A smooth version of such a result was already stated in [@dela Theorem 2.2] under the validity of the Palais–Smale condition. However, the key point in the proof of [@fri Theorem 3.1] is the possibility of defining deformations between sublevels, as it is possible under the validity of the Cerami condition. For this reason we recall that $f$ satisfies the $(C)_c$ condition, $c\in {\mathbb{R}}$, if
for every $(u_n)_n$ such that $f(u_n)\to c$ and $(1+\|u_n\|) f'(u_n) \to 0 $ in $X'$, then, up to a subsequence, $u_n \to u$ in $X$.
Hence, we will need the following version of [@fri Theorem 3.1]:
\[critp\] Let $X$ be a complete Finsler manifold of class $C^1$ and let $f:X\to {\mathbb{R}}$ be a function of class $C^1$. Let $D,S,A,B$ be four subsets of $X$, with $S \subseteq D$ and $B\subseteq A$, such that $(D,S)$ links $(A,B)$ and such that $$\sup_S f< \inf_A f, \quad \quad \sup_D f< \inf_B f$$ $($with $\sup \emptyset=-\infty$ and $\inf \emptyset=+\infty)$. Define $$c=\inf_{\eta \in \mathcal{N}}\sup f(\eta(D\times \{1\})),$$ where $\mathcal{N}$ is the set of deformations $\eta:D\times[0,1] \to X\setminus B$ with $\eta(S\times[0,1])\cap A=\emptyset$. Then we have $$\inf_A f\leq c \leq \sup_D f.$$ Moreover, if $f$ satisfies $(C)_c$, then $c$ is a critical value of $f$.
Let $D,S,A,B$ be four subsets of $X$ with $S \subseteq D$ and $B\subseteq A$; let $m$ be a nonnegative integer and let $\mathbb{K}$ be a field. We say that $(D,S)$ *links* $(A,B)$ *cohomologically in dimension $m$ over* $\mathbb{K}$ if $S\cap A= B\cap D =\emptyset$ and the restriction homomorphism $H^m(X\setminus B,X\setminus A;\mathbb{K})\to H^m(D,S;\mathbb{K})$ is not identically zero.
The geometry we are interested in is described by the following
\[geom\] Let $X$ be a real normed space and let ${\mathcal C}_-$, ${\mathcal C}_+$ be two cones such that ${\mathcal C}_+$ is closed in $X$, ${\mathcal C}_- \cap {\mathcal C}_+=\lbrace 0 \rbrace$ and such that $(X,{\mathcal C}_-\setminus \lbrace 0 \rbrace)$ links ${\mathcal C}_+$ cohomologically in dimension $m$ over $\mathbb{K}$. Let $r_-,r_+>0$ and let $$D_-=\lbrace u\in {\mathcal C}_-:\,\|u\|\leq r_-\rbrace, \quad \quad S_-=\lbrace u\in {\mathcal C}_-:\,\|u\|= r_-\rbrace,$$ $$D_+=\lbrace u\in {\mathcal C}_+:\,\|u\|\leq r_+\rbrace, \quad \quad S_+=\lbrace u\in {\mathcal C}_+:\,\|u\|= r_+\rbrace.$$ Then the following facts hold:
- $(D_-,S_-)$ links ${\mathcal C}_+$ cohomologically in dimension $m$ over $\mathbb{K}$;
- $(D_-,S_-)$ links $(D_+,S_+)$ cohomologically in dimension $m$ over $\mathbb{K}$;
Moreover, let $e\in X$ with $-e\notin {\mathcal C}_-$, let $$Q=\lbrace u+te:\, u\in {\mathcal C}_-,\, t\geq 0,\, \|u+te\|\leq r_- \rbrace,$$ $$H=\lbrace u+te:\, u\in {\mathcal C}_-,\, t\geq 0,\, \|u+te\|= r_- \rbrace,$$ and assume that $r_->r_+$. Then the following facts hold:
- $(Q,D_-\cup H)$ links $S_+$ cohomologically in dimension $m+1$ over $\mathbb{K}$;
- $D_-\cup H$ links $(D_+,S_+)$ cohomologically in dimension $m$ over $\mathbb{K}$;
In order to prove our existence result, we shall use assertion (c) in Section \[seclinking\] and assertion (a) in Section \[secsaddle\], that correspond to the classical linking and saddle geometry, respectively.
We will also take advantage of the following result
\[corgeom\] Let $X$ be a real normed space and let ${\mathcal C}_-,{\mathcal C}_+$ be two symmetric cones in $X$ such that ${\mathcal C}_+$ is closed in $X$, ${\mathcal C}_-\cap {\mathcal C}_+ = \lbrace 0 \rbrace$ and such that $$i({\mathcal C}_-\setminus \lbrace0\rbrace)= i(X\setminus {\mathcal C}_+)<\infty.$$ Then the assertion (a)-(d) of Theorem $\ref{geom}$ hold for $m=i({\mathcal C}_-\setminus \lbrace0\rbrace)$ and $\mathbb{K}=\mathbb{Z}_2$.
Going back to definitions and , we have the following result, which is the transcription in our setting of [@dela Theorem 3.2], and whose proof follows that one step-by-step.
\[index\] Let $m\geq 1$ be such that $\lambda_m < \lambda_{m+1}$, then we have $$i(C_m^- \setminus \lbrace 0 \rbrace) =i(X\setminus C_m^+)=m$$
Finally, in order to use Theorem \[critp\], the crucial tool is
\[collegamento\] If $(D, S)$ links $(A, B)$ cohomologically (in some dimension), then $(D, S)$ links $(A, B)$.
Linking-like problems {#seclinking}
=====================
Now, let us go back to problem , that is $$\begin{cases}
(-\Delta)^s_p u =\lambda |u|^{p-2}u + g(x,u) & $ in $ \Omega,
\\
\mathscr{N}_{s,p}u=0 & $ in $ {\mathbb{R}}^N \setminus \overline{\Omega}.
\end{cases}$$ We recall that $p\in(1,\infty)$, $\Omega$ is a bounded domain with Lipschitz boundary, $\lambda \geq 0$ and $g:\Omega\times {\mathbb{R}}\to {\mathbb{R}}$ is a Carathéodory function, that is the map $x\mapsto g(x,t)$ is measurable for every $t\in {\mathbb{R}}$ and the map $t\mapsto g(x,t)$ is continuous for a.e. $x\in \Omega$.
Of course, we shall assume growth conditions on $g$ which will ensure that any critical point of the $C^1$ functional $I:X\to {\mathbb{R}}$ defined as $$\label{I}
I(u)=\frac{1}{2p}\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy-\frac{\lambda}{p}\int_\Omega |u|^p\,dx
-\int_\Omega G(x,u)\,dx$$ is a weak solution of .
Notice that, quite strangely, the coefficient $\frac{1}{2}$ appears in front of the expected $\frac{1}{p}[u]^p$. This is related to symmetry properties of the double integral in the definition of $I$, and it justifies the fact that $u$ solves if and only if $I'(u)=0$, see [@BMPS; @mupli].
We first we give the following result, which will be useful in any case and which makes precise the statement in [@mupli] related to the $(S)$ property.
\[s+\] Set $A(u)=[u]^p$. Then the functional $A':X\to X'$ satisfies the $(S)_+$ property, that is for every sequence $(u_n)_n$ such that $u_n \rightharpoonup u$ in $X$ as $n\to \infty$ and $$\label{lsup}
\limsup_{n\to \infty}\langle A'(u_n),u_n-u\rangle_{X',X} \leq 0,$$ then $u_n \to u$ in $X$ as $n\to \infty$.
Assume that $u_n \rightharpoonup u$ in $X$ and $\limsup \langle A'(u_n),u_n-u\rangle_{X',X} \leq 0$. First of all, $A$ is convex, of class $C^1$ and weakly lower semicontinuous in $X$, so that $A(u)\leq \liminf A(u_n)$.
Moreover, the linear functional $\langle A'(u),\cdot \rangle_{X',X}$ is in $X'$. So, since $u_n \rightharpoonup u$ in $X$, $$\label{A'to0}
\langle A'(u),u_n-u \rangle_{X',X} \to 0$$ as $n\to \infty$. By the convexity of $A$, we get that $A'$ is a monotone operator, so that $$\langle A'(u_n)-A'(u),u_n-u\rangle_{X',X}\geq0.$$ By we get $$0\leq \limsup_{n\to \infty} \langle A'(u_n)-A'(u),u_n-u\rangle_{X',X}\leq 0,$$ and so $$\label{Alim0}
\lim_{n\to \infty} \langle A'(u_n)-A'(u),u_n-u\rangle_{X',X}=0.$$ Hence, and imply that $$\label{Alim02}
\lim_{n\to \infty} \langle A'(u_n),u_n-u\rangle_{X',X}=0.$$ Again by the convexity of $A$ we have that $$A(u)\geq \langle A'(u_n),u-u_n\rangle_{X',X}\geq A(u_n).$$ By , $A(u)\geq \limsup A(u_n)$, and so $$A(u)=\lim_{n\to \infty}A(u_n).$$ By the compact embedding of $X$ into $L^p(\Omega)$ we also have $u_n \to u$ in $L^p(\Omega)$. In the end, $\|u_n\|\to \|u\|$. Hence, by the uniform convexity of $X$ (recall that $1<p<\infty$) , we obtain that $u_n$ converges strongly to $u$ in $X$ as $n\to \infty$.
With the Ambrosetti-Rabinowitz condition
----------------------------------------
This case is the easy one, which we present just to show the extension of the approach in [@dela] to the nonlocal case.
Here we will further assume the following hypotheses on $g$:
- there exist constants $a_1,a_2>0$ and $q>p$ such that for every $t\in {\mathbb{R}}$ and for a.e. $x\in \Omega$ $$|g(x,t)|\leq a_1+a_2 |t|^{q-1},$$ where $q<\frac{pN}{N-ps}$ if $N>ps$;
- $g(x,t)=o(|t|^{p-1})$ as $t \to 0$ uniformly a.e. in $\Omega$;
- denoting $G(x,t)=\int_0^t g(x,\tau)\, d\tau $, there exist $\mu>p$ and $R\geq 0$ such that for every $t$ with $|t|>R$ and for a.e. $x\in \Omega$ $$0<\mu G(x,t) \leq g(x,t)t,$$ and there exist $\tilde \mu>p$, $a_3>0$ and $a_4\in L^1(\Omega)$ such that for every $t\in {\mathbb{R}}$ and a.e. $x\in \Omega$, $$\label{add}
G(x,t)\geq a_3|t|^{\tilde \mu} -a_4(x);$$
- if $R>0$, then $G(x,t)\geq 0$ for every $t\in {\mathbb{R}}$ and a.e. $x\in \Omega$.
Condition was introduced in [@addendum] to complete the Ambrosetti-Rabinowitz condition in presence of a Carathéodory functions.
Our first existence result is
\[teolink\] If hypotheses $(g_1)-(g_4)$ hold, then problem admits a nontrivial weak solution.
In order to prove Theorem \[teolink\] it will be enough to apply Theorem \[critp\] to the functional $I$ defined in under the validity of the Palais-Smale condition (of course, if the Cerami condition holds, the Palais-Smale condition holds, as well); hence, we will apply Theorem \[critp\] in the version of [@dela Theorem 2.2], where the Palais-Smale condition is assumed.
Thus, now we prove that $I$ satisfies the Palais-smale condition at any level $c\in{\mathbb{R}}$ - $(PS)_c$ for short -, that is
for every sequence $(u_n)_n$ in $X$ such that $I(u_n)\to c$ and $I'(u_n)\to 0$ in $X'$, there exists a strongly converging subsequence of $(u_n)_n$.
\[pslink\] Under the assumptions of Theorem $\ref{teolink}$, $I$ satisfies $(PS)_c$ for every $c\in {\mathbb{R}}$.
Let $(u_n)_n$ in $X$ be such that $I(u_n)\to c$ and $I'(u_n)\to 0$ and fix $k\in (p,\mu)$. We re-write the functional in the following way: $$\begin{aligned}
I(u)&=\frac{1}{2p}\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy+\frac{1}{2p}\int_\Omega |u|^pdx\\
&-\left(\frac{\lambda}{p}+\frac{1}{2p}\right)\int_\Omega |u|^p\,dx
-\int_\Omega G(x,u)\,dx\\
&=\frac{1}{2p}\|u\|^p-\left(\frac{\lambda}{p}+\frac{1}{2p}\right)\int_\Omega |u|^p\,dx
-\int_\Omega G(x,u)\,dx.
\end{aligned}$$
We observe that $$\label{psbound}
k I(u_n)- \langle I'(u_n),u_n \rangle \leq M + N \|u_n\|$$ for some $M,N>0$ and all $n\in {\mathbb{N}}$. On the other hand, by $(g_3)$ and $(g_1)$ we have $$\begin{aligned}
k&I(u_n)-\langle I'(u_n),u_n \rangle \\
&=\left(\frac{k}{2p}-\frac{1}{2}\right)\|u_n\|^p-\left(\frac{k}{p}-1\right)\left(\lambda+\frac{1}{2}\right)\int_\Omega |u_n|^p\,dx \\
&+\int_\Omega \big(g(x,u_n)u_n-kG(x,u_n)\big)\,dx \\
&\geq \left(\frac{k}{2p}-\frac{1}{2}\right)\|u_n\|^p-\left(\frac{k}{p}-1\right)\left(\lambda+\frac{1}{2}\right)\int_\Omega |u_n|^p\,dx \\
&+(\mu -k)\int_\Omega G(x,u_n)\,dx -C_R\end{aligned}$$ for some constant $C_R\geq 0$. By , we get $$\begin{aligned}
k&I(u_n)-\langle I'(u_n),u_n \rangle \\
&\geq \left(\frac{k}{2p}-\frac{1}{2}\right)\|u_n\|^p-\left(\frac{k}{p}-1\right)\left(\lambda+\frac{1}{2}\right)\int_\Omega |u_n|^p\,dx \\
&+(\mu -k)a_3\int_\Omega |u_n|^{\tilde\mu} \,dx -C\end{aligned}$$ for some constant $C\geq 0$. By the Hölder and the Young inequalities, we get that for any $\varepsilon>0$ we have that for every $u\in X$ $$\|u\|_p^p \leq \varepsilon \|u\|_{\tilde\mu}^{\tilde\mu} + C_\varepsilon.$$ Thus, we obtain $$\begin{aligned}
k&I(u_n)-\langle I'(u_n),u_n \rangle \\
&\geq \left(\frac{k}{2p}-\frac{1}{2}\right)\|u_n\|^p
+\left[(\mu -k)a_3- \varepsilon \left( \frac{k}{p}-1\right)\left(\lambda+\frac{1}{2}\right)\right] \int_\Omega |u_n|^{\tilde \mu}\,dx -\tilde{C_\varepsilon}\end{aligned}$$ for some $\tilde{C_\varepsilon}>0$. Taking $\varepsilon$ small enough, we get $$kI(u_n)-\langle I'(u_n),u_n \rangle \geq \left(\frac{k}{2p}-\frac{1}{2}\right)\|u_n\|^p -\tilde{C_\varepsilon}.$$ This together with implies that $(u_n)_n$ is bounded in $X$. Up to a subsequence, we can assume that $u_n\rightharpoonup u$ in $X$ and $u_n \to u$ in $L^p(\Omega)$ as $n\to \infty$. By assumption, we have $$\langle I'(u_n),u_n-u \rangle \to 0.$$ On the other hand $$\begin{aligned}
\langle& A'(u_n),u_n-u \rangle \\
&=\langle I'(u_n),u_n-u \rangle
+\lambda\int_\Omega |u_n|^{p-2}u_n(u_n-u)\,dx+\int_\Omega g(x,u_n)(u_n-u)\,dx.\end{aligned}$$ Since $u_n \to u$ in $L^p(\Omega)$, from ($g_1$) we obtain that $$\int_\Omega |u_n|^{p-2}u_n(u_n-u)\,dx \to 0$$ and $$\int_\Omega g(x,u_n)(u_n-u)\,dx \to 0;$$ so $\langle A'(u_n),u_n-u \rangle_{X',X} \to 0$ as $n\to \infty$. By Proposition \[s+\] we get that $u_n \to u$ in $X$, as desired.
Now we are ready to prove Theorem \[teolink\].
Let $(\lambda_m)_m$ be the sequence of eigenvalues defined in . Since this sequence is divergent, there exists $m\geq 1$ such that $\lambda_m \leq 2\lambda +1< \lambda_{m+1}$. Defining $C_m^-$ and $C_m^+$ as in and , we have that $C_m^-$,$C_m^+$ are two symmetric closed cones in $X$ with $C_m^-\cap C_m^+ =\lbrace0\rbrace$. We recall that by Theorem \[index\] we have $$i(C_m^-\setminus \lbrace0\rbrace)= i(X\setminus C_m^+)=m.$$
Now, by $(g_1)$ and $(g_2)$ it is standard to see that for any ${\varepsilon}>0$ there exists $C_{\varepsilon}>0$ such that $$|G(x,t)|\leq \frac{{\varepsilon}}{2p} |t|^p+C_{\varepsilon}|t|^q$$ for a.e. $x\in \Omega$ and all $t\in {\mathbb{R}}$. As a consequence, taking $u\in C_m^+$, by the inequality in and the Sobolev inequality, we have that $$\begin{aligned}
I(u)&\geq\frac{1}{2p}\|u\|^p -\frac{2\lambda+1}{2p}\int_\Omega |u|^pdx-\frac{{\varepsilon}}{2p} \int_\Omega |u|^pdx-C_{\varepsilon}\int_\Omega |u|^qdx\\
&\geq\frac{1}{2p}\|u\|^p -\frac{1}{2p\lambda_{m+1}}\left(2\lambda+1+{\varepsilon}\right)[u]^p-C_{\varepsilon}\int_\Omega |u|^qdx\\
& \geq\frac{1}{2p}\left(1-\frac{2\lambda+1+{\varepsilon}}{\lambda_{m+1}}\right)\|u\|^p -C\|u\|^q
\end{aligned}$$ for some $C>0$.
Hence, choosing ${\varepsilon}$ small enough, there exists $r_+>0$ and $\alpha>0$ such that, if $\|u\|=r_+$, then $I(u)\geq \alpha$.
On the other hand, taking $u\in C_m^-$, $e\in X\setminus C_m^-$ and $t>0$, by we get that $$\begin{aligned}
I(u+te)&\leq \frac{2^{p-2}}{p}\left( \int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy
+t^p\int\int_{\mathcal{Q}}\frac{|e(x)-e(y)|^p}{|x-y|^{N+ps}}\, dxdy \right)\\
&-\frac{\lambda}{p} \int_\Omega |u+te|^p\,dx -a_3t^{\tilde \mu}\int_\Omega \left|\frac{u}{t}+e\right|^{\tilde\mu} \,dx+\|a_4\|_1 \to -\infty\end{aligned}$$ as $t \to +\infty$. In conclusion, there exists $r_->r_+$ such that $I(v)\leq 0$ when $v\in C_m^-+({\mathbb{R}}^+e)$ and $\|v\|\geq r_-$.
Defining $D_-,S_+,Q$ and $H$ as in Theorem \[geom\], by Corollary \[corgeom\] we have that $(Q,D_-\cup H)$ links $S_+$ cohomologically in dimension $m+1$ over $\mathbb{Z}_2$. In particular, $(Q,D_-\cup H)$ links $S_+$ by Proposition \[collegamento\]. In addition, $I$ is bounded on $Q$, $I(u)\leq 0$ for every $u\in D_-\cup H$ and $I(u)\geq \alpha>0$ for every $u\in S_+$. By Proposition \[pslink\] $(PS)_c$ holds. Finally, by applying Theorem \[critp\] with $S=D_-\cup H$, $D=Q$, $A=S_+$ and $B=\emptyset$, $I$ admits a critical value $c\geq \alpha$, hence there exists a critical point $u$ with $I(u)=c>0$. It follows that $u$ is a nontrivial weak solution of .
Without the Ambrosetti-Rabinowitz condition
-------------------------------------------
In this section we consider the problem
$$\label{plinkc}
\begin{cases}
(-\Delta)^s_p u =\lambda |u|^{p-2}u + f(x,u) \quad $ in $ \Omega,
\\
\mathscr{N}_{s,p}u=0 \quad \quad $ in $ {\mathbb{R}}^N \setminus \overline{\Omega},
\end{cases},$$
where $\lambda \geq 0$ and $f:\Omega\times {\mathbb{R}}\to {\mathbb{R}}$ is a Carathéodory function such that $f(x,0)=0$ for almost every $x\in \Omega$. This time, we assume the following hypotheses on $f$, first introduced in [@MP]:
- there exists $a\in L^q(\Omega)$, $a\geq 0$, with $q\in ((p^*_s)',p)$, $c>0$ and $r\in (p,p^*_s)$ such that $$|f(x,t)|\leq a(x)+c|t|^{r-1}$$ for a.e. $x \in \Omega$ and for all $t \in {\mathbb{R}}$;
- denoting $F(x,t)=\int_0^t f(x,\tau)d\tau $, we have $$\lim_{t\to \pm \infty}\frac{F(x,t)}{|t|^p}=+\infty$$ uniformly for a.e. $ x\in \Omega$;
- if $\sigma(x,t):=f(x,t)t-pF(x,t)$, then there exist $\vartheta\geq1$ and $\beta^* \in L^1(\Omega)$, $\beta^*\geq0$, such that $$\sigma(x,t_1)\leq \vartheta\sigma(x,t_2)+\beta^*(x)$$ for a.e. $x\in \Omega$ and all $0\leq t_1 \leq t_2$ or $t_2\leq t_1 \leq 0$;
- $$\lim_{t\to 0} \frac{f(x,t)}{|t|^{p-2}t}=0$$ uniformly for a.e. $x\in \Omega$.
In $(f_1)$ we have denoted by $p^*_s$ the fractional Sobolev exponent of order $s$, that is $$p^*_s=\begin{cases}
\dfrac{pN}{N-ps}& \mbox{ if }ps<N,\\
\infty &\mbox{ if }ps\geq N.
\end{cases}$$ In this way, the embedding in $L^q(\Omega)$ of $W^{s,p}(\Omega)$ (and thus of $X$) is compact for every $q<p^*_s$.
As before, we give the definition of a weak solution.
Let $u\in X$. We say that $u$ is a weak solution of problem if $$\frac{1}{2}\int\int_{\mathcal{Q}}\frac{J_p(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\, dxdy
= \lambda\int_\Omega |u|^{p-2}uv\,dx+\int_\Omega f(x,u)v\,dx$$ for every $v\in X$.
Again, any critical point of the $C^1$ functional ${\mathscr{E}}:X\to {\mathbb{R}}$ defined as $${\mathscr{E}}(u)=\frac{1}{2p}\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy-\frac{\lambda}{p}\int_\Omega |u|^p\,dx
-\int_\Omega F(x,u)\,dx$$ is a weak solution of .
The main result of this section is the following.
\[thlc\] If hypotheses $(f_1)$-$(f_4)$ hold, then problem admits two nontrivial constant sign solutions. More precisely, one solution is strictly positive and the other one is strictly negative in ${\mathbb{R}}^N$.
First of all, we introduce the functionals $$\begin{aligned}
{\mathscr{E}}_\pm(u)&=\frac{1}{2p}\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy
+\frac{1}{p}\int_\Omega |u|^pdx \\
&-\frac{\lambda+1}{p}\int_\Omega |u^\pm|^pdx -\int_\Omega F(x,u^\pm)\,dx ,\end{aligned}$$ where $u^+:=\max\{u,0\}$ and $u^-:=\max\{-u,0\}$ are the classical positive part and negative part of $u$, respectively. Notice that ${\mathscr{E}}_+(u)={\mathscr{E}}(u)$ for every $u\geq0$ and ${\mathscr{E}}_-(u)={\mathscr{E}}(u)$ for every $u\leq0$.
The following algebraic inequalities will be very useful in the following: $$\label{disug}
|x^--y^-|^p \leq |x-y|^{p-2}(x-y)(y^--x^-),$$ $$\label{disug1}
|x^+-y^+|^p \leq |x-y|^{p-2}(x-y)(x^+-y^+),$$ $$\label{disug2}
|x-y|^p\leq 2^{p-1}(|x^+-y^+|^p+|x^--y^-|^p)$$ and $$\label{11}
|x^\pm-y^\pm|\leq |x-y|$$ for any $x,y\in {\mathbb{R}}$. The proofs are obvious.
\[C\] Under the assumptions of Theorem $\ref{thlc}$, ${\mathscr{E}}_\pm$ satisfies $(C)_c$ for every $c\in {\mathbb{R}}$.
We do the proof for ${\mathscr E}_+$, the proof for ${\mathscr E}_-$ being analogous.
Let $(u_n)_n$ in $X$ be such that $$\label{cer1}
|{\mathscr E}_+(u_n)|\leq M_1$$ for some $M_1>0$ and all $n\geq 1$, and $$\label{cer2}
(1+\|u_n\|){\mathscr E}_+'(u_n)\to 0$$ in $X'$ as $n\to \infty$. From we have $$|{\mathscr E}_+'(u_n)(h)|\leq \frac{\varepsilon_n\|h\|}{1+\|u_n\|}$$ for every $h\in X$ and with $\varepsilon_n \to 0$ as $n \to \infty$, that is $$\label{vs}
\begin{aligned}
\left|\frac{1}{2}\int \int_{\mathcal{Q}}\right.&\frac{J_p(u_n(x)-u_n(y))(h(x)-h(y))}{|x-y|^{N+ps}}\,dxdy
+\int_\Omega |u_n|^{p-2}u_nh\,dx \\
&- (\lambda +1) \int_\Omega |u_n^+|^{p-2}u_n^+ h\,dx \left. -\int_\Omega f(x,u_n^+)h\,dx\right|
\leq \frac{\varepsilon_n\|h\|}{1+\|u_n\|}.
\end{aligned}$$ Taking $h=-u_n^-$ in , we obtain $$\label{a0}
\frac{1}{2}\int \int_{\mathcal{Q}}\frac{J_p(u_n(x)-u_n(y))(u_n^-(y)-u_n^-(x))}{|x-y|^{N+ps}}\,dxdy +\lambda\int_\Omega |u_n^-|^pdx\leq \varepsilon_n,$$ and by we get $$\int \int_{\mathcal{Q}}\frac{|u_n^-(x)-u_n^-(y)|^p}{|x-y|^{N+ps}}\,dxdy+2\lambda\int_\Omega |u_n^-|^pdx \leq 2\varepsilon_n.$$ As a consequence, we get that $$\label{vs-}
u_n^-\to 0 \mbox{ in $X$ as $n\to \infty$}.$$ In particular, $(u_n^-)_n$ is bounded in $X$.
On the other hand, taking $h=-u_n^+$ in , we get $$\label{vs+}
\begin{aligned}
-\frac{1}{2}\int \int_{\mathcal{Q}}&\frac{J_p(u_n(x)-u_n(y))(u_n^+(x)-u_n^+(y))}{|x-y|^{N+ps}}\,dxdy \\
&+ \lambda \int_\Omega |u_n^+|^p\,dx
+\int_\Omega f(x,u_n^+)u_n^+\,dx\leq \varepsilon_n.
\end{aligned}$$
From we know that $$\label{maggio}
\frac{1}{2}[u_n]^p + \int_\Omega |u_n|^p\,dx
- (\lambda+1) \int_\Omega |u_n^+|^p\,dx
-p\int_\Omega F(x,u_n^+)\,dx\leq pM_1$$ for all $n\geq 1$. Now, by and , we have that $$\int \int_{\mathcal{Q}}\frac{J_p(u_n(x)-u_n(y))(u_n^-(x)-u_n^-(y))}{|x-y|^{N+ps}}\,dxdy \to 0,$$ and so from we get $$\label{m2}
\begin{aligned}
\frac{1}{2}\int \int_{\mathcal{Q}}&\frac{J_p(u_n(x)-u_n(y))(u_n^+(x)-u_n^+(y))}{|x-y|^{N+ps}}\,dxdy \\
&+ \int_\Omega |u_n|^p\,dx - (\lambda+1) \int_\Omega |u_n^+|^p\,dx
-p\int_\Omega F(x,u_n^+)\,dx\leq M_2
\end{aligned}$$ for some $M_2>0$ and all $n\geq 1$. Adding to we obtain $$\int_\Omega |u_n|^p\,dx
- \int_\Omega |u_n^+|^p\,dx +\int_\Omega f(x,u_n^+)u_n^+ \,dx -p \int_\Omega F(x,u_n^+)\,dx \leq M_3$$ for some $M_3>0$ and all $n\geq 1$, which clearly implies $$\label{m3}
\int_\Omega \sigma(x,u_n^+)\,dx \leq M_3.$$
Now we claim that $(u_n^+)_n$ is bounded in $X$, as well. We argue by contradiction. Up to a subsequence, we assume that $\|u_n^+\|\to \infty$ as $n\to \infty$. Defining $y_n=u_n^+/\|u_n^+\|$, we can assume that $$\label{cdeb}
y_n \rightharpoonup y \quad \text{in } X \quad \text{and } y_n\to y \quad \text{in } L^q(\Omega)$$ for every $q\in (p,p^*_s)$ with $y\geq 0$ in $\Omega$.
First we deal with the case $y\not \equiv 0$. We define $Z(y)=\lbrace x\in\Omega : y(x)=0\rbrace$, and so we have $\left|\Omega \setminus Z(y)\right|>0$ and $u_n^+\to \infty$ for almost every $x\in \Omega \setminus Z(y)$ as $n\to \infty$. By ($f_2$), we have $$\frac{F(x,u_n^+(x))}{\|u_n^+\|^p}=\frac{F(x,u_n^+(x))}{u_n^+(x)^p}y_n(x)^p \to \infty$$ for almost every $x\in \Omega \setminus Z(y)$. From Fatou’s Lemma we get that $$\int_\Omega \liminf_{n\to \infty} \frac{F(x,u_n^+(x))}{\|u_n^+\|^p}\,dx \leq
\liminf_{n\to \infty}\int_\Omega \frac{F(x,u_n^+(x))}{\|u_n^+\|^p}\, dx,$$ and so $$\label{fatu}
\int_\Omega \frac{F(x,u_n^+(x))}{\|u_n^+\|^p}\, dx \to \infty$$ as $n\to \infty$.
Again from we have $$-\frac{1}{2p}[u_n]^p -\frac{1}{p} \int_\Omega |u_n|^p\,dx
+ \frac{\lambda +1}{p} \int_\Omega |u_n^+|^p\,dx+\int_\Omega F(x,u_n^+)\,dx \leq M_4$$ for some $M_4>0$ and $n\geq 1$. From we get $$-\frac{2^{p-2}}{p}([u_n^+]^p+[u_n^-]^p)-\frac{1}{p} \int_\Omega |u_n|^p\,dx
+ \frac{\lambda +1}{p} \int_\Omega |u_n^+|^p\,dx +\int_\Omega F(x,u_n^+)\,dx \leq M_4,$$ and from $$-\frac{2^{p-2}}{p}[u_n^+]^p
+ \frac{\lambda }{p} \int_\Omega |u_n^+|^p\,dx +\int_\Omega F(x,u_n^+)\,dx \leq M_5,$$ for some $M_5>0$ and all $n\geq1$, so that $$\begin{aligned}
\int_\Omega F(x,u_n^+)\,dx \leq M_5+c\|u_n^+\|^p\end{aligned}$$ for some $c>0$ and all $n\geq1$. Dividing by $\|u_n^+\|^p$ and passing to the limit we obtain $$\limsup_{n\to \infty}\int_\Omega \frac{F(x,u_n^+(x))}{\|u_n^+\|^p}\, dx \leq M_6$$ for some $M_6$, which is in contradiction with , and this concludes the case $y\neq 0$.
Now, we deal with the case $y\equiv 0$. We consider the continuous functions $\gamma_n:[0,1]\to{\mathbb{R}}$, defined as $$\gamma_n(t):={\mathscr E}_+(tu_n^+)$$ for any $n\geq 1$. So, there exists $t_n\in[0,1]$ such that $$\label{max}
\gamma_n(t_n)=\max_{t\in[0,1]}\gamma_n(t).$$ Now, fixed $\mu>0$, we define $v_n:=(p\mu)^\frac{1}{p}y_n\in X$. From we get that $v_n \to 0$ in $L^q(\Omega)$ for all $q\in (p,p^*_s)$. From ($f_1$) we know that $$\int_\Omega F(x,v_n(x))\,dx \leq \int_\Omega a(x)|v_n(x)|\,dx + C\int_\Omega |v_n(x)|^r\,dx,$$ and so $$\label{to0}
\int_\Omega F(x,v_n(x))\,dx \to 0$$ as $n\to \infty$. Since $\|u_n^+\|\to \infty$, there exists $n_0\geq 1$ such that $(p\mu)^\frac{1}{p} /\|u_n^+\| \in(0,1)$ for all $n\geq n_0$. Then, from , we have $$\gamma_n(t_n)\geq \gamma_n\left(\frac{(p\mu)^\frac{1}{p}}{\|u_n^+\|} \right)$$ for all $n\geq n_0$. Thus, we get $$\begin{aligned}
&{\mathscr E}_+(t_nu_n^+) \geq {\mathscr E}_+((p\mu)^\frac{1}{p}y_n)={\mathscr E}_+(v_n) \\
& =\frac{1}{2}\mu \int \int_{\mathcal{Q}}\frac{|y_n(x)-y_n(y)|^p}{|x-y|^{N+ps}}\,dxdy
- \frac{\lambda}{p} \int_\Omega v_n^p\,dx - \int_\Omega F(x,v_n(x))\,dx \\
&=\frac{\mu}{2} \|y_n\|^p -\frac{\mu}{2}\int_\Omega y_n^pdx- \frac{2\lambda +1}{2p} \int_\Omega v_n^p\,dx - \int_\Omega F(x,v_n(x))\,dx\\
&=\frac{\mu}{2} - \frac{2\lambda +1}{2p} \int_\Omega v_n^p\,dx - \int_\Omega F(x,v_n(x))\,dx\end{aligned}$$ From and the fact that $v_n\to 0$ in $L^p(\Omega)$, we get that $${\mathscr E}_+(t_nu_n^+)\geq \frac{\mu}{2} + o(1),$$ where $o(1)\to 0$ as $n\to \infty$. Since $\mu$ is arbitrary, we have $$\label{toinf}
\lim_{n\to \infty}{\mathscr E}_+(t_nu_n^+)=+ \infty.$$
On the other hand, since $0\leq t_nu_n^+\leq u_n^+$ for all $n\leq 1$, from ($f_3$) we get $$\label{sigma}
\int_\Omega \sigma(x,t_nu_n^+)\,dx \leq \vartheta \int_\Omega \sigma(x,u_n^+)\,dx + \|\beta^*\|_1$$ for all $n\geq 1$.
In addition, we have that ${\mathscr E}_+(0)=0$; moreover, from we get that $${\mathscr{E}}_+(u_n^+)\leq {\mathscr{E}}_+(u_n)\leq M_1$$ for all $n\ge1$ by . Together with , these two facts imply the existence of $ n_1\geq n_0$ such that $t_n\in (0,1)$ for all $n\geq n_1$, namely $t_n\neq0$ and $t_n\neq1$. Since $t_n$ is a maximum point for $\gamma_n$, we have $$\label{zero}
\begin{aligned}
0&= t_n\gamma_n'(t_n) \\
&=\frac{1}{2}\int \int_{\mathcal{Q}}\frac{|t_nu_n^+(x)-t_nu_n^+(y)|^p}{|x-y|^{N+ps}}\,dxdy \\
&- \lambda \int_\Omega |t_nu_n^+|^p\,dx- \int_{\Omega} f(x,t_nu_n^+(x))t_nu_n^+(x)\,dx.
\end{aligned}$$ Adding to , we get $$\begin{aligned}
\frac{1}{2}\int \int_{\mathcal{Q}}& \frac{|t_nu_n^+(x)-t_nu_n^+(y)|^p}{|x-y|^{N+ps}}\,dxdy \\
&- \lambda \int_\Omega |t_nu_n^+|^p\,dx - p \int_\Omega F(x,t_nu_n^+(x))\,dx \\
&\leq \vartheta \int_\Omega \sigma(x,u_n^+)\,dx + \|\beta^*\|_1,\end{aligned}$$ which is $$p{\mathscr E}_+(t_nu_n^+)\leq \vartheta \int_\Omega \sigma(x,u_n^+)\,dx + \|\beta^*\|_1.$$ So, from , we get $$\label{toinf2}
\lim_{n\to\infty}\int_\Omega \sigma(x,u_n^+)\,dx = \infty.$$ Comparing and we obtain a contradiction, and so the claim follows.
In conclusion, we have proved that $(u_n^+)_n$ is bounded in $X$, so from and we have that $(u_n)_n$ is bounded in $X$. Hence, we can assume that $$\label{cdeb2}
u_n \rightharpoonup u \quad \text{in } X \quad \text{and } u_n\to u \quad \text{in } L^q(\Omega)$$ for every $q\in (p,p^*_s)$ as $n\to \infty$. Taking $h=u_n-u$ in , we have $$\label{vss}
\begin{aligned}
&\left|\frac{1}{2}\int \int_{\mathcal{Q}}\frac{|u_n(x)-u_n(y)|^p}{|x-y|^{N+ps}}\,dxdy \right. \\
&-\frac{1}{2}\int \int_{\mathcal{Q}}\frac{J_p(u_n(x)-u_n(y))(u(x)-u(y))}{|x-y|^{N+ps}}\,dxdy
+\int_\Omega |u_n|^{p-2}u_n(u_n-u) \\
&\left.- (\lambda+1)\int_\Omega |u_n^+|^{p-2}u_n^+(u_n-u) \,dx
- \int_{\Omega} f(x,u_n^+)(u_n-u)\,dx\right|\leq\varepsilon_n.
\end{aligned}$$ From ($f_1$) and , we have $$\int_{\Omega} f(x,u_n^+(x))(u_n(x)-u(x))\,dx \to 0,$$ $$\int_\Omega |u_n|^{p-2}u_n(u_n-u)\to 0$$ and $$\int_\Omega |u_n^+|^{p-2}u_n^+(u_n-u)\to 0$$ as $n\to \infty$. Passing to the limit in , we get $$\begin{aligned}
&\int \int_{\mathcal{Q}}\frac{|u_n(x)-u_n(y)|^p}{|x-y|^{N+ps}}\,dxdy \\
&-\int \int_{\mathcal{Q}}\frac{J_p(u_n(x)-u_n(y))(u(x)-u(y))}{|x-y|^{N+ps}}\,dxdy \to 0\end{aligned}$$ as $n\to \infty$. From Proposition \[s+\] we can conclude that $u_n \to u$ in $X$ and this concludes the proof that ${\mathscr E}_+$ satisfies $(C)_c$ for every $c\in {\mathbb{R}}$.
Proceeding analogously, we have that ${\mathscr{E}}_-$ satisfies $(C)_c$ for every $c\in {\mathbb{R}}$, as well.
Now we are ready to give the proof Theorem \[thlc\].
First, we want to apply Theorem \[critp\] to ${\mathscr{E}}_+$. So, as before, let $(\lambda_m)_m$ be the sequence of eigenvalues defined in . As in the proof of Theorem \[teolink\], there exists $m\geq 1$ such that $\lambda_m \leq 2\lambda+1 < \lambda_{m+1}$, and we use the same two symmetric closed cones $C_m^-$ and $C_m^+$ with $C_m^-\cap C_m^+ =\lbrace0\rbrace$. By Theorem \[index\] we also have $$i(C_m^-\setminus \lbrace0\rbrace)= i(X\setminus C_m^+)=m.$$ In a similar way to the proof of Theorem \[teolink\], by ($f_1$), ($f_4$) and taking $u\in C_m^+$ we have $$\begin{aligned}
{\mathscr{E}}_+(u)&\geq\frac{1}{2p}\|u\|^p -\frac{2\lambda+1}{2p}\int_\Omega |u^+|^pdx
-\frac{{\varepsilon}}{2p} \int_\Omega |u^+|^pdx-C_{\varepsilon}\int_\Omega |u^+|^qdx\\
&\geq\frac{1}{2p}\|u\|^p -\frac{2\lambda+1}{2p}\int_\Omega |u|^pdx
-\frac{{\varepsilon}}{2p} \int_\Omega |u|^pdx-C_{\varepsilon}\int_\Omega |u|^qdx\\
&\geq\frac{1}{2p}\|u\|^p -\frac{1}{2p\lambda_{m+1}}\left(2\lambda+1+{\varepsilon}\right)[u]^p-C_{\varepsilon}\int_\Omega |u|^qdx\\
& \geq\frac{1}{2p}\left(1-\frac{2\lambda+1+{\varepsilon}}{\lambda_{m+1}}\right)\|u\|^p -C\|u\|^q
\end{aligned}$$ for some $C>0$. So there exists $r_+>0$ and $\alpha>0$ such that, if $\|u\|=r_+$ then ${\mathscr{E}}_+(u)\geq \alpha$.
On the other hand, taking $u\in C_m^-$, $e\in X\setminus C_m^-$ with $e^+\neq 0$ and $t>0$, from ($f_2$) we get $$\begin{aligned}
&{\mathscr{E}}_+(u+te) \leq \frac{1}{2p}\|u+te\|^p -\frac{2\lambda+1}{2p}\int_\Omega |(u+te)^+|^pdx -\int_ \Omega F(x,(u+te)^+)\,dx\\
&\leq \frac{1}{2p}\|u+te\|^p \left( 1- \int_ \Omega \frac{F(x,(u+te)^+)}{((u+te)^+)^p} \frac{((u+te)^+)^p}{\|u+te\|^p}\,dx \right)
\to -\infty\end{aligned}$$ as $t \to +\infty$. So, there exists $r_->r_+$ such that ${\mathscr{E}}_+(u)\leq 0$ when $u\in C_m^-+{\mathbb{R}}^+e$ and $\|u\|\geq r_-$.
Again, we define $D_-,S_+,Q$ and $H$ as in Theorem \[geom\]. By Corollary \[corgeom\] we have that $(Q,D_-\cup H)$ links $S_+$ cohomologically in dimension $m+1$ over $\mathbb{Z}_2$. In particular, $(Q,D_-\cup H)$ links $S_+$. In addition, ${\mathscr{E}}_+$ is bounded on $Q$, ${\mathscr{E}}_+(u)\leq 0$ for every $u\in D_-\cup H$ and ${\mathscr{E}}_+(u)\geq \alpha>0$ for every $u\in S_+$. Moreover, by Proposition \[C\] $(C)_c$ holds as well.
By Theorem \[critp\], ${\mathscr{E}}_+$ admits a critical value $c\geq \alpha$, hence a critical point $u$ with ${\mathscr{E}}_+(u)>0$. In particular, we have $$\begin{aligned}
0&=-\frac{1}{2} \int \int_{\mathcal{Q}}\frac{J_p(u(x)-u(y))(u^-(x)-u^-(y))}{|x-y|^{N+ps}}\,dxdy - \int_\Omega |u|^{p-2}uu^-\,dx\\
&+(\lambda+1) \int_\Omega |u^+|^{p-2}u^+u^-\,dx +\int_ \Omega f(x,u^+)u^-\,dx \\
&= -\frac{1}{2}\int \int_{\mathcal{Q}}\frac{J_p(u(x)-u(y))(u^-(x)-u^-(y))}{|x-y|^{N+ps}}\,dxdy + \int_\Omega (u^-)^pdx.\end{aligned}$$ From we get $$0\geq \int\int_{\mathcal{Q}}\frac{|u^-(x)-u^-(y)|^p}{|x-y|^{N+ps}}\, dxdy + \int_\Omega (u^-)^pdx$$ so that $u^-\equiv 0$ and $u\geq 0$. As a consequence, ${\mathscr{E}}_+(u)={\mathscr{E}}(u)$, and so $u\geq 0$ is a nontrivial solution of .
Arguing in the same way for ${\mathscr{E}}_-$, we can find a nontrivial negative solution $v$ for .
By the maximum principle (see, for instance, [@DPRS] and [@MPV] for the Robin problem and also [@musina] for some linear cases), we can conclude that $u>0$ and $v<0$ a.e. in ${\mathbb{R}}^N$.
A problem with linear growth {#secsaddle}
============================
In this section we consider the problem $$\label{psad}
\begin{cases}
(-\Delta)^s_p u = g(x,u) \quad $ in $ \Omega,
\\
\mathscr{N}_{s,p}u=0 \quad \quad $ in $ {\mathbb{R}}^N \setminus \overline{\Omega},
\end{cases},$$ where $\Omega$ is as before and $g:\Omega\times {\mathbb{R}}\to {\mathbb{R}}$ is a Carathéodory function with $p-$linear growth; namely, there exist $a\in L^{p'}(\Omega)$ and $b\in {\mathbb{R}}$ such that $$\label{gbound}
|g(x,t)|\leq a(x)+b|t|^{p-1}$$ for every $t\in {\mathbb{R}}$ and for a.e. $x\in \Omega$.
As usual, we define the functional $$I(u):=\frac{1}{2p}\int\int_{\mathcal{Q}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dxdy -\int_\Omega G(x,u)\,dx$$ so that every critical point of $I$ is a weak solution of .
In order to state our result, we need to introduce $$\label{asu}
\overline{\alpha}(x):=\limsup_{|t|\to \infty}\frac{g(x,t)}{|t|^{p-2}t}$$ for a.e. $x \in \Omega$. Then we have:
\[thsad\] Assume . If $\overline{\alpha}(x)<\lambda_1=0$, then problem admits a weak solution.
In this case we shall obtain one solution by applying the Weierstrass Theorem to $I$.
First, we claim that $$\label{lsup1}
\limsup_{|t|\to \infty}\frac{G(x,t)}{|t|^p}\leq \frac{\overline{\alpha}(x)}{p},$$ where $G(x,t):= \int_0^t g(x,\tau)\, d\tau$. By , for every $\varepsilon>$ there exists $K>0$ such that $$\frac{g(x,t)}{t^{p-1}}<\overline{\alpha}(x)+\varepsilon$$ for $t\geq K$ and a.e. $x\in \Omega$. Reasoning in a similar way for $t<0$ and integrating gives $$G(x,t)\leq \frac{\overline{\alpha}(x)+\varepsilon}{p}(|t|^p-K^p)+\max\left\{G(x,K),G(x,-K)\right\}$$ for $|t|\geq K$. Hence, $$\limsup_{|t|\to \infty}\frac{G(x,t)}{|t|^p}\leq \frac{\overline{\alpha}(x)}{p}$$ as claimed.
Now we want to prove that implies that $$\label{toinfs}
\liminf_{\|u\|\to \infty}\frac{I(u)}{\|u\|^p}>0.$$ Indeed, take a sequence $(u_n)_n$ in $X$ such that $\|u_n\|\to \infty$. Up to a subsequence, we can assume that $v_n:=\frac{u_n}{\|u_n\|}$ converges to some $u$ weakly in $X$ and strongly in $L^p(\Omega)$. Moreover, $\|u\|\leq 1$, and also $$\frac{G(x,u_n)}{\|u_n\|^p}\leq \frac{a(x)|u_n|+b|u_n|^p/p}{\|u_n\|^p}\to \frac{b}{p}|u|^p$$ in $L^1(\Omega)$ as $n\to \infty$. By the generalized Fatou Lemma we have $$\label{FL}
\limsup_{n\to \infty}\int_\Omega \frac{G(x,u_n)}{\|u_n\|^p}\, dx
\leq \int_\Omega \limsup_{n\to \infty}\frac{G(x,u_n)}{\|u_n\|^p}\, dx.$$ If $(u_n(x))_n$ is bounded, $$\frac{G(x,u_n(x))}{\|u_n\|^p}\to 0,$$ while if $|u_n(x)|\to \infty $, $$\limsup_{n\to \infty} \frac{G(x,u_n(x))}{\|u_n\|^p}
=\limsup_{n\to \infty} \frac{G(x,u_n(x))}{|u_n(x)|^p} \frac{|u_n(x)|^p}{\|u_n\|^p}\leq \frac{\overline{\alpha}(x)}{p}|u(x)|^p\leq 0.$$ In both cases $$\limsup_{n\to \infty}\int_\Omega \frac{G(x,u_n)}{\|u_n\|^p}\,dx\leq 0,$$ but when $u\ne0$, we have $$\label{aggiunta}
\limsup_{n\to \infty}\int_\Omega \frac{G(x,u_n)}{\|u_n\|^p}\,dx<0.$$ Therefore, if $u\neq0$ in $\Omega$, we have $$\liminf_{n\to \infty}\frac{I(u_n)}{\|u_n\|^p}\geq -\int_\Omega \frac{G(x,u_n)}{\|u_n\|^p}\,dx,$$ and so by we get $$\label{aggiunta2}
\liminf_{n\to \infty}\frac{I(u_n)}{\|u_n\|^p}>0.$$ On the other hand, if $u\equiv0$ in $\Omega$, $$\frac{I(u_n)}{\|u_n\|^p}=\frac{1}{2p}-\int_\Omega\frac{|u_n|^p}{\|u_n\|^p}dx -\int_\Omega\frac{G(x,u_n)}{\|u_n\|^p}dx,$$ and so holds also in this case.
Since holds for every diverging sequence, holds, as well.
In conclusion, it is easy to show that $I$ is lower semicontinuous, while it is coercive from . So we can apply the Weierstrass Theorem to find a minimum for $I$, which is a solution of problem .
Acknowledgments {#acknowledgments .unnumbered}
===============
The first author is Member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) “F. Severi”. He is supported by the MIUR National Research Project [*Variational methods, with applications to problems in mathematical physics and geometry*]{} (2015KB9WPT009) and by the FFABR “Fondo per il finanziamento delle attività base di ricerca” 2017.
The second author is is Member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) “F. Severi”.
[99]{}
B. Barrios, L. Montoro, I. Peral, F. Soria. Neumann conditions for the higher order $s-$fractional Laplacian $(−\Delta)^su$ with $s > 1$. *Nonlinear Anal.*, to appear.
M. Degiovanni, S. Lancellotti. Linking over cones and nontrivial solutions for $p-$Laplace equations with $p-$superlinear nonlinearity. [*Ann. I. H. Poincaré Anal. Non Linéaire*]{} [**24**]{} (2007), 907-919.
L.M. Del Pezzo, J.D. Rossi, A.M. Salort. Fractional Eigenvalue problems that approximate Steklov eigenvalue problems. [*Proc. Roy. Soc. Edinburgh Sect.*]{} A [**148**]{} (2018), no. 3, 499-516.
S. Di Pierro, X. Ros-Oton, E. Valdinoci. Nonlocal problems with Neumann boundary conditions. *Rev. Mat. Iberoam.* [**[33]{}**]{} (2017), no. 2, 377–416.
E.R. Fadell, P.H. Rabinowitz. Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. *Invent. Math.* [**[45]{}**]{} (1978), no. 2, 139-174.
M. Frigon. On a new notion of linking and application to elliptic problems at resonance. [*J. Differential Equations* ]{} [**153**]{} (1999),no. 1, 96-120.
C. Liu, Y. Zheng. Linking solutions for $p-$Laplace equations with nonlinear boundary conditions and indefinite weight, *Calc. Var. Partial Differential Equations* [**41**]{} (2011), 261–284.
J.M. Mazón, J.D. Rossi, J. Toledo. Fractional $p-$Laplacian evolution equations, *J. Math. Pures Appl. (9)* [**105**]{} (2016), 810–844.
G. Molica Bisci, V. Radulescu, R. Servadei. *Variational methods for nonlocal fractional problems*. Encyclopedia Math. Appl. 162. Cambridge University Press, Cambridge, 2016.
D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3, 379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition. *NoDEA Nonlinear Differential Equations Appl.* [**19**]{} (2012), no. 3, 299-301.
D. Mugnai, N.S. Papageorgiou. Wang’s multiplicity result for superlinear $(p,q)-$equations without the Ambrosetti-Rabinowitz condition. *Trans. Amer. Math. Soc.* [**366**]{} (2014), 4919-4937.
D. Mugnai, A. Pinamonti, E. Vecchi. Towards a Brezis-Oswad-Type result for fractional problems with Robin boundary conditions, to appear in *Calc. Var. Partial Differential Equations*.
D. Mugnai, E. Proietti Lippi. Neumann fractional $p-$Laplacian: eigenvalues and existence results. [*Nonlinear Anal.*]{} [**188**]{} (2019), 455-474
R. Musina, A.I. Nazarov. Strong maximum principles for fractional Laplacians, [*Proc. Roy. Soc. Edinburgh Sect.*]{} A [**149**]{} (2019), 1223–1240.
K. Perera, A. Szulkin. $p-$Laplacian problems where the nonlinearity crosses an eigenvalue, *Discrete Contin. Dyn. Syst.* [**13**]{} (2005), 743–753.
K. Perera, Y. Yang, Z. Zhang. Asymmetric critical $p-$Laplacian problems, *Calc. Var. Partial Differential Equations* [**57**]{} (2018), no.5 Art. 131, 18 pp.
P. Piersanti, P. Pucci. Existence theorems for fractional $p-$Laplacian problems. *Anal. Appl. (Singap.)* [**15**]{} (2017), 607–640.
R. Servadei, E. Valdinoci. Variational methods for non-local operators of elliptic type. *Discrete Contin. Dyn. Syst.* [**33**]{} (2013), 2105–2137.
M. Warma. The fractional Neumann and Robin boundary condition for the fractional $p-$Laplacian on open sets, *NoDEA Nonlinear Differential Equations Appl.* [**23**]{} (2016), 1–46.
|
---
abstract: 'In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit everywhere defined unbounded non-closable $nth$ roots of the identity operator as well as the zero operator. We also show a non-closable unbounded operator without any non-closable square root. Among other consequences, we have a way of finding everywhere defined bijective operators, everywhere defined operators which are surjective without being injective and everywhere defined operators which are injective without being surjective. Some related results on nilpotence are also given.'
address: 'Department of Mathematics, Laboratoire d’analyse mathématique et applications, University of Oran 1, Ahmed Ben Bella, B.P. 1524, El Menouar, Oran 31000, Algeria.'
author:
- Mohammed Hichem Mortad
title: 'Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results'
---
[^1]
Introduction {#introduction .unnumbered}
============
First, while we will recall most of the needed notions for readers’ convenience, we also assume readers have some familiarity with other very standard concepts and results of operator theory. Some useful references are [@Conway-OPERATOR-TH-AMS-GSM-2000], [@Mortad-Oper-TH-BOOK-WSPC], [@SCHMUDG-book-2012], [@tretetr-book-BLOCK] and [@Weidmann].
Let $H$ be a Hilbert space and let $B(H)$ be the algebra of all bounded linear operators defined from $H$ into $H$.
If $S$ and $T$ are two linear operators with domains $D(S)\subset H$ and $D(T)\subset H$ respectively, then $T$ is said to be an extension of $S$, written as $S\subset T$, if $D(S)\subset D(T)$ and $S$ and $T$ coincide on $D(S)$. The restriction of some operator $T$ to some subspace $M$ is denoted by $T_M$.
The product $ST$ and the sum $S+T$ of two operators $S$ and $T$ are defined in the usual fashion on the natural domains:
$$D(ST)=\{x\in D(T):~Tx\in D(S)\}$$ and $$D(S+T)=D(S)\cap D(T).$$
When $\overline{D(T)}=H$, we say that $T$ is densely defined. In this case, the adjoint $T^*$ exists and is unique.
An operator $T$ is called closed if its graph is closed in $H\oplus
H$. $T$ is called closable if it has a closed extension. Equivalently, this signifies that for each sequence $(x_n)$ in $D(T)$ such that $x_n\to 0$ and $Tx_n\to y$, then $y=0$. If $T$ is densely defined, then $T$ is closable if and only if $D(T^*)$ is dense. The smallest closed extension of $T$ is called its closure, and it is denoted by $\overline{T}$. When $T$ is closable, then $\overline{T}=(T^*)^*$. Recall also that if $T$ is a bounded operator on some domain $D(T)$, then $T$ is closed if and only if $D(T)$ is closed (see e.g. Theorem 5.2 in [@Weidmann]).
If $T$ is densely defined, we say that $T$ is self-adjoint when $T=T^*$; symmetric if $T\subset T^*$; normal if $T$ is *closed* and $TT^*=T^*T$. A symmetric operator $T$ is called positive if $$<Tx,x>\geq 0, \forall x\in D(T).$$ Notice that unlike positive operators in $B(H)$, unbounded positive operators need not be self-adjoint.
In the event of the density of all of $D(S)$, $D(T)$ and $D(ST)$, then $$T^*S^*\subset (ST)^*,$$ with the equality occurring when $S\in B(H)$. Also, when $S$, $T$ and $S+T$ are densely defined, then $$S^*+T^*\subset (S+T)^*,$$ and the equality holding again if $S\in B(H)$.
The real and imaginary parts of a densely defined operator $T$ are defined respectively by $${\operatorname{Re}}T=\frac{T+T^*}{2} \text{ and } {\operatorname{Im}}T=\frac{T-T^*}{2i}.$$ Clearly, if $T$ is closed, then ${\operatorname{Re}}T$ is symmetric but it is not always self-adjoint (it may even fail to be closed).
Let $T$ be a densely defined operator with domain $D(T)\subset H$. If there exist densely defined symmetric operators $A$ and $B$ with domains $D(A)$ and $D(B)$ respectively and such that $$T=A+iB \text{ with } D(A)=D(B),$$ then $T$ is said to have a Cartesian decomposition ([@Ota-normal; @extensions-BPAS]).
A densely defined operator $T$ admits a Cartesian decomposition if and only if $D(T)\subset D(T^*)$. In this case, $T=A+iB$ where $$A={\operatorname{Re}}T \text{ and } B={\operatorname{Im}}T.$$
Let $A$ be an injective operator (not necessarily bounded) from $D(A)$ into $H$. Then $A^{-1}: {\operatorname{ran}}(A)\rightarrow H$ is called the inverse of $A$ with domain $D(A^{-1})={\operatorname{ran}}(A)$.
If the inverse of an unbounded operator is bounded and everywhere defined (e.g. if $A:D(A)\to H$ is closed and bijective), then $A$ is said to be boundedly invertible. In other words, such is the case if there is a $B\in B(H)$ such that $$AB=I\text{ and } BA\subset I.$$ Clearly, if $A$ is boundedly invertible, then it is closed. Recall also that $T+S$ is closed if $S\in B(H)$ and $T$ is closed, and that $ST$ is closed if $S^{-1}\in B(H)$ and $T$ is closed or if $S$ is closed and $T\in B(H)$.
Based on the bounded case and the previous definition, we say that an unbounded $A$ with domain $D(A)\subset H$ is right invertible if there exists an everywhere defined $B\in B(H)$ such that $AB=I$; and we say that $A$ is left invertible if there is an everywhere defined $C\in B(H)$ such that $CA\subset I$. Clearly, if $A$ is left and right invertible simultaneously, then $A$ is boundedly invertible.
The spectrum of unbounded operators is defined as follows: Let $A$ be an operator on a complex Hilbert space $H$. The resolvent set of $A$, denoted by $\rho(A)$, is defined by $$\rho(A)=\{\lambda\in{\mathbb{C}}:~\lambda I-A\text{ is bijective and }(\lambda I-A)^{-1}\in B(H)\}.$$ The complement of $\rho(A)$, denoted by $\sigma(A)$, i.e. $$\sigma(A)={\mathbb{C}}\setminus \rho(A)$$ is called the spectrum of $A$.
Clearly, $\lambda\in \rho(A)$ iff there is a $B\in B(H)$ such that $$(\lambda I-A)B=I\text{ and } B(\lambda I-A)\subset I.$$
Also, recall that if $A$ is a linear operator which is not closed, then $\sigma(A)={\mathbb{C}}$.
Kulkrani et al. showed using simple arguments in [@Kulkrani; @et; @al-2008] that $$\sigma(A^2)=[\sigma(A)]^2$$ when $A$ is closed.
Next, we recall the definition of unbounded nilpotent operators. We choose to use Ôta’s definition in [@Ota-nilpotent-idempotent] of nilpotence (S. Ôta gave the definition in the case $n=2$).
\[Ota-nilpotent-idempotent DEFINITIONNN\] Let $T$ be a non necessarily bounded operator with a dense domain $D(T)$. We say that $T$ is nilpotent if $T^n$ is well defined and $$T^n=0 \text{ on }D(T)$$ for some $n\in{\mathbb{N}}$ (hence necessarily $D(T^n)=D(T^{n-1})=\cdots
D(T)$).
Recall the following lemma:
\[ghghhgghghghghghghghghghghghghghghgh\]([@Sebestyen-Stochel-JMAA] or [@Tarcsay-2012-bounded; @D(T)=D(T2)]) If $H$ and $K$ are two Hilbert spaces and if $T:D(T)\subset H\to K$ is a densely defined closed operator, then $$D(T)=D(T^*T)\Longleftrightarrow T\in B(H,K).$$
Thanks to the previous lemma, if $T$ is some densely defined closed nilpotent operator with domain $D(T)\subset D(T^*)\subset H$, then $T\in B(H)$. In particular, if $T$ is a closed densely defined nilpotent symmetric or hyponormal operator, then $T=0$ everywhere on $H$. See [@Frid-Mortad-Dehimi-nilpotence] for a proof and some closely related results. See also [@Tarcsay-2012-bounded; @D(T)=D(T2)].
Now, we give a definition of square roots for general linear operators.
Let $A$ and $B$ be linear operators. Say that $B$ is a square root of $A$ if $B^2=A$ (and so $D(B^2)=D(A)$). More generally, say that $B$ is an $nth$ root of $A$ (where $n\in{\mathbb{N}}$) if $B^n=A$ (and so $D(B^n)=D(A)$).
Notice that for the case $A=0$ on $D(A)$, this includes Definition \[Ota-nilpotent-idempotent DEFINITIONNN\] above. An objection, why not use a definition like $B^2\subset A$? The issue then is that it is quite conceivable to have $D(B^2)=\{0\}$ (or higher powers as well). In such case, $B^2\subset A$ holds trivially whilst $A$ can be anything in $B(H)$! Given the diversity of classes of examples such that $D(B^n)=\{0\}$ for some $n$ (as may be seen in [@Arlinski-Kovalev], [@Brasche-Neidhardt], [@CH], [@Dehimi-Mortad-CHERNOFF], [@Mortad-TRIVIALITY; @POWERS; @DOMAINS] and [@SCHMUDG-1983-An-trivial-domain]), a definition like $B^n\subset A$ would not therefore yield too informative conclusions in many situations.
It is well known that self-adjoint positive operators have a unique positive square root. This does not exclude the fact that a self-adjoint positive operator may well have other square roots. Self-adjoint *positive* square roots of self-adjoint *positive* operators play an important role. For instance, they intervene in the definition of the absolute value of unbounded operators (cf. [@Boucif-Dehimi-Mortad]), and so in the polar decomposition. Recall here that the absolute value of a closed $A$ is given by $|A|=\sqrt{A^*A}$ where $\sqrt{\cdot}$ designates the unique positive square root of $A^*A$, which is positive by the closedness of $A$.
Positive self-adjoint square roots are also present in abstract wave and Schrödinger’s equations (see e.g. [@SCHMUDG-book-2012]). See also [@Mc-Intosh-square-hyperbolic; @PDE]. They, of course, have other utilizations. Here, we confine our attention to arbitrary square (or other) roots.
Finding counterexamples using matrices of non necessarily bounded operators has been a success as demonstrates the recent papers: [@Dehimi-Mortad-CHERNOFF], [@Mortad-TRIVIALITY; @POWERS; @DOMAINS], [@Mortad-Commutators-CEXAMPLES], [@Mortad-paranormal-Daniluk] and [@Mortad-Paranormal-all; @cexas]. Let us recall their definition briefly:
Let $H$ and $K$ be two Hilbert spaces and let $A:H\oplus K\to
H\oplus K$ (we may equally use $H\times K$ instead of $H\oplus K$) be defined by $$\label{matrix reprenstation UNBD EQU}
A=\left(
\begin{array}{cc}
A_{11} & A_{12} \\
A_{21} & A_{22} \\
\end{array}
\right)$$ where $A_{11}\in L(H)$, $A_{12}\in L(K,H)$, $A_{21}\in L(H,K)$ and $A_{22}\in L(K)$ are not necessarily bounded operators. If $A_{ij}$ has a domain $D(A_{ij})$ with $i,j=1,2$, then $$D(A)=(D(A_{11})\cap D(A_{21}))\times (D(A_{12})\cap D(A_{22}))$$ is the natural domain of $A$. So if $(x_1,x_2)\in D(A)$, then $$A\left(
\begin{array}{c}
x_1 \\
x_2 \\
\end{array}
\right)=\left(
\begin{array}{c}
A_{11}x_1+A_{12}x_2\\
A_{21}x_1+A_{22}x_2 \\
\end{array}
\right).$$ As is customary, we tolerate the abuse of notation $A(x_1,x_2)$ instead of $A\left(
\begin{array}{c}
x_1 \\
x_2 \\
\end{array}
\right)$. The generalization to $n\times n$ matrices of operators is also clear.
Note that unlike matrices of everywhere defined bounded operators, not all unbounded operators admit such a decomposition (cf. [@Taylor-Lay-Functional-Analysis]). Readers should also be wary when dealing with products of matrices of (unbounded) operators as they are not always well defined. However, when dealing with everywhere defined (unbounded) operators, all products are possible in this setting.
Recall that the adjoint of $\left(
\begin{array}{cc}
A_{11} & A_{12} \\
A_{21} & A_{22} \\
\end{array}
\right)$ is not always $\left(
\begin{array}{cc}
A^*_{11} & A^*_{21} \\
A^*_{12} & A^*_{22} \\
\end{array}
\right)$ (even when all domains are dense including the main domain $D(A)$) as many known counterexamples show. Nonetheless, e.g. $$\left(
\begin{array}{cc}
A & 0 \\
0 & B \\
\end{array}
\right)^*=\left(
\begin{array}{cc}
A^* & 0 \\
0 & B^* \\
\end{array}
\right)\text{ and } \left(
\begin{array}{cc}
0 & C \\
D & 0 \\
\end{array}
\right)^*=\left(
\begin{array}{cc}
0 & D^* \\
C^* & 0 \\
\end{array}
\right)$$ if $A$, $B$, $C$ and $D$ are all densely defined. See e.g. [@Moller-Szafanriac-matri-unbounded] or [@Wu-CHEN-ADJOINTS!!] for more about the adjoint’s operation of general operator matrices.
The special case of the matrix $\left(
\begin{array}{cc}
A & 0 \\
0 & B \\
\end{array}
\right)$ may be denoted by $A\oplus B$.
Finally, since we will often be dealing with everywhere defined unbounded operators, we give an example (which is well known to most readers):
Let $f:H\to{\mathbb{C}}$ be a *discontinuous* linear functional (this requires the axiom of choice as is known to readers). Let $x_0$ be any non-zero vector in $H$ and define $A:H\to H$ by $$Ax=f(x)x_0.$$
Then, $A$ is clearly unbounded and everywhere defined. Obviously, $A$ is not closable for if it were, the Closed Graph Theorem would give $A\in B(H)$, which is impossible.
As observed above, an operator $A$ is closable iff $D(A^*)$ is dense. There are well known examples in the literature of densely defined unbounded operators $A$ such that even $D(A^*)=\{0\}$. See [@Mortad-TRIVIALITY; @POWERS; @DOMAINS] for some “recent example”. Even a non closable $A$ with $D(A)=H$ could be such that $D(A^*)=\{0\}$. This is a famous example by Berberian which may be consulted on Page 53 in [@Goldberg; @BOOK-UNBD].
Main Results
============
First, we give some examples of square roots as regards closedness. Let $\mathcal{F}_0$ be the restriction of the $L^2({\mathbb{R}})$-Fourier transform to the dense subspace $C_0^{\infty}({\mathbb{R}})$ which denotes here the space of infinitely differentiable functions with compact support. Then, it is well known that $$D(\mathcal{F}_0^2)=\{0\}$$ because any function $f\in C_0^{\infty}({\mathbb{R}})$ such that $\hat f\in
C_0^{\infty}({\mathbb{R}})$ is null. Hence $\mathcal{F}_0$ is an unclosed square root of the trivially closed operator $0$ on $\{0\}$.
On the other hand, there are unclosed operators having closed square roots. For example, on $\ell^2$ define the linear operator $A$ by $$Ax=A(x_n)=(x_2,0,2x_4,0,\cdots,\underbrace{nx_{2n}}_{2n-1},\underbrace{0}_{2n},\cdots)$$ on the domain $$D(A)=\{x=(x_n)\in \ell^2:~(nx_{2n})\in\ell^2\}.$$ We may check that $D(A)$ is dense in $\ell^2$, that $A$ is unbounded and closed.
Then, it can readily be checked that $$A^2=0\text{ on } D(A^2)=D(A)$$ and so $A^2$ is bounded on $D(A)$. Finally, since we know that $A^2=0$ on $D(A)$ and that $D(A)$ is not closed, it follows that $A^2$ is a non closed operator. This example first appeared in [@Ota-range-in-domain-1984].
As another example, take any unbounded closed operator $T$ on a dense domain $D(T)\subset H$. Then setting $$A=\left(
\begin{array}{cc}
0 & T \\
0 & 0 \\
\end{array}
\right),$$ we see that $D(A)=H\oplus D(T)$. Hence $$A^2=\left(
\begin{array}{cc}
0 & 0_{D(T)} \\
0 & 0_{D(T)} \\
\end{array}
\right)=\left(
\begin{array}{cc}
0 & 0_{D(T)} \\
0 & 0 \\
\end{array}
\right)$$ and so $A^2=0$ on $D(A^2)=D(A)$. Hence $A^2$ is unclosed.
Next, we provide an invertible bounded operator without any closable square root.
There is a bounded subnormal invertible operator without any closable square root.
Let $D$ be the annulus $\{z\in{\mathbb{C}}:~r<|z|<R\}$ where $r,R>0$. Let $\mu$ be a planar Lebesgue measure in $D$. Let $L^2(D)$ be the collection of all complex-valued functions which are analytic throughout $D$ and square-integrable w.r.t. $\mu$ (the Bergman space). That is, $f\in L^2(D)$ if $f$ is analytic in $D$ and $\int_D|f(z)|^2d\mu(z)<\infty$. Then $L^2(D)$ is a Hilbert space w.r.t. the inner product $$<f,g>=\int_Df(z)\overline{g(z)}d\mu(z).$$
Define now an analytic position operator $A: L^2(D)\to L^2(D)$ by $$Af(z)=zf(z).$$ Then $A$ is bounded, subnormal, invertible and without any (bounded) square root. In addition, $A$ does not have any bounded square root. This is utterly non trivial and was established by Halmos et al. in [@Halmos; @et; @al; @SQ; @ROOT; @Invert; @NO!].
Assume now that there is a closable operator $B$ such that $B^2=A$. Then $$D(B^2)=D(A)=L^2(D)\subset D(B)$$ whereby $D(B)=L^2(D)$. The Closed Graph Theorem then tells us that $B$ is bounded and so $A$ would possess a square root, which contradicts the first part of the proof. Therefore, $A$ does not possess any closable square root.
As is known to readers, there are finite square matrices which are rootless, i.e. not having any root of any order. Such is the case for instance with the matrix $\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)$. Next, we present two examples of everywhere defined, unclosable and unbounded nilpotent operators. In other words, we supply two non-closable square roots of $0\in B(H)$.
\[stochel’s example\] Let $f:H\to{\mathbb{C}}$ be a *discontinuous* linear functional (where we allow $\dim H\geq
\aleph_0$). Let $e$ be a normalized vector in $\ker f$. Now, define a linear operator $A$ on $H$ by $D(A)=H$ and $Ax=f(x)e$ for each $x\in H$. Then for $x\in H$ $$A^2x=A(f(x)e)=f(x)f(e)e=0.$$
Thus, $A^2=0$ everywhere on the whole of $H$. Accordingly, $A^2$ is self-adjoint! Now, $A$ cannot be closable. A way of seeing this, is that if $A$ were closable, the Closed Graph Theorem would make it bounded.
An alternative way of seeing that the operator $A$ is not closable is to invoke Proposition 4.5 of [@Stochel-TWO; @STOCHELS!]. There, the writers showed that $D(A^{*})=\{e\}^{\perp}$ (and $A^*$ is the zero operator on $\{e\}^{\perp}$), that is, $A^*$ is not densely defined and consequently, $A$ is not closable.
The second example is simple once readers are familiar with matrices of operators.
\[unclosable A A2=0 matrices of oper EXA\] Consider any unbounded non-closable operator $B$ with domain $D(B)=H$ (and so $D(B^*)$ is not dense). Then set $$A=\left(
\begin{array}{cc}
0 & B \\
0 & 0 \\
\end{array}
\right)$$ and so $D(A)=H\oplus H$. Clearly, $A$ is unbounded. Since $$A^*=\left(
\begin{array}{cc}
0 & 0 \\
B^* & 0 \\
\end{array}
\right),$$ we see that $D(A^*)=D(B^*)\oplus H$ is not dense in $H\oplus H$, making $A$ non-closable. Moreover, $$D(A^2)=\{(x,y)\in H\times H: A(x,y)=(By,0)\in H\times H\}=H\oplus H.$$ Hence, we see that $$A^2=\left(
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}
\right)$$ (everywhere on $H\oplus H$).
As observed above, the foregoing two examples constitute non-closable unbounded square roots of $0\in B(H)$. Incidently, some compact operators may have unbounded square roots. Indeed, once we have seen one, non-zero compact operators may also have unbounded square roots. More precisely, we have:
There are many compact operators having unbounded square roots.
Let $T$ be an unbounded square root of $0\in B(H)$. Letting $B\in B(H)$ to be a square root of some compact operator $C$, we see that $T\oplus B$ is an unbounded square root of the non-zero compact operator $0\oplus C$.
Now, we treat the general case.
\[Tn=0 everywhere on H T unclosable\] Let $n\in{\mathbb{N}}$ be given. There are infinitely many everywhere defined non-closable unbounded operators $T$ such that $T^n=0$ *everywhere* on $H$ while $T^{n-1}\neq 0$.
Let $B$ be an everywhere defined unbounded unclosable operator such that $B^2\neq0$. Perhaps some more details are desirable. Let $A$ be a non-closable operator such that $D(A)=H$ and $A^2=0$ as in Examples \[stochel’s example\] & \[unclosable A A2=0 matrices of oper EXA\]. Then set $$B=\left(
\begin{array}{cc}
A & 0 \\
0 & I \\
\end{array}
\right)$$ where $I$ is the identity operator on $H$ (hence $D(B)=H\oplus H$). It is seen that $B$ is unclosable, unbounded and $$B^2=\left(\begin{array}{cc}
A & 0 \\
0 & I \\
\end{array}
\right)\left(
\begin{array}{cc}
A & 0 \\
0 & I \\
\end{array}
\right)=\left(
\begin{array}{cc}
A^2 & 0 \\
0 & I \\
\end{array}
\right)=\left(
\begin{array}{cc}
0 & 0 \\
0 & I \\
\end{array}
\right)\neq \left(
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}
\right).$$
Now, define $$T=\left(
\begin{array}{ccc}
0 & B & B \\
0 & 0 & B \\
0 & 0 & 0 \\
\end{array}
\right)$$ and so $D(T)=H\oplus H\oplus H$. Clearly, $T$ is unbounded and not closable. Then $$T^2=\left(
\begin{array}{ccc}
0 & 0 & B^2 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}
\right)$$ and $$T^3=\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}
\right)$$ where all zeros are in $B(H)$.
To deal with the general case, define on $H\oplus H\oplus
\cdots\oplus H$ ($n$ copies of $H$) the unbounded non-closable $$T=\left(
\begin{array}{cccccc}
0 & B & B & \cdots & \cdots & B \\
0 & 0& B & B & & \vdots\\
\vdots & & 0 & B & \ddots & \vdots \\
\vdots & & & \ddots & \ddots & B \\
0 & & & & 0 & B \\
0 &0 & \cdots & \cdots & 0 & 0 \\
\end{array}
\right).$$ Clearly, $D(T)=H\oplus H\oplus \cdots \oplus H$, and as above, it may be checked that $$T^{n-1}\neq 0 \text{ whereas }T^n=0$$ everywhere on $D(T^n)=H\oplus H\oplus \cdots \oplus H$. To obtain infinitely many of them, just change each $B$ by $\alpha B$ where $\alpha\in{\mathbb{R}}$, say.
In an interesting preprint, I. D. Mercer [@Mercer] gave a way of constructing $n\times n$ matrices $B$ such that none of $B,B^2,\cdots, B^{n-1}$ has any zero entry yet $B^n=0$. The general form (though not explicitly indicated in that preprint) is given by:
$$B=\left(
\begin{array}{ccccccc}
2 & 2 & \cdots & \cdots & 2& 1-n \\
n+2 & 1 & \cdots & \cdots &1& -n \\
1 & n+2 & 1 & \cdots & 1 &\vdots \\
\vdots & 1 & \ddots & \ddots & \vdots &\vdots\\
\vdots & \vdots & \ddots & \ddots & 1&-n \\
1& 1 & \cdots & 1& n+2& -n\\
\end{array}
\right)$$
By way of an example, consider the $6\times 6$ matrix: $$B=\left(
\begin{array}{cccccc}
2 & 2 & 2 & 2 & 2 &-5 \\
8 & 1 & 1 & 1 &1 &-6 \\
1 & 8 & 1 & 1 & 1&-6 \\
1 & 1 & 8 & 1 & 1&-6\\
1 & 1 & 1 & 8 &1 &-6 \\
1 & 1 & 1 & 1 & 8& -6\\
\end{array}
\right).$$
Then it may be checked that $B^p\neq 0_{\mathcal{M}_6}$ and none of their entries is null for all $1\leq p\leq 5$, yet $B^6=0_{\mathcal{M}_6}$.
Let $n\in{\mathbb{N}}$. There is a matrix of operators $A$ of size $n\times n$ whose entries are either all in $B(H)$ or all unbounded and unclosable and defined on all of $H$, such that all entries of all $A^p$ ($1\leq p\leq n-1$) are non zero operators, yet $A^n=0$ everywhere on $H\oplus H\oplus \cdots \oplus H$.
The proof remains unchanged whether the entries are all in $B(H)$ or are all unbounded, unclosable and everywhere defined on $H$. So let $T$ be any linear operator defined on all of $H$ such that $T^{p}\neq 0$ for $1\leq p\leq n-1$ (as in Theorem \[Tn=0 everywhere on H T unclosable\]). Set $$A=\left(
\begin{array}{ccccccc}
2T & 2T & \cdots & \cdots & 2T& (1-n)T \\
(n+2)T & T & \cdots & \cdots &T& -nT \\
T & (n+2)T & T & \cdots & T &\vdots \\
\vdots & T & \ddots & \ddots & \vdots &\vdots\\
\vdots & \vdots & \ddots & \ddots & T &-nT \\
T& T & \cdots & T& (n+2)T & -nT\\
\end{array}
\right)$$ which is defined on all $H\oplus H\oplus \cdots \oplus H$ ($n$ copies of $H$). Then none of the entries of $A^p$ with $1\leq p\leq
n-1$ is the zero operator yet $A^n=0$ everywhere on $H\oplus H\oplus
\cdots \oplus H$.
By borrowing an idea from [@Conway-Morrel] used for bounded operators, we may give a way of finding non-closable roots of some particular non-closable operators:
Let $T$ be a non-closable operator such that $D(T)=H$. Then $T\oplus
T\oplus\cdots \oplus T$, defined on $H\oplus H\oplus\cdots \oplus H$ ($n$ times), has always non-closable $nth$ roots.
Let $I$ be the identity operator on $H$. An unclosable everywhere defined $nth$ root of $T\oplus T\oplus\cdots \oplus T$ is given by the $n\times n$ matrix of operators: $$S:=\left(
\begin{array}{ccccc}
0 & 0 & \cdots & 0 & T \\
I & 0 & \cdots & 0 & 0 \\
0 & I & \cdots & 0 & 0 \\
\vdots & \ddots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & I & 0 \\
\end{array}
\right).$$ Indeed, $S$ is clearly unclosable on $D(S)=H\oplus H\oplus\cdots
\oplus H$. Moreover, $$S^n=T\oplus T\oplus\cdots \oplus T,$$ as wished.
Now, we give a non-closable operator without any closable square root.
There exists a non-closable operator without any closable square root whatsoever.
Let $A$ be a non-closable operator such that $D(A)=H$ and $A^2=0$ everywhere on $H$ as in Example \[stochel’s example\] (or Example \[unclosable A A2=0 matrices of oper EXA\]). Assume now that $B$ is a closable square root of $A$, that is, $B^2=A$. Hence $B^4=A^2=0$ everywhere on $H$. Therefore, $$H=D(A^2)=D(B^4)\subset D(B).$$
This means that $B$ would be everywhere defined on $H$, and by remembering that $B$ is closable, it would ensue that $B\in B(H)$ and so $B^2\in B(H)$ too. Hence $A$ would equally be in $B(H)$, and this is the sought contradiction. Accordingly, the non-closable $A$ does not possess any closable square root.
It is known that $B\in B(H)$ is a square root of some $A\in B(H)$ if and only if $B^*$ is a square root of $A^*$. This is not always the case for unbounded operators.
There is a densely defined linear operator having a square root $T$ but $T^*$ is not a square root of its adjoint.
We provide two examples, one unclosable and one closed.
Consider a non closable $A$ with $D(A)=H$ and $D(A^*)=\{0\}$ (Berberian’s example recalled above). Set $$T=\left(
\begin{array}{cc}
0 & A \\
0 & 0 \\
\end{array}
\right)$$ where $D(T)=H\oplus H$. Then $T$ is unbounded and $T^2=0$ on $H\oplus H$, i.e. $T$ is a square root of $0\in B(H\oplus H)$. However, $T^*$ is not a square root of $0^*=0$ for $$T^*=\left(
\begin{array}{cc}
0 & 0 \\
A^* & 0 \\
\end{array}
\right)$$ is defined on $D(T^*)=\{0\}\oplus H$. Clearly $$D[(T^*)^2]=\{(0,y)\in D(T^*):T^*(0,y)\in D(T^*)\}=D(T^*).$$ Therefore, $T^*$ cannot be a square root of $0\in B(H\oplus H)$ for $$D[(T^*)^2]\neq B(H\oplus H).$$
Another example is to let $A$ to be an unbounded closed operator with domain $D(A)\subset H$ and let $I$ be the identity operator in $H$. Then set $$T=\left(
\begin{array}{cc}
I & A \\
0 & -I \\
\end{array}
\right)$$ and so $T$ is closed. Hence $$T^2=\left(
\begin{array}{cc}
I & A \\
0 & -I \\
\end{array}
\right)\left(
\begin{array}{cc}
I & A \\
0 & -I \\
\end{array}
\right)=\left(
\begin{array}{cc}
I & 0 \\
0 & I_{D(A)} \\
\end{array}
\right):=S.$$ Since $$T^*=\left(
\begin{array}{cc}
I & 0 \\
A^* & -I \\
\end{array}
\right),$$ it ensues that $$T^{*2}=\left(
\begin{array}{cc}
I_{D(A^*)} & 0 \\
0 & I \\
\end{array}
\right)$$ meaning that $T^*$ is not a square root of $S^*=\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right)$.
What about closures? As is guessable, this is not the case either.
There is a densely defined linear operator having a square root $T$ but $\overline{T}$ is not a square root of its closure.
Consider the same example as before. Then, by considering $(T^*)^*$ or else, it is seen that $$\overline{T}=\left(
\begin{array}{cc}
I & \overline{A} \\
0 & -I \\
\end{array}
\right).$$ Therefore, $$\overline{T}^2=\left(
\begin{array}{cc}
I & 0 \\
0 & I_{D(\overline{A})} \\
\end{array}
\right)$$ and so $\overline{T}$ cannot be a square root of $\overline{S}=\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right)$.
Non-closable operators may have closed square roots. This is maybe known to some readers, however, the approach here is different.
There is a densely defined non-closable operator defined formally on $L^2({\mathbb{R}})\oplus L^2({\mathbb{R}})$ possessing a densely defined closed square root.
First, consider two self-adjoint operators $A$ and $B$ defined on $L^2({\mathbb{R}})$ and such that $$D(AB)=\{0\}\text{ and } D(BA)=D(A)$$ where also $B\in B[L^2({\mathbb{R}})]$. This is highly non-trivial and most probably original. Let us proceed to obtain such a pair. Consider the operators $C$ and $A$: $$Cf(x)=e^{\frac{x^2}{2}}f(x)$$ defined on $D(C)=\{f\in L^2({\mathbb{R}}):~e^{\frac{x^2}{2}}f\in L^2({\mathbb{R}})\}$ and $A:=\mathcal{F}^*C\mathcal{F}$, where $\mathcal{F}$ designates the usual $L^2({\mathbb{R}})$-Fourier transform. Clearly $C$ is boundedly invertible (hence so is $A$) and $$Bf(x):=C^{-1}f(x)=e^{\frac{-x^2}{2}}f(x)$$ is defined from $L^2({\mathbb{R}})$ onto $D(C)\subset L^2({\mathbb{R}})$.
We also know that $D(AB)$ is trivial if $D(A)\cap {\operatorname{ran}}(B)$ is so and if $B$ is further assumed to be one-to-one (which is our case here). But, $$D(A)\cap
{\operatorname{ran}}(B)=D(A)\cap D(C)=\{0\},$$ because this is already available to us from [@KOS]. Accordingly $$D(AB)=\{0\}.$$
Since $B$ is everywhere defined and bounded, clearly $$D(BA)=D(A)$$ which is actually dense in $L^2({\mathbb{R}})$.
Now, define $$T=\left(
\begin{array}{cc}
B^2 & BA \\
0 & 0 \\
\end{array}
\right)$$ on $L^2({\mathbb{R}})\oplus D(A)$. Obviously, $T$ is densely defined. Since $B\in B[L^2({\mathbb{R}})]$ and $A$ and $B$ are self-adjoint, it follows that $$T^*=\left[\left(
\begin{array}{cc}
B^2 &0 \\
0 & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
0 & BA \\
0 & 0 \\
\end{array}
\right)\right]^*=\left(\begin{array}{cc}
B^2 &0 \\
0 & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
0 & 0 \\
AB & 0 \\
\end{array}
\right),$$ that is, $$T^*=\left(
\begin{array}{cc}
B^2 & 0 \\
AB & 0 \\
\end{array}
\right).$$ Thus, $$D(T^*)=\{0\}\oplus L^2({\mathbb{R}})$$ which is not dense, i.e. $T$ is not closable.
Let us now exhibit a densely defined closed square root of $T$. Let $$R=\left(
\begin{array}{cc}
B & A \\
0 & 0 \\
\end{array}
\right)$$ be defined on $D(R):=L^2({\mathbb{R}})\oplus D(A)$. Then $R$ is closed on $D(R)$. In addition, $$R^2=\left(
\begin{array}{cc}
B^2 & BA \\
0 & 0 \\
\end{array}
\right)$$ because $$\begin{aligned}
D(R^2)&=\{(f,g)\in L^2({\mathbb{R}})\times D(A):(Bf+Ag,0)\in L^2({\mathbb{R}})\times D(A)\}\\
&=L^2({\mathbb{R}})\oplus D(A)\\
&=D(T).\end{aligned}$$
S. Ôta [@Ota-nilpotent-idempotent] introduced the concept of an unbounded projection or idempotent. Recall that if $T$ is a non necessarily bounded operator with a dense domain $D(T)$, then $T$ is said to be idempotent if $T^2$ is well defined and $$T^2=T \text{ on }D(T).$$
S. Ôta gave an example of an unclosable idempotent but did not show any closed idempotent operator. Here we give a different example of a non-closable idempotent as well as a closed idempotent. These two examples are in a close relationship to the main topic of the paper.
There are non-closable unbounded idempotent operators as well as closed ones.
Let $A$ be an unbounded closed operator with domain $D(A)\subset H$ and let $I$ be the identity operator on all of $H$. Set $$T=\left(
\begin{array}{cc}
I & A \\
0 & 0 \\
\end{array}
\right)$$ and so $D(T)=H\times D(A)$. Then $T$ is densely defined, closed and unbounded. Since $$D(T^2)=\{(x,y)\in H\times D(A):(x+Ay,0)\in H\times D(A)\}=D(T),$$ we see that $$T^2=\left(
\begin{array}{cc}
I & A \\
0 & 0 \\
\end{array}
\right)\left(
\begin{array}{cc}
I & A \\
0 & 0 \\
\end{array}
\right)=\left(
\begin{array}{cc}
I & A \\
0 & 0 \\
\end{array}
\right)=T.$$
In other words $T$ is idempotent. Once we have seen one example, others come to mind (e.g. $A$ may be replaced by $\alpha A$ say where $\alpha\neq0$). For example, let $T$ be such that $T^2=T$. If $U$ is unitary, then $U^*TU$ too is a densely defined closed idempotent. The density of $D(U^*TU)$ is easily seen. Since $TU$ is closed and $U^*$ is invertible, it follows that $U^*TU$ remains closed. Finally, observe that $$(U^*TU)^2=U^*TUU^*TU=U^*T^2U=U^*TU.$$
A similar idea is used to the non closable case. Indeed, define $$T=\left(
\begin{array}{cc}
I & A \\
0 & 0 \\
\end{array}
\right)$$ on $D(T)=H\times D(A)$ where this time $A$ is *not closable*. Then $T$ too is not closable. Indeed, there is a sequence $(x_n)$ in $D(A)$ such that $x_n\to 0$, $Ax_n\to y$ yet $y\neq0$ (by the non closability of $A$). Next, $(0,x_n)\in D(T)$, $(0,x_n)\to (0,0)$ and $T(0,x_n)=(Ax_n,0)\to (y,0)\neq (0,0)$. This proves the non closability of $T$. That $T^2=T$ may be checked as above. Therefore, $T$ is a densely defined non closable idempotent operator.
Now, we treat some related results to nilpotence and invertibility. Let $N\in B(H)$ be nilpotent and let $I\in B(H)$ be the identity operator. Then, it is known that $I\pm N$ are invertible. For example, the inverse of $I-N$ is given by $I+N+N+\cdots+N^p$ if $p+1$ is the index of nilpotence of $N$.
What about unbounded nilpotent operators?
\[with DEHIMI I\] There exist closed as well as non closable nilpotent unbounded operators $N$ such that $I+N$ is not boundedly invertible.
We start with the case of non closable nilpotent operators. Let $N$ be an unbounded non closable operator such that $D(N)=H$ and $N^2=0$ everywhere on $H$. Then, $I+N$ cannot be boundedly invertible for it were, it would ensue that $(I+N)^2$ too is boundedly invertible. However, $$(I+N)^2=I+2N+N^2=I+2N$$ (all full equalities) is not even closable while boundedly invertible operators must be closed.
Consider now the case of a closed nilpotent operator. The simplest example to think of perhaps is: $$N=\left(
\begin{array}{cc}
0 & A \\
0 & 0 \\
\end{array}
\right)$$ defined on $D(N)=H\oplus D(A)$, where $A$ is an unbounded closed operator with domain $D(A)$. If $I_{H\oplus H}$ is the identity on $H\oplus H$, then $$I_{H\oplus H}+N=\left(
\begin{array}{cc}
I & A \\
0 & I \\
\end{array}
\right)$$ is not boundedly invertible as it is not surjective (observe that it is injective).
In both cases above, $I+N$ are invertible.
Let us remain in the context of nilpotence a little longer. Recall that if $N\in B(H)$ is nilpotent, then $$\sigma(N)=\{0\}.$$ Such is not always the case in case of unbounded closed operators.
\[pmlopmlopmlopmlopmlopmlokjhyutyghfgtrft\] There are nilpotent closed operators $N$ such that $\sigma(N)\neq\{0\}$.
Let $T$ be any (unbounded) closed operator with a domain $D(T)\subset H$. Set $N=\left(
\begin{array}{cc}
0 & T \\
0 & 0 \\
\end{array}
\right)$ with $D(N)=H\oplus D(T)$.
Since $N$ is closed, we know that $\sigma(N^2)=[\sigma(N)]^2$. But, $$N^2=\left( \begin{array}{cc}
0 & 0_{D(T)} \\
0 & 0 \\
\end{array}
\right)$$ with $D(N^2)=D(N)$ and so $N$ is nilpotent. Since $N^2$ is clearly unclosed, it results that $\sigma(N^2)={\mathbb{C}}$. If $\sigma(N)=\{0\}$, then we would have $[\sigma(N)]^2=\{0\}$ as well, and this is absurd. Therefore, $\sigma(N)\neq\{0\}$, as needed.
It is known that if $A,N\in B(H)$ are such that $AN=NA$ and $N$ is nilpotent, then $\sigma(A+N)=\sigma(A)$ (see e.g. Exercise 7.3.29 in [@Mortad-Oper-TH-BOOK-WSPC] for an interesting proof).
Is the previous result valid in the context of one unbounded operator? Since commutativity in this case means $NA\subset AN$, we need to treat the case of the nilpotence of $N$ as well as that of $A$.
First, we have:
\[nilpotent NA subset AN THM spc(A+N)=spec A\] Let $N\in B(H)$ be nilpotent and let $A$ be a densely defined closed operator such that $NA\subset AN$. Then $$\sigma(A+N)=\sigma(A).$$
For the proof, we must fall back on the following auxiliary result:
\[arendt et al. THM\]([@Arendt-et; @al; @AX+XB=Y]) If $B\in B(H)$ and commutes with an unbounded $A$, i.e. $BA\subset AB$, then $$\sigma(A+B)\subset \sigma(A)+\sigma(B)$$ holds.
1. In fact, the writers in [@Arendt-et; @al; @AX+XB=Y] established the above result under the condition $\sigma(A)\neq {\mathbb{C}}$ which was also imposed for other subsequent results. However, the inclusion $\sigma(A+B)\subset \sigma(A)+\sigma(B)$ is trivial when $\sigma(A)={\mathbb{C}}$.
2. This result generalizes a well known result stating that if $A$ and $B$ are in $B(H)$ and $AB=BA$, then $\sigma(A+B)\subset \sigma(A)+\sigma(B)$ holds.
3. How about the case of two unbounded operators? Without digging too much into the difficult notion of strong commutativity, we give a simple example by assuming readers have the necessary means to understand it. Let $A$ be an unbounded self-adjoint operator with domain $D(A)$ and such that $\sigma(A)={\mathbb{R}}$, and let $B=-A$. Then $A$ commutes strongly with $B$ yet $$\sigma(A+B)\not\subset \sigma(A)+\sigma(B)$$ because $A+B$ is unclosed and hence $\sigma(A+B)={\mathbb{C}}$ whereas $\sigma(A)+\sigma(B)={\mathbb{R}}$.
So much for the digression, now we prove Theorem \[nilpotent NA subset AN THM spc(A+N)=spec A\].
The proof is not difficult. By Theorem \[arendt et al. THM\], we know that: $$\sigma(A+N)\subset \sigma(A)+\sigma(N)=\sigma(A).$$ Conversely, $$\sigma(A)=\sigma(A+N-N)\subset \sigma(A+N)+\sigma(-N)=\sigma(A+N)$$ for $N$ is nilpotent and commutes with $A+N$. Therefore, $$\sigma(A+N)=\sigma(A).$$
What about the case when the nilpotent operator is the unbounded one?
There are linear operators $A$ and $N$, where $A\in B(H)$ and $A$ is densely defined and closed, obeying $NA\subset AN$ and yet $$\sigma(A+N)\neq\sigma(A).$$
The simplest example is to take $A=0$ and $N$ as in Proposition \[pmlopmlopmlopmlopmlopmlokjhyutyghfgtrft\]. Then trivially $AN\subset NA$ holds. Besides, $\sigma(N)\neq \{0\}$. Hence $$\sigma(A+N)=\sigma(N)\neq \{0\}=\sigma(A).$$
Another “richer” example is based on one which appeared in [@Hardt-Konstantinov-Spectrum-product] with a different aim. Let $T$ be a closed, unbounded and boundedly invertible operator with domain $D(T)\subset H$. Define on $H\oplus H$ $$A=\left(
\begin{array}{cc}
I & T \\
0 & T \\
\end{array}
\right)\text{ and }B=\left(
\begin{array}{cc}
I & 0 \\
0 & T^{-1} \\
\end{array}
\right)$$ with $D(A)=H\oplus D(T)$ and $D(B)=H\oplus H$. Then $A$ is closed on $D(A)$ and $B$ is everywhere defined and bounded on $H\oplus H$. Then $$BA=\left(
\begin{array}{cc}
I & T \\
0 & I \\
\end{array}
\right)$$ and $\sigma(BA)={\mathbb{C}}$ (see [@Hardt-Konstantinov-Spectrum-product] for further details).
Now, write $$\left(
\begin{array}{cc}
I & T \\
0 & I \\
\end{array}
\right)=\underbrace{\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right)}_{=\tilde{I}}+\underbrace{\left(
\begin{array}{cc}
0 & T \\
0 & 0 \\
\end{array}
\right)}_{=N}.$$ Accordingly, $$\sigma(\tilde{I}+N)={\mathbb{C}}\neq \{1\}=\sigma(\tilde{I})$$ yet $\tilde{I}$ is everywhere defined and bounded, and it plainly commutes with $N$.
Now, we give some more results as regards invertibility.
\[Main THM\] Let $T$ be a non necessarily bounded, closed and densely defined operator on a Hilbert space $H$. Assume that $T$ has a densely defined and closed square root $S$, that is, $S^2=T$. Then $T$ is boundedly invertible if and only if $S$ is boundedly invertible. In such case, $S^{-1}$ is always a square root of $T^{-1}$.
If $S$ has an everywhere defined bounded inverse, then $S^2$ or $T$ too has an everywhere defined bounded inverse.
Conversely, assume that $T$ is boundedly invertible. Since $T$ is one-to-one and $S^2=T$, it follows that $S$ is also one-to-one. By passing to adjoints and taking into account the “one-to-oness” of $T^*$, we easily see that $S^*$ is one-to-one. Now, clearly $$S^2=T\Longrightarrow S(ST^{-1})=S^2T^{-1}=I$$ where $I$ the identity on $H$. The aim is to show that $ST^{-1}\in
B(H)$. By the general theory, $ST^{-1}$ is closed for $T^{-1}\in
B(H)$. Besides, $$H=D(S^2T^{-1})\subset D(ST^{-1})$$ and so $$D(ST^{-1})=H.$$
By the Closed Graph Theorem, $ST^{-1}$ is in effect in $B(H)$. Therefore, $S$ is right invertible. As $\ker S=\ker(S^*)$, then Theorem 2.3 in [@Dehimi-Mortad-2018] tells us that $S$ is (fully) invertible, marking the end of the proof.
A related result is the following:
Let $T$ and $S$ be two densely defined linear operators such that $S^2=T$. If $T$ is right invertible, so is $\overline{S}$ whenever $S$ is closable.
Since $S^2=T$ and $S\subset \overline{S}$, it follows that $T\subset \overline{S}^2$. By the right invertibility of $T$, we obtain $I\subset \overline{S}^2B$ for some $B\in B(H)$. That is, $$\overline{S}~\overline{S}B=\overline{S}^2B=I.$$
Since $\overline{S}B$ is closed and $H=D(\overline{S}B)$, clearly $\overline{S}B\in B(H)$. Therefore, $\overline{S}$ is right invertible.
The converse being untrue as seen by taking $T=S=I_D$ (the identity restricted to some domain $D$). Then $S^2=T$ yet $\overline{S}=I$ is right invertible whilst $T$ is not.
The next result is easily shown and so we omit its proof.
Let $S$ and $T$ be two linear operators such that $S^2=T$. If $S$ is right invertible, so is $T$. If $T$ is right invertible, so is $S$ if $S$ is closable. If $S$ is left invertible, then so is $T$.
Before stating and proving a result about normal and self-adjoint square roots, we give some auxiliary result whose proof is very simple and so it is omitted. It is worth noticing in passing that there are unbounded self-adjoint operators $A$ and $B$ such that $A+iB\subset 0$ (where 0 designates the zero operator on all of $H$), yet $A\not\subset 0$ and $B\not\subset 0$. For example, let $A$ and $B$ be unbounded self-adjoint operators such that $D(A)\cap
D(B)=\{0_H\}$ (see e.g. [@KOS]). Assuming $D(A)=D(B)$ makes the whole difference. Indeed:
\[kkkkkkkkkkkkkkkkkkkkkkkkkkk\] Let $A$ and $B$ be two densely defined symmetric operators with domains $D(A),D(B)\subset H$ respectively. Assume that $D(A)=D(B)$. If $A+iB\subset 0$, then $A\subset 0$ and $B\subset 0$. If $A$ (or $B$) is further taken to be closed, then $A=B=0$ everywhere on $H$.
The following result generalizes one in [@Frid-Mortad-Dehimi-nilpotence].
\[main THMMMMMMMMMMMMMMMM SQ RT\] Let $T=A+iB$ where $A$ and $B$ are self-adjoint (one of them is also positive), $D(A)=D(B)$ and $D(AB)=D(BA)$. If $T^2=S$, where $S$ is symmetric, then $T$ is normal. In particular, if $S$ is self-adjoint and positive, then $T$ is self-adjoint and positive, i.e. $T$ is the unique square root of $S$.
Assume that $A$ is positive (the proof in the case of the positiveness of $B$ is similar). Let $T=A+iB$. Clearly, $$A^2-B^2+i(AB+BA)\subset (A+iB)A+i(A+iB)B=T^2\subset S$$ thereby $$A^2-B^2-S+i(AB+BA)\subset 0.$$ Since $D(A)=D(B)$, it is seen that $D(A^2)=D(BA)$ and that $D(B^2)=D(AB)$. Thus, $$D(A^2-B^2)=D(AB+BA).$$ Since $D(AB)=D(BA)$, we have $$D(A^2-B^2-S)=D(AB+BA)=D(A^2)=D(B^2).$$
Since $A$ is self-adjoint, so is $A^2$ and in particular $A^2$ is necessarily densely defined. Thus, $A^2-B^2$ and $AB+BA$ are both densely defined. Now, by the symmetricity (only) of both $A$ and $B$ we have $$AB+BA\subset A^*B^*+B^*A^*\subset (BA)^*+(AB)^*\subset (AB+BA)^*.$$ Similarly, $A^2-B^2\subset (A^2-B^2)^*$. Therefore, both $AB+BA$ and $A^2-B^2$ are symmetric. By Proposition \[kkkkkkkkkkkkkkkkkkkkkkkkkkk\], we get $AB+BA\subset 0$. Hence $AB=-BA$ (for $D(AB)=D(BA)$) and so $$A^2B=-ABA=BA^2.$$ As $A$ is positive, we obtain $AB=BA$ by [@Bernau; @JAusMS-1968-square; @root]. Hence $AB+B=BA+B$. But $AB+B=(A+I)B$ and $BA+B\subset B(A+I)$. Hence $(A+I)B\subset B(A+I)$. But $$D[B(A+I)]=\{x\in D(A):Ax+x\in D(B)\}.$$
So, if $x\in D[B(A+I)]$, it follows that $x\in D(A)=D(B)$ and $$Ax=Ax+x-x\in D(B),$$ i.e. $Ax\in D(B)$, i.e. $x\in D(BA)$. Since $D(AB)=D(BA)$, we equally have $x\in D(AB)=D[(A+I)B]$. This actually means that $$(A+I)B=B(A+I).$$
Since $A$ is self-adjoint and positive, it results that $A+I$ is boundedly invertible. Right and left multiplying by $(A+I)^{-1}$ yield $(A+I)^{-1}B\subset B(A+I)^{-1}$. By Proposition 5.27 in [@SCHMUDG-book-2012], this means that $A$ commutes strongly with $B$. Accordingly $T$ is normal.
Finally, we show the last statement. Assume that $S$ is self-adjoint and positive (remember that $T$ is still normal). Let $\lambda\in\sigma(T)$. Then $$\lambda^2\in
[\sigma(T)]^2=\sigma(T^2)=\sigma(S).$$ That is, $\lambda^2\geq0$ and so the only possible outcome is $\lambda\in{\mathbb{R}}$. Therefore, $T$ is self-adjoint. Since in this case $$0\leq A={\operatorname{Re}}T=\frac{T+T^*}{2}=T,$$ it follows that $T$ is also positive. This marks the end of the proof.
Let $T=A+iB$ where $A$ and $B$ are self-adjoint (one of them is also positive)where $D(A)=D(B)$. If $T^2=0$ on $D(T)$, then $T\in B(H)$ is normal and so $T=0$ everywhere on $H$.
What prevents us a priori from using Theorem \[main THMMMMMMMMMMMMMMMM SQ RT\] is that the condition $D(AB)=D(BA)$ is missing. But, writing $A=(T+T^*)/2$ and $B=(T-T^*)/{2i}$ (and so $D(T)\subset D(T^*)$), we see that if $x\in D(T)$, then $$Tx+T^*x\in D(T)\Longleftrightarrow Tx-T^*x\in D(T)$$ for $Tx\in D(T)$ (because $D(T^2)=D(T)$). In other language, $D(AB)=D(BA)$, as needed.
It is shown in ([@Weidmann], Theorem 9.4) that if $A$ and $B$ are two self-adjoint positive operators with domains $D(A)$ and $D(B)$ respectively, then $$D(A)=D(B)\Longrightarrow D(\sqrt{A})=D(\sqrt{B}).$$ It is therefore natural to wonder whether this property remains valid for arbitrary square roots? That is, if $A$ and $B$ are square roots of some $S$, i.e. $A^2=B^2=S$, is it true that $D(A)=D(B)$?
There are square roots of self-adjoint operators $S$ having different domains. However, if the square roots are self-adjoint then they necessarily have equal domains.
Let $T$ be any unbounded self-adjoint operator with domain $D(T)\subsetneq H$ and set $S=\left(
\begin{array}{cc}
T & 0 \\
0 & T \\
\end{array}
\right)$ where $D(S)=D(T)\oplus D(T)$. Then both $A:=\left(
\begin{array}{cc}
0 & T \\
I & 0 \\
\end{array}
\right)$ and $B:=\left(
\begin{array}{cc}
0 & I \\
T & 0 \\
\end{array}
\right)$ are square roots of $S$ yet $$D(A)=H\oplus D(T)\neq D(T)\oplus H=D(B).$$ By taking $T$ to be further positive, it is seen that $\left(
\begin{array}{cc}
\sqrt{T} & 0 \\
0 & \sqrt{T} \\
\end{array}
\right)$ (where $\sqrt{T}$ represents here the unique positive square root of $T$) is yet another square root of $S$ whose domain is different from both $D(A)$ and $D(B)$.
To deal with the second statement, remember first that if $T$ is closed and densely defined, then $D(T)=D(|T|)$. Now, let $A$ and $B$ be two self-adjoint square roots of some (necessarily self-adjoint and positive) $S$, i.e. $A^2=B^2=S$. Then $D(A^2)=D(B^2)$ and so $$D(A)=D(|A|)=D(\sqrt{A^2})=D(\sqrt{B^2})=D(|B|)=D(B),$$ as needed.
It is well known that if $S$ is a positive self-adjoint which commutes with some $R\in B(H)$, i.e. $RS\subset SR$, then $R\sqrt{S}\subset \sqrt S R$ where $\sqrt S$ designates the unique positive self-adjoint square root of $S$. See e.g. [@Sebestyen-Tarcsay-self-adjoint; @square; @ROOT] for a new proof.
What about arbitrary roots? The answer is again negative. For instance, take again $S=\left(
\begin{array}{cc}
T & 0 \\
0 & T \\
\end{array}
\right)$ as in the previous proof and set $U=\left(
\begin{array}{cc}
0 & I \\
I & 0 \\
\end{array}
\right)$. Hence $U\in
B(H\oplus
H)$, in fact $U$ is a fundamental symmetry (it is both self-adjoint and unitary). Then $US\subset
SU=\left(
\begin{array}{cc}
0 & T \\
T & 0 \\
\end{array}
\right)$. However, $U$ does not commute with $A$ for $$UA=\left(
\begin{array}{cc}
I & 0 \\
0 & T \\
\end{array}
\right)\text{ while }AU=\left(
\begin{array}{cc}
T & 0 \\
0 & I \\
\end{array}
\right).$$
In fact, the previous question does not even hold on finite dimensional spaces. Just consider: $$S=\left(
\begin{array}{cc}
a & 0 \\
0 & a \\
\end{array}
\right),~U=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)\text{ and }A=\left(
\begin{array}{cc}
0 & a \\
1 & 0 \\
\end{array}
\right)$$ where $a\in{\mathbb{C}}$.
Let $A$ and $B$ be two (closed) quasinormal operators such that $A^2=B^2$. Then $D(A)=D(B)$.
For the definition of quasinormality in the unbounded case, we refer readers to [@Jablonski; @et; @al; @2014] or [@Uchiyama-1993-QUASINORMAL]. From either of the previous two references, we know that if $T$ is a (closed) quasinormal operator, then $|T|^n=|T^n|$ for any $n\in{\mathbb{N}}$.
Since $A$ and $B$ are quasinormal, we have $$A^2=B^2\Longrightarrow |A|^2=|A^2|=|B^2|=|B|^2.$$ Upon passing to the unique positive self-adjoint square root implies that $|A|=|B|$. Hence $D(A)=D(B)$ by the closedness of both $A$ and $B$.
It is unknown to me whether the previous result is valid for the weaker classes of subnormal (see e.g. [@McDonald-Sundberg:; @unbounded; @subnormal] for its definition) or hyponormal closed operators. Recall that a densely defined operator $A$ with domain $D(A)$ is called hyponormal if $$D(A)\subset D(A^*)\text{ and } \|A^*x\|\leq\|Ax\|,~\forall x\in D(A).$$
However, we have:
\[20/07/2020\] Let $A$ and $B$ be two (closed) hyponormal operators such that $A^2=B^2$. Assume further that $A^2$ is self-adjoint. Then $D(A)=D(B)$.
The proof relies on the following lemma:
\[012125454878796963223656554549898\] If $A$ is a closed hyponormal operator such that $A^2$ (resp. $-A^2$) is positive, then $A$ is self-adjoint (resp. skew-adjoint, i.e. $A^*=-A$).
In view of the proof of Theorem 8 in [@Dehimi-Mortad-BKMS], closed hyponormal operators having a real spectrum are self-adjoint. This result may be used to show that closed hyponormal operators with a purely imaginary spectrum are skew-adjoint. Indeed, let $A$ be a closed hyponormal operator such that $\sigma(A)$ is purely imaginary. Then set $B=iA$ and so $B$ remains hyponormal. Hence $\sigma(B)\subset {\mathbb{R}}$ since by hypothesis $\sigma(A)\subset i{\mathbb{R}}$. Thus $B$ is self-adjoint, i.e. $$-iA^*=B^*=B=iA,$$ i.e. $A$ is clearly skew-adjoint.
Now, let $\lambda\in\sigma(A)$. Since $A$ is closed, we have that $\lambda^2\in\sigma(A^2)$, i.e. $\lambda^2\geq0$ as $A^2$ is positive. But, this forces $\lambda$ to be real. Accordingly, $A$ is self-adjoint. When $-A^2$ is positive, it may be shown that $A$ is skew-adjoint, and the proof is over.
Now we prove Proposition \[20/07/2020\]:
Since $A^2=B^2$ are self-adjoint and $A$ and $B$ are hyponormal, Lemma \[012125454878796963223656554549898\] says that $A$ and $B$ are self-adjoint or skew-adjoint. In all possible cases, we may obtain $D(A)=D(B)$.
What about $$D(A)=D(B)\Longrightarrow D(A^2)=D(B^2)?$$ This is not true even when $A$ and $B$ are self-adjoint. Let us give a counterexample.
\[19/07/2020\] There exist unbounded self-adjoint positive operators $A$ and $B$ such that $D(A)=D(B)$ yet $D(A^2)\neq D(B^2)$.
There could be simpler counterexamples, but here we may construct lots of them. Indeed, let $T$ be a closed and densely defined operator such that $D(T)=D(T^*)$ but $D(TT^*)\neq D(T^*T)$.
Let $A$ be a densely defined closed and *unbounded* operator with domain $D(A)$ such that $D(A)=D(A^*)\subset H$. Define $T$ on $H\oplus H$ by $$T=\left(
\begin{array}{cc}
A & I \\
0 & 0 \\
\end{array}
\right)$$ with domain $D(T)=D(A)\oplus H$. It is plain that $T$ is closed. $$T^*=\left[\left(
\begin{array}{cc}
A & 0 \\
0 & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
0 & I \\
0 & 0 \\
\end{array}
\right)\right]^*=\left(
\begin{array}{cc}
A^* & 0 \\
0 & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
0 & 0 \\
I & 0 \\
\end{array}
\right)=\left(
\begin{array}{cc}
A^* & 0 \\
I & 0 \\
\end{array}
\right).$$
Since $D(A)=D(A^*)$, it results that $D(T)=D(T^*)$. In addition $$D(TT^*)=\{(x,y)\in D(A)\times H:(A^*x,x)\in D(A)\times H\}=D(AA^*)\times H$$ and also $$D(T^*T)=\{(x,y)\in D(A)\times H:(Ax+y,0)\in D(A^*)\times H\}.$$
To see explicitly why $D(TT^*)\neq D(T^*T)$, let $\alpha$ be in $H$ such that $\alpha\not\in D(A^*)$. If $x_0\in D(AA^*)\subset
D(A^*)=D(A)$, then clearly $-Ax_0\in H$. Set $y_0=-Ax_0+\alpha$. Then $(x_0,y_0)\in D(AA^*)\times H=D(TT^*)$. Nonetheless, $(x_0,y_0)\not\in
D(T^*T)$ for $$Ax_0+y_0=Ax_0-Ax_0+\alpha=\alpha\not\in D(A^*).$$
To finish the proof, observe that $TT^*$ and $T^*T$ are both self-adjoint and positive. In particular, $|T|$ and $|T^*|$ are both self-adjoint. Moreover, $$D(|T|)=D(T)=D(T^*)=D(|T^*|).$$ However, $$D(|T|^2)=D(T^*T)\neq D(TT^*)=D(|T^*|^2),$$ as needed.
We are aware now that $D(A)=D(B)$ does not entail $D(A^2)=D(B^2)$ even when $A$ and $B$ are self-adjoint. It is worth noting that the condition $D(A)=D(B)$ does not even have to imply that $D(A^2-B^2)$ (or $D(AB+BA)$) is dense (cf. Theorem \[main THMMMMMMMMMMMMMMMM SQ RT\]). Before giving a counterexample, we give a simple lemma:
\[18/07/2020\] Let $A$ and $B$ two linear operators such that $D(A)=D(B)$. Then $$D(AB+BA)=D(A^2-B^2)\subset D[(A-B)^2]\text{ or }D[(A+B)^2].$$
Write $$A^2-B^2+AB-BA\subset (A+B)(A-B).$$ Since $D(A)=D(B)$, it follows that $D(A^2)=D(BA)$ and that $D(B^2)=D(AB)$. Hence $$D(AB+BA)=D(A^2-B^2)\subset D[(A+B)(A-B)].$$ But $$D[(A+B)(A-B)]=D[(A-B)^2]$$ for $D(A+B)=D(A-B)$. The other inclusion can be shown analogously.
Now, we give the promised counterexample.
There are self-adjoint positive unbounded operators $A$ and $B$ such that $D(A)=D(B)$ yet neither $A^2-B^2$ nor $AB+BA$ is densely defined.
First, observe that $D(A)=D(B)$ yields $D(AB+BA)=D(A^2-B^2)$. So, it suffices to exhibit $A$ and $B$ with the claimed properties such that $D(A^2-B^2)$ is not dense.
Consider a closed densely defined positive symmetric operator $T$ such that $D(T^2)=\{0\}$ (as in e.g. [@CH]), then set $A=T/2+|T|$ and $B=|T|$. That $A$ and $B$ are positive is plain. Also, $D(A)=D(B)$ and $B$ is self-adjoint. As for the self-adjointness of $A$ one needs to call on the Kato-Rellich theorem (see e.g. [@Weidmann]).
By Lemma \[18/07/2020\], if $A^2-B^2$ were densely defined, so would be $D[(A-B)^2]$. However, $$D[(A-B)^2]=D(T^2)=\{0\},$$ and so $A^2-B^2$ is not densely defined.
Let us pass now to unclosable square (or other types of) roots of the identity operator $I:H\to H$.
\[T\^2=I UNCLOSABLE PRO\] There exists an everywhere defined non-closable unbounded operator $T$ such that $$T^2=I.$$
Let $A$ be a non-closable unbounded operator defined on all of $H$ such that $A^2=0$ everywhere. Then, set $$T=\left(
\begin{array}{cc}
A & I \\
I & -A \\
\end{array}
\right),$$ which is defined fully on $H\oplus H$. Then $T$ is unclosable and besides $D(T^2)=H\oplus H$. Since $A-A=0$ and $A^2=0$ both everywhere on $H$, we may write $$T^2=\left(
\begin{array}{cc}
A & I \\
I & -A \\
\end{array}
\right)\left(
\begin{array}{cc}
A & I \\
I & -A \\
\end{array}
\right)=\left(
\begin{array}{cc}
A^2+I & A-A \\
A-A & A^2+I \\
\end{array}
\right)=\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right),$$ i.e. $T^2=I_{H\oplus H}$, as needed.
The equation $T^2=I$ says that $T:H\to H$ is a bijective or invertible (not boundedly though) non-closable operator which is everywhere defined.
There are two everywhere defined unbounded non-closable operators $A$ and $B$ such that $AB=BA=I$ everywhere on some Hilbert space $K$, that is, $$ABx=BAx=x,~\forall x\in K.$$
From Proposition \[T\^2=I UNCLOSABLE PRO\], we have a non-closable operator $T$ such that $T^2=I$ everywhere on $H\oplus H$. Setting $$A=\left(
\begin{array}{cc}
0 & T \\
I & 0 \\
\end{array}
\right)\text{ and }B=\left(
\begin{array}{cc}
0 & I \\
T & 0 \\
\end{array}
\right),$$ which are everywhere defined on $H\oplus H\oplus H\oplus H$, we see that $$AB=\left(
\begin{array}{cc}
T^2 & 0 \\
0 & I \\
\end{array}
\right)=\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right)=\left(
\begin{array}{cc}
I & 0 \\
0 & T^2 \\
\end{array}
\right)=BA$$ everywhere.
Let us give a second example. Let $T$ be any unbounded non-closable everywhere defined operator on $H$ and let $$A=\left(
\begin{array}{cc}
I & T \\
0 & I \\
\end{array}
\right).$$
It is seen that $A$, which is defined on all of $H\oplus H$, is bijective and so it is invertible (not boundedly) with an inverse given by $B=\left(
\begin{array}{cc}
I & -T \\
0 & I \\
\end{array}
\right)$ for $$AB=BA=\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right).$$
Let $A,B,T$ be all everywhere defined and not closable. Note by $I$ the identity operator which need not act on the same space in each case. The existence of a $T$ such that $T^2=I$ gave rise to two different operators $A$ and $B$ such that $AB=BA=I$.
Conversely the availability of a pair of two different operators $A$ and $B$ such that $AB=BA=I$ in turn leads to $T^2=I$. This is easily seen by taking $$T=\left(
\begin{array}{cc}
0 & A \\
B & 0 \\
\end{array}
\right)$$ which is defined on $D(T):=D(B)\oplus D(A)=H\oplus H$. Hence $$T^2=\left(
\begin{array}{cc}
AB & 0 \\
0 & BA \\
\end{array}
\right)=\left(
\begin{array}{cc}
I & 0 \\
0 & I \\
\end{array}
\right),$$ as desired.
If we have a non-closable operator $T$ such that $D(T)=H$ and $T^2=I$, then we can always manufacture a non-closable $S$ such that $S^2=0$. Just consider $$S=\left(
\begin{array}{cc}
I & T \\
-T & -I \\
\end{array}
\right)$$ on $D(S)=H\oplus H$. Then $$S^2=\left(
\begin{array}{cc}
I-T^2 & T-T \\
-T+T & -T^2+I \\
\end{array}
\right)=\left(
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}
\right)$$ everywhere on $H\oplus H$ as all the resulting operations are carried out on all of $H$.
As alluded above, boundedly invertible operators are necessarily closed while invertible operators might even be unclosable in some cases (as when $T^2=I$). What about the weaker notion of left or right invertibility?
\[left-right invert not closed THM\] There is a left (resp. right) invertible operator which is not closed.
The simplest example in the left invertibility case is to restrict the identity operator on $H$ (noted $I_H$) to some non-closed domain $D\subset H$ and denote this restriction by $I_D$. Then $I_D$ is left invertible for $$I_HI_D=I_D\subset I_H.$$
Since $I_D$ is bounded on a non closed domain, it follows that $I_D$ is unclosed.
As for the right invertibility case, there is an example of such an $A$ which is even everywhere defined in $H$ (there might not be any more explicit one). Start with $B$ in $B(H)$ such that its range ${\operatorname{ran}}(B)$ is dense but is not all of $H$. Let $E$ be a linear subspace of $H$ which is complementary to ${\operatorname{ran}}(B)$ in the algebraic sense (i.e. ${\operatorname{ran}}(B)+E=H$, without taking closure, while the intersection is $\{0\}$). Then define $A$ on ${\operatorname{ran}}(B)$ by $$ABx=x,$$ and define $A$ on $E$ to be an arbitrary linear mapping of $E$ to $H$. $A$ then extends by linearity to all of $H$, and $AB=I$, but $A$ is not bounded (as it is not bounded on ${\operatorname{ran}}(B)$ as if it were, then ${\operatorname{ran}}(B)$ would be closed), so it cannot be closable.
We have given a way of finding everywhere defined bijective operators. A similar ideas applies to injectivity and surjectivity independently.
There is an everywhere defined unbounded operator which is injective but not surjective, and there is an everywhere defined unbounded operator which is surjective but not injective.
Let $T$ be an everywhere defined operator such that $T^2=I$. Hence $T$ is bijective.
1. Let $S\in B(H)$ be any injective operator which is not surjective. Set $$A:=T\oplus S=\left(
\begin{array}{cc}
T & 0 \\
0 & S \\
\end{array}
\right),$$ and so $D(A)=H\oplus H$. Then $A$ is unbounded and not closable. That $A$ is injective is plain. As ${\operatorname{ran}}S\neq H$, it results that $${\operatorname{ran}}A=H\oplus {\operatorname{ran}}S\neq H\oplus H,$$ that is, $A$ is not surjective.
2. Consider a surjective $R\in B(H)$ which is not injective. Then $$B:=\left(
\begin{array}{cc}
T & 0 \\
0 & R \\
\end{array}
\right)$$ is unbounded, $D(B)=H\oplus H$, ${\operatorname{ran}}B=H\oplus H$ and $$\ker B\neq \{(0,0)\},$$ as needed.
Now, we deal with the general case. First, we provide a finite dimensional example:
\[israel example\] Let $n\in {\mathbb{N}}$ be given. There is an $n\times n$ matrix such that $A^n=I$ with $A^{n-1}\neq I$ (in fact, $A^p\neq I$ for $p=1,2,\cdots, n-1$).
There are many types of counterexamples. The simplest one is to take the following circulant permutation $n\times n$ matrix
$$A=\left(
\begin{array}{cccccc}
0 & 1 & 0 & \cdots & \cdots & 0 \\
0 & 0& 1 & 0 & & \vdots\\
\vdots & & 0 & 1 & \ddots & \vdots \\
\vdots & & & \ddots & \ddots & 0 \\
0 & & & & 0 & 1 \\
1 &0 & \cdots & \cdots & 0 & 0 \\
\end{array}
\right)$$ where the corresponding permutation being $p(i)=i+1$. Then it is well known that $A^n=I$ and $A^p\neq I$ for $p=1,2,\cdots, n-1$.
In order to carry over this type of examples to matrices of unbounded operators, we need to place some parameter inside the previous matrix, and still obtain the same conclusions. So, a more general form of the previous example reads: $$A=\left(
\begin{array}{cccccc}
0 & 1 & \alpha & \cdots & \cdots & 0 \\
0 & 0& 1 & 0 & & \vdots\\
\vdots & & 0 & 1 & \ddots & \vdots \\
\vdots & & & \ddots & \ddots & 0 \\
0 & & & & 0 & 1 \\
1 &-\alpha & \cdots & \cdots & 0 & 0 \\
\end{array}
\right),$$ where it can be again checked that $A^n=I$ and that $A^p\neq I$ for $p=1,2,\cdots, n-1$ (all that holding for any $\alpha$).
\[26/06/2020\] Let $n\in{\mathbb{N}}$ be given. There are infinitely many everywhere defined non-closable unbounded operators $T$ such that $T^n=I$ *everywhere* on some Hilbert space while $T^p\neq I$ for $p=1,2,\cdots, n-1$.
Let $A$ be a non-closable unbounded operator which is everywhere defined, i.e. $D(A)=H$ and let $I\in B(H)$ be the identity operator. Inspired by the example above, let $$T=\left(
\begin{array}{cccccc}
0 & I & A & 0 & \cdots & 0 \\
0 & 0& I & 0 & & \vdots\\
\vdots & & 0 & I & \ddots & \vdots \\
\vdots & & & \ddots & \ddots & 0 \\
0 & & & & 0 & I \\
I &-A & 0 & \cdots & 0 & 0 \\
\end{array}
\right)$$ be defined on $D(T)=H\oplus H\oplus\cdots \oplus H$ ($n$ times). This means that $T$ is everywhere defined. Notice also that $T$ is clearly unbounded and not closable.
Readers may check that $T^n=I$ on $D(T^n)=H\oplus H\oplus \cdots
\oplus H$ whereas $T^p\neq I$ for $p=1,2,\cdots, n-1$. As an illustration, we treat the special case $n=3$. In this case, $$T=\left(
\begin{array}{ccc}
0 & I & A \\
0 & 0 & I \\
I & -A & 0 \\
\end{array}
\right).$$ Then $$T^2=\left(
\begin{array}{ccc}
A & -A^2 & I \\
I & -A & 0 \\
0 & I & 0 \\
\end{array}
\right)\neq I\oplus I\oplus I$$ $$T^3=\left(
\begin{array}{ccc}
I & 0 & 0 \\
0 & I & 0 \\
0 & 0 & I \\
\end{array}
\right)=I\oplus I\oplus I,$$ as wished.
To obtain an infinite family of such roots, just replace $A$ by $\alpha A$ where $\alpha$ is real, say.
We finish with a digression which is in the spirit of the paper. In the case of matrices of operators, readers have already observed here an apparent resemblance to usual matrices with real or complex coefficients. In view of many examples treated here and elsewhere, it seems therefore reasonable to conjecture that:
If $T$ is a matrix of operators defined formally on $H\oplus H\oplus
\cdots \oplus H$ ($n$ times), that is, on $H\times H\times \dots
\times H=H^n$ whether the entries are all in $B(H)$ or not, and $T^p=0$ for some integer $p\geq n$, then necessarily $T^n=0$.
The answer to this conjecture is negative. A counterexample is available on finite dimensional spaces!
Let $H={\mathbb{C}}^2$ and let $$A=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)\text{ and }B=\left(
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}
\right)$$ be both defined on $H$. Then $$AB=\left(
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}
\right)\text{ and } BA=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)$$ and so $ABA=BAB=\left(
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}
\right)$. Finally, set $$T=\left(
\begin{array}{cc}
\mathbf{0} & A \\
B & \mathbf{0} \\
\end{array}
\right)$$ which is defined on $H\times H$ (where $\mathbf{0}\in B({\mathbb{C}}^2)$). Thus, $$T^2=\left(
\begin{array}{cc}
\mathbf{0} & \mathbf{0} \\
\mathbf{0} & BA \\
\end{array}
\right)\text{ and }T^3=\left(
\begin{array}{cc}
\mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} \\
\end{array}
\right)$$ and so $T^2\neq 0$ whilst $T^3=0$, marking the end of the proof.
An open question
================
A closable operator $A$ such that $\overline{A}^2$ is self-adjoint but $A^2$ is not self-adjoint exists. A simple example is to take $A$ to be the restriction of the identity operator $I$ (on $H$) to some dense (non closed) subspace $D$ of $H$. Then $\overline{A}^2=I$ fully on $H$ and so $\overline{A}^2$ is self-adjoint. However, $A^2$ is not self-adjoint for $A^2=I_D$ and so $A^2$ is not even closed.
What about the converse, i.e. if $A$ is closable and $A^2$ is self-adjoint, then could it be true that $\overline{A}^2$ is self-adjoint? A positive answer can be obtained if one comes to show that if $A$ is a closable operator with a self-adjoint square $A^2$, then $A$ is closed.
Let us posit that we have in effect shown that a closable $A$ such that $A^2$ is self-adjoint is necessarily closed. Another natural question would then follow: What about when $A^2$ is normal? Another more general question is to see whether the self-adjointness or the normality of $A^n$ entails the closedness of $A$ whenever it is closable?
Acknowledgements {#acknowledgements .unnumbered}
================
The second example in the proof of Theorem \[left-right invert not closed THM\] was communicated to me by Professor A. M. Davie (The University of Edinburgh, UK) a while ago.
Thanks go to Dr S. Dehimi (University of Mohamed El Bachir El Ibrahimi, Algeria) with whom I discussed e.g. Proposition \[with DEHIMI I\] and Theorem \[nilpotent NA subset AN THM spc(A+N)=spec A\].
Thanks are also due to Professor Robert B. Israel (University of British Columbia, Canada). Indeed, for the purpose of Theorem \[26/06/2020\], I asked him whether we can place a parameter inside the first matrix in Example \[israel example\] and still obtain the same conclusion? He kindly suggested to conjugate with some nonsingular matrix which does not commute with $A_n$ but contains a parameter $\alpha$, leading to the second example.
Finally, Example \[stochel’s example\] is due to Professor Jan Stochel (Uniwersytet Jagielloński, Poland), who communicated it to me some time ago.
[1]{}
W. Arendt, F. Räbiger, A. Sourour. Spectral properties of the operator equation $AX+XB=Y$, *Quart. J. Math. Oxford Ser. (2)*, **45/178** (1994) 133-149.
Y. Arlinskiĭ, Y. Kovalev. Factorizations of nonnegative symmetric operators, *Methods Funct. Anal. Topology*, **19/3** (2013) 211-226.
S. J. Bernau. The square root of a positive self-adjoint operator, *J. Austral. Math. Soc.*, **8** (1968) 17-36.
I. Boucif, S. Dehimi and M. H. Mortad. On the absolute value of unbounded operators, *J. Operator Theory*, **82/2** (2019) 285-306.
J. F. Brasche, H. Neidhardt. Has every symmetric operator a closed symmetric restriction whose square has a trivial domain?, *Acta Sci. Math. (Szeged)*, **58/1-4** (1993) 425-430.
P. R. Chernoff. A semibounded closed symmetric operator whose square has trivial domain, *Proc. Amer. Math. Soc.,* [**89/2**]{} [(1983) 289-290]{.nodecor}.
J. B. Conway. *A course in operator theory*. Graduate Studies in Mathematics, **21**, American Mathematical Society, Providence, RI, 2000.
J. B. Conway, B. B. Morrel. Roots and logarithms of bounded operators on Hilbert space, *J. Funct. Anal.*, **70/1** (1987) 171-193.
S. Dehimi and M. H. Mortad, Bounded and Unbounded Operators Similar to Their Adjoints, *Bull. Korean Math. Soc.*, **54/1** (2017) 215-223.
S. Dehimi, M. H. Mortad. Right (or left) invertibility of bounded and unbounded operators and applications to the spectrum of products, *Complex Anal. Oper. Theory*, **12/3** (2018) 589-597.
S. Dehimi, M. H. Mortad. Chernoff like counterexamples related to unbounded operators, *Kyushu J. Math.*, **74/1** (2020) 105-108.
N. Frid, M. H. Mortad, S. Dehimi. On nilpotence of bounded and unbounded linear operators, (submitted).
S. Goldberg. *Unbounded linear operators. Theory and applications*, Reprint of the 1985 corrected edition, Dover Publications, Inc., Mineola, NY, 2006.
P. R. Halmos, G. Lumer, J. J. Schäffer. Square roots of operators, *Proc. Amer. Math. Soc.*, **4** (1953) 142-149.
V. Hardt, A. Konstantinov, R. Mennicken. On the spectrum of the product of closed operators, *Math. Nachr.*, **215**, (2000) 91-102.
Z. J. Jab[ł]{}oński, Il B. Jung, J. Stochel. Unbounded quasinormal operators revisited, *Integral Equations Operator Theory,* **79/1** (2014) 135-149.
H. Kosaki. On intersections of domains of unbounded positive operators, *Kyushu J. Math.*, [**60/1**]{} (2006) 3-25.
S. H. Kulkarni, M. T. Nair, G. Ramesh. Some properties of unbounded operators with closed range, *Proc. Indian Acad. Sci. Math. Sci.*, **118/4** (2008) 613-625.
G. McDonald, C. Sundberg. On the spectra of unbounded subnormal operators, *Canad. J. Math.*, **38/5** (1986) 1135-1148.
A. McIntosh. Square roots of operators and applications to hyperbolic PDEs. *Miniconference on operator theory and partial differential equations (Canberra, 1983)*, 124-136, Proc. Centre Math. Anal. Austral. Nat. Univ., 5, *Austral. Nat. Univ., Canberra*, 1984.
I. D. Mercer, Finding “nonobvious” nilpotent matrices (2005). http://people.math.sfu.ca/ idmercer/nilpotent.pdf.
M. Möller, F. H. Szafraniec. Adjoints and formal adjoints of matrices of unbounded operators, *Proc. Amer. Math. Soc.*, **136/6** (2008) 2165-2176.
M. H. Mortad. *An operator theory problem book*, World Scientific Publishing Co., (2018). ISBN: 978-981-3236-25-7 (hardcover).
M. H. Mortad. On the triviality of domains of powers and adjoints of closed operators, *Acta Sci. Math. (Szeged)*, **85** (2019) 651-658.
M. H. Mortad. Counterexamples related to commutators of unbounded operators, *Results Math.*, **74** (2019), no. 4, Paper No. 174.
M. H. Mortad. Simple examples of non closable paranormal operators. arXiv:2002.06536.
M. H. Mortad. Counterexamples related to unbounded paranormal operators, (submitted).
S. Ôta. Closed linear operators with domain containing their range, *Proc. Edinburgh Math. Soc.*, (2) **27/2** (1984) 229-233.
S. Ôta. Unbounded nilpotents and idempotents, *J. Math. Anal. Appl.*, **132/1** (1988) 300-308.
S. Ôta. On normal extensions of unbounded operators, *Bull. Polish Acad. Sci. Math.*, **46/3** (1998) 291-301.
K. Schmüdgen. On domains of powers of closed symmetric operators, *J. Operator Theory*, [**9/1**]{} (1983) 53-75.
K. Schmüdgen. *Unbounded self-adjoint operators on Hilbert space*, Springer. GTM [**265**]{} (2012).
Z. Sebestyén, J. Stochel. On suboperators with codimension one domains, *J. Math. Anal. Appl.*, **360/2** (2009) 391-397.
Z. Sebestyén, Zs. Tarcsay. On the square root of a positive selfadjoint operator, *Period. Math. Hungar.*, **75/2** (2017) 268-272.
J. Stochel, J. B. Stochel. Composition operators on Hilbert spaces of entire functions with analytic symbols, *J. Math. Anal. Appl.*, **454/2** (2017) 1019-1066.
Zs. Tarcsay. Operator extensions with closed range, *Acta Math. Hungar.*, **135** (2012) 325-341.
A. E. Taylor, D. C. Lay. Introduction to functional analysis. Reprint of the second edition. *Robert E. Krieger Publishing Co.*, Inc., Melbourne, FL, 1986.
Ch. Tretter. *Spectral Theory of Block Operator Matrices and Applications*. Imperial College Press, London, 2008.
M. Uchiyama. Operators which have commutative polar decompositions. Contributions to operator theory and its applications, 197-208, *Oper. Theory Adv. Appl.*, **62**, Birkhäuser, Basel, 1993.
J. Weidmann. *Linear Operators in Hilbert Spaces*, Springer, 1980.
D. Y. Wu, A. Chen. On the adjoint of operator matrices with unbounded entries II, *Acta Math. Sin. (Engl. Ser.)*, **31/6** (2015) 995-1002.
[^1]: Supported in part by PRFU project: C00L03UN310120200003
|
---
abstract: |
=0.6 cm
[**Abstract**]{}
We have investigated the absorption cross section and the Hawking radiation of electromagnetic field with Weyl correction in the background of a four-dimensional Schwarzschild black hole spacetime. Our results show that the properties of the absorption cross section and the Hawking radiation depend not only on the Weyl correction parameter, but also on the parity of the electromagnetic field, which is quite different from those of the usual electromagnetic field without Weyl correction in the four-dimensional spacetime. With increase of Weyl correction parameter, the absorption probability, the absorption cross section, the power emission spectra and the luminosity of Hawking radiation decreases with Weyl correction parameter for the odd-parity electromagnetic field and increases with the event-parity electromagnetic field.
author:
- 'Hao Liao, Songbai Chen[^1], Jiliang Jing [^2]'
title: '**Absorption cross section and Hawking radiation of the electromagnetic field with Weyl corrections**'
---
=0.8 cm
Since generalized Einstein-Maxwell theories contain higher derivative interactions and carry more information about the electromagnetic field, a lot of attention have been recently focused on studying such kind of generalized Einstein-Maxwell theories in order to probe the full properties and effects of the electromagnetic fields. There are two main classes of the generalized Einstein-Maxwell theories. The first class is minimally coupled gravitational-electromagnetism in which in the Lagrangian there is no coupling between the Maxwell part and the curvature part, but the form of Lagrangian of electromagnetic field is changed. The well-known Born-Infeld theory [@Born] belongs to such class of generalized Einstein-Maxwell theory. Born-Infeld theory removes the divergence of the electron’s self-energy in the classical Maxwell electrodynamics and possesses good physical properties including the absence of shock waves and birefringence phenomena [@Boillat]. Moreover, Born-Infeld theory has also received special attention because it enjoys an electric-magnetic duality [@Gibbons] and can describe gauge fields in the low-energy regime of string and D-Brane physics [@Fradkin]. In the second class of generalized Einstein-Maxwell theory, there exist the nonminimal coupling terms between the gravitational and electromagnetic fields in the Lagrangian [@Balakin; @Faraoni; @Hehl]. These nonminimal coupling terms modify the coefficients of the second-order derivatives in the Maxwell and Einstein equations and change behavior of gravitational and electromagnetic waves in the spacetime, which may result in time delays in the arrival of those waves [@Balakin]. Moreover, it is also find that such a kind of coupled terms may affect evolution of the early Universe because they may modify electromagnetic quantum fluctuations, which could affect the inflation [@Turner; @Mazzitelli; @Lambiase; @Raya; @Campanelli]. Furthermore, these cross-terms can be used as attempt to explain the large scale magnetic fields observed in clusters of galaxies [@Bamba; @Kim; @Clarke].
The theory of electromagnetic field with Weyl corrections contains a coupling between the Maxwell field and the Weyl tensor [@Weyl1; @Drummond]. Since Weyl tensor is actually related to the curvature tensors $R_{\mu\nu\rho\sigma}$, $R_{\mu\nu}$ and the Ricci scalar $R$, the theory of electromagnetic field with Weyl corrections can be treated as a special kind of generalized Einstein-Maxwell theory with the coupling between the gravitational and electromagnetic fields. These special coupling terms could be obtained from a calculation in QED of the photon effective action from one-loop vacuum polarization on a curved background [@Drummond]. Moreover, it was found that these couplings could exist near classical compact astrophysical objects with high mass density and strong gravitational field such as the supermassive black holes at the center of galaxies [@Dereli1; @Solanki]. Considering that black hole is an important subject in the modern physics, a lot of efforts have been recently focused on probing the effects of Weyl correction on black hole physics. In Ref.[@Weyl1], the authors studied the effects of Weyl correction on holographic conductivity and charge diffusion in the anti-de Sitter spacetime and found that the presence of Wely correction changes the universal relation with the $U(1)$ central charge observed at leading order. Moreover, the dependence of the holographic superconductors on Weyl corrections are also studied in [@Wu2011; @Ma2011; @Momeni]. It is shown that Weyl corrections modify the critical temperature at which holographic superconductors occur [@Wu2011] and in the Stückelberg mechanism [@Ma2011] Weyl corrections also change the order of the phase transition of the holographic superconductor. In the AdS soliton background, it is find that Weyl corrections also affect the phase transition between the holographic insulator and superconductor [@zhao2013]. Recently, we [@sb2013] studied the dynamical evolution of the electromagnetic field with Weyl corrections in the Schwarzschild black hole spacetime and analyze the effect of the Weyl corrections on the stability of the black hole. In this letter we are going to study the Hawking radiation of electromagnetic field with Weyl corrections in the background of a Schwarzschild black hole. We will calculate the absorption probability, absorption cross section and the luminosity of Hawking radiation and show physics brought by the Weyl corrections and the parity of electromagnetic field.
Let us now first review briefly the wave equations of the electromagnetic field with Weyl corrections in the background of a black hole [@sb2013]. For the electromagnetic field with Weyl corrections, the action in the black hole spacetime can be modified as $$\begin{aligned}
\label{acts}
S=\int d^4x \sqrt{-g}\bigg[\frac{R}{16\pi
G}-\frac{1}{4}\bigg(F_{\mu\nu}F^{\mu\nu}-4\alpha
C^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\bigg)\bigg].\end{aligned}$$ where $F_{\mu\nu}$ is the usual electromagnetic tensor, which is related to the electromagnetic vector potential $A_{\mu}$ by $F_{\mu\nu}=A_{\nu;\mu}-A_{\mu;\nu}$. The coefficient $\alpha$ is a coupling constant with dimensions of length squared and the tensor $C_{\mu\nu\rho\sigma}$ is so-called Weyl tensor, which can be expressed as $$\begin{aligned}
C_{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}-\frac{2}{n-2}(
g_{\mu[\rho}R_{\sigma]\nu}-g_{\nu[\rho}R_{\sigma]\mu})+\frac{2}{(n-1)(n-2)}R
g_{\mu[\rho}g_{\sigma]\nu},\end{aligned}$$ where $n$ and $g_{\mu\nu}$ are the dimension and metric of the spacetime, and brackets around indices refers to the antisymmetric part. Obviously, Weyl tensor $C_{\mu\nu\rho\sigma}$ is a function of the Riemann tensor $R_{\mu\nu\rho\sigma}$, the Ricci tensor $R_{\mu\nu}$ and the Ricci scalar $R$. Therefore, the Weyl correction to electromagnetic field can be treated as a kind of special couplings between the gravitational and electromagnetic fields.
Varying the action (\[acts\]) with respect to $A_{\mu}$, one can find the corresponding Maxwell equation becomes $$\label{WE}
\nabla_{\mu}(F^{\mu\nu}-4\alpha C^{\mu\nu\rho\sigma}F_{\rho\sigma})=0.$$ It is well known that the metric of a Schwarzschild black hole spacetime has a form $$\label{fd}
ds^2=fdt^2-\frac{1}{f}dr^2-r^2d\theta^2-r^2sin^2\theta d\phi^2,$$ where $f=1-\frac{2M}{r}$. In such kind of static and spherical symmetric black hole background, one can expand $A_{\mu}$ in vector spherical harmonics [@rr] $$A_{\mu}
=\sum_{l,m}
\left( \left[
\begin{matrix}
0 \\
0 \\
\frac{a^{lm}(t,r)}{sin\theta}\partial_{\phi}Y_{lm} \\
-a^{lm}(t,r)sin\theta\partial_{\theta}Y_{lm}
\end{matrix}
\right]
+
\left[
\begin{matrix}
j^{lm}(t,r)Y_{lm} \\
h^{lm}(t,r)Y_{lm} \\
k^{lm}(t,r)\partial_{\theta}Y_{lm} \\
k^{lm}(t,r)\partial_{\phi}Y_{lm}
\end{matrix}
\right]\right),\label{Au}$$ where the first term in the right side has parity $(-1)^{l+1}$ and the second term has parity $(-1)^{l}$, $l$ is the angular quantum number and $m$ is the azimuthal number. Making use of the following form $$\begin{aligned}
\label{w5}
a^{lm}(t,r)&=&a^{lm}(r)e^{-i\omega t},~~~~~h^{lm}(t,r)=h^{lm}(r)e^{-i\omega t},\nonumber\\
j^{lm}(t,r)&=&j^{lm}(r)e^{-i\omega t},~~~~~k^{lm}(t,r)=k^{lm}(r)e^{-i\omega t},\end{aligned}$$ and inserting the above expansion (\[Au\]) into the generalized Maxwell equation (\[WE\]), we can obtain three independent coupled differential equations. Eliminating $k^{lm}(r)$, we find that equations of motion of electromagnetic field can be decoupled into a single second order differential equation $$\begin{aligned}
\frac{d^2\Psi(r)}{dr^2_*}+[\omega^2-V(r)]\Psi(r)=0,\label{radial}\end{aligned}$$ where the tortoise coordinate $r_{*}$ is defined as $dr_*=\frac{
r}{r-2M}dr$. The wavefunction $\Psi(r)$ is a linear combination of the functions $j^{lm}(r)$, $h^{lm}(r)$, and $a^{lm}(r)$, which appeared in the expansion (\[w5\]). Both of the forms of wavefunction $\Psi(r)$ and effective potential $V(r)$ depend on the parity of electromagnetic field. For the odd parity $(-1)^{l+1}$, $\Psi(r)$ and $V(r)$ are given by $$\begin{aligned}
\Psi(r)_{odd}&=&\sqrt{1-\frac{8\alpha M}{r^3}} ~a^{lm}(r),\label{podd}
\\
V(r)_{odd}&=&(1-\frac{2M}{r})\bigg[\frac{l(l+1)}{r^2}(\frac{r^3+16\alpha
M}{r^3-8\alpha M})-\frac{24\alpha M(2r^4-5Mr^3-10\alpha Mr+28\alpha
M^2)}{r^3 (r^3-8\alpha M)^2}\bigg].\label{vodd}\end{aligned}$$ For the even parity $(-1)^{l}$, the forms of $\Psi(r)$ and $V(r)$ become $$\begin{aligned}
\Psi(r)_{even}&=&\frac{r^{\frac{7}{2}}}{l(l+1)}\bigg(-i\omega
h^{lm}(r)-\frac{dj^{lm(r)}}{dr}\bigg)
\frac{\sqrt{r^3+8\alpha M}}{
r^3+16\alpha M},\label{peven}
\\
V(r)_{even}&=&(1-\frac{2M}{r})\bigg[\frac{l(l+1)}{r^2}(\frac{r^3-8\alpha
M}{r^3+16\alpha M})+\frac{24\alpha M(2r^4-5Mr^3+2\alpha Mr+4\alpha
M^2)}{r^3 (r^3-8\alpha M)^2}\bigg].\label{veven}\end{aligned}$$ It is obvious that the Weyl corrections change the behavior of effective potentials $V(r)_{odd}$ and $V(r)_{even}$. Moreover, the change of effective potential originating from Weyl corrections is different for the electromagnetic fields with different parities. For fixed $l$, we find in Ref.[@sb2013] that the peak height of the potential barrier increases with the coupling constant $\alpha$ for $V(r)_{odd}$ and decreases for $V(r)_{even}$. It implies that the effects of Weyl corrections on the absorption cross section and Hawking radiation for the electromagnetic field with the odd parity are different from those of the field with the even parity. From Eq.(\[vodd\]), one can find that the effective potential $V(r)_{odd}$ has a discontinuity of the second kind at the point $r_d=(8\alpha M)^{1/3}$ for the positive $\alpha$ and near the discontinuity point the wave function $\Psi(r)$ is not well-defined. However, if the discontinuity point is located in the region inside the event horizon, the above problem can be avoided in the physical region of the black hole (i.e., $r>r_H$) and then one can study the Hawking radiation of electromagnetic field with Weyl corrections by the standard methods in this case. Therefore, we here must impose a constraint on the value of the coupling constant, $r_H>r_d$ (i.e., $\alpha<M^2$), to keep the continuity of the effective potential and the well-defined behavior of the wave function $\Psi(r)$ in the physical region of black hole for the electromagnetic field with the odd parity. After similar analyses, we can find that the coupling constant $\alpha$ must be limited in the range $r^3_H-8\alpha M>0$ and $r^3_H+16\alpha M>0$ (i.e., $-\frac{M^2}{2}<\alpha<M^2$) for the field with the even parity.
In order to calculate the absorption cross section and the luminosity of Hawking radiation for the electromagnetic field with Weyl corrections, we must solve the radial equation (\[radial\]) numerically. Near the horizon regime and at infinity, one can find that the radial wavefunction $\Psi(r)$ satisfies the boundary conditions $$\begin{aligned}
\label{bc}
\Psi(r)\approx \bigg\{
\begin{array}{rrr}
A^{tr}_{\omega l}(\omega)e^{-i\omega r_*},\ \ \ \ \ \ \ \ \ \ \ & & r_*\rightarrow -\infty;\\ \\
A^{in}_{\omega l}(\omega)e^{-i\omega r_*}+A^{out}_{\omega l}(\omega)e^{i\omega r_*}& , & r_*\rightarrow +\infty,
\end{array}\end{aligned}$$ respectively. The coefficients $A^{tr}_{l}(\omega)$, $A^{out}_{l}(\omega)$ and $A^{in}_{l}(\omega)$ obey the conservation relationship $|A^{tr}_{l}(\omega)|^2+|A^{out}_{l}(\omega)|^2=|A^{in}_{l}(\omega)|^2$. With help of this relationship and equation (\[bc\]), we can calculate the absorption probability $$\begin{aligned}
A_{l}=\bigg|\frac{A^{tr}_{\omega l}(\omega)}{A^{in}_{\omega l}(\omega)}\bigg|^2=1-\bigg|\frac{A^{out}_{\omega l}(\omega)}{A^{in}_{\omega l}(\omega)}\bigg|^2.\end{aligned}$$ The absorption cross section $\sigma_{abs}$ is related to the absorption probability $A_{l}$ by $$\begin{aligned}
\sigma_{abs}=\sum^{\infty}_{l=0}\sigma_{l}
=\frac{\pi}{\omega^2}\sum^{\infty}_{l=0}(2l+1)A_{l}.\end{aligned}$$
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
In Figs.(1) and (2), we present the effects of Weyl correction on the absorption probability of electromagnetic field for fixed angular index $l$. For the odd parity electromagnetic field, one can easily see that the absorption probability decreases with the increase of the Weyl correction parameter $\alpha$. The main reason behind this phenomenon is that the larger $\alpha$ yields the higher peak of the effective potential so that less electromagnetic wave from infinity can be transmitted to the black hole. For the even parity electromagnetic field, we find that the absorption probability increases with the Weyl correction parameter $\alpha$, which means that the dependence of the absorption probability on the Weyl correction parameter $\alpha$ is different entirely from that of in the case of the electromagnetic field with odd parity. It can be attributed to the difference in the change of the effective potential with the parameter $\alpha$ for the electromagnetic fields with different types of parities. With increase of the angular index $l$, we find that the absorption probability decreases for the two different parities, which is similar to that in the case without Weyl corrections.
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
{width="6cm"}
In Figs. (3) and (4), we plot the change of the partial and total absorption cross sections on the Weyl correction parameter $\alpha$ for the odd-parity and even-parity electromagnetic fields, respectively. With increase of the Weyl correction parameter $\alpha$, we find both of the partial and total absorption cross sections decrease for the odd-parity electromagnetic field, but increase for the even-parity one. As the frequency $\omega\rightarrow 0$, one can easily obtain that the absorption cross section tend to zero for both of different electromagnetic fields, which is the same as that in the case without Weyl corrections. Figs. (3) and (4) also tell us that in the high-energy region the total absorption cross section oscillate around the geometric-optical limit $\sigma_{\text{geo}}$, which is similar to that in the case without Weyl corrections [@ns; @ad1]. However, in the case with Weyl corrections, we find that the geometric-optical limit $\sigma_{\text{geo}}$ decreases with $\alpha$ for the odd-parity electromagnetic field and increases for the even-parity one. It is not surprising because the geometric-optical limit $\sigma_{geo}$ can be approximated as a function of impact parameter $b_c$, i.e., $\sigma_{geo}\sim\pi b^2_c$ and $b_c$ is related to the radius of photon sphere $r_{ps}$, which depends on behavior of effective potential of electromagnetic field in the background of black hole spacetime. For the electromagnetic field with Weyl corrections, the change of $b_c$ with $\alpha$ is shown in Fig. (5). It is clear that with increase of the Weyl correction parameter $\alpha$ the impact parameter $b_c$ decreases for the odd-parity electromagnetic field and increases for the even-parity one.
Now let us turn to study the effects of Weyl corrections on Hawking radiation of electromagnetic field in the background of a Schwarzschild black hole. The Hawking power emission spectrum and the Hawking luminosity of electromagnetic field are [@pk; @pk1; @cmh] $$\frac{d^2E}{dtd\omega}=\frac{1}{2\pi}\sum_{l}\frac{(2l+1)A_l\omega}
{e^{\omega/T_{H}}-1},$$ and $$\label{le1}
L=\sum_{l}\int_{0}^{\infty}\frac{(2l+1)A_l \omega }{e^{\omega/T_{H}}-1}\frac{d\omega}{2\pi},$$ respectively, where $T_H$ is the Hawking temperature of Schwarzschild black hole.
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
{width="5.5cm"} {width="5.5cm"} {width="5.5cm"}
In Figs. (6) and (7), we present the power emission spectra of the electromagnetic field with Weyl corrections in the Schwarzschild black hole spacetime for fixed $l$. With increase of Weyl correction parameter $\alpha$, we find that the power emission spectra decreases for the electromagnetic field with odd parity and increases for the electromagnetic field with even parity. The dependence of power emission spectra on Weyl correction parameter is similar to the dependence of absorption probability on Weyl correction parameter. In Fig. (8), we show the dependence of the luminosity of Hawking radiation on parameter $\alpha$ for fixed angular index $l$. It is clear that as $\alpha$ increases, the luminosity $L$ of partial wave decreases for the odd parity electromagnetic field and increases for the even parity one. It is shown again that Weyl corrections modifies the standard results of Hawking radiation of electromagnetic field in the black hole spacetime.
In summary, we have investigated numerically the absorption cross section and the Hawking radiation of an electromagnetic field with Weyl correction in the background of a four-dimensional Schwarzschild black hole spacetime. Our results show that Weyl correction modifies the standard results of the absorption cross section and the Hawking radiation for the electromagnetic field. Due to the presence of Weyl corrections, the properties of the absorption cross section and the Hawking radiation depend not only on the Weyl correction parameter $\alpha$, but also on the parity of the electromagnetic field. With increase of Weyl correction parameter $\alpha$, we find that both of the absorption probability and the absorption cross section, decreases with Weyl correction parameter for the odd-parity electromagnetic field and increases with the event-parity electromagnetic field. In the low frequency limit $\omega\rightarrow 0$, we find that both of the absorption probability and the absorption cross section tend to zero, which is similar to those in the case of electromagnetic field without Weyl correction. In high-energy region we also find that the total absorption cross section oscillates around the geometric-optical limit $\sigma_{\text{geo}}$. However, in the case with Weyl corrections, the geometric-optical limit $\sigma_{\text{geo}}$ also depend on the Weyl correction parameter $\alpha$ and the parity of the electromagnetic field. Moreover, we also find that the power emission spectra and the luminosity of Hawking radiation decreases with Weyl correction parameter for the odd-parity electromagnetic field and increases with the event-parity electromagnetic field. Our results show again that Weyl corrections modifies the properties of the absorption cross section and the Hawking radiation for the electromagnetic field in the black hole spacetime.
[**Acknowledgments**]{}
This work was partially supported by the National Natural Science Foundation of China under Grant No.11275065, the NCET under Grant No.10-0165, the PCSIRT under Grant No. IRT0964, the Hunan Provincial Natural Science Foundation of China (11JJ7001) and the construct program of key disciplines in Hunan Province. J. Jing’s work was partially supported by the National Natural Science Foundation of China under Grant Nos. 11175065, 10935013; 973 Program Grant No. 2010CB833004.
[99]{} =0.6 cm
M. Born and L. Infeld, Proc. R. Soc. A [**144**]{}, 425 (1934) G. Boillat, J. Math. Phys. [**11**]{}, 941 (1970); [**11**]{}, 1482 (1970). G. W. Gibbons and D. A. Rasheed, Nucl. Phys. B [**454**]{}, 185 (1995).
E. Fradkin and A. A. Tseytlin, Phys. Lett. B [**163**]{}, 123 (1985); A. Abouelsaood, C. G. Callan Jr., C. R. Nappi, and S.A. Yost, Nucl. Phys. B [**280**]{}, 599 (1987); R. G. Leigh, Mod. Phys. Lett. A [**4**]{}, 2767 (1989); D. Brecher, Phys. Lett. B [**442**]{}, 117 (1998); D. Brecher and M. J. Perry, Nucl. Phys. B [**527**]{}, 121 (1998); A. A. Tseytlin, Nucl. Phys. B [**501**]{}, 41 (1997).
A. B. Balakin and J. P. S. Lemos, Class. Quantum Grav. [**22**]{}, 1867 (2005). V. Faraoni, E. Gunzig and P. Nardone, Fundamentals of Cosmi Physis [**20**]{}, 121 (1999). F. W. Hehl and Y. N. Obukhov, Lect. Notes Phys. [**562**]{}, 479 (2001).
M. S. Turner and L. M. Widrow , Phys. Rev. D [**37**]{} 2743 (1988). F. D. Mazzitelli and F. M. Spedalieri, Phys. Rev. D [**52**]{} 6694 (1995). G. Lambiase and A. R. Prasanna, Phys. Rev. D [**70**]{}, 063502 (2004). A. Raya, J. E. M. Aguilar and M. Bellini, Phys. Lett. B [**638**]{}, 314 (2006). L. Campanelli, P. Cea, G. L. Fogli and L. Tedesco, Phys. Rev. D [**77**]{}, 123002 (2008).
K. Bamba and S. D. Odintsov, JCAP [**0804**]{}, 024, (2008). K. T. Kim, P. P. Kronberg, P. E. Dewdney and T. L. Landecker, Astrophys. J. [**355**]{} 29 (1990); K.T. Kim, P. C. Tribble and P. P. Kronberg, Astrophys. J. [**379**]{} 80 (1991). T. E. Clarke, P. P. Kronberg and H. Boehringer, Astrophys. J. [**547**]{}, L111 ( 2001).
A. Ritz and J. Ward, Phys. Rev. D [**79**]{} 066003 (2009). I. T. Drummond and S. J. Hathrell, Phys. Rev. D [**22**]{}, 343 (1980).
T. Dereli1 and O. Sert, Eur. Phys. J. C [**71**]{}, 1589 (2011).
S. K. Solanki, O. Preuss, M. P. Haugan, A. Gandorfer, H. P. Povel, P. Steiner, K. Stucki, P. N. Bernasconi, and D. Soltau, Phys. Rev. D [**69**]{}, 062001 (2004); O. Preuss, M. P. Haugan, S. K. Solanki, and S. Jordan, Phys. Rev. D [**70**]{}, 067101 (2004); Y. Itin and F. W. Hehl, Phys. Rev. D [**68**]{}, 127701 (2003).
J. P. Wu, Y. Cao, X. M. Kuang, and W. J. Li, Phys. Lett. B [**697**]{}, 153 (2011). D. Z. Ma, Y. Cao, and J. P. Wu, Phys. Lett. B [**704**]{}, 604 (2011). D. Momeni, N. Majd, and R. Myrzakulov, Europhys. Lett. 97, 61001 (2012); D. Roychowdhury, Phys. Rev. D [**86**]{}, 106009 (2012); D. Momeni, M. R. Setare, and R. Myrzakulov, Int. J. Mod. Phys. A [**27**]{}, 1250128 (2012); D. Momeni and M. R. Setare, Mod. Phys. Lett. A [**26**]{}, 2889 (2011). Z. X. Zhao, Q. Y. Pan, J. L. Jing, Phys. Lett. B [**719**]{}, 440 (2013).
S. Chen and J. Jing, Phys. Rev. D [**88**]{}, 064058 (2013).
R. Ruffini, [*The Mathematical Theory of Blck Holes*]{}, (Oxford University Press, New York, 1983). N. S$\acute{a}$nchez, [*Phys. Rev.*]{} D[**16**]{} 973(1977). L. C. B. Crispino, E. S. Oliveira, A. Higuchi and G. E. A. Matsas, Phys. Rev. D [**75**]{}, 104012 (2007); L. C. B. Crispino, E. S. Oliveira and A. Higuchi, Phys. Rev. D [**84**]{}, 084048 (2011); C. F. B. Macedo1, L. C. S. Leite1, E. S. Oliveira1, S. R. Dolan and L. C. B. Crispino, Phys. Rev. D [**88**]{}, 064033 (2013); L. C. B. Crispino, A. Higuchi and E. S. Oliveira, Phys. Rev. D [**80**]{}, 104026 (2009).
P. Kanti, [*Int.J.Mod.Phys.*]{} A[**19**]{} 4899(2004). P. Kanti and J. March-Russell, [*Phys. Rev.*]{} D[**66**]{}, 024023(2002); [*Phys. Rev.*]{} D [**67**]{}, 104019(2003). C. M. Harris and P. Kanti, JHEP [**0310**]{} 014(2003).
[^1]: csb3752@hunnu.edu.cn
[^2]: jljing@hunnu.edu.cn
|
---
abstract: 'The magnetic character of the ground-state of two electrons on a double quantum dot, connected in series to left and right single-channel leads, is considered. By solving exactly for the spectrum of the two interacting electrons, it is found that the coupling to the continuum of propagating states on the leads, in conjunction with the electron-electron interactions, may result in a delocalization of the bound state of the two electrons. This, in turn, reduces significantly the range of the Coulomb interaction parameters over which singlet-triplet transitions can be realized. It is also found that the coupling to the leads favors the singlet ground-state.'
address:
- 'Center of Advanced Studies, Norwegian Academy of Sciences, Oslo 0271, Norway'
- |
*School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences,\
Tel Aviv University, Tel Aviv 69978, Israel*
- '*Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel*'
author:
- 'O. Entin-Wohlman[@OEW]'
- Amnon Aharony
- 'and Y. Levinson'
title: 'Quantum dots with two electrons: Singlet-triplet transitions'
---
[2]{}
Introduction
============
The spin state of the many electron ground-state is determined by the interplay between the kinetic and the electrostatic interactions. According to Hund’s law the ground-state of electrons in a partially filled shell of an atom has the maximal possible spin, in order to minimize the electrostatic repulsion. On the other hand, Anderson’s super-exchange antiferromagnetic interaction, which favors zero total spin, arises from the reduction in the ground-state energy brought about by the hopping of electrons between adjacent ions. Another example is realized in gases made of homonuclear diatomic molecules, in which the distance between the nuclei determines the magnetic properties of the gas; in other words, the ground-state of the molecule can be either a triplet or a singlet.
The possibility to study these rules as function of controlled parameters, and in particular to observe deviations from them, has been opened up in recent years with the intensive experimental and theoretical investigations of small confined systems. Devices based on quantum dots formed in GaAs heterostructures allow to probe the electronic states in situations where there are only a few electrons in the system, as function of, e.g., the number of electrons in the sample, the spacing of the single-electron energy levels, the internal structure of the dots (e. g. the distance and the coupling between internal parts) or the coupling of the system to external leads. Thus, modulating the single-particle spectrum by a magnetic field, applied perpendicularly to the plane of the electron gas, [@tarucha1; @tarucha2] has produced a structure of the conductance peaks that has been interpreted in terms of singlet-triplet transitions of the last two electrons in the dot [@pustilnik1; @pustilnik2] (see also Ref. ). This has been observed on vertical quantum dots, [@tarucha2] and also in lateral ones. [@schmid] The ground-state spin of [*chaotic*]{} quantum dots has been studied by tracing the conductance peak spacing as function of a weak parallel magnetic field (which couples primarily to the spins). [@folk] A numerical investigation of such a configuration analyzed the influence of the exchange interaction on the peak structure. [@berkovits] In a similar fashion, the singlet-triplet transitions in such dots have been attriubted to avoided-crossings in the many-electron states, and the relation between those and kinks in the conductance pattern has been explored. [@baranger]
When the device consists of two or more coupled dots, another controllable parameter comes into the play: the interdot coupling. Experimentally, [@lee] it has been found that this coupling shows up in the conductance peak positions. Theoretically, the effect of the interdot distance on the states of a few electrons confined in a parabolic potential has been analyzed using mean-field methods, [@palacios] the Kohn-Sham equation, [@partoens] and numerical diagonalization.[@martin] Different spin states of double-quantum-dots systems have been also studied using the numerical renormalization group method. [@izumida]
Here we present an exact analytical solution for the eigenvalues and eigenfunctions of two electrons on a double-dot system, which is coupled to two single-channel leads. The two electrons interact while they are on the dots, and we include in the calculation the direct Coulomb and the exchange interactions between the dots, and the on-site Hubbard interactions on the dots. We obtain the condition for the singlet-triplet transitions of the ground-state energies, and in particular examine the role of the [*delocalization*]{} effect of the interactions: It has been found in our previous works, [@we1; @we2] as well as in other studies, [@vidal; @weinmann] that the interplay between the hybridization of the localized single-particle states on the dot with the propagating states on the leads, and the electron-electron interactions, may promote one of the electrons to the continuum. If the system had only one doubly bound state, then the above ‘promotion’ results in a ‘delocalization’ of the ground state. Once delocalized, the singlet and the triplet states become degenerate.
Our earlier work included only the case with on-site Coulomb repulsion. In that case, the bound ground state is a singlet. As the coupling of the dots with the leads was increased, the parameter range where this bound state exists was decreased, and the delocalization was accompanied by a transition from the singlet to a degenerate singlet-triplet ‘metallic’ state. In the present paper we add the effects of direct and exchange interactions. In the absence of the coupling to the leads, it is well known [@harrison] that the ground state changes from a singlet to a triplet upon increasing the exchange interaction. Here we investigate what happens to that singlet-triplet transition in the presence of the coupling to the leads.
After discussing our model Hamiltonian in Sec. II, we present the exact two-electron solution in Sec. III. Section IV then discusses the singlet-triplet transition, mainly for a special choice of the parameters where it is easiest to explore the solution analytically. Our results are then summarized in Sec. V.
Hamiltonian
===========
The Hamiltonian of the model reads $$\begin{aligned}
{\cal H}={\cal H}_{\rm sp}+{\cal H}_{\rm c}.\label{ham}
\end{aligned}$$ The single-particle tight binding Hamiltonian is $$\begin{aligned}
{\cal H}_{\rm sp}=\sum_{{\rm i}\sigma}
\epsilon_{{\rm i}}c^{\dagger}_{{\rm i}\sigma}
c_{{\rm i}\sigma}-\sum_{{\rm ij}\sigma}t_{\rm ij}
c^{\dagger}_{{\rm i}\sigma}c_{{\rm j}\sigma},
\label{hamsp}
\end{aligned}$$ where $c^{\dagger}_{{\rm i}\sigma}$ creates an electron with spin $\sigma$ on site i. The site energies $\epsilon_{{\rm i}}$ are different from zero only on the ‘quantum dots’, which can be viewed as ‘impurities’ on the lattice. In what follows, we specifically consider two neighboring quantum dots, which will be denoted by $\ell$ and $r$, and study the symmetric case where $\epsilon_{\ell ,r}=\epsilon_{0}$. The hopping matrix elements $t_{\rm ij}$ are divided into three kinds: the hopping among the ‘dots’ is denoted by $t_D$ ($\equiv t_{\ell r}$), the hopping between a dot and a neighboring site on the ‘lead’ is denoted by $t_0$ (e. g. $t_{\ell {\rm i}}$ for i $\neq r$), and the hopping between neighboring ‘lead’ sites is denoted by $t = 1$, setting the units of energy.
The Coulomb interactions are assumed to exist only among electrons which sit on the dots. Generally, this interaction has the form $$\begin{aligned}
{\cal H}_{\rm c}=\sum_{\rm ijmn}\Gamma_{\rm ij}^{\rm mn}
\sum_{\sigma\sigma '}c^{\dagger}_{{\rm i}\sigma}c^{\dagger}_{{\rm j}\sigma '}
c_{{\rm m}\sigma '}c_{{\rm n}\sigma},\end{aligned}$$ where $$\begin{aligned}
\Gamma_{{\rm ij}}^{\rm mn}=\int d{\bf r}\int d{\bf r}'
v({\bf r}-{\bf r}')\varphi^{\ast}_{\rm i}({\bf r})
\varphi^{\ast}_{\rm j}({\bf r}')\varphi_{\rm m}({\bf r}')
\varphi_{\rm n}({\bf r}),
\label{general}\end{aligned}$$ while $\varphi_{\rm i}({\bf r})$ is the Wannier wave function of an electron localized at site i. It is usually assumed that the dominant terms will be those in which the indices are equal pair-wise. [@kurland] Here we follow the parametrization used in magnetic coupling studies, [@entin] in which the case ${\rm i=j=m=n}$ is treated separately. Neglecting the coefficient with ${\rm i}={\rm j} \neq {\rm m}={\rm n}$, which (when negative) leads to the superconductivity vertex, we are left with three possible parameters: $$\begin{aligned}
2\Gamma_{\rm ii}^{\rm ii}&=&{\rm U},\nonumber\\
2\Gamma_{\rm ij}^{\rm ji}&=&{\rm V},\ \
2\Gamma_{\rm ij}^{\rm ij}={\rm K},\ \ \ {\rm i} \neq {\rm j},
\label{me}\end{aligned}$$ where ${\rm U},{\rm V}$ and ${\rm K}$ are the (intra-dot) Hubbard, (inter-dot) direct, and (inter-dot) exchange interactions, respectively. Then $$\begin{aligned}
{\cal H}_{\rm int}&=&{\rm U}\sum_{\rm i}\hat{n}_{{\rm i}\uparrow}
\hat{n}_{{\rm i}\downarrow}\nonumber\\
&+&\frac{1}{4}(2{\rm V}-{\rm K})\sum_{{\rm i}\neq {\rm j}}
\hat{n}_{{\rm i}}\hat{n}_{{\rm j}}
-{\rm K}\sum_{{\rm i}\neq{\rm j}}{\bf S}_{{\rm i}}\cdot{\bf S}_{{\rm j}},
\label{Hwe}\end{aligned}$$ where $$\begin{aligned}
\hat{n}_{{\rm i}\sigma}&=&c^{\dagger}_{{\rm i}\sigma}c_{{\rm i}\sigma},\ \
\hat{n}_{{\rm i}}=\sum_{\sigma}\hat{n}_{{\rm i}\sigma},\nonumber\\
{\bf S}_{{\rm i}}&=&\sum_{\sigma\sigma '}c^{\dagger}_{{\rm i}\sigma}
{\vec \sigma}_{\sigma\sigma '}c_{{\rm i}\sigma '}\end{aligned}$$ and ${\vec \sigma}_{\sigma\sigma '}$ is the vector of Pauli’s spin matrices. It is important to note that with the above approximation for the Coulomb interaction vertex, the interaction Hamiltonian becomes [*spin-dependent*]{}.
A similar Hamiltonian has been used to study the effects of the spin states on the conducting properties of confined mesoscopic quantum dots. [@pustilnik1; @pustilnik2; @folk; @berkovits] In these studies, ${\rm i}$ represents some orbital state on the dot. In Ref. it has been argued that when the Thouless conductance of the confined (mesoscopic) system is large, the Coulomb vertices can be parametrized in terms of two coupling constants, independent of the orbital indices. In terms of our parameters, these are the charging energy, ${\rm E}_{c}=(2{\rm
V}-{\rm K})/4$, and the exchange energy, ${\rm J}={\rm K}$, while ${\rm U} ={\rm V}+{\rm K}$. It is straightforward to apply our results to that case.
In our special case of the double dot, the energies $\Gamma$ in Eq. (\[me\]) differ from zero only for ${\rm i},\ {\rm j}=
\ell,r$, and thus $$\begin{aligned}
{\cal H}_{\rm int}&=&{\rm U}(\hat{n}_{\ell\uparrow}\hat{n}_{\ell\downarrow}+
\hat{n}_{r\uparrow}\hat{n}_{r\downarrow})
\nonumber\\
&+&({\rm V}-\frac{{\rm K}}{2})\hat{n}_{\ell}
\hat{n}_{r}-2{\rm K}{\bf S}_{\ell}\cdot{\bf S}_{r}.\label{hamint}
\end{aligned}$$
The two-electron ’molecule’
===========================
We will confine ourselves to a double-dot system (a ‘molecule’), containing two electrons. The system is modeled by two identical single-level impurities, each having an on-site single-particle energy level $\epsilon_{\ell}=\epsilon_r=\epsilon_{0}$. The two impurities are coupled to one another by the interdot matrix element $t_{D}$. When the molecule is isolated ($t_0=0$), there are three degenerate triplet states, of energy $2\epsilon_{0}+{\rm V}-{\rm K}$, and three singlet states. One of the latter has the energy $2\epsilon_{0}+{\rm U}$, while the other two energies are [@harrison] $$\begin{aligned}
{\rm E}=2\epsilon_{0}+\frac{{\rm U}+{\rm V}+{\rm K}}{2}\pm\sqrt{4t_{D}^{2}
+\Bigl (\frac{{\rm U}-{\rm V}-{\rm K}}{2}\Bigr )^{2}}.\label{Es}
\end{aligned}$$ Examination of the interaction matrix elements, Eqs. (\[general\]) and (\[me\]), shows that ${\rm U}>{\rm V}>0$ and ${\rm U}>{\rm K}$. Moreover, the exchange interaction ${\rm K}$ involves the square of the overlap matrix element of the two impurities, and hence is of the order of $t_{D}^{2}$. Therefore one concludes that the lowest singlet energy is given by the minus sign in (\[Es\]). The triplet states energy is below the lowest singlet one provided that $$\begin{aligned}
2t_{D}^{2}<{\rm K}({\rm U}-{\rm V}+{\rm K}).\label{st0}
\end{aligned}$$ One notes that when the direct and exchange Coulomb vertices V and K are disregarded, the energy of the singlet state is lowered by the interdot kinetic energy, producing the Anderson super-exchange antiferromagnetic coupling, [@anderson] $\sim 4t_{D}^{2}/{\rm U}$. The triplet state becomes the ground-state once the exchange energy ${\rm K}$ wins over this kinetic energy. [@brouwer; @eto] Finally, with the choice of parameters in which ${\rm U}={\rm V}+{\rm K}$ (see above), the condition becomes $t_{D}^{2}<{\rm K}^2$ (see also Ref. ). Both $t_D$ and K decay exponentially with the interdot distance. However, Eq. (\[general\]) indicates that K is roughly of order $t_D^2$, and thus decays faster. Therefore, one might expect a transition from the triplet to the singlet ground state as this distance increases.
The main purpose of the present paper is to study how the criterion for the singlet-triplet transitions, Eq. (\[st0\]), is modified when the double quantum dot is connected to external leads, on which it is assumed that there are no electron-electron interactions. In other words, we study the changes in these transitions due to coupling the dot to a continuum of propagating states. A similar problem was discussed in Ref. , in the framework of the mean-field approximation, within a single-lead geometry. Here we first derive [*exactly*]{} the ground-state energy of [*two*]{} electrons, and then compare the singlet and the triplet ground-state energies.
For further simplification, we describe the external leads by single channel one dimensional chains, with nearest neighbor hopping $t=1$. In our previous work, [@we1; @we2] we have shown that the spectrum and the wave functions of two interacting electrons can be obtained in terms of the energy spectrum and the wave functions of the single-particle Hamiltonian. For our model, cf. Eq. (\[hamsp\]), the latter can be divided into even and odd solutions, denoted by “e" and “o". Of particular interest here will be the regions in the parameter plane $\{\epsilon_{0}-\gamma\}$, where $\gamma =t_{0}^{2}$, where the spectrum has bound states, see Fig. \[fig1\]. For $\epsilon_{0}<\gamma-2+t_{D}$, there appears one (even) bound (single-particle) state below the band of propagating states, of energy $\epsilon_{\rm B}^{\rm e}<-2$; For even smaller values of $\epsilon_{0}$, such that $\epsilon_{0}<\gamma -2 -t_{D}$ there appears a second (odd) bound state below the band, of energy $\epsilon_{\rm B}^{\rm o}$, with $$\begin{aligned}
\epsilon_{\rm B}^{\rm e,o}&=&\frac{2-\gamma}{1-\gamma}
\Bigl (\frac{\epsilon_{0}^{\rm e,o}}{2}
\Bigr )
+\frac{\gamma}{1-\gamma}\sqrt{\Bigl (\frac{\epsilon_{0}^{\rm e,o}}{2}
\Bigr )^{2}+\gamma -1},
\nonumber\\
\epsilon_{0}^{\rm e}&=&\epsilon_{0}- t_{D},\ \ \epsilon_{0}^{\rm o}=\epsilon_{0}+ t_{D}.\label{eb}
\end{aligned}$$ Similarly, for $\epsilon_{0}
>2-\gamma -t_{D}$ there appears the first bound state above the band, while for $\epsilon_{0}
>2-\gamma +t_{D}$ there are two bound states above the band. Clearly, a necessary condition to have the two electrons bound in a triplet state is the existence of two distinct single-electron bound states, so that each electron occupies a different ‘state’. For simplicity, we shall confine ourselves to the case in which both occur below the continuum, i.e., to the lowest region II in Fig. \[fig1\].
= 8cm
1truecm
It has been shown in Refs. and that when interactions between the two electrons are allowed, they may lead to the [*delocalization*]{} of one (or both) of the electrons from the doubly bound states. Thus, for example, [@we2] when V=K=0, in part of region I in Fig. \[fig1\] one finds that the doubly bound ground state is replaced by a ground state in which one of the electrons is shifted to the band. In these parameter regions there are no two-electron bound states, and the system may be called ‘metallic’. Indeed, such interaction-induced delocalization effects have been recently observed in quantum dots containing two potential minima. [@brodsky] This effect has also been found for two interacting electrons moving in a one-dimensional periodic structure, [@vidal] and for spinless fermions on strongly disordered chains. [@weinmann] In contrast to region I, in region II there potentially exist two doubly bound states, i. e. the singlet and the triplet, and our goal is to study which of these states is the ground state in the presence of this interaction-induced delocalization.
To solve for the spectrum of the two interacting electrons we proceed as follows. Let us denote the eigenstates and the eigenvalues of the single-particle Hamiltonian (\[hamsp\]) by $\phi_{a}(n)$ and $\epsilon_{a}$, respectively, where $n$ is the site index. Consider first the two-electron states with spin $S_{z}=\pm 1$, $$\begin{aligned}
|\Psi^{\sigma}\rangle &=&\sqrt{\frac{1}{2}}\sum_{ab}
{\rm X}^{\sigma}_{ab}c^{\dagger}_{a\sigma}c^{\dagger}_{b\sigma}
|0\rangle\nonumber\\
&=&\sqrt{\frac{1}{2}}\sum_{ab}{\rm X}^{\sigma}_{ab}
\sum_{nn'}\phi^{\ast}_{a}(n)\phi^{\ast}_{b}(n')
c^{\dagger}_{n\sigma}c^{\dagger}_{n'\sigma}|0\rangle\nonumber\\
&&\equiv \sqrt{\frac{1}{2}}\sum_{nn'}
{\rm X}_{nn'}^{\sigma}c^{\dagger}_{n\sigma}c^{\dagger}_{n'\sigma}|0\rangle,
\ \ \sigma =\pm 1,
\end{aligned}$$ where the amplitudes ${\rm X}^{\sigma}_{nn'}=-{\rm X}^{\sigma}_{n'n}$ are antisymmetric in the coordinates (the origin is half way between the dots), and $\sum_{nn'}|{\rm X}^{\sigma}_{nn'}|^{2}=1$. The Schrödinger equation for the two electrons then yields $$\begin{aligned}
{\rm X}^{\sigma}_{nn'}&=&{\rm X}^{\sigma}_{r\ell}({\rm V-K})\nonumber\\
&&\times \sum_{ab}\frac{\phi_{a}(r)\phi_{b}(\ell )-\phi_{a}(\ell )
\phi_{b}(r)}{{\rm E}-\epsilon_{a}-\epsilon_{b}}
\phi^{\ast}_{a}(n)\phi^{\ast}_{b}(n').
\end{aligned}$$ Hence, the two-particle energies, E, are given by the solutions of the equation $$\begin{aligned}
\frac{1}{\rm V-K}&=&\frac{1}{2}\sum_{ab}\frac{|\phi_{a}(\ell )
\phi_{b}(r)-\phi_{a}(r)\phi_{b}(\ell )|^2}
{{\rm E}-\epsilon_{a}-\epsilon_{b}}.
%\nonumber\\
%&&\times \Bigl (\phi^{\ast}_{a}(\ell )\phi^{\ast}_{b}(r)-
%\phi^{\ast}_{a}(r)\phi^{\ast}_{b}(\ell )\Bigr ).
\end{aligned}$$ Making now use of the symmetry properties of $\phi_{a}$, i.e., $\phi^{\rm e}_{a}(r)=\phi^{\rm e}_{a}(\ell)$, and $\phi^{\rm o}_{a}(r)=-\phi^{\rm o}_{a}(\ell )$, we see that the only contributions to the sum come from the cases in which $a$ is even, and $b$ is odd, and [*vice versa*]{}. For the sake of brevity, in the following expressions we replace $|\phi^i_{a}(\ell)|=|\phi^i_{a}(r)|$ ($i=e,o$) by $|\phi^i_{a}|$. Consequently, we may write the result in the form $$\begin{aligned}
\frac{1}{\rm V-K}=G_{\rm eo}({\rm E}),\label{v-k}
\end{aligned}$$ where $G_{\rm eo}$ is a [*noninteracting*]{} two-electron Green’s function, $$\begin{aligned}
G_{\rm eo}({\rm E})&=&4\frac{|\phi^{\rm e}_{\rm B}|^{2}
|\phi^{\rm o}_{\rm B}|^{2}}{{\rm E}-\epsilon^{\rm e}_{\rm B}
-\epsilon^{\rm o}_{\rm B}}
+2\sum_{\stackrel{i,j={\rm e,o}}{i\neq j}}
\sum_{k}\frac{|\phi^{i}_{\rm B}|^{2}|\phi^{j}_{k}|^{2}}
{{\rm E}-\epsilon^{i}_{\rm B}-\epsilon_{k}}
\nonumber\\
&+&\sum_{kk'}\frac{|\phi^{\rm e}_{k}|^{2}|\phi^{\rm o}_{k'}|^{2}}
{{\rm E}-\epsilon_{k}-\epsilon_{k'}}.\label{geo}
\end{aligned}$$ (In writing down this equation, we have assumed the existence of two single-particle bound states, as mentioned above.) Here the subscript B denotes a single-particle bound state, and $k$ refers to a band state, with $\epsilon_{k}=-2\cos k$. In the continuum of the band energies, there is no need to distinguish between $\epsilon_{k}^{\rm e}$ and $\epsilon_{k}^{\rm o}$. Also, the sum over all states $k$ is divided into the sum over the even propagating states, and the sum over the odd ones. We give in the Appendix the explicit expressions for the eigenstates required for the calculation of $G$.
Equation (\[v-k\]) is an implicit equation for the exact two-electrons eigenenergies E. We will postpone the discussion of these solutions, and consider now the two-electron states with $S_{z}=0$, $$\begin{aligned}
|\Psi^{0}\rangle &=&\sum_{ab}{\rm X}^{0}_{ab}
c^{\dagger}_{a\uparrow}c^{\dagger}_{b\downarrow}
|0\rangle
\nonumber\\
&=&\sum_{ab}{\rm X}^{0}_{ab}\sum_{nn'}
\phi^{\ast}_{a}(n)\phi^{\ast}_{b}(n')c^{\dagger}_{n\uparrow}
c^{\dagger}_{n'\downarrow}|0\rangle\nonumber\\
&&\equiv \sum_{nn'}{\rm X}^{0}_{nn'}
c^{\dagger}_{n\uparrow}c^{\dagger}_{n'\downarrow}|0\rangle ,
\label{sz0}
\end{aligned}$$ with $\sum_{nn'}|{\rm X}^{0}_{nn'}|^{2}=1$. For the triplet state, the amplitudes ${\rm X}_{nn'}^{0}$ are antisymmetric in the site indices. Then (inserting Eq. (\[sz0\]) in the Schrödinger equation) the energies are again given by Eqs. (\[v-k\]) and (\[geo\]). For the singlet states, ${\rm X}^{0}_{nn'}={\rm X}^{0}_{n'n}$ and the Schrödinger equation yields $$\begin{aligned}
&&{\rm X}^{0}_{nn'}=\sum_{ab}\Biggl ({\rm U}
\sum_{i=\ell ,r}{\rm X}^{0}_{ii}\frac{\phi_{a}(i)\phi_{b}(i)}{{\rm E}
-\epsilon_{a}-\epsilon_{b}}
+({\rm V+K}){\rm X}^{0}_{\ell r}\nonumber\\
&&\times
\frac{\phi_{a}(\ell )\phi_{b}(r)+\phi_{a}(r)
\phi_{b}(\ell )}{{\rm E}-\epsilon_{a}-\epsilon_{b}}\Biggr )
\phi^{\ast}_{a}(n)\phi^{\ast}_{b}(n').
\end{aligned}$$ We use this equation for $n,n'=\ell ,\ell$, $n,n'=r,r$, and $n,n'=\ell ,r$, and find two families of singlet solutions: (i) ${\rm X}^{0}_{\ell\ell}=-{\rm X}^{0}_{rr}$, ${\rm X}^{0}_{\ell r}=0$, for which $$\begin{aligned}
\frac{1}{\rm U}=G_{\rm eo}({\rm E});\label{u}
\end{aligned}$$ (ii) ${\rm X}^{0}_{\ell\ell}={\rm X}^{0}_{rr}$, with $$\begin{aligned}
1&-&({\rm U+V+K})(G_{\rm ee}({\rm E})+G_{\rm oo}({\rm E}))\nonumber\\
&+&4{\rm U(V+K)}G_{\rm ee}({\rm E})G_{\rm oo}({\rm E})=0.\label{2s}
\end{aligned}$$ \[Note that the last equation includes twice the number of solutions as Eqs. (\[v-k\]) and (\[u\]).\] Here, $G_{\rm ee,oo}$ are noninteracting, two-particle Green’s functions, which consist of the even and odd (with respect to interchanging the dots) solutions of the single-particle spectrum, respectively, with $$\begin{aligned}
G_{\rm ee(oo)}({\rm E})&=&2\frac{|\phi^{\rm e(o)}_{\rm B}|^{4}}
{{\rm E}-2\epsilon^{\rm e(o)}_{\rm B}}
+2\sum_{k}\frac{|\phi^{\rm e(o)}_{\rm B}|^{2}
|\phi^{\rm e(o)}_{k}|^{2}}
{{\rm E}-\epsilon^{\rm e(o)}_{\rm B}-\epsilon_{k}}
\nonumber\\
&+&\frac{1}{2}\sum_{kk'}\frac{|\phi^{e(o)}_{k}|^{2}|\phi^{\rm e(o)}_{k'}|^{2}}
{{\rm E}-\epsilon_{k}-\epsilon_{k'}}.\label{geeoo}
\end{aligned}$$
We next determine the ground-state energy, starting with the triplet states, whose energies are given by Eq. (\[v-k\]). We assume that we are in the lower region II of Fig. \[fig1\], where there are two bound states below the band. (Note that in the regions marked 0 and I in Fig. \[fig1\], where there is at most one single-particle bound state, the triplet ‘bound’ state will always lie in the continuum.) The function $G_{\rm eo}({\rm E})$, Eq. (\[geo\]), has the following behavior.[@we1; @we2] As E approaches $-\infty $, it goes to zero from below. At E$=\epsilon^{\rm e}_{\rm B}+\epsilon^{\rm o}_{\rm B}$, it diverges to $-\infty$, jumps to $+\infty $ as E crosses that value, and then decreases, as E approaches the bottom of the two-electron continuum states, located at $-2+\epsilon^{\rm e}_{\rm B}$. As discussed in our earlier work [@we1; @we2], in the thermodynamic limit of infinite ‘leads’ $G_{\rm eo}$ has a finite value at this band threshold, due to the $k$-dependence of $\phi^{\rm e(o)}_{k}$ at the impurities. The triplet ground-state energy is where $G_{\rm eo}$ crosses $({\rm V-K)}^{-1}$. Hence, there will be a two-electron bound state only when $({\rm V-K})^{-1}<G_{\rm eo}(-2+\epsilon^{\rm e}_{\rm B})$. It follows that there are values of the direct and exchange Coulomb couplings such that the ground triplet state is not bound, but lies in the continuum.
We now turn to the singlet states, again assuming the existence of both $\epsilon^{\rm e}_{\rm B}$ and $\epsilon^{\rm o}_{\rm B}$. Consider first the solutions given by Eq. (\[u\]). Since U$>$V (and V–K), the lowest solution of this equation lies [*above*]{} the lowest solution of the triplet state, (which is given by the same function $G_{\rm eo}$). Hence, we need not consider anymore the states given by (\[u\]). To explore the other family of singlet solutions, it is convenient to re-write Eq. (\[2s\]) in the form $$\begin{aligned}
&&\frac{1}{4}\Biggl (\frac{1}{G_{\rm ee}({\rm E})}+
\frac{1}{G_{\rm oo}({\rm E})}\Biggr )=
\frac{\rm U+V+K}{2}\nonumber\\
&&\pm
\sqrt{\frac{1}{16}\Biggl (\frac{1}{G_{\rm ee}({\rm E})}-\frac{1}{G_{\rm oo}({\rm E})}
\Biggr )^{2}+\Biggl (\frac{\rm U-V-K}{2}\Biggr )^{2}}.\label{2s2}
\end{aligned}$$ Since the behavior of $G_{\rm ee}$ and $G_{\rm oo}$ as function of E is similar to that of $G_{\rm eo}$ described above, it follows from Eq. (\[2s2\]) that the lowest singlet state energy obeys that equation with the minus sign.
Singlet-triplet transitions
===========================
In order to decide when the lowest bound state of the two electrons is a singlet or a triplet, we need to (i) determine for which values of the Coulomb parameters Eqs. (\[v-k\]) and (\[2s2\]) have bound solutions; and (ii) to compare these two solutions, when they exist. Consider as an example the case in which there is only the on-site Hubbard interaction, that is, V=K=0. Then the triplet bound state has the energy ${\rm E}_{\rm T}=
\epsilon^{\rm e}_{\rm B}+\epsilon^{\rm o}_{\rm B}$, cf. Eq. (\[v-k\]). The singlet energy, E$_{\rm S}$, in that case is given by the lowest solution of $$\begin{aligned}
\frac{1}{\rm U}=G_{\rm ee}({\rm E})+G_{\rm oo}({\rm E}).\label{onlyu}\end{aligned}$$ Similarly to the behavior of $G_{\rm eo}({\rm E})$, the right-hand-side of this equation starts at very small negative values when E tends to $-\infty$. It then diverges to $-\infty$ as E approaches $2\epsilon^{\rm e}_{\rm B}$, jumps to $+\infty$ as E crosses that value, diverges again to $-\infty $ as E$\rightarrow 2\epsilon^{\rm o}_{\rm B}$, then jumps to $+\infty$, and finally decreases towards a finite value as E approaches the bottom of the two-electron continuum. It follows that Eq. (\[onlyu\]) has always a bound energy solution. Moreover, if $G_{\rm ee}+G_{\rm oo}$ is [*negative*]{} at E$={\rm E}_{\rm T}\equiv \epsilon^{\rm e}_{\rm B}
+\epsilon^{\rm o}_{\rm B}$, then that solution E$_{\rm S}$ lies below E$_{\rm T}$, i.e., the ground-state is a singlet. This is indeed the case, as is shown in the Appendix \[Eq. (\[neg\])\]. This is in accordance with the general rule, which states that in order for the ground-state to be a triplet, the exchange Coulomb energy has to overcome the kinetic energy.
For the sake of clarity of the presentation, we will carry the rest of the analysis to lowest order in the coupling to the leads, $\gamma $. In that case, it is possible to derive simple expressions for the two-particle Green’s functions, see Eqs. (\[sg\]). Using those equations, we find that the singlet energies are given by $$\begin{aligned}
&&{\rm E} -2\epsilon_{0}+\gamma (e^{-\alpha^{\rm e}}+e^{-\alpha^{\rm o}})=
\frac{\rm U+V+K}{2}\nonumber\\
&-&\sqrt{
4t_{D}^{2}+\Bigl (\frac{\rm U-V-K}{2}\Bigr )^{2}
+4t_{D}\gamma (e^{-\alpha^{\rm e}}-e^{-\alpha^{\rm o}})},
\label{ES}\end{aligned}$$ and the triplet energies are given by $$\begin{aligned}
{\rm E}-2\epsilon_{0} +\gamma (e^{-\alpha^{\rm e}}+e^{-\alpha^{\rm o}})={\rm V-K},
\label{tt}\end{aligned}$$ where $\alpha^{\rm e,o}$ is related to the corresponding ${\rm E}$ (with indices S or T) via $$\begin{aligned}
{\rm E}-\epsilon^{\rm e,o}_{\rm B}=-2{\rm cosh}\alpha^{\rm e,o}.
\label{al}\end{aligned}$$ Let us examine the case in which in the absence of the coupling to the leads, the singlet and the triplet ground-state energies are equal, i.e., $4t_{D}^{2}=2$K(U–V+K), see Eq. (\[st0\]). Then $$\begin{aligned}
{\rm E}_{\rm S}-{\rm E}_{\rm T}&=&\gamma (e^{-\alpha^{\rm e}_{\rm T}}
+e^{-\alpha^{\rm o}_{\rm T}}-e^{-\alpha^{\rm e}_{\rm S}}
-e^{-\alpha^{\rm o}_{\rm S}})
\nonumber\\
&-&
\frac{4t_{D}}{{\rm U-V+3K}}\gamma (e^{-\alpha^{\rm e}_{\rm S}}
-e^{-\alpha^{\rm o}_{\rm S}}).
\end{aligned}$$ The last term on the right-hand-side of this equation is negative. This follows from Eq. (\[al\]) and the fact that $\epsilon^{\rm e}_{\rm B}<\epsilon^{\rm o}_{\rm B}$. As for the first term, we use again Eq. (\[al\]), to write it in the form $({\rm E}_{\rm T}-{\rm E}_{\rm S})\gamma
((e^{2\alpha^{\rm e}_{\rm S}} -1)^{-1}+(
e^{2\alpha^{\rm o}_{\rm S}}-1)^{-1})$. Hence, E$_{\rm S}<{\rm E}_{\rm T}$ and the singlet is prefered.
The above discussion shows that the coupling to the continuum of propagating states enhances the tendency of the two electrons to form a singlet state, in the situation where in the absence of that coupling, the singlet and the triplet states are degenerate. In order to investigate whether this tendency persists for other choices of parameters (and at the same time to keep the calculations tractable) we will now confine ourselves to the choice U=V+K. In this case, again to leading order in $\gamma$, the equation for the singlet energies (\[ES\]) reads $$\begin{aligned}
f_{\rm S}({\rm E})&=&{\rm V+K},\nonumber\\
f_{\rm S}({\rm E})&=&{\rm E}-2\epsilon_{0}+2t_{D}
+2\gamma e^{-\alpha^{\rm e}}.
\label{ss}
\end{aligned}$$ Similarly, Eq. (\[tt\]) for the triplet energies can be written as $$\begin{aligned}
f_{\rm T}({\rm E})&=&{\rm V-K},\nonumber\\
f_{\rm T}({\rm E})&=&{\rm E}-2\epsilon_{0}+\gamma (e^{-\alpha^{\rm e}}+
e^{-\alpha^{o}}).
\label{ttt}
\end{aligned}$$ Let us first determine for which parameters these equations yield bound, two-electron energies. To this end, we consider Eqs. (\[ss\]) and (\[ttt\]) at the bottom of the two-electron continuum, E=$-2+\epsilon^{\rm e}_{\rm B}$. The first of these equations will have a bound state for $f_{\rm S}(-2+\epsilon^{\rm e}_{\rm B})>{\rm V+K}$; the second will have such a solution when $f_{\rm T}(-2+\epsilon^{\rm e}_{\rm B})>{\rm V-K}$. These conditions are plotted in Fig. \[fig2\] as the thick lines there. A bound triplet state exists in regions I+III, below the heavy line of positive slope. A singlet bound state exists in I+II, below the heavy line of negative slope. In region IV, there are no bound states; both the triplet and the singlet states are in the continuum, and their energy is about the same. Crossing the line between regions II and IV (or III and IV) thus corresponds to the delocalization transition discussed above, from a singlet (triplet) bound ground state to a degenerate ‘metallic’ state. This transition is the most striking effect of the coupling to the leads, and we expect it to appear irrespective of the quantitative approximation used in Fig. \[fig2\].
In region I one has to compare the singlet energy with the triplet one. These two become degenerate along the diamond curve, whose equation is derived from (\[al\]), (\[ss\]), and (\[ttt\]) $$\begin{aligned}
{\rm V-K}&=&-\frac{\epsilon^{\rm o}_{\rm B}}{2}({\rm B}-1)
+\frac{\epsilon^{\rm e}_{\rm B}}{2}({\rm B}+1)-2\epsilon_{0}
-(1-\gamma ){\rm AB},\nonumber\\
{\rm A}&=&\frac{2}{\gamma}({\rm K}-t_{D}),\nonumber\\
{\rm B}&=&\sqrt{1+4/[{\rm A}({\rm A}+\epsilon^{\rm o}_{\rm B}-
\epsilon^{\rm e}_{\rm B})]}.\end{aligned}$$ This line is almost vertical, with ${\rm E}_{\rm S}<{\rm E}_{\rm
T}$ to its left, and ${\rm E}_{\rm S}>{\rm E}_{\rm T}$ to its right. The conclusion is that, as long as there exists a bound state of the two electrons, then this line moves slightly to the right as $\gamma$ is increased from zero (when the line was at K$=t_D$). However, the coupling to the continuum states delocalizes the electrons, making the two states degenerate over a large part of the parameter plane $\{{\rm
V-K}\}$, i. e. region IV in Fig. \[fig2\]. With all other parameters fixed, one might expect that increasing the distance between the dots causes a decrease in $t_D$, in V and in K, thus causing a shift towards to lower left side of Fig. \[fig2\], towards a bound singlet ground state.
= 8cm
1truecm
discussion
==========
We have derived analytical expressions for the spectrum of two interacting electrons on a simplified model for a double quantum-dot. When the dot is decoupled from the external leads, it is straightforward to obtain this spectrum, and discuss the criterion for its ground-state to be a singlet or a triplet. The question we have addressed is how this criterion is modified when the single-particle states which are localized on the dot are coupled to the continuum of extended states on the leads. A typical example of our results is shown in Fig. \[fig2\]: as long as the the electron-electron interactions do not delocalize the ground-state, then the location for the singlet-triplet transition shifts continuously with the coupling to the leads, $\gamma$. In that case, one can still say that the singlet state is the ground one provided that the kinetic energy dominates over the exchange energy. In a way, the coupling to the leads enhances the kinetic energy, and therefore it slightly favors the singlet ground state. However, the coupling to the leads has a much more drastic effect: over a significant part of the parameter space (which consists of the Coulomb couplings, the single-particle energies on the dots, and the coupling to the leads), e. g. region IV in Fig. \[fig2\], the interplay between the coupling to the leads and the electron-electron interactions delocalizes one or both electrons. Then, the singlet and the triplet states are degenerate. In such a situation, the bound state (which may be either a singlet or a triplet) disappears and the ground state becomes a degenerate singlet-triplet ‘metallic’ state. We believe that this delocalization effect should be taken into consideration in the analyzes of experimental data related to this question.
We enjoyed many discussions with Yoseph Imry. This project was supported by grants from the Israel Science Foundation and from the Israeli Ministry of Science and the French Ministry of Research and Technology, AFIRST.
The two-electron Green’s functions
==================================
We first list the eigenfunctions required for the calculation of the noninteracting, two-particle Green’s functions. These are needed only on the double quantum dot, that is, on the sites $r$ and $\ell $. Writing the even and odd bound-state energies, Eqs. (\[eb\]), in the form $$\begin{aligned}
\epsilon_{\rm B}^{\rm e,o}=-2{\rm cosh}\kappa^{\rm e,o},\end{aligned}$$ the wave functions on the dot sites is $$\begin{aligned}
|\phi^{\rm e,o}_{\rm B}|^{2}=\frac{1}{2}\Biggl (1+\frac{\gamma}
{e^{2\kappa^{\rm e,o}}-1}\Biggr )^{-1}.\end{aligned}$$ The band states have been calculated assuming periodic boundary conditions for a system of N sites (these include the leads). Then $$\begin{aligned}
|\phi_{k}^{\rm e,o}|^{2}=\frac{2}{{\rm N}}
\frac{\gamma\sin^{2}k}{\gamma^{2}\sin^{2}k +(\epsilon_{k}-\epsilon^{\rm e,o}_{0}
+\gamma\cos k)^{2}},\label{phik}\end{aligned}$$ with $\epsilon_{k}=-2\cos k$, for both the even and odd states. In calculating the sums over $k$ in the two-particle Green’s functions, we shall use the continuum limit, dividing the $k$-integrations on the even (odd) functions by 2.
We next derive $G_{\rm ee}$, $G_{\rm oo}$, and $G_{\rm eo}$. It is convenient first to calculate the function $$\begin{aligned}
Q^{\rm e,o}(\omega )=\sum_{k}\frac{|\phi^{\rm e,o}_{k}|^{2}}{\omega -\epsilon_{k}},
\ \ \ \omega <-2.
\end{aligned}$$ Writing $$\begin{aligned}
\omega =-2{\rm cosh}\alpha ,
\end{aligned}$$ and using Eq. (\[phik\]), we find $$\begin{aligned}
Q^{\rm e,o}&=&-\gamma \frac{e^{\kappa^{\rm e,o}}}{e^{2\kappa^{\rm e,o}}-1+\gamma}
\nonumber\\
&&\times
\frac{e^{\alpha +\kappa^{\rm e,o}}}{(e^{\alpha +\kappa^{\rm e,o}}-1+\gamma )(e^{\alpha +
\kappa^{\rm e,o}}-1)},
\end{aligned}$$ where we have also used $$\begin{aligned}
\sum_{k}|\phi^{\rm e,o}_{k}|^{2}=1-2|\phi^{\rm e,o}_{\rm B}|^{2}.
\end{aligned}$$ Exploiting this result, we now introduce the function $F$, $$\begin{aligned}
F^{\rm e,o}(\omega )&=&|\phi^{\rm e,o}_{\rm B}|^{2}+
(\omega -\epsilon^{\rm e,o}_{\rm B})\frac{1}{2}Q^{\rm e,o}(\omega )\nonumber\\
&=&\frac{1}{2}\times\frac{1}{1+(\gamma /(e^{\alpha +\kappa^{\rm e,o}}-1))}.\label{F}
\end{aligned}$$ Note that $$\begin{aligned}
F^{\rm e}(\epsilon^{\rm e}_{\rm B})&=&|\phi^{\rm e}_{\rm B}|^{2},\ \ \
F^{\rm o}(\epsilon^{\rm o}_{\rm B})=|\phi^{\rm o}_{\rm B}|^{2},\nonumber\\
F^{\rm e}(\epsilon^{\rm o}_{\rm B})=F^{\rm o}(\epsilon^{\rm e}_{\rm B})&=&
\frac{1}{2}\times \frac{1}{1+(\gamma /(e^{\kappa^{\rm e}+\kappa^{\rm o}}-1))}.
\label{Fs}
\end{aligned}$$ It is now straightforward to show, using Eqs. (\[geeoo\]), (\[geo\]), and (\[F\]) that $$\begin{aligned}
&&G_{\rm ee}({\rm E})=2\frac{(F^{\rm e}({\rm E}-\epsilon^{\rm e}_{\rm B}))^{2}}
{{\rm E}-2\epsilon^{\rm e}_{\rm B}}\nonumber\\
&+&\sum_{kk'}
\frac{|\phi^{\rm e}_{k}|^{2}|\phi^{\rm e}_{k'}|^{2}
(\epsilon^{\rm e}_{\rm B}-\epsilon_{k})(\epsilon^{\rm e}_{\rm B}-\epsilon_{k'})/2}
{({\rm E}-\epsilon_{k}-\epsilon_{k'})
({\rm E}-\epsilon^{\rm e}_{\rm B}-\epsilon_{k})
({\rm E}-\epsilon^{\rm e}_{\rm B}-\epsilon_{k'})},\label{gee}
\end{aligned}$$ with an analogous result for $G_{\rm oo}$, with e replaced by o, and $$\begin{aligned}
&&G_{\rm eo}({\rm E})=4\frac{F^{\rm e}({\rm E}-\epsilon^{\rm o}_{\rm B})F^{\rm o}
({\rm E}-\epsilon^{\rm e}_{\rm B})}{{\rm E}-
\epsilon^{\rm e}_{\rm B}-\epsilon^{\rm o}_{\rm B}}
\nonumber\\
&+&\sum_{kk'}
\frac{|\phi^{\rm o}_{k}|^{2}|\phi^{\rm e}_{k'}|^{2}
(\epsilon^{\rm o}_{\rm B}-\epsilon_{k})(\epsilon^{\rm e}_{\rm B}-\epsilon_{k'})}
{({\rm E}-\epsilon_{k}-\epsilon_{k'})
({\rm E}-\epsilon^{\rm e}_{\rm B}-\epsilon_{k})
({\rm E}-\epsilon^{\rm o}_{\rm B}-\epsilon_{k'})}.\label{Geo}
\end{aligned}$$ For energies E below the bottom of the two-electron continuum, i.e., E$<-2+\epsilon^{\rm e}_{\rm B}$, the double sum on $k$ and $k'$ in (\[gee\]) is negative. Using Eq. (\[Fs\]) for the first terms in the equations for $G_{\rm ee}$ and $G_{\rm oo}$, it follows that $$\begin{aligned}
G_{\rm ee}(\epsilon^{\rm e}_{\rm B}
+\epsilon^{\rm o}_{\rm B})+G_{\rm oo}(\epsilon^{\rm e}_{\rm B}
+\epsilon^{\rm o}_{\rm B})<0.\label{neg}
\end{aligned}$$ This result is used to show that when the only Coulomb coupling is the Hubbard U, the singlet is always the ground state.
Up to this point, the results were given for general $\gamma$. To lowest order in the coupling to the leads, $\gamma $, we may discard the double sums in (\[gee\]) and (\[Geo\]). Then, using (\[F\]), we find $$\begin{aligned}
\frac{1}{G_{\rm eo}({\rm E})}&&\sim {\rm E}-2\epsilon_{0}
+\gamma (e^{-\alpha^{\rm e}}+e^{-\alpha^{\rm o}}),\nonumber\\
\frac{1}{2G_{\rm ee}({\rm E})}&&\sim {\rm E}-2\epsilon_{0}+2t_{D}
+2\gamma e^{-\alpha^{\rm e}},\nonumber\\
\frac{1}{2G_{\rm oo}({\rm E})}&&\sim {\rm E}-2\epsilon_{0}-2t_{D}
+2\gamma e^{-\alpha^{\rm o}},\label{sg}
\end{aligned}$$ where we have used Eq. (\[al\]).
Permanent address: School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel.
S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage, and L. P. Kouwenhoven, Phys. Rev. Lett. [**77**]{}, 3613 (1996).
S. Tarucha, D. G. Austing, Y. Tokura, W. G. van der Wiel, and L. P. Kouwenhoven, Phys. Rev. Lett. [**84**]{}, 2485 (2000).
M. Pustilnik and L. I. Glazman, Phys. Rev. Lett. [**85**]{}, 2993 (2000).
M. Pustilnik, L. I. Glazman, D. H. Cobden, and L. P. Kouwenhoven, cond-mat/0010336.
M. Eto and Y. V. Nazarov, Phys. Rev. Lett. [**85**]{}, 1306 (2000).
J. Schmid, J. Weis, K. Eberl, and K. v. Klitzing, Phys. Rev. Lett. [**84**]{}, 5824 (2000).
J. A. Folk, C. M. Marcus, R. Berkovits, I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, cond-mat/0010441.
I. L. Kurland, R. Berkovits, and B. L. Altshuler, cond-mat/0005424.
H. U. Baranger, D. Ullmo, and L. I. Glazman, Phys. Rev. [**B61**]{}, R2425 (2000).
S. D. Lee, K. S. Park, J. W. Park, J. B. Choi, S.-R. E. Yang, K.-H. Yoo, J. Kim, S. I. Park, and K. T. Kim, Phys. Rev. [**B62**]{}, R7735 (2000).
J. J. Palacios and P. Hawrylak, Phys. Rev. [**B51**]{}, 1769 (1995).
B. Partoens and F. M. Peeters, Phys. Rev. Lett. [**84**]{}, 443 (2000).
L. Martin-Moreno, L. Brey, and C. Tejedor, Phys. Rev. [**B62**]{}, R10 633 (2000).
W. Izumida and O. Sakai, Phys. Rev. [**B62**]{}, 10 260 (2000).
A. Aharony, O. Entin-Wohlman, and Y. Imry, Phys. Rev. [**B61**]{}, 5452 (2000).
A. Aharony, O. Entin-Wohlman, Y. Imry, and Y. Levinson, Phys. Rev. [**B**]{}, in press (cond-mat/0005241).
J. Vidal, B. Douçot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. [**85**]{}, 3906 (2000).
D. Weinmann, P. Schmitteckert, R. A. Jalabert, and J-L Pichard, cond-mat/0011335.
W. A. Harrison, Phys. Rev. [**B29**]{}, 2917 (1984).
I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, cond-mat/00004205.
See, e.g., O. Entin-Wohlman, A. Aharony, and A. B. Harris, Phys. Rev. [**B53**]{}, 11661 (1996).
P. W. Anderson, Phys. Rev. [**115**]{}, 2 (1959).
P. W. Brouwer, Y. Oreg, and B. I. Halperin, Phys. Rev. [**B60**]{}, R13 977 (1999).
A. L. Chudnovskiy and S. E. Ulloa, cond-mat/0009281.
M. Brodsky, N. B. Zhitenev, R. C. Ashoori, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**85**]{}, 2356 (2000).
|
---
abstract: 'Due to collision singularities, the Lagrange action functional of the N-body problem in general is not differentiable. Because of this, the usual critical point theory can not be applied to this problem directly. Following ideas from [@BR91], [@Tn93a] and [@ABT06], we introduce a notion called weak critical point for such an action functional, as a generalization of the usual critical point. A corresponding definition of Morse index for such a weak critical point will also be given. Moreover it will be shown that the Morse index gives an upper bound of the number of possible binary collisions in a weak critical point of the $N$-body problem with weak force potentials including the Newtonian potential.'
address: 'Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Italy'
author:
- Guowei Yu
bibliography:
- 'RefMountPass.bib'
title: 'Application of Morse index in weak force $N$-body problem'
---
[^1]
Introduction {#sec:intro}
============
The motion of $N$ point masses, $m_i>0, i \in {\mathbf{N}}:=\{1, \dots, N\}$, under the universal gravitational force is a classic problem that has been studied by many authors since the time of Newton. Let $q_i \in {\mathbb{R}}^d$ be the position of mass $m_i$, then it satisfies the following equation $$\label{eq: N body} m_i \ddot{q}_i = \frac{\partial U(q)}{\partial q_i} = - {\alpha}\sum_{j \in {\mathbf{N}}\setminus \{i\}} m_i m_j\frac{q_i -q_j}{|q_i-q_j|^{{\alpha}+2}}, \;\; \forall i \in {\mathbf{N}},$$ where $q=(q_i)_{i \in {\mathbf{N}}} \in {\mathbb{R}}^{dN}$ and $U(q)= \sum_{\{i < j\} \subset {\mathbf{N}}} \frac{m_im_j}{|q_i-q_j|^{{\alpha}}}$ is the potential function (the negative potential energy). ${\alpha}$ here is a positive constant.
Traditionally $U$ is called a *strong force potential*, when ${\alpha}\ge 2$, and a *weak force potential*, when $0 < {\alpha}<2$. The Newtonian potential is a weak force potential corresponding to ${\alpha}=1$.
This is a singular Lagrange system with the Lagrangian $$L(q, {\dot{q}}):= K({\dot{q}}) + U(q), \; \text{ where } K({\dot{q}}) = {\frac{1}{2}}\sum_{i \in {\mathbf{N}}} m_i |{\dot{q}}_i|^2.$$ The singularities are caused by collisions between two or more masses $$\label{eq: collision} \Delta := \{q \in {\mathbb{R}}^{dN}: \; q_i = q_j, \text{ for some } \{i \ne j \} \subset {\mathbf{N}}\}.$$ For any ${\mathbf{I}}\subset {\mathbf{N}}$ with $|{\mathbf{I}}| \ge 2$ ($|{\mathbf{I}}|$ is the cardinality of ${\mathbf{I}}$), we say $q$ has an *${\mathbf{I}}$-cluster collision* at the moment $t$, when $$\forall i \in {\mathbf{I}}, \;\;
\begin{cases}
q_i(t)= q_j(t), \; & \text{ if } j \in {\mathbf{I}}, \\
q_i(t) \ne q_j(t), \; & \text{ if } j \in {\mathbf{N}}\setminus {\mathbf{I}},
\end{cases}$$ An ${\mathbf{I}}$-cluster collision is a *binary collision*, when $|{\mathbf{I}}|=2$.
Let $H^1([T_1, T_2], {\mathbb{R}}^{dN})$ be the space of Sobolev paths defined on $[T_1, T_2]$, and $\hat{{\mathbb{R}}}^{dN} := {\mathbb{R}}^{dN} \setminus \Delta$ the set of collision-free configurations, we say a path $q \in H^1([T_1, T_2], {\mathbb{R}}^{dN})$ is *collision-free*, if $q(t) \in \hat{{\mathbb{R}}}^{dN}$, for any $t \in [T_1, T_2]$. It is well known the Lagrange action functional $$\label{eq: action functional} {\mathcal{A}}(q; T_1, T_2):= \int_{T_1}^{T_2} L(q, {\dot{q}}) \,dt, \;\; \forall q \in H^1([T_1, T_2], {\mathbb{R}}^{dN}),$$ is $C^2$ on $H^1([T_1, T_2], \hat{{\mathbb{R}}}^{dN})$ (see [@AZ93]), and a critical point of ${\mathcal{A}}$ in $H^1([T_1, T_2], \hat{{\mathbb{R}}}^{dN})$ is a classical solution of .
The solution of is invariant under linear translations, in many cases it will be more convenient to fix the center of mass at the origin, so we set $${\mathcal{X}}:=\{ q \in {\mathbb{R}}^{dN}: \; \sum_{i \in {\mathbf{N}}} m_i q_i = 0 \}, \;\; \hat{{\mathcal{X}}} : = {\mathcal{X}}\setminus \Delta.$$ As the action functional is also invariant under linear translation, a critical point of ${\mathcal{A}}$ in $H^1([T_1, T_2], \hat{{\mathcal{X}}})$ will be a classical solution of as well.
In general it is much easier to apply variational methods to the $N$-body, when the potential is a *strong force*, i.e. ${\alpha}\ge 2$, as in this case any path with a finite action value must be collision-free, see [@CGMS02]. It is not so, when the potential is a *weak force*, i.e. ${\alpha}\in (0, 2)$, as the attracting force between the masses are too weak and the action value of a path with collision may still be finite, see [@Go77].
Because of this, for the strong force $N$-body problem, many results have been obtained by different authors using both minimization and non-minimization variational methods, see [@BR91], [@AZ93], [@MT93a], [@MT93b], [@Mo98], [@CGMS02] and the references within. However the problem is much more difficult for weak force potentials due to the possibility of collision. Set back by this, Bahri and Rabinowitz introduced the so called *generalized solution* in [@BR89] and [@BR91] (see Definition \[dfn: generalized solution\]), where such a solution is allowed to have a non-empty set of collision moments with zero Lebesgue measure. Here we are only interested in the weak force $N$-body problem, so we assume ${\alpha}\in (0, 2)$ in the rest of the paper.
For action minimization methods, the breakthrough followed the proofs of the Figure-Eight solution of the three body by Chenciner and Montgomery [@CM00] and the Hip-Hop solution of the four body by Chenciner and Venturelli [@CV00], where both solutions are found as collision-free minimizers of the action functional under proper symmetric constraints. Since then action minimization methods have thrived in the study of the $N$-body problem with Newtonian potential as well as other weak force potentials. We refer the interesting readers to [@C02], [@FT04], [@Ch08], [@Y15b], [@Yu17] and the references within.
Now it is more or less well understood, when we can show an action minimizer is collision-free.
\[dfn: local minimizer\] $q \in H^1([T_1, T_2], {\mathcal{X}})$ is a **local action minimizer of ${\mathcal{A}}$ with fixed-ends** in $H^1([T_1, T_2], {\mathcal{X}})$ , if there is a ${\delta}>0$ small enough, such that $${\mathcal{A}}(q; T_1, T_2) \le {\mathcal{A}}(q+{\tilde{q}}; T_1, T_2), \;\; \forall {\tilde{q}}\in H^1_0([T_1, T_2], {\mathcal{X}}) \text{ with } \|{\tilde{q}}\|_{H^1}\le {\delta},$$ where $H^1_0([T_1, T_2], {\mathcal{X}})$ is the space of Sobolev paths with compact support in $[T_1, T_2]$.
The following fundamental result is due to Marchal [@Mc02] and Chenciner [@C02], when ${\alpha}=1$, and Ferrario and Terracini [@FT04], when ${\alpha}\in (0, 2)$.
\[thm: fixed ends minimizer\] For any ${\alpha}\in (0, 2)$, when $d \ge 2$, if $q$ is a local action minimizer of ${\mathcal{A}}$ with fixed-ends in $H^1([T_1, T_2], {\mathcal{X}})$, then $q(t)$ is collision-free, for any $t \in (T_1, T_2)$.
Despite of the above progress, by our knowledge, for the $N$-body problem no result seems to be available regarding how to rule out collision when the corresponding path is obtained through non-minimization methods, like minimax or mountain-pass. Meanwhile for a special type of singular Lagrange systems with weak force potentials (essentially equivalent to perturbations of the $N$ center problem), using *Morse index theory*, in a series of papers ([@Tn93a], [@Tn93b], [@Tn94a], [@Tn94b]), Tanaka showed how to rule out collisions when a critical point was obtained by the minimax approach of Bahri and Rabinowitz [@BR89]. The main purpose of our paper is to generalize Tanaka’s idea to the $N$-body problem and show that the Morse index of a critical point can be used to give an upper bound of the number of binary collisions that could occur in it.
To give the precise statement, we recall the following definition according to [@ABT06].
\[dfn: morse index\] Let $X$ be a Hilbert space and $\Omega$ an open and dense subspace of it, i.e. $\bar{\Omega} = X$. If a functional ${\mathcal{F}}$ is lower semi-continuous in $X$ and $C^2$ in $\Omega$, then $c \in {\mathbb{R}}$ will be called a **critical value** of ${\mathcal{F}}$, if there is $q \in \Omega$, such that ${\mathcal{F}}(q)=c$ and the first derivative of ${\mathcal{F}}$ vanishes at $q$, i.e. $d {\mathcal{F}}(q)=0$. Moreover such a $q$ will be called a **critical point** of ${\mathcal{F}}$.
If $q \in \Omega$ is a critical point of ${\mathcal{F}}$, we define its **Morse index** (with respect to ${\mathcal{F}}$), $m^-_{\Omega}(q, {\mathcal{F}})$, as the dimension of the largest subspace of $\Omega$, where the second derivative $d^2 {\mathcal{F}}(q)$ is negative definite.
The above definition suits the study of the $N$-body problem, as $H^1([T_1, T_2], \hat{{\mathcal{X}}})$ is an open and dense subspace of $H^1([T_1, T_2], {\mathcal{X}})$ with the action functional ${\mathcal{A}}$ being $C^2$ in $H^1([T_1, T_2], \hat{{\mathcal{X}}})$ and lower semi-continuous in $H^1([T_1, T_2], {\mathcal{X}})$. Since ${\mathcal{A}}$ is generally not differentiable at a collision path, such a path can not be a *critical point* and moreover it does not have a well-defined Morse index for a collision path. Although a collision path can still be *a local minimizer*.
To deal with the above problem, following ideas from [@BR89], [@Tn93a] and [@ABT06], let’s perturb the weak force $N$-body problem by a strong force potential $$\label{eq; V} {\varepsilon}{\mathfrak{U}}(q) := {\varepsilon}\sum_{ \{ i < j \} \subset {\mathbf{N}}} \frac{m_i m_j}{|q_i -q_j|^2}, \; \text{ for } \; {\varepsilon}> 0 \text{ small enough}.$$ Then the motion of masses satisfies $$\label{eq: N body strong} m_i \ddot{q}_i = - {\alpha}\sum_{ j \in {\mathbf{N}}\setminus \{i\}} \frac{ m_i m_j(q_i -q_j)}{|q_i-q_j|^{{\alpha}+2}} - 2 {\varepsilon}\sum_{ j \in {\mathbf{N}}\setminus \{i\}} \frac{ m_i m_j(q_i -q_j)}{|q_i-q_j|^4} , \; \forall i \in {\mathbf{N}},$$ which is the Euler-Lagrange equation of the action functional $$\label{eq: action fun strong perturb} {\mathcal{A}^{{\varepsilon}}}(q; T_1, T_2) : = \int_{T_1}^{T_2} L^{{\varepsilon}}(q, {\dot{q}}) \,dt, \; \text{ where } \; L^{{\varepsilon}}(q, {\dot{q}}):= L(q, {\dot{q}})+ {\varepsilon}{\mathfrak{U}}(q).$$ Furthermore we set ${\mathcal{A}}^{0}(q; T_1, T_2) = {\mathcal{A}}(q; T_1, T_2)$. When ${\varepsilon}>0$, any path with a finite action value of ${\mathcal{A}^{{\varepsilon}}}$ must be collision-free.
Given an arbitrary path $q \in H^1([T_1, T_2], {\mathcal{X}})$, set $$H^1(q): = \{ {\tilde{q}}\in H^1([T_1, T_2], {\mathcal{X}}): \; {\tilde{q}}(T_i) = q(T_i), i = 1, 2 \};$$ $$\hat{H}^1(q): = H^1(q) \cap H^1([T_1, T_2], \hat{{\mathcal{X}}} \}.$$ Then $\hat{H}^1(q)$ is an open and dense subset of $H^1(q)$, where ${\mathcal{A}^{{\varepsilon}}}$ is lower semi-continuous in $H^1(q)$ and $C^2$ in $\hat{H}^1(q)$. Hence if $q$ is collision-free and $d {\mathcal{A}}^{{\varepsilon}}(q)=0$, then it is a critical point of ${\mathcal{A}}^{{\varepsilon}}$, and we will denote its Morse index in $\hat{H}^1(q)$ by $m^-_{T_1, T_2}(q, {\mathcal{A}^{{\varepsilon}}})$. Now we introduce a notion called *weak critical points* as a generalization of the usual critical points.
\[dfn: weak critical pt Morse index\] We say a path $q \in H^1([T_1, T_2], {\mathcal{X}})$ (which may contain collision) with finite action value, ${\mathcal{A}}(q; T_1, T_2) < \infty$, is a **weak critical point** of ${\mathcal{A}}$, if there exists a sequence of positive numbers ${{\varepsilon}_n}\to 0$ and a sequence of ${q^n}\in H^1([T_1, T_2], {\mathcal{X}})$, such that
1. ${\mathcal{A}^{{\varepsilon}_n}}({q^n}; T_1, T_2) < C$, $\forall n $, for some finite constant $C$;
2. ${q^n}$ is a critical point of ${\mathcal{A}^{{\varepsilon}_n}}$, for any $n$;
3. ${q^n}\to q$ weakly in $H^1$-norm and strongly in $L^{\infty}$-norm.
$c={\mathcal{A}}(q; T_1, T_2)$ will be called a **weak critical value** of ${\mathcal{A}}$, and the *Morse index* of such a weak critical point $q$ in $H^1(q)$ (with respect to ${\mathcal{A}}$) will be defined as $$\label{eq: morse index A} {m^{-}}_{T_1, T_2}(q, {\mathcal{A}}) = \inf \liminf_{n \to \infty} {m^{-}}_{T_1, T_2}({q^n}, {\mathcal{A}^{{\varepsilon}_n}}),$$ where the infimum is taken over all sequences ${{\varepsilon}_n}$ and ${q^n}$ satisfying the above conditions.
A similar notation was introduced in [@ABT06], where it was called **generalized critical point**.
With the above definition, we have the following result, which can be seen as a partial generalization of Theorem \[thm: fixed ends minimizer\].
\[thm: fix end not minimizer\] When $d \ge 3$, given a weak critical point $q \in H^1([T_1, T_2], {\mathcal{X}})$ of ${\mathcal{A}}$, let $\mathcal{B}(q)$ represent the number of binary collisions occurring in $q(t)$, $t \in (T_1, T_2)$ (when there are more than one binary collision at a given moment, each of them should be counted separately), then $$\label{eq; bound number of binary coll} (d-2)i({\alpha})\mathcal{B}(q) \le {m^{-}}_{T_1, T_2}(q, {\mathcal{A}}),$$ where $$\label{eq: i al} i({\alpha}) = \max\{k \in {\mathbb{Z}}: \; k < \frac{2}{2-{\alpha}} \}.$$ In particular $q(t), t \in (T_1, T_2)$, is free of binary collision, i.e. $\mathcal{B}(q)=0$, if $$m^-_{T_1, T_2}(q, {\mathcal{A}}) < (d-2) i({\alpha}).$$
\[rem; i al\] Notice that by , $i({\alpha})=1$, if ${\alpha}\in (0, 1]$ and $i({\alpha}) \ge 2$, if ${\alpha}\in (1, 2)$. Moreover $i({\alpha})$ goes to infinity, as ${\alpha}$ goes to $2$.
It seems the above result is the best we can get based on Tanaka’s idea. In particular, we are unable to obtain any nontrivial result, when $d=2$. For an explanation see Remark \[rem d=2 no result\].
The idea of using Morse index to rule out collision should work even when a collision cluster has more than two masses, although the technical difficulty seems very challenging. This is because when two masses approach to a binary collision, they behaves more and more like the two body problem, where the solutions are well understood and their Morse indices are relatively easy to compute. However when the collision cluster has more than two masses, as they approach to collision, the dynamics is much more complicate (see [@Mg74]) and the computation of Morse indices of the relevant solutions is also much more difficult. Despite of this, some progresses have been made recently in [@BS08], [@BHPT17] and [@HY17].
Theorem \[thm: fix end not minimizer\] has the following obvious corollaries.
\[cor: index 0\] When ${m^{-}}_{T_1, T_2}(q, {\mathcal{A}}) =0$ and $d \ge 3$, $q(t)$, $t \in (T_1, T_2),$ is free of binary collision.
\[cor: index 1\] When ${m^{-}}_{T_1, T_2}(q, {\mathcal{A}}) = 1$, the following results hold.
1. If $d \ge 4$ and ${\alpha}\in (0, 2)$, then $q(t)$, $t \in (T_1, T_2)$ is free of binary collision.
2. If $d = 3$ and ${\alpha}\in (1, 2)$, then $q(t)$, $t \in (T_1, T_2)$ is free of binary collision.
3. If $d = 3$ and ${\alpha}\in (0, 1]$, then $q(t)$, $t \in (T_1, T_2)$ has at most one binary collision, i.e. $\mathcal{B}(q) \le 1$.
Notice that in Corollary \[cor: index 1\], when $d=3$ and ${\alpha}\in (0, 1]$, the weak critical point may very well contains a binary collision and in this case we have the following result.
\[thm: regularization\] When $d=3$ and ${\alpha}=1$, let $q \in H^1([T_1, T_2], {\mathcal{X}})$ be a weak critical point of ${\mathcal{A}}$ with $m^-_{T_1, T_2}(q, {\mathcal{A}})=1$, if there is a binary collision between $m_{i_1}$ and $m_{i_2}$ at a moment $t_0 \in (T_1, T_2)$, then both limits $\lim_{t \to t_0^{\pm}} \frac{q_{i_2}(t)-q_{i_1}(t)}{|q_{i_2}(t)-q_{i_1}(t)|}$ exist and equal to each other.
The above result is interesting, because it is well-known if a solution of the spatial $N$-body problem has a single binary collision at a moment (no other partial collision exists at the same moment), then it can be regularized by Kustaanheimo-Stiefel regularization [@KS65]. With the result from the above theorem, under the assumption that there is no other partial collisions, one can show the generalize solution corresponding to the weak critical point is actually a classical solution in the regularized system.
Results similar to Theorem \[thm: regularization\] can be obtained for potentials with ${\alpha}\ne 1$, see Lemma \[lem: asymp angle\]. However the problem of regularizing a binary collision is more complicate for non-Newtonian potentials, see [@MG81].
Since the Morse index of a critical point obtained by the mountain pass theorem must be less than or equal to one (see [@Hf85]), we believe Corollary \[cor: index 0\] and \[cor: index 1\] and Theorem \[thm: regularization\] could be useful, when mountain pass methods are used in the study the $N$-body problem. This shall be discussed in a forthcoming paper.
Our paper is organized as follows: in Section \[sec: generalized\] we show a weak critical point is a generalized solution, in Section \[sec: Tanaka\] the proofs of the main results will be given, and in Section \[sec: lem 1\] and \[sec: lm asymp angle\] we give the proofs of some technical lemmas.
Generalized solutions {#sec: generalized}
=====================
Consider the perturbed $N$-body problem , for any subset of indices ${\mathbf{I}}\subset {\mathbf{N}}$, we define the *Lagrangian* and *energy* of the ${\mathbf{I}}$-cluster as $$L^{{\varepsilon}}_{{\mathbf{I}}}(q, {\dot{q}}): = K_{{\mathbf{I}}}({\dot{q}})+U_{{\mathbf{I}}}(q)+{\varepsilon}{\mathfrak{U}}_{{\mathbf{I}}}(q),$$ $$E^{{\varepsilon}}_{{\mathbf{I}}}(q, {\dot{q}}):= K_{{\mathbf{I}}}({\dot{q}})-U_{{\mathbf{I}}}(q)-{\varepsilon}{\mathfrak{U}}_{{\mathbf{I}}}(q),$$ where $$K_{{\mathbf{I}}}({\dot{q}}):= {\frac{1}{2}}\sum_{i \in {\mathbf{I}}} m_i|{\dot{q}}_i|^2, \;\; U_{{\mathbf{I}}}(q): = \sum_{\{i<j\}\subset {\mathbf{I}}}\frac{m_i m_j}{|q_i-q_j|^{{\alpha}}}, \; \; {\mathfrak{U}}_{{\mathbf{I}}}(q): = \sum_{\{i < j \} \subset {\mathbf{I}}} \frac{m_i m_j}{|q_i-q_j|^2}.$$
Let ${\mathbf{I}}':= {\mathbf{N}}\setminus {\mathbf{I}}$ denote the complement of ${\mathbf{I}}$ in ${\mathbf{N}}$, then $$L^{{\varepsilon}}(q, {\dot{q}}) = L^{{\varepsilon}}_{{\mathbf{I}}}(q, {\dot{q}}) + L^{{\varepsilon}}_{{\mathbf{I}}'}(q, {\dot{q}}) + U_{{\mathbf{I}}, {\mathbf{I}}'}(q) +{\varepsilon}{\mathfrak{U}}_{{\mathbf{I}}, {\mathbf{I}}'}(q),$$ $$E^{{\varepsilon}}(q, {\dot{q}}) = E^{{\varepsilon}}_{{\mathbf{I}}}(q, {\dot{q}}) + E^{{\varepsilon}}_{{\mathbf{I}}'}(q, {\dot{q}}) - U_{{\mathbf{I}}, {\mathbf{I}}'}(q) - {\varepsilon}{\mathfrak{U}}_{{\mathbf{I}}, {\mathbf{I}}'}(q),$$ where $$U_{{\mathbf{I}}, {\mathbf{I}}'}(q): = \sum_{i \in {\mathbf{I}}, j \in {\mathbf{I}}'} \frac{m_im_j}{|q_i -q_j|^{{\alpha}}}, \; \; {\mathfrak{U}}_{{\mathbf{I}}, {\mathbf{I}}'}(q): = \sum_{i \in {\mathbf{I}}, j \in {\mathbf{I}}'} \frac{m_im_j}{|q_i -q_j|^{2}}.$$
\[dfn: generalized solution\] $q \in H^1([T_1, T_2], {\mathcal{X}})$ is a **generalized solution** of , if it satisfies the following conditions:
1. $\Delta^{-1}(q): = \{t \in [T_1, T_2]: \; q(t) \in \Delta\}$ has measure $0$ in $[T_1, T_2]$;
2. $q \in C^2$ on $[T_1, T_2] \setminus \Delta^{-1}(q)$ and satisfies ;
3. the total energy of $q(t)$, $E(t)=E(q(t), {\dot{q}}(t))$, is a constant, for all $t \in [T_1, T_2] \setminus \Delta^{-1}(q)$;
4. for any subset ${\mathbf{I}}\subset {\mathbf{N}}$ and sub-interval $(t_1, t_2) \subset [T_1, T_2]$, if $$\label{eq: I cluster collision} q_i(t) \ne q_j(t), \;\; \forall i \in {\mathbf{I}}, \forall j \in {\mathbf{I}}', \text{ and } \forall t \in (t_1, t_2),$$ then $E_{{\mathbf{I}}}(t)= E_{{\mathbf{I}}}(q(t), {\dot{q}}(t)) \in H^1((t_1, t_2), {\mathbb{R}})$. In particular, $E_{{\mathbf{I}}}(t)$ is continuous in $(t_1, t_2)$.
Condition (iv) in the above definition shows the energy of a ${\mathbf{I}}$-cluster is continuous, as long as the masses from the ${\mathbf{I}}$-cluster do not collide with masses outside of the cluster, even when there are collisions among the masses inside the cluster. This condition was not required by in the original definition of a generalized solution introduced by Bahri and Rabinowitz, see [@BR91] and [@AZ93]. Our definition here is stronger and follows from [@FT04 Definition 4.6].
\[prop: weak solution to generalized\] A weak critical point $q \in H^1([T_1, T_2], {\mathcal{X}})$ of ${\mathcal{A}}$ is a generalized solution of .
Let ${{\varepsilon}_n}>0$ and ${q^n}\in H^1(T_1, T_2, \mathcal{X})$ be two sequences satisfying the conditions given in Definition \[dfn: weak critical pt Morse index\]. Then there is a finite constant $C>0$, such that $${\frac{1}{2}}\int_{t_1}^{t_2} \sum_{i \in {\mathbf{N}}} m_i |\dot{{q^n}_i}(t)|^2 \,dt \le {\mathcal{A}^{{\varepsilon}_n}}({q^n}; T) \le C, \;\; \forall n.$$
The fact that $q$ satisfies the first three conditions given in Definition \[dfn: generalized solution\] is a standard result, for details see [@BR91] or [@AZ93]. In the following, we will show $q$ also satisfies condition (iv). Given an arbitrary ${\mathbf{I}}\subset {\mathbf{N}}$, recall that $$E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)=E^{{{\varepsilon}_n}}_{{\mathbf{I}}}({q^n}(t), \dot{{q^n}}(t)) = K_{{\mathbf{I}}}(\dot{{q^n}}(t))- U_{{\mathbf{I}}}({q^n}) -{{\varepsilon}_n}{\mathfrak{U}}_{{\mathbf{I}}}({q^n}).$$ By a direct computation, $$\label{eq derivative E I}
\frac{d E^{{{\varepsilon}_n}}_{{\mathbf{I}}}}{dt} = \sum_{i \in {\mathbf{I}}} \left \langle \frac{\partial U_{{\mathbf{I}}, {\mathbf{I}}'}({q^n})}{\partial {q^n}_i} + {{\varepsilon}_n}\frac{\partial {\mathfrak{U}}_{{\mathbf{I}}, {\mathbf{I}}'}({q^n})}{\partial {q^n}_i} , \dot{{q^n}_i} \right \rangle. \\$$ Let’s assume $q$ satisfies for the above ${\mathbf{I}}$ and an arbitrary sub-interval $(t_1, t_2) \subset [T_1, T_2]$. Since ${q^n}(t)$ converges to $q(t)$ uniformly on $[0, T]$, $$\label{eq: bound partial U V}
\left |\frac{\partial U_{{\mathbf{I}}, {\mathbf{I}}'}({q^n}(t))}{\partial {q^n}_i} \right |, \left |\frac{\partial {\mathfrak{U}}_{{\mathbf{I}}, {\mathbf{I}}'}({q^n}(t))}{\partial {q^n}_i} \right| \le C_1, \;\; \forall t \in [t_1, t_2], \; \forall i \in {\mathbf{I}}.$$ Here and in the rest of the proof $C_i, i \in {\mathbb{Z}}^+$, always represents some positive constant independent of $n$. With and , the Cauchy-Schwarz inequality tells us $$|\dot{E^{{{\varepsilon}_n}}_{{\mathbf{I}}}}(t)|^2 \le C_2 \sum_{i \in {\mathbf{I}}} m_i |\dot{{q^n}_i}(t)|^2 \le C_2 \sum_{i \in {\mathbf{N}}} m_i|\dot{{q^n}_i}(t)|^2, \; \; \forall t \in [t_1, t_2].$$ Then $$\label{eq: dot E I L2 norm} \int_{t_1}^{t_2} |\dot{E^{{{\varepsilon}_n}}_{{\mathbf{I}}}}(t)|^2 \,dt \le C_2 \int_{t_1}^{t_2} \sum_{i \in {\mathbf{N}}} m_i |\dot{{q^n}_i}(t)|^2 \,dt \le C_3.$$
Since $U_{{\mathbf{I}}}$, ${\mathfrak{U}}_{{\mathbf{I}}}$ are always positive, $E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t) \le K_{{\mathbf{I}}}(\dot{{q^n}}(t))$, $\forall t$. Then $$\label{eq: E I L1 norm} \int_{t_1}^{t_2} E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t) \,dt \le {\frac{1}{2}}\int_{t_1}^{t_2} \sum_{i \in {\mathbf{N}}} m_i |\dot{{q^n}_i}(t)|^2 \,dt \le C_4.$$ Meanwhile by Poincaré inequality and , $$\label{eq: E I L2 norm} \int_{t_1}^{t_2} |E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t) - \int_{t_1}^{t_2} E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(s)\,ds|^2 \,dt \le C_5 \int_{t_1}^{t_2} |\dot{E^{{{\varepsilon}_n}}_{{\mathbf{I}}}}|^2 \,dt \le C_6.$$ Then implies $$\int_{t_1}^{t_2} |E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)|^2 \,dt \le C_7.$$ The above inequality and implies $E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)$ is a bounded sequence in $H^1([t_1, t_2], {\mathbb{R}})$. After passing to a subsequence, it converges to a $\hat{E}_{{\mathbf{I}}}(t) \in H^1([t_1, t_2], {\mathbb{R}})$ weakly in $H^1$ norm and strongly in $L^{\infty}$ norm.
Since ${\dot{q}}(t)$ and $E_{{\mathbf{I}}}(t)=E_{{\mathbf{I}}}(q(t), {\dot{q}}(t))$ are well defined for any $t \notin \Delta^{-1}(q)$, and $${q^n}(t) \to q(t), \;\; \dot{{q^n}}(t) \to {\dot{q}}(t), \; \text{ as } n \to \infty, \;\; \forall t \notin \Delta^{-1}(q),$$ we have $E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t) \to E_{{\mathbf{I}}}(t)$, for any $t \notin \Delta^{-1}(q)$. As a result, $E_{{\mathbf{I}}}(t)= \hat{E}_{{\mathbf{I}}}(t)$, for any $t \in [t_1, t_2] \setminus \Delta^{-1}(q)$. Since $\Delta^{-1}(q)$ is a set of measure zero, $E_{{\mathbf{I}}}(t) = \hat{E}_{{\mathbf{I}}}(t)$ as a $H^1$-Sobolev function, and it is continuous in $(t_1, t_2)$.
\[dfn: I cluster coll\] Given a path $q \in H^1([T_1, T_2], {\mathcal{X}})$ with an ${\mathbf{I}}$-cluster collision at a moment $t_0 \in (T_1, T_2)$, we say it is **isolated**, if there is a constant $a>0$ small enough, such that for any $i \in {\mathbf{I}}$, $$q_i(t) \ne q_j(t), \; \forall t \in [t_0-a, t_0+a] \setminus \{t_0\}, \; \forall j \in {\mathbf{N}}\setminus \{i \}.$$
\[prop: isolated binary coll\] Given a weak critical point $q \in H^1([T_1, T_2], {\mathcal{X}})$, if there is a binary collision at the moment $t_0 \in (T_1, T_2)$, then it must be isolated.
By Proposition \[prop: weak solution to generalized\], $q$ is a generalized solution of . In particular it satisfies condition (iv) in Definition \[dfn: generalized solution\], then the desired result was already proven in [@FT04 Corollary 5.12]. Once the reader notices that every binary collision is a so called *locally minimal collision* defined in [@FT04 Definition 5.2].
Proof of Theorem \[thm: fix end not minimizer\] and \[thm: regularization\] {#sec: Tanaka}
===========================================================================
To prove the main theorems, three technical lemmas will be needed. We present them as Lemma \[lm: lim xin\], \[lem: asymp angle\] and \[lm: lim zt\] in this section and postpone their proofs until the next two sections.
Let $q \in H^1([T_1, T_2], {\mathcal{X}})$ be a weak critical point of ${\mathcal{A}}$ with a binary collision at the moment $t_0 \in (T_1, T_2)$ and ${q^n}\in H^1([T_1, T_2], {\mathcal{X}})$ a sequence of critical points of ${\mathcal{A}}^{{{\varepsilon}_n}}$ satisfying the conditions required in Definition \[dfn: weak critical pt Morse index\]. Without loss of generality, we may assume such a binary collision is between $m_1$ and $m_2$, i.e. $$q_{1}(t_0)= q_{2}(t_0) \ne q_i(t_0), \; \forall i \in {\mathbf{N}}\setminus \{1, 2 \}.$$ By Proposition \[prop: isolated binary coll\], such an binary collision must be isolated, so we may choose an $a>0$ small enough, such that $[t_0-2a, t_0+2a] \subset (T_1, T_2)$ and $$q_1(t) \ne q_2(t), \;\; \forall t \in [t_0-2a, t_0 +2a] \setminus \{t_0\};$$ $$\label{eq: away from binary coll q}
q_i(t) \ne q_j(t), \;\; \forall t \in [t_0 -2a, t_0 +2a], \; \forall i \in \{1, 2\}, \; \forall j \in {\mathbf{N}}\setminus \{1, 2 \}.$$
For each $n$, we can always find a $t_n \in [t_0-2a, t_0+2a]$ such that $${{\delta}_n}:=|{q^n}_1(t_n)-{q^n}_2(t_n)|= \min \{|{q^n}_1(t)-{q^n}_2(t)|: \; t \in [t_0-2a, t_0+2a] \}.$$ Obviously ${\delta}_n$ converges to $0$, as $n$ goes to infinity. After passing to a subsequence, we may assume the limit of $t_n$ exists. Since $q_1(t_0)=q_2(t_0)$ is an isolated binary collision at the moment $t_0$, $t_n$ must converge to $t_0$. Then for $n$ large enough, $[t_n-a, t_n+a] \subset [t_0-2a, t_0+2a]$. As a result, $$\label{eq: dn} {{\delta}_n}=\min \{|{q^n}_1(t)-{q^n}_2(t)|: \; t \in [t_n-a, t_n+a] \}.$$
By Definition \[dfn: weak critical pt Morse index\], ${q^n}(t)$ converges to $q(t)$ uniformly on $[T_1, T_2]$. According to , there is constant $C_1>0$ independent of $n$, such that $$\label{eq: away from bi coll} |{q^n}_i(t)-{q^n}_j(t)| \ge C_1, \;\; \forall t \in [t_n-a, t_n+a], \;\; \forall i \in \{1, 2\}, \; \forall j \in {\mathbf{N}}\setminus \{1, 2\}.$$ Let $${\widetilde{U}}({q^n}): = U({q^n}) - \frac{m_1 m_2}{|{q^n}_1 - {q^n}_2|^{{\alpha}}}, \;\; {\widetilde{{\mathfrak{U}}}}({q^n}):= {\mathfrak{U}}({q^n})- \frac{m_1 m_2}{|{q^n}_1 - {q^n}_2|^2}.$$ There are constants $C_2, C_3>0$ independent of $n$, such that $$\label{eq: bound partial Ut Vt} \left|\frac{\partial{\widetilde{U}}({q^n}(t))}{\partial {q^n}_i} \right|, \left|\frac{\partial {\widetilde{{\mathfrak{U}}}}({q^n}(t))}{\partial {q^n}_i} \right|\le C_2, \;\; \forall t \in [t_n-a, t_n+a], \; \forall 1 \le i \le 2,$$ $$\label{eq: bound second partial Ut Vt} \left|\frac{\partial^2 {\widetilde{U}}({q^n}(t))}{\partial{q^n}_i \partial {q^n}_i} \right|, \left|\frac{\partial^2 {\widetilde{{\mathfrak{U}}}}({q^n}(t))}{\partial{q^n}_i \partial {q^n}_i} \right| \le C_3, \;\; \forall t \in [t_n-a, t_n+a], \; \forall 1 \le i \le 2.$$
We introduce a new function ${\eta^n}(t)=({\eta^n}_i(t))_{i=1}^N$ by $$\label{eq: etn} \begin{cases}
{\eta^n}_1(t) & = {q^n}_2(t)-{q^n}_1(t), \\
{\eta^n}_2(t) & = \frac{1}{m_1+m_2}(m_1 {q^n}_1(t) + m_2 {q^n}_2(t)) , \\
{\eta^n}_i(t) & = {q^n}_i(t), \; \text{ if } i =3, \dots, N.
\end{cases}$$ By a direct computation, ${\eta^n}(t)$ is a solution of $$\begin{aligned}
\frac{m_1m_2}{m_1+m_2} \ddot{{\eta^n}_1} &= -{\alpha}m_1 m_2 \frac{{\eta^n}_1}{|{\eta^n}_1|^{{\alpha}+2}}-2m_1 m_2 \frac{{{\varepsilon}_n}{\eta^n}_1}{|{\eta^n}_1|^4} + \frac{\partial{\widetilde{U}}({\eta^n})}{\partial{{\eta^n}_1}}+{{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial{{\eta^n}_1}}; \label{eq: etn 1}\\
(m_1+m_2) \ddot{{\eta^n}_2} &= \frac{\partial {\widetilde{U}}({\eta^n})}{\partial {\eta^n}_2} + {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial {\eta^n}_2}; \label{eq: etn 2} \\
m_i \ddot{{\eta^n}_i} & = \frac{\partial {\widetilde{U}}({\eta^n})}{\partial {\eta^n}_i} + {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial {\eta^n}_i}, \;\; i = 3, \dots, N. \label{eq: etn 3}\end{aligned}$$ This is the Euler-Lagrange equation of the following Lagrangian $$\label{eq: L etn} \begin{split}
L({\eta^n}, \dot{{\eta^n}}) & = \frac{m_1m_2}{2(m_1+m_2)}|\dot{{\eta^n}_1}|^2 + \frac{m_1 +m_2}{2} |\dot{{\eta^n}_2}|^2 + {\frac{1}{2}}\sum_{i=3}^N m_i |\dot{{\eta^n}}|^2 \\
& +\frac{m_1m_2}{|{\eta^n}_1|^{{\alpha}}}+ \frac{{{\varepsilon}_n}m_1 m_2}{|{\eta^n}_1|^2} + {\widetilde{U}}({\eta^n}) + {\widetilde{{\mathfrak{U}}}}({\eta^n}).
\end{split}$$
To study the behaviors of the solutions as they approach to the binary collision, Tanaka’s *blow-up* technique will be used. The precise argument depends on the limit of ${{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$. After passing to subsequence, we may assume such a limit ${\lambda}= \lim_{n \to \infty}{{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$ always exists. Then two different cases need to be considered: *Case 1*, ${\lambda}\in [0, \infty)$; *Case 2*, ${\lambda}=\infty$.
For *Case 1*, we blow up ${\eta^n}(t)$ according to $$\label{eq: blow up xin} {\xi^n}(s)= ({\xi^n}_i(s))_{i=1}^N: = ({{\delta}_n}^{-1} {\eta^n}_i(t(s)))_{i=1}^N, \; \text{ where } t(s) = {{\delta}_n}^{1+\frac{{\alpha}}2}s + t_n.$$ By changing the time parameter from $t$ to $s$, the time interval $[t_n-a, t_n+a]$ is mapped onto $[-a/{{\delta}_n}^{1+{\alpha}/2}, a/{{\delta}_n}^{1+{\alpha}/2}]$. Notice that the latter interval converges to ${\mathbb{R}}$, as $n$ goes to infinity. Let $'$ denote derivatives with respect $s$, then $\xi_1(s)$ satisfies $$\label{eq: xin 1} \begin{split}
\frac{1}{m_1+m_2} ({\xi^n}_1)'' & = -{\alpha}\frac{{\xi^n}_1}{|{\xi^n}_1|^{{\alpha}+2}}- 2\frac{{{\varepsilon}_n}}{{{\delta}_n}^{2-{\alpha}}}\frac{{\xi^n}_1}{|{\xi^n}_1|^4} \\
& + \frac{{{\delta}_n}^{1+{\alpha}}}{m_1m_2} \left(\frac{\partial {\widetilde{U}}({\eta^n})}{\partial {\eta^n}_1} + {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial {\eta^n}_1}\right).
\end{split}$$
\[lm: lim xin\] If ${\lambda}= \lim_{n \to \infty} {{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$ is finite, then the following results hold.
1. After passing to a subsequence, ${\xi^n}_1(s)$ converges to a $\xi_1(s)$ in $C^2([-\ell, \ell], {\mathbb{R}}^d)$, for any $\ell >0$, where $\xi_1(s)$ is a solution of $$\label{eq: xih} \frac{1}{m_1+m_2}{\hat{\xi}}_1''= -{\alpha}\frac{{\hat{\xi}}_1}{|{\hat{\xi}}_1|^{{\alpha}+2}}-2 \frac{{\lambda}{\hat{\xi}}_1}{|{\hat{\xi}}_1|^4};$$ $$\label{eq: xih energy} \frac{1}{2(m_1+m_2)}|{\hat{\xi}}_1'|^2-\frac{1}{|{\hat{\xi}}_1|^{{\alpha}}}- \frac{{\lambda}}{|{\hat{\xi}}_1|^2} = 0;$$ $$\label{eq: xih orthogonal} \langle {\hat{\xi}}_1(0), {\hat{\xi}}_1'(0) \rangle =0.$$
2. $\xi_1(t) \in W(\xi_1), \forall t \in {\mathbb{R}}$, where $ W(\xi_1)= \text{span}\{\xi_1(0), \xi_1'(0)\}$ is a $2$-dim subspace of ${\mathbb{R}}^d$. Moreover the following limits exist $$\lim_{s \to \pm \infty} |\xi_1(s)| = +\infty, \; \; \lim_{t \to \pm \infty} \frac{\xi_1(s)}{|\xi_1(s)|} = u^{\pm},$$ and $$\angle(u^-, u^+)= 2\pi \frac{\sqrt{1+{\lambda}}}{2-{\alpha}}.$$ For any two unit vectors $u, v \in {\mathbb{R}}^d$, $\angle(u,v)$ represents the angle between them.
3. Let $ W^{\perp}(\xi_1)$ be the orthogonal complement of $W(\xi_1)$ in ${\mathbb{R}}^d$ and $H(\xi_1)$ the largest subspace of $H^1_0({\mathbb{R}}, W^{\perp}(\xi_1))$, such that $$d^2 {\mathcal{I}}(\xi_1)(\phi, \phi)<0, \;\; \forall \phi \in H(\xi_1),$$ where ${\mathcal{I}}$ is the Lagrange action functional corresponding to equation : $$\label{eq: I} {\mathcal{I}}(\hat{\xi}_1) = \int \frac{1}{2(m_1+m_2)}|\hat{\xi}_1'|^2 +\frac{1}{|\hat{\xi}_1|^{{\alpha}}} + \frac{{\lambda}}{|\hat{\xi}_1|^2} \,dt ,$$ then $\dim(H(\xi_1)) \ge (d-2)i({\alpha}, {\lambda})$, where $$\label{eq: i al lmd} i({\alpha}, {\lambda}):= \max \{k \in {\mathbb{Z}}: \; k < \frac{2\sqrt{1+{\lambda}}}{2-{\alpha}} \}.$$
\[lem: asymp angle\] If ${\lambda}= \lim_{n \to \infty} {{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$ is finite, then $$\lim_{t \to t_0^{\pm}} \frac{q_2(t)-q_1(t)}{|q_2(t)-q_1(t)|}= u^{\pm},$$ where $u^{\pm}$ are the two unit vectors given in property (b), Lemma \[lm: lim xin\].
For *Case 2*, we define a blow-up of ${\eta^n}$ according to $$\label{eq: blow up ztn} {{\zeta}^n}(s)=({{\zeta}^n}_i(s))_{i=1}^N = ({{\delta}_n}^{-1} {\eta^n}_i(t(s)))_{i=1}^N, \; \text{ where } t(s)= {{\varepsilon}_n}^{-{\frac{1}{2}}} {{\delta}_n}^2 s + t_n.$$ Like the previous case, after changing the time parameter from $t$ to $s$, the time interval $ [t_n-a, t_n+a]$ is mapped onto $ [-a{{\varepsilon}_n}^{{\frac{1}{2}}}/{{\delta}_n}^2, a {{\varepsilon}_n}^{{\frac{1}{2}}}/{{\delta}_n}^2]$, which converges to ${\mathbb{R}}$, as $n$ goes to infinity. Again if we let $'$ represents derivatives with respect to $s$, then ${{\zeta}^n}_1(s)$ satisfies $$\label{eq: ztn 1} \begin{split}
\frac{1}{m_1 +m_2} ({{\zeta}^n}_1)'' & = -2 \frac{{{\zeta}^n}_1}{|{{\zeta}^n}_1|^4} - {\alpha}\frac{{{\delta}_n}^{2-{\alpha}}}{{{\varepsilon}_n}} \frac{{{\zeta}^n}_1}{|{{\zeta}^n}_1|^{{\alpha}+1}} \\
& +\frac{{{\varepsilon}_n}}{{{\delta}_n}^3 m_1m_2} \left(\frac{\partial{\widetilde{U}}({\eta^n})}{\partial {\eta^n}_1} + {{\varepsilon}_n}\frac{\partial{\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial {\eta^n}_1}\right).
\end{split}$$
\[lm: lim zt\] If ${\lambda}= \lim_{n \to \infty} {{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}=\infty$, then the following results hold.
1. After passing to a subsequence, ${{\zeta}^n}_1(s)$ converges to a ${\zeta}_1(s)$ in $C^2([-\ell, \ell], {\mathbb{R}}^d)$, for any $\ell >0$, where ${\zeta}_1(s)$ is a solution of $$\label{eq: zth} \frac{1}{m_1+m_2} {\hat{{\zeta}}}_1''= - 2 \frac{{\hat{{\zeta}}}_1}{|{\hat{{\zeta}}}_1|^4};$$ $$\label{eq: zth energy} \frac{1}{2(m_1+m_2)} |{\hat{{\zeta}}}_1'|^2 - \frac{1}{|{\hat{{\zeta}}}_1|^2} = 0;$$ $$\label{eq: zth orthogonal} \langle {\hat{{\zeta}}}_1(0), {\hat{{\zeta}}}_1'(0) \rangle =0.$$
2. ${\zeta}_1(t) \in W({\zeta}_1), \forall t \in {\mathbb{R}}$, where $ W({\zeta}_1)= \text{span}\{{\zeta}_1(0), {\zeta}_1'(0)\}$ is a $2$-dim subspace of ${\mathbb{R}}^d$, and $\lim_{s \to \pm \infty} |{\zeta}_1(s)|= +\infty$.
3. Let $ W^{\perp}({\zeta}_1)$ be the orthogonal complement of $W({\zeta}_1)$ in ${\mathbb{R}}^d$ and $H({\zeta}_1)$ the largest subspace of $H^1_0({\mathbb{R}}, W^{\perp}({\zeta}_1))$, such that $$d^2 {\mathcal{J}}({\zeta}_1)(\phi, \phi)<0, \;\; \forall \psi \in H({\zeta}_1),$$ where ${\mathcal{J}}$ is the Lagrange action functional corresponding to equation , $$\label{eq: J} {\mathcal{J}}({\hat{{\zeta}}}) : = \int \frac{1}{2(m_1+m_2)}|{\hat{{\zeta}}}'|^2 + \frac{1}{|{\hat{{\zeta}}}|^2} \,ds,$$ then $\dim(H({\hat{{\zeta}}})) = +\infty$.
\[prop: one binary coll\] Under the above notation,
1. if ${\lambda}= \lim_{n \to \infty} {{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$ is finite, then $$\label{eq: one binary coll} \liminf_{n \to \infty} {m^{-}}_{t_n-a, t_n+a}({q^n}, {\mathcal{A}^{{\varepsilon}_n}}) \ge (d-2)i({\alpha}, {\lambda});$$
2. if ${\lambda}= \lim_{n \to \infty} {{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}= +\infty$, then $$\liminf_{n \to \infty} {m^{-}}_{t_n-a, t_n+a}({q^n}, {\mathcal{A}^{{\varepsilon}_n}}) = +\infty.$$
(a). Given an arbitrary $f=(f_i)_{i=1}^N \in H^1_0([t_n-a, t_n+a], {\mathbb{R}}^{dN})$ satisfying $$\label{eq: f} f_i(t) \equiv 0, \;\; \forall t, \; \text{ and } \; \forall i \ne 1,$$ by , for any $c >0$, the center of mass corresponding to the path ${\eta^n}(t)+c f(t)$ will always be at the origin. In the following, we shall compute the second variation of ${\mathcal{A}^{{\varepsilon}_n}}$ at ${\eta^n}$ among all $ f \in H^1_0([t_n-a, t_n+a], {\mathbb{R}}^{dN})$ satisfying . By a direct computation, $$\label{eq: sec var aen} \begin{split}
d^2{\mathcal{A}^{{\varepsilon}_n}}({\eta^n})[f, f] & = \int_{t_n-a}^{t_n+a} \frac{m_1m_2}{m_1+m_2}|\dot{f}_1|^2 -{\alpha}m_1m_2 \frac{|f_1|^2}{|{\eta^n}_1|^{{\alpha}+2}}- 2m_1m_2 \frac{{{\varepsilon}_n}|f_1|^2}{|{\eta^n}_1|^4} \\
& + {\alpha}({\alpha}+2)m_1m_2 \frac{\langle {\eta^n}_1, f_1 \rangle}{|{\eta^n}_1|^{{\alpha}+4}} + 8m_1m_2{{\varepsilon}_n}\frac{\langle {\eta^n}_1, f_1 \rangle}{|{\eta^n}_1|^6} \\
& + \langle \frac{\partial^2{\widetilde{U}}({\eta^n})}{\partial {\eta^n}_1 \partial {\eta^n}_1}f_1, f_1 \rangle+ {{\varepsilon}_n}\langle \frac{\partial^2{\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial {\eta^n}_1 \partial {\eta^n}_1}f_1, f_1 \rangle \,dt.
\end{split}$$ Define the linear operators $T_n: H^1_0({\mathbb{R}}, W^{\perp}(\xi_1)) \to H^1_0([t_n-a, t_n+a], {\mathbb{R}}^{dN})$ by $f^n(t)=(f^n_i(t))_{i+1}^N = (T_n\phi)(s(t))$, where $s(t)= {{\delta}_n}^{-\frac{2+{\alpha}}{2}}(t-t_n)$ and $$f^n_1(t)= {{\delta}_n}\phi(s(t)) \; \text{ and } \; f^n_i(t) \equiv 0, \; \forall i \ne 1.$$ Then $$\label{eq: sec var aen blow up} \begin{split}
{{\delta}_n}^{-\frac{2-{\alpha}}{2}} &d^2 {\mathcal{A}^{{\varepsilon}_n}}({\eta^n})[f^n, f^n] = \int_{\text{supp}(\phi)} \frac{m_1m_2}{m_1+m_2}|\phi'|^2- {\alpha}m_1m_2 \frac{|\phi|^2}{|{\xi^n}_1|^{{\alpha}+2}} \\
& -2m_1m_2 \frac{{{\varepsilon}_n}}{{{\delta}_n}^{2-{\alpha}}} \frac{|\phi|^2}{|{\xi^n}_1|^4} +{\alpha}({\alpha}+2)m_1m_2 \frac{\langle {\xi^n}_1, \phi \rangle^2}{|{\xi^n}_1|^{{\alpha}+4}} + 8m_1m_2 \frac{{{\varepsilon}_n}}{{{\delta}_n}^{2-{\alpha}}}\frac{\langle {\xi^n}_1, \phi \rangle^2}{|{\xi^n}_1|^6}\\
& + {{\delta}_n}^{2+{\alpha}}\left( \langle \frac{\partial^2 {\widetilde{U}}({\eta^n})}{\partial {\eta^n}_1 \partial {\eta^n}_1}\phi, \phi\rangle^2 + {{\varepsilon}_n}\langle \frac{\partial^2 {\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial {\eta^n}_1 \partial {\eta^n}_1}\phi, \phi\rangle^2 \right) \,ds
\end{split}$$ By , there is a constant $C>0$ independent of $n$, such that $$\label{eq: upper bound partial 2 U V}
\left|\frac{\partial^2 {\widetilde{U}}({\eta^n}(t(s)))}{\partial {\eta^n}_1 \partial {\eta^n}_1}\right|, \left|\frac{\partial^2 {\widetilde{U}}({\eta^n}(t(s)))}{\partial {\eta^n}_1 \partial {\eta^n}_1}\right| \le C, \; \forall s \in [-\frac{a}{{{\delta}_n}^{1+{\alpha}/2}}, \frac{a}{{{\delta}_n}^{1+{\alpha}/2}}], \; \forall 1 \le i \le 2.$$ As $\xi^n_1$ converges to $\xi_1$ in $C^2([\ell, \ell], {\mathbb{R}}^d)$, for any $\ell >0$, the right hand side of converges to $$m_1m_2 \left(\int_{\text{supp}(\phi)} \frac{1}{m_1+m_2}|\phi'|^2- {\alpha}\frac{|\phi|^2}{|\xi_1|^{{\alpha}+2}} -2 {\lambda}\frac{|\phi|^2}{|\xi_1|^4} \,dt \right) =m_1 m_2 \big(d^2 {\mathcal{I}}(\xi_1)[\phi, \phi] \big).$$ Then Lemma \[lm: lim xin\] implies, $${{\delta}_n}^{-\frac{2-{\alpha}}{2}} d^2 {\mathcal{A}^{{\varepsilon}_n}}({\eta^n})[f^n, f^n] < 0, \; \text{ for } n \text{ large enough, if } \phi \in H(\xi_1).$$ As $\dim(H(\xi_1)) \ge (d-2)i({\alpha}, {\lambda})$, it immediately implies .
(b). If ${\lambda}$ is infinity, with Lemma \[lm: lim zt\], the desired property following from a similar argument as above. We will not repeat it again.
With the above results, we can now prove Theorem \[thm: fix end not minimizer\] and \[thm: regularization\].
By Proposition \[prop: isolated binary coll\], if $q(t)$ has a binary collision, then it is isolated. Then $\mathcal{B}(q)$ must be finite.
Assume $q$ has a binary collision at the moment $t_0 \in (T_1, T_2)$, let the corresponding sequences ${{\varepsilon}_n}$, ${q^n}(t)$ and ${\delta}_n$ be defined as before. If $\lim_{n \to \infty} {{\varepsilon}_n}/ {{\delta}_n}^{2-{\alpha}} =+\infty$, then property (b) in Proposition \[prop: one binary coll\] implies $m_{T_1, T_2}^-(q, {\mathcal{A}}) = +\infty$, which obviously implies .
If the corresponding limit of ${{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$ is finite for each binary collision, then follows from the sub-additivity of the Morse index and property (a) in \[prop: one binary coll\]. Once we notice $i({\alpha}, {\lambda}) \ge i({\alpha})$, for any ${\lambda}\ge 0$.
\[Proof of Theorem \[thm: regularization\]\] Without loss of generality let’s assume the binary collision is between $m_1$ and $m_2$ with the sequences ${{\varepsilon}_n}$, ${q^n}(t)$ and ${{\delta}_n}$ defined as before, and ${\lambda}= \lim_{n \to +\infty}{{\varepsilon}_n}/{{\delta}_n}^{2-{\alpha}}$. Obviously ${\lambda}$ must be finite, as otherwise by Proposition \[prop: one binary coll\], $m^-_{T_1, T_2}(q, {\mathcal{A}})= +\infty$, which is absurd.
Since ${\lambda}$ is finite, ${\alpha}=1$ and $d=3$, by Proposition \[prop: one binary coll\], $$1 = m^-_{T_1, T_2}(q, {\mathcal{A}}) \ge i(1, {\lambda}) = \max \{k \in {\mathbb{Z}}: \; k < 2 \sqrt{1+{\lambda}} \}.$$ This implies ${\lambda}=0$. Then by Lemma \[lem: asymp angle\], both limits $\lim_{t \to t_0^{\pm}} \frac{q_2(t)-q_1(t)}{|q_2(t)-q_1(t)|}$ exist and the angle between them is $ 2\pi$, as ${\alpha}=1$ and ${\lambda}=0$.
Proof of Lemma \[lm: lim xin\] and \[lm: lim zt\] {#sec: lem 1}
=================================================
\[Proof of Lemma \[lm: lim xin\]\] (a). Recall that and imply, $$|{\eta^n}_1(t_n)|= {{\delta}_n}= \min \{ |{\eta^n}_1(t)|: \; t \in [t_n-a, t_n+a]\}.$$ Then by the definition of ${\xi^n}_1(s)$, $$|{\xi^n}_1(s)| \ge |{\xi^n}_1(0)| = 1, \; \forall s \in [-a/{{\delta}_n}^{1+\frac{{\alpha}}{2}}, a/{{\delta}_n}^{1+\frac{{\alpha}}{2}}].$$ As a result, $$\langle {\xi^n}_1(0), ({\xi^n}_1)'(0) \rangle =0.$$ Let $\xi_1(s)$ be a solution of , with initial condition $$\xi_1(0)= \lim_{n \to \infty} {\xi^n}_1(0), \;\; \xi_1'(0) = \lim_{n \to \infty} ({\xi^n}_1)'(0).$$ Upon passing to a subsequence, we may assume the above limits always exist. Then $\langle \xi_1(0), \xi_1'(0) \rangle = \lim_{n \to \infty} \langle {\xi^n}_1(0), ({\xi^n}_1)'(0) \rangle =0$.
Meanwhile by and , there is a constant $C_1>0$ independent of $n$ with $$\left|\frac{\partial{\widetilde{U}}({\eta^n}(t))}{\partial {\eta^n}_1} \right|, \left|\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n}(t))}{\partial {\eta^n}_1} \right|\le C_1, \;\; \forall t \in [t_n-a, t_n+a].$$ This means the second line in equation converges to zero, which gives us equation , as $n$ goes to infinity. Then by the continuous dependence of solutions on initial conditions and coefficients of differential equations, we have ${\xi^n}_1(s)$ converges to $\xi_1(s)$ in $C^2([-\ell, \ell], {\mathbb{R}}^d)$, for any $\ell >0$.
To show that $\xi_1(s)$ satisfies , consider the energy $E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)=E^{{{\varepsilon}_n}}_{{\mathbf{I}}}({q^n}(t), \dot{{q^n}}(t))$ and $E_{{\mathbf{I}}}(t)= E_{{\mathbf{I}}}(q(t), {\dot{q}}(t))$ of the ${\mathbf{I}}$-cluster with ${\mathbf{I}}=\{1, 2\}$ corresponding to ${q^n}$ and $q$. By the proof of Proposition \[prop: weak solution to generalized\], $E_{{\mathbf{I}}}(t)$ is continuous on $[t_0-a, t_0+a]$ and $E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)$ converges to it under the $L^{\infty}$ norm, after passing to a subsequence. Therefore for $n$ large enough, there is a constant $C_2>0$ independent of $n$, such that $$\label{eq: E en I bound} |E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)| \le C_2, \;\; \forall t \in [t_n -a, t_n+a].$$
Meanwhile with and , a direct computation shows $$\label{eq: energy xin I} \frac{1}{2(m_1 +m_2)} |({\xi^n}_1)'|^2 - \frac{1}{|{\xi^n}_1|^{{\alpha}}} - \frac{{{\varepsilon}_n}}{{{\delta}_n}^{2-{\alpha}}} \frac{1}{|{\xi^n}_1|^2} = \frac{{{\delta}_n}^{2-{\alpha}}}{m_1m_2} ( E^{{{\varepsilon}_n}}_{{\mathbf{I}}} - {\frac{1}{2}}|\dot{{\eta^n}_2}|^2)$$ To prove that $\xi_1(s)$ satisfies , it is enough to show the right hand side of the above equation converges to zero, as $n$ goes to infinity.
To see this, notice that ${\eta^n}_2(t)$ converges to $\frac{m_1q_1(t)+m_2q_2(t)}{m_1 + m_2}$, which is the center of mass of $m_1$ and $m_2$. Although they collide at the moment $t_0$, the path of their center of mass, is actually $C^2$ on $[t_0-a, t_0+a]$ (see [@FT04 Remark 4.10]). Hence the convergence of ${\eta^n}_2(t)$ holds at least under the $C^1$ norm, and as a result, there is a constant $C_3>0$ independent of $n$, such that $$\label{eq: etn bound}
|\dot{{\eta^n}_2}(t)| \le C_3, \;\; \forall t \in [t_n-a, t_n+a].$$ Now our claim follows directly from and .
(b). Notice that describes the motion of a point mass under the attraction of a isotropic central force. As a result, $\langle \xi_1(0), \xi_1'(0) \rangle =0$ implies $\xi_1(t) \in W(\xi_1)= \text{span} \{\xi_1(0), \xi'_1(0) \}$, for all $t \in {\mathbb{R}}$. By property (a), $\xi_1(s)$ is a collision-free zero energy solution of . Then the rest of the property is well known and a detailed proof can be found in [@Tn93a Section 4].
(c). Let $\{ {\mathbf{e}}_i: i=1, \dots, d \}$ be an orthogonal basis of ${\mathbb{R}}^d$, such that $W(\xi_1)=\text{span}\{{\mathbf{e}}_1, {\mathbf{e}}_2 \}$. Then for any $\varphi \in H^1_0({\mathbb{R}}, {\mathbb{R}})$ and ${\mathbf{e}}_i$, $i=3, \dots, d$, a simple computation shows $$\label{eq; sec var I varphi}
d^2 {\mathcal{I}}(\xi_1)[\varphi {\mathbf{e}}_i, \varphi {\mathbf{e}}_i] = \int_{-\infty}^{\infty} \frac{ |\varphi'|^2}{2(m_1+m_2)}-{\alpha}\frac{|\varphi|^2}{|\xi_1|^{{\alpha}+2}}-2 {\lambda}\frac{|\varphi|^2}{|\xi_1|^4} \,dt$$
Let ${\Lambda}(\xi_1)$ be the largest subspace of $H^1_0({\mathbb{R}}, {\mathbb{R}})$, such that the value in is negative for any $\varphi \in {\Lambda}(\xi_1)$. Using Sturm Comparison Theorem, in [@Tn93a Section 4], [@Tn93b Section 4] and [@Tn94a Proposition 1.1], Tanaka showed the dimension of ${\Lambda}(\xi_1)$ is related to the winding number of $\xi_1(s)$, $s \in {\mathbb{R}}$, in the plane $W(\xi_1)$ with respect to the origin. More precisely $$\label{eq: Lmd i} \text{dim}({\Lambda}(\xi_1)) \ge i({\alpha}, {\lambda}),$$ and this proves property (c).
\[rem d=2 no result\] Although Tanaka only state , a slight modification of his proof should show this is in fact an equality. As a result, if one wants to get a better estimate of the Morse index near a binary collision, one has to compute the Morse index of $\xi_1$ inside $W(\xi_1)$. However a result in [@HY17 Corollary 5.1] by Hu and the author shows this is actually zero. Because of this we believe with Tanaka’s approach, one can not get any nontrivial result for the planar $N$-body problem.
\[Proof of Lemma \[lm: lim zt\]\] With the results given by Tanaka in [@Tn93b Section 4], the lemma can be proven following the same argument given in Lemma \[lm: lim xin\]. The only difference is the *blow up* should follow , instead of , and correspondingly needs to be replaced by $$\label{eq: energy ztn I} \frac{1}{2(m_1 + m_2)} |({{\zeta}^n}_1)'|^2- \frac{1}{|{{\zeta}^n}_1|^2} -\frac{{{\delta}_n}^{2-{\alpha}}}{{{\varepsilon}_n}} \frac{1}{|{{\zeta}^n}_1|^{{\alpha}}} = \frac{{{\delta}_n}^2}{{{\varepsilon}_n}m_1m_2} ( E^{{{\varepsilon}_n}}_{{\mathbf{I}}} - {\frac{1}{2}}|\dot{{\eta^n}_2}|^2).$$ Nevertheless the right hand side of the equation still goes to zero, because ${\lambda}= +\infty$ implies ${{\delta}_n}^2/{{\varepsilon}_n}$ converge to zero, as $n$ goes to infinity.
Proof of Lemma \[lem: asymp angle\] {#sec: lm asymp angle}
===================================
Our proof follows the approach given by Tanaka in [@Tn94a]. First we establish a lemma that corresponds to Lemma 1.3 in [@Tn94a].
\[lm: bound energy and convexity\]
1. There is a constant $C_1>0$ independent of $n$, such that $$\label{eq: up bound dot etn 1} |\dot{{\eta^n}_1}(t)|^2 \le \frac{2(m_1+m_2)}{|{\eta^n}_1(t)|^{{\alpha}}} + \frac{2{{\varepsilon}_n}(m_1+m_2)}{|{\eta^n}_1(t)|^2}+C_1, \;\; \forall t \in [t_n-a, t_n+a].$$
2. There is a constant $\ell_0>0$, such that for $n$ large enough, if $t \in [t_n-a, t_n+a]$ and ${\eta^n}_1(t) \in B_{\ell_0}$, then $\frac{d^2}{dt^2}|{\eta^n}_1(t)|^2 >0$, where $B_{\ell}=\{ x \in {\mathbb{R}}^d: \; |x| \le \ell \}$ for any $\ell >0$.
(a). Let ${\mathbf{I}}= \{1,2\}$. Recall that $E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t)= E^{{{\varepsilon}_n}}_{{\mathbf{I}}}({q^n}(t), \dot{{q^n}}(t))$ is the energy of the ${\mathbf{I}}$-cluster. By , $$\label{eq: energy 12 etn} E^{{{\varepsilon}_n}}_{{\mathbf{I}}}(t) = \frac{m_1m_2}{2(m_1+m_2)}|\dot{{\eta^n}_1}|^2+\frac{m_1+m_2}{2} |\dot{{\eta^n}_2}|^2-\frac{m_1m_2}{|{\eta^n}_1|^{{\alpha}}}-{{\varepsilon}_n}\frac{m_1m_2}{|{\eta^n}_1|^2}.$$ As a result, $$|\dot{{\eta^n}_1}(t)|^2 \le \frac{2(m_1+m_2)}{|{\eta^n}_1(t)|^{{\alpha}}}+2{{\varepsilon}_n}\frac{m_1+m_2}{|{\eta^n}_1(t)|^2}-\frac{(m_1+m_2)^2}{m_1m_2}|\dot{{\eta^n}_2}(t)|^2 + \frac{2(m_1+m_2)}{m_1m_2} E_{{\mathbf{I}}}^{{{\varepsilon}_n}}(t).$$ Then property (a) following from and .
(b). By a direct computation, $$\begin{split}
{\frac{1}{2}}\frac{d^2}{dt^2} |{\eta^n}_1|^2 &= |\dot{{\eta^n}_1}|^2+ \langle {\eta^n}_1, \ddot{{\eta^n}_1} \rangle = |\dot{{\eta^n}_1}|^2-{\alpha}\frac{m_1+m_2}{|{\eta^n}_1|^{{\alpha}}}-2{{\varepsilon}_n}\frac{m_1+m_2}{|{\eta^n}_1|^2} \\
& \quad + \frac{m_1+m_2}{m_1m_2} \langle {\eta^n}_1, \frac{\partial {\widetilde{U}}}{\partial {\eta^n}_1}+ {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}}{\partial {\eta^n}_1} \rangle.
\end{split}$$ Use and , we can find a positive constant $C_2$, such that $${\frac{1}{2}}\frac{d^2}{dt^2} |{\eta^n}_1|^2 \ge (2-{\alpha})\frac{m_1+m_2}{|{\eta^n}_1|^{{\alpha}}}-\frac{m_1+m_2}{2}|\dot{{\eta^n}_2}|^2 + \frac{m_1+m_2}{m_1m_2} \langle {\eta^n}_1, \frac{\partial {\widetilde{U}}}{\partial {\eta^n}_1}+ {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}}{\partial {\eta^n}_1} \rangle- C_2,$$ which clearly implies property (b).
Again following [@Tn94a], we introduce the following functions $$e_n(t)= \sqrt{|{\eta^n}_1(t)|^2 |\dot{{\eta^n}_1}(t)|^2 - \langle {\eta^n}_1(t), \dot{{\eta^n}_1}(t)\rangle^2}, \;\; {\omega}(t)= \frac{e_n(t)}{|{\eta^n}_1(t)| |\dot{{\eta^n}_1}(t)|}.$$ Notice that ${\omega}_n(t)$ is well defined, when ${\eta^n}_1(t) \ne 0$ and $\dot{{\eta^n}_1}(t) \ne 0$. In particular, ${\omega}_n(t) = \sin (\angle({\eta^n}_1(t)/|{\eta^n}_1(t)|, \dot{{\eta^n}_1}(t))/|\dot{{\eta^n}_1}(t)|)$ and $|{\omega}_n(t)| \le 1$. By , $$\frac{d e_n}{dt}= \frac{m_1+m_2}{m_1m_2} \left \langle \frac{|{\eta^n}_1|^2 |\dot{{\eta^n}_1}|^2-\langle {\eta^n}_1, \dot{{\eta^n}_1} \rangle {\eta^n}}{\sqrt{|{\eta^n}_1|^2 |\dot{{\eta^n}_1}|^2-\langle {\eta^n}_1, \dot{{\eta^n}_1} \rangle^2}}, \frac{\partial{\widetilde{U}}({\eta^n})}{\partial{{\eta^n}_1}}+{{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n})}{\partial{{\eta^n}_1}} \right \rangle.$$ According to , there are positive constants $C_3, C_4$ independent of $n$, such that $$\label{eq: bound dot e} |\dot{e}_n(t)|\le C_3 |{\eta^n}_1(t)| \cdot \left|\frac{\partial {\widetilde{U}}({\eta^n}_1(t))}{\partial {\eta^n}_1}+ {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}({\eta^n}_1(t))}{\partial {\eta^n}_1} \right| \le C_4 |{\eta^n}_1(t)|, \;\; \forall t \in [t_n-a, t_n+a].$$
Again by , a direct computation shows $$\begin{split}
\frac{d {\omega}_n}{dt} &= \frac{\dot{e}_n}{|{\eta^n}_1||\dot{{\eta^n}_1}|}-\frac{{\omega}_n}{|{\eta^n}_1|^2 |\dot{{\eta^n}_1}|^2} \Big( |\dot{{\eta^n}_1}|^2 \langle {\eta^n}_1, \dot{{\eta^n}_1}\rangle + |{\eta^n}_1|^2 \langle \dot{{\eta^n}_1}, \ddot{{\eta^n}_1} \rangle \Big) \\
&=\frac{\dot{e}_n}{|{\eta^n}_1||\dot{{\eta^n}_1}|} -\frac{m_1+m_2}{m_1m_2} \frac{{\omega}_n}{|\dot{{\eta^n}_1}|^2}\langle \dot{{\eta^n}_1}, \frac{\partial {\widetilde{U}}}{\partial {\eta^n}_1}+ {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}}{\partial {\eta^n}_1} \rangle \\
& \quad - {\omega}_n \frac{\langle {\eta^n}_1, \dot{{\eta^n}_1} \rangle}{|{\eta^n}_1|^2 |\dot{{\eta^n}_1}|^2} \Big(|\dot{{\eta^n}_1}|^2-{\alpha}\frac{m_1+m_2}{|{\eta^n}_1|^{{\alpha}}}-2{{\varepsilon}_n}\frac{m_1+m_2}{|{\eta^n}_1|^2} \Big) \\
\end{split}$$ Then and implies, $$\label{eq: om dot}
\begin{split}
\dot{{\omega}}_n(t) & \le \frac{C_5}{|\dot{{\eta^n}_1}|} \left|\frac{\partial {\widetilde{U}}}{\partial {\eta^n}_1}+ {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}}{\partial {\eta^n}_1} \right|- {\omega}_n \frac{\langle {\eta^n}_1, \dot{{\eta^n}_1} \rangle}{|{\eta^n}_1|^2 |\dot{{\eta^n}_1}|^2} \Big( (2-{\alpha})\frac{m_1+m_2}{|{\eta^n}_1|^{{\alpha}}}-C_6 \Big) \\
& \le \left(C_5|{\eta^n}_1|\left|\frac{\partial {\widetilde{U}}}{\partial {\eta^n}_1}+ {{\varepsilon}_n}\frac{\partial {\widetilde{{\mathfrak{U}}}}}{\partial {\eta^n}_1}\right| -{\omega}_n \sqrt{1-{\omega}_n^2} \Big((2-{\alpha})\frac{m_1+m_2}{|{\eta^n}_1|^{{\alpha}}}-C_6\Big) \right) \frac{1}{|{\eta^n}_1||\dot{{\eta^n}_1}|},
\end{split}$$ where $C_5$ is positive constant and $C_6$ is a constant, whose sign depends on the sign of $\langle {\eta^n}_1, \dot{{\eta^n}_1} \rangle$), independent of $t \in [t_n-a, t_n+a]$ and $n$.
With Lemma \[lm: bound energy and convexity\] and (this corresponds to ($1.8$) in [@Tn94a]), the next result can be proven following the argument given in Proposition 1.4 and 1.5 in [@Tn94a] line by line, and we will not repeat it here.
\[lm: eta 1 cauchy\] Let $\ell_0$ be the constant given in Lemma \[lm: bound energy and convexity\], for any $b>0$ small enough, there exist constants $0 < \ell_2 <\ell_0 <\ell_1$, such that when $n$ is large enough, for any $$t_n < t < t^* < t_n +a, \; \text{ or } \; t_n-a < t < t^* < t_n,$$ if ${\eta^n}_1(t)$ and ${\eta^n}_1(t^*) \in B_{\ell_2} \setminus B_{{{\delta}_n}\ell_1} $, then $ \Big| {\eta^n}_1(t)/|{\eta^n}_1(t)| - {\eta^n}_1(t^*)/|{\eta^n}_1(t^*)| \Big| < b.$
Now we give a proof of Lemma \[lem: asymp angle\].
\[Proof of Lemma \[lem: asymp angle\]\] By property (c) in Lemma \[lm: lim xin\], $\lim_{s \to +\infty} \xi_1(s)/|\xi_1(s)|= u^+$. Fix an arbitrary small $b>0$, let $\ell_1, \ell_2$ be the constants given in Lemma \[lm: eta 1 cauchy\]. We can choose an $s^*>0$ large enough, such that for $n$ large enough, $$\ell_1 < |\xi_1(s^*)| < {\delta}_n^{-1} \ell_2, \; \text{ and } \; \left| \frac{\xi_1(s^*)}{|\xi_1(s^*)|}- u^+ \right| < b.$$ By Lemma \[lm: lim xin\], the same inequalities hold for ${\xi^n}_1(s^*)$, when $n$ is large enough.
Let $t^*= t_n+{{\delta}_n}^{1+\frac{{\alpha}}{2}}s^*$, by equation , $${\eta^n}_1(t^*) \in B_{\ell_2} \setminus B_{{{\delta}_n}\ell_1}, \text{ and } |{\eta^n}_1(t^*)/|{\eta^n}_1(t^*)| -u^+| < b.$$ Since $\ell_2 < \ell_0$, for any $t \in (t^*, t_n +a)$, we claim if ${\eta^n}_1(t) \in B_{\ell_2}$, then $|{\eta^n}_1(t)| > {{\delta}_n}\ell_1.$ Indeed this follows from the fact $d |{\eta^n}_1(t_n)|/dt=0$ and property (b) in Lemma \[lm: bound energy and convexity\].
As a result, for any $t \in (t^*, t_n +a)$, ${\eta^n}_1(t) \in B_{\ell_2}$ implies ${\eta^n}_1(t) \in B_{\ell_2} \setminus B_{{{\delta}_n}\ell_1}$. Then by Lemma \[lm: eta 1 cauchy\], $$\left| \frac{{\eta^n}_1(t)}{|{\eta^n}_1(t)|} - u^+ \right| \le \left| \frac{{\eta^n}_1(t)}{|{\eta^n}_1(t)|}-\frac{{\eta^n}_1(t^*)}{|{\eta^n}_1(t^*)|} \right| + \left| \frac{{\eta^n}_1(t^*)}{|{\eta^n}_1(t^*)|}-u^+ \right| \le 2b.$$ This means $$(\{ {\eta^n}_1(t): \; t \in (t_n+ {{\delta}_n}^{1+\frac{{\alpha}}{2}}s^*, t_n+a) \} \cap B_{\ell_2}) \subset (\{x \in B_{\ell_2}: \; |x/x|-u^+| < 2b \} \cup B_{{{\delta}_n}\ell_1}).$$ Recall that when $n$ goes to infinity, $t_n$ converges to $t_0$, ${{\delta}_n}$ converges to zero, and ${\eta^n}_1(t)$ converges uniformly to $q_2(t)-q_1(t)$. Then $$(\{q_2(t)-q_1(t): \; t \in (t_0, t_0+a) \} \cap B_{l_2}) \subset \{ x \in B_{l_2}:\; |x/x|-u^+| < 2b \}.$$ Since the above result hold for any $b>0$ small enough, we get $$\lim_{t \to +\infty} \frac{q_2(t)-q_1(t)}{|q_2(t)-q_1(t)|} = u^+.$$ A similar argument shows $$\lim_{t \to -\infty} \frac{q_2(t)-q_1(t)}{|q_2(t)-q_1(t)|} = u^-.$$
*Acknowledgements.* The author thanks Alain Chenciner for a careful reading of an early draft of the paper and Richard Montgomery for pointing out a mistake. Valuable discussions with Vivina Barutello, Jacques Féjoz and Xijun Hu are also appreciated. The main part of the work was done when the author was a postdoc and a visitor at Ceremade in University of Paris-Dauphine, IMCCE in the Paris Observatory and School of Mathematics in Shandong University. He thanks the hospitality of these institutes and the financial support of FSMP and NSFC(No.11425105).
[^1]: The author acknowledges the support of the ERC Advanced Grant 2013 No. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”
|
---
abstract: 'PKS 1510-089 is one of the most variable blazars in the third *Fermi*-LAT source catalog. During 2015, this source has shown four flares identified as flare A, B, C, and D in between three quiescent states Q1, Q2, and Q3. The multiwavelength data from *Fermi*-LAT, Swift-XRT/UVOT, OVRO, and SMA Observatory are used in our work to model these states. Different flux doubling times have been observed in different energy bands which indicate there could be multiple emission zones. The flux doubling time from the gamma-ray and X-ray light curve are found to be 10.6 hr, 2.5 days, and the average flux doubling time in the optical/UV band is 1 day. It is possible that the gamma-ray and optical/UV emission are produced in the same region whereas X-ray emission is coming from a different region along the jet axis. We have also estimated the discrete correlations functions (DCFs) among the light curves of different energy bands to infer about their emission regions. However, our DCF analysis does not show significant correlation in different energy bands though it shows peaks in some cases at small time lags. We perform a two-zone multi-wavelength time-dependent modeling with one emission zone located near the outer edge of the broad line region (BLR) and another further away in the dusty/molecular torus (DT/MT) region to study this high state.'
author:
- 'Raj Prince$^1$, Nayantara Gupta$^1$, Krzysztof Nalewajko$^2$'
title: 'Two-zone emission modeling of PKS 1510-089 during the high state of 2015 '
---
Introduction
============
Being one of the most variable flat spectrum radio quasars (FSRQs) in the third *Fermi*-LAT source catalog (3FGL) PKS 1510-089 has been well observed during its high states in the past. It is located at a redshift of z = 0.361 [@Tanner; @et; @al.; @(1996)] with black hole mass 5.4$\times 10^8$ M$_{\sun}$ and accretion rate approximately 0.5 M$_{\sun}$/yr (@Abdo [@et; @al.; @(2010)]). A long term analysis of the light curve of PKS 1510-089 with the eight year *Fermi*-LAT data has been done by @Prince [@et; @al.; @(2017)] earlier. They have observed five major flares during 2008-2016, and their temporal and spectral features have been studied in detail. During its high activity period between September 2008 and June 2009 its gamma-ray emission showed a weak correlation with the UV, strong correlation with the optical and no correlation with the X-ray emission [@Abdo; @et; @al.; @(2010)]. PKS 1510-089 was also studied by @Nalewajko [@(2013)], where he used the first four years of *Fermi*-LAT data and observed 14 flares with a minimum and maximum flux of 7.4 and 26.6 ($\times$10$^{-6}$) ph cm$^{-2}$ s$^{-1}$, respectively. Detection of high energy gamma-rays up to 300-400 GeV has been reported by the H.E.S.S. collaboration (@Abramowski [@et; @al.; @(2013)]) during March-April 2009 and by the MAGIC collaboration (@Aleksic [@et; @al.; @(2014)]) between February 3 and April 3, 2012.
In the second half of 2011 this source was active in several energy bands and the optical, gamma-ray and radio flares were detected. The gamma-ray variability down to 20 minutes indicated the highly variable nature of this source. @Aleksic [@et; @al.; @(2014)] did a detailed multiwavelength modeling for the period January-April 2012 covering radio to very high energy gamma-rays. They explained the multiwavelength emission as the result of turbulent plasma flowing at a relativistic speed down the jet and crossing a standing conical shock. In modeling the spectral energy distributions (SEDs) from PKS 1510-089 it is most commonly thought that the low energy (radio, optical) emission is from synchrotron radiation of relativistic electrons and high energy emission (X-ray, gamma-ray) is from external Compton (EC) scattering of the seed photons in the broad line region (BLR) and dusty torus region (@Kataoka [@et; @al.; @(2008)], @Abdo [@et; @al.; @(2010)], @Brown [@(2013)], @Barnacka [@et; @al.; @(2014)]). The gamma-ray emission region could also be located at a radio knot, far away from the black hole as suggested by @Marscher [@et; @al.; @(2010)]. They modeled the eight major gamma-ray flares of PKS 1510-089 that happened in 2009. During optical and gamma-ray flare a bright radio knot travelled through the core/stationary feature at 43 GHz seen by VLBA (Very Long Baseline Array) images. The knot continued to propagate down the length of the jet at an apparent speed of 22c. A strong emission in gamma-ray energy accompanied by a month long emission in X-ray/radio emission which gradually intensified, represented the complex nature of the flares.
The hadronic scenario of high energy photon emission (X-ray, gamma-ray) by $p-\gamma$ interactions and proton synchrotron emission has been studied before (@Bottcher [@et; @al.; @(2013)], ). Hadronic models require super-Eddington luminosities to explain the gamma-ray flux.
A two zone modeling was considered earlier by @Nalewajko [@et; @al.; @(2012)] after including Herschel observations, *Fermi*-LAT, Swift, SMARTS and Submillimeter Array data for explaining the spectral and temporal features of activities of PKS 1510-089 in 2011. From March to August 2015 this source was again very active. Optical R-band monitoring with ATOM, supporting H.E.S.S. observations, detected very high flux of optical photons (@Zacharias [@et; @al.; @(2016)]).
Its enhanced activity in very high energy gamma-rays was also observed by MAGIC telescope (@Ahnen [@et; @al.; @(2017)]) in May, 2015. In the middle of a long high state in optical and gamma-rays, for the first time they detected a fast variability in very high energy gamma-rays. Their observation periods MJD 57160-57161(Period A) and MJD 57164-57166 (Period B) overlap with one of the flares identified as flare-B (MJD 57150-57180) in our work. They collected simultaneous data in radio, optical, UV, X-ray, and gamma-ray frequencies for multiwavelength modeling. They noted most of the flux variation happened in *Fermi*-LAT and MAGIC energy bands.
The complex nature of multiwavelength emission indicated a single zone model is not suitable for explaining the flares of PKS 1510-089. In the present work several months of observational data have been studied for multiwavelength modeling of the high state of PKS 1510-089 in 2015.
The paper is organised as follows: in Section 2, we have provided the details about the multiwavelength data used in our study. In Section 3, we have presented our results, in Section 4, we have discussed our results and compared with the previous studies on this source.
Multiwavelength Data Analysis
=============================
The *Fermi*-LAT and Swift-XRT/UVOT observations together cover optical, ultraviolet, X-ray and gamma-ray energy bands which allow us to do multiwavelength variability analysis and modeling of blazar flares.
*Fermi*-LAT
-----------
After the successful launch of *Fermi* Gamma-ray Space Telescope in 2008, thousands of sources have been revealed in the gamma-ray sky by the on-board instrument, Large Area Telescope (LAT), in the past eleven years. With a field of view of about 2.4 sr (@Atwood [@et; @al.; @(2009)]) LAT covers 20% of the sky at any time and scans the whole sky every three hours. It is sensitive to photons having energy between 20 MeV to higher than 500 GeV. The third Fermi source catalog (3FGL; @Acero [@et; @al.; @(2015)]) shows that the extragalactic sky is dominated by active galactic nuclei (AGN) emitting high energy gamma-rays. The FSRQ PKS 1510-089 has been continuously monitored by *Fermi*-LAT since August 2008. We collected the data for the year 2015 and analyzed it for energy range 0.1–300 GeV. A circular region of radius 10$\degree$ is chosen around the source of interest and the circular region is known as region of interest (ROI). The detailed procedure to analyze the *Fermi*-LAT data is given in @Prince [@et; @al.; @(2018)].
The data analysis also takes care of contamination from Earth’s limb gamma-rays by rejecting the events having zenith angle higher than 90$\degree$. In this analysis, we have used the latest instrument Response Function “P8R2$\_$SOURCE$\_$V6" provided by the Fermi Science Tools.
Swift-XRT/UVOT
--------------
Swift data for PKS 1510-089 has been collected from *HEASARC* webpage[^1] for a period of one year during 2015, which is part of archived data. In total 44 observations were reported during 2015 . A task *‘xrtpipeline’* version 0.13.2 has been run for every observation to get the cleaned event files. The latest version of calibration files (CALDB version 20160609) and standard screening criteria have been used to re-process the raw data. Cleaned event files corresponding to the Photon Counting (PC) mode have been considered for further analysis. A circular region of radius 20 arc seconds around the source and away from the source has been chosen for the source and the background respectively while analyzing the XRT data. The X-ray light curve and spectra have been extracted by a tool called *xselect*. The spectrum has been obtained and fitted in “xspec” using simple power law model with the galactic absorption column density $n_H$ = 6.89$\times$10$^{20}$ cm$^{-2}$ [@Kalberla; @et; @al.; @(2005)]. The Swift Ultraviolet/Optical Telescope (UVOT, @Roming [@et; @al.; @(2005)]) also observed PKS 1510-089 in all the six filters U, V, B, W1, M2 and W2. The source image has been extracted by choosing a circular region of 5 arc seconds around the source. Similarly, the background region has also been chosen with a radius of 10 arcseconds away from the source. The task ‘uvotsource’ has been used to extract the source magnitudes and fluxes. Magnitudes are corrected for galactic extinction (E(B-V) = 0.087 mag; @Schlafly [@and; @Finkbeiner; @(2011)]) and converted into a flux using the zero points (@Breeveld [@et; @al.; @(2011)]) and conversion factors (@Larionov [@et; @al.; @(2016)]).
Radio data at 15 and 230 GHz
----------------------------
PKS 1510-089 was also observed in radio wavelength by OVRO [@Richards; @et; @al.; @(2011)][^2] at 15 GHz and by Sub-millimeter array (SMA)[^3] at 230 GHz [@Gurwell; @et; @al.; @(2007)] as a part of the Fermi monitoring program. We have collected the data for the year 2015 from both the observatories.
Results
=======
In this section, we have presented the results obtained from temporal and spectral analysis, and we have discussed the importance of these results in multi-wavelength SED modeling.
Multiwavelength Light Curves
----------------------------
Multiwavelength light curves are shown in Figure 1, where they show indication of flares in various wavebands during the year of 2015 for PKS 1510-089.
The topmost panel of Figure 1 represents the gamma-ray light curve. The gamma-ray light curve is divided into different states on the basis of the fluxes observed during different time periods. We have also estimated the fractional rms variability amplitudes (@Fossati [@et; @al.; @(2000)]; @Vaughan [@et; @al.; @(2003)]) to identify the different states. If the value of the fractional variability during a time period is more than 0.5 (50$\%$) then it is considered as a flaring state. During the time period MJD 57023–57100, the average flux of the source is found to be 2.49$\pm$0.10($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$) and it does not change significantly. The fractional variability amplitude is found to be 0.50$\pm$0.04. This period has been identified as quiescent state Q1.
The source started showing high activity from MJD 57100 and continued for almost 50 days till MJD 57150. The average flux measured during this period is 10.06$\pm$0.19 ($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$) which is five times higher than that in the quiescent state Q1. The fractional variability found during this period is 0.67$\pm$0.02 which confirms that the source is more variable than in state Q1. This period is defined as flare A in our Figure 1.
After the end of flare A, in 2-3 days the flux again started rising and it lasted for a month. This period is noted as MJD 57150–57180. The average flux estimated during this period is 12.17$\pm$0.26 ($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$). The fractional variability measured during this period is 0.57$\pm$0.02, which is higher than that in the state Q1. This period of high flux and high variability amplitude is referred as flare B in our work.
After flare B, the source flux became lower compared to flare A and B and the source remained for a month ( MJD 57180–57208) in this low state. The average flux obtained during this low state is 5.37$\pm$0.21 ($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$) and the fractional variability amplitude measured as 0.28$\pm$0.04. The flux variability is not significant during this period as it is below 30$\%$. We have named this period as quiescent state Q2.
During the time period MJD 57208–57235, the source again went to a high state compared to Q1 and Q2, with average flux 7.20$\pm$0.23($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$). The flux variability amplitude measured during this period is 0.51$\pm$0.03, which is just above the limit we have set (50$\%$). The large average flux and 51$\%$ fractional variability suggest that this period is different from the state Q1 and Q2. Therefore, this state is identified as flare-C in gamma-ray which also have a strong flaring counter-part in optical/UV.
A higher state surpassing all the observed flares and quiescent states was observed during the time period MJD 57235–57260. The average flux estimated during this period is 12.67$\pm$0.36 ($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$), which is much higher than the average flux observed during any of the other states. A huge flux variation can be seen from Figure 1, and the fractional variability amplitude measured during this period is 1.16$\pm$0.03 ($>$ 100$\%$). This period of high state is defined as flare D.
After flare D, the flux decreased sharply and attained an average value of 2.93$\pm$0.12 ($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$). The source continued to have this average flux over a long period of time from MJD 57260–57320. A small variation in flux was seen during this period (see Figure 1), for which the fractional variability amplitude has been estimated as 0.50$\pm$0.05. Since the average flux in this state is very low compare to the flaring states, we have considered this period as quiescent state Q3.
The maximum flux observed during flare A, B, C, and D are 38.4, 27.92, 15.78, and 55.05 ($\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$) at MJD 57115.5, 57167.5, 57220.5, and 57244.5 respectively. Flare D has been identified as the brightest gamma-ray flare of the year 2015. In Figure 1, the gamma-ray light curve is binned in one day time bin. The other light curves do not have an equally spaced binning because different observations were carried out at different times. We have estimated the average time between two consecutive observations for X-ray and optical/UV light curves. In X-ray it is found to be 3.4 days and in optical/UV band it is estimated as 4.1, 4.3, 3.8, 3.8, 4.4, 3.5 days for filters B, V, U, W1, M2, W2 respectively.
$\gamma$-ray variability
------------------------
During 2015, the source was very active as it had been also seen earlier. The maximum flux attained at this time is (5.50$\pm$0.34)$\times$10$^{-6}$ ph cm$^{-2}$ s$^{-1}$ with photon spectral index 1.86 at MJD 57244.5. The variability of the source can be seen from the gamma-ray light curve in Figure 1, which represents all the flares along with the photon spectral index in the second panel.
It is seen that as the flux increases in gamma-ray the photon spectral index becomes harder and harder. The flux doubling/halving time is estimated during the flaring episodes by using the following equation (@Brown [@(2013)]; @Saito [@et; @al.; @(2013)]; @Paliya [@(2015)]), $$F_2 = F_1.2^{({t_2-t_1})/\tau_d}
%t_{var} = \frac{F_1 + F_2}{2} \frac{t_2 - t_1}{|F_2 - F_1|}$$ where, F$_1$ and F$_2$ are the fluxes measured at two consecutive time t$_1$ and t$_2$, and $\tau_d$ represent the doubling /halving times scale. One day binned gamma-ray light curve, shown in Figure 1, revealed the flux doubling time of 10.6 hr, when the source flux is changing from 1.15$\times$10$^{-6}$ to 5.50$\times$10$^{-6}$ between MJD 57243.5 to MJD 57244.5.
X-ray variability
-----------------
The source is also followed by the Swift-XRT/UVOT telescope to unveil the behavior in X-ray, UV, and optical bands. In the third panel of Figure 1, we have shown the X-ray light curve in the energy range of 0.3–10 keV. X-ray light curve is scanned by equation 1 and the flux doubling time is estimated for consecutive time interval and it is found that the source is less variable in X-rays, moreover flaring states cannot be clearly identified. The flux doubling time estimated by using equation 1 from X-ray light curve is 2.5 days for F$_1$ = 1.38$\times$10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ at t$_1$ = MJD 57156.41 and F$_2$ = 1.07$\times$10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ at t$_2$ = MJD 57157.34.
Optical and UV variability
--------------------------
The Swift-UVOT light curve is plotted in the fourth and fifth panels of Figure 1. The source variability is significant during flare C while flare A, B, and D are less variable. In these three flares (A, B, $\&$ D) the variability of the source is constrained by the number of observations. Equation 1 applied to the entire light curve of optical and UV band and the flux doubling times estimated for U, B, and V band light curves are 1.0, 0.7, and 1.1 days respectively (see Table 1). Similar behaviour has also been seen in UV band (W1, M2, W2). The flux doubling times estimated in these three bands of UV (W1, M2, W2) are 0.8, 1.4, and 1.1 days.\
Radio Light curves
------------------
The last panel of Figure 1 represents the radio light curve in two different frequencies. Owens Valley Radio Observatory (OVRO) and Sub-millimeter array (SMA) telescope radio data at 15 GHz and 230 GHz are plotted in the last panel of Figure 1. The radio light curves during 2015 clearly show that the radio fluxes in both the bands are increasing towards the end of the year. The maximum radio flux in 2015 has been recorded as 4.77 Jy and 4.55 Jy at 15 GHz and 230 GHz respectively. The flux doubling time for the radio light curve is not estimated because of the poorly sparse data points.\
\
The fractional variability (F$_{\rm var}$) in different wavebands are also estimated following the [@Vaughan; @et; @al.; @(2003)]. For the gamma-ray light curve F$_{\rm var}$ is found to be 1.04$\pm$0.01, which corresponds to more than 100$\%$ variability. The F$_{\rm var}$ estimated for X-ray light curve is 0.14$\pm$0.04, which is the lowest among all the wavebands. A good amount of fractional variability is noticed from the optical light curve shown in Figure 1. The F$_{\rm var}$ in optical U, B, and V bands are found to be 0.55$\pm$0.01, 0.59$\pm$0.01, and 0.64$\pm$0.01 respectively, and in UV bands for W1, M2, $\&$ W2 filters it is found to be 0.56$\pm$0.01, 0.53$\pm$0.01, $\&$ 0.48$\pm$0.01. The fractional variability is also computed for the radio light curve at 15 GHz and 230 GHz. From OVRO light curve at 15 GHz the F$_{\rm var}$ is found to be 0.34$\pm$0.01. The SMA light curve shown in Figure 1 shows the large fractional variability compared to the OVRO light curve and the F$_{\rm var}$ is noticed as 0.60$\pm$0.02.
[cccccc p[1cm]{}]{}\
Telescope/Bands & F$_1$ & F$_2$ & t$_1$ & t$_2$ & $\tau_d$\
\
Fermi-LAT &&&& & (hr)\
$\gamma$-rays & 1.15 & 5.50 & 57243.5 & 57244.5 & 10.6\
\
Swift-XRT/UVOT &&&&& (days)\
X-rays & 1.38 & 1.07 & 57156.41 & 57157.34 & 2.5\
U & 2.27 & 2.78 & 57166.72 & 57167.01 & 1.0\
B & 2.35 & 3.09 & 57166.72 & 57167.01 & 0.7\
V & 2.24 & 2.70 & 57166.72 & 57167.01 & 1.1\
W1 & 1.91 & 2.47 & 57166.72 & 57167.01 & 0.8\
M2 & 2.40 & 2.76 & 57166.72 & 57167.01 & 1.4\
W2 & 2.07 & 2.47 & 57166.72 & 57167.01 & 1.1\
\
Cross-Correlation
-----------------
A cross-correlation study between different energy bands can be done to find out the location of different emission regions responsible for multi-wavelength emission along the jet axis. The Discrete Correlation Function (DCF) formulated by @Edelson [@and; @Krolik; @(1988)] can be used to estimate cross and auto-correlations of the unevenly sampled light curves. We have made a few different combinations to show the DCFs. The combinations are $\gamma$-Swift B band, $\gamma$-X-rays, $\gamma$-OVRO, $\gamma$-SMA, and OVRO-SMA. DCFs for all these combinations are shown in Figure 2. When the two light curves LC1 and LC2 are cross-correlated, a positive time lag between them implies that the light curve LC1 is leading with respect to LC2, and a negative time lag implies the opposite.
    
\
\
**$\gamma$-ray vs optical B-band DCF:**\
\
The left most plot of Figure 2 upper panel shows the DCF between $\gamma$-ray and Swift optical B-band, and it is found that there are different peaks at different time lags. We select the peak near zero time lag to constrain the location of the emission region. The peaks at +52 days and -20 days and the other two outer peaks could be due to strong gamma-ray flare correlating with strong optical flare within the total period used for DCF analysis. A peak is observed at time lag 3.9 days, though the correlation coefficient is not much significant. We have estimated the average time resolution of the worst light curve and the DCF time bin is chosen as three times of this average time resolution (@Edelson [@and; @Krolik; @(1988)]; @Castignani [@et; @al.; @(2014)]). In case of gamma-ray vs optical B-band the DCF time bin is 12.2 days. The peak found within the DCF time bin is not considered as a time lag. The multiple peaks in DCF are also observed by @Kushwaha [@et; @al.; @(2017)] for 3C 454.3 during segment 4 as mentioned in their paper. The zero time lag observed by @Castignani [@et; @al.; @(2017)] for PKS 1510-089 and the small time lag observed in our case are consistent with the results obtained by @Abdo [@et; @al.; @(2010)]; and @Nalewajko [@et; @al.; @(2012)] for other epochs. A zero or small time lag between two different emissions suggests their co-spatial origin. Similar results were also found for different sources (@Prince [@(2019)], @Kaur [@and; @Baliyan; @(2018)]). The inference of a small time lag between gamma-ray and optical B band emission has been used to assume that the gamma-ray and optical photons are produced in the same region by the inverse Compton and synchrotron emission of the same population of electrons respectively.\
\
\
**$\gamma$-ray vs X-ray DCF:**\
\
The gamma-ray vs. X-ray DCF is shown in the middle plot of the upper panel of Figure 2. A peak is observed at time lag 4.99 days with a correlation coefficient 0.36$\pm$0.17. The DCF time bin 10.2 days is chosen on the basis of the average time resolution of the X-ray light curve. The observed peak is within the DCF time bin and hence is not considered as the time lag between gamma-ray and X-ray emission. The peak observed at the edge of the DCF can be discarded. A time lag of 50 days between gamma-rays and X-rays has been seen by @Castignani [@et; @al.; @(2017)] in PKS 1510-089. This time lag between gamma-rays and X-rays suggests that the X-rays might have been produced far away from the region of gamma-ray emission in the jet. A small correlation coefficient found in our case makes our results consistent with the result obtained by @Abdo [@et; @al.; @(2010)], where they have also not found any robust evidence of cross-correlation between gamma-ray and X-ray at zero time lag.\
\
**$\gamma$-ray vs OVRO and SMA DCF:**\
\
The rightmost plot of the upper panel and left plot of the lower panel of Figure 2 represent the gamma-ray vs. OVRO (15 GHz) and gamma-ray vs. SMA (230 GHz) DCFs respectively. In gamma-ray vs OVRO, a peak is observed at time lag 75 days which is almost equal to one third of the length of the OVRO light curve. Hence, it cannot be considered as a DCF peak. Similar behavior is also seen in gamma-ray vs SMA DCF, where a peak is observed at the time lag between 60–100 days. This peak also lies at one third of the length of the SMA light curve and hence cannot be considered as DCF peak.\
\
**OVRO vs SMA DCF:**\
\
We have also tried to estimate the DCF between OVRO (15 GHz) and SMA (230 GHz) and the result is shown in the lower panel of Figure 2. The DCF analysis does not show any significant peak, hence it is difficult to comment anything about the correlation between these two emissions.\
\
From the DCF analysis, it is clear that no good correlation is observed in any of the pairs. One of the reasons behind this is non-availability of good quality data and a significant number of observations in X-ray, optical, and radio wavelengths for this particular time of period. Hence, it would not be justified to conclude anything about the locations of different emission regions from this analysis.
Multiwavelength SED Modeling with GAMERA
----------------------------------------
Our analysis shows the source went in long and bright flaring episodes in 2015. The four bright flares are identified as Flare-A, Flare-B, Flare-C, and Flare-D. The quiescent states (Q1 and Q3) were observed before and after the flares and the quiescent state Q2 in between Flare-B and Flare-C. We have produced the gamma-ray SED in the energy range 0.1–300 GeV, for all the four gamma-ray flares along with one of the quiescent states Q2, by using the unbinned likelihood analysis. The observed gamma-ray spectrum are fitted with four different functional forms Power Law (PL), Log Parabola (LP), Broken Power Law (BPL) and Power Law with Exponential cut-off (PLEC) as discussed in @Prince [@et; @al.; @(2018)]. We have found that the gamma-ray SED data points for all the flares and state Q2 are well described by log-parabola (LP) distribution function. A LP photon spectrum can be produced by radiative losses of a LP electron spectrum (@Massaro [@et; @al.; @(2004)]). Due to this reason we have considered LP distribution for the injected electron spectrum in our multiwavelength SED modeling. In X-rays and UV/Optical, the SED data points are also produced. All the spectral data points are plotted together in Figure 3 and modeled using the publicly available code GAMERA[^4] [@Prince; @et; @al.; @(2018)]. GAMERA solves the time-dependent transport equation for input injected electron spectrum, and it calculates the propagated electron spectrum, and further, it uses this propagated electron spectrum as an input and estimates the synchrotron, synchrotron self-Compton (SSC), and inverse-Compton (IC) emission.\
The following continuity equation we have used in our study, $$\label{8}
\frac{\partial N(E,t)}{\partial t}=Q(E,t)-\frac{\partial}{\partial E}\Big(b(E,t) N(E,t)\Big)$$ $Q(E,t)$ is the injected electron (electron and positron) spectrum, $N(E,t)$ is the propagated electron spectrum after the radiative loss, and $b(E,t)$ covers the energy loss rate of electrons due to the synchrotron, synchrotron self-Compton (SSC) and external Compton (EC) emission. The code GAMERA estimates the inverse Compton emission using the full Klein-Nishina cross-section from @Blumenthal [@and; @Gould; @(1970)].
The flux doubling times (see Table 1) estimated in different wavelengths suggest different emission zones. The doubling times found in gamma-ray and UV/Optical bands are closer to each other which suggests they might have been produced in the same region. Two emission regions are considered in this work, one is responsible for optical/UV and gamma-ray emission and another for the X-ray emission.
The locations of the emitting blobs along the jet axis are estimated by using the flux doubling time scales. We have used the following relation, $$d = \frac{c t_{d} \delta}{(1+z)\theta_{jet}}$$ where $t_{d}$ is the flux doubling time and $\theta_{jet}$ is the half opening angle of the jet (@Kaur [@and; @Baliyan; @(2018)]), d is the distance of the emitting region from the central supermassive black holes (SMBH), $c$ is the speed of light in vacuum, $z$=0.361 is the redshift of the source, and $\delta$=25 is the Doppler factor. The jet opening angle was estimated from the radio observations by using the relation $\theta_{jet} = \theta_p sin \langle \Theta_0 \rangle$, where $\theta_p$ = 4.8$\degree$ is the projected half opening angle, and $\langle \Theta_0 \rangle$ is the angle between jet axis and the line of sight. With the values of $\theta_p$ and $\langle \Theta_0 \rangle$ from @Jorstad [@et; @al.; @(2005)] the jet opening angle is found to be 0.12$\degree$. The observed flux doubling times for gamma-ray and X-rays are 10.6 hr and 2.5 days respectively, which are used to estimate the distances of the emission regions by using equation 3. The distance of gamma-ray emitting blob from the central SMBH is estimated at 1.76$\times$10$^{17}$ cm and the location of the X-rays emitting blob is estimated as 1.0$\times$10$^{18}$ cm along the jet axis. The exact boundary of the broad line region (BLR) is not known, but we have some idea about the radius of the BLR. To estimate the size of BLR and Dusty torus (DT), a simple scaling law is given by @Ghisellini [@and; @Tavecchio; @(2009)]. It only depends on the disk luminosity (L$_{disk}$). The relations are R$_{BLR}$ = 10$^{17}$L$_{d,45}^{1/2}$ and R$_{DT}$ = 2.5$\times$10$^{18}$L$_{d,45}^{1/2}$, where L$_{d,45}$ is the disk luminosity in units of 10$^{45}$ erg/s. The disk luminosity has been estimated earlier by several authors (@Celotti [@et; @al.; @(1997)]; @Nalewajko [@et; @al.; @(2012)]) in the range of 3 – 7$\times$10$^{45}$ erg s$^{-1}$. Using the typical value of disk luminosity (L$_{disk}$ = 6.7$\times$10$^{45}$ erg s$^{-1}$), we have found that the radius of BLR (R$_{BLR}$) is 2.6$\times$10$^{17}$ cm and the size of the DT (R$_{DT}$) region is 6.47$\times$10$^{18}$ cm. From this calculation we conclude that the gamma-ray emitting blob is located within the edge of the BLR whereas the X-ray emitting blob lies outside the BLR, in the DT region. We use these inferences in the SED modeling with the time dependent code GAMERA.
All the different flares and quiescent state Q2 are modeled with GAMERA as shown in Figure 3. The model parameters are presented in Table 2. The energy density of the external radiation field in the comoving jet frame is given as, $$\label{5}
U'{_{ext}} = \frac{\Gamma^{2} {\xi_{ext}} L_{disk}} {4 \pi c R_{ext}^{2}}$$ where “ext" represents the BLR or DT. The values of $\xi_{BLR}$ = 0.06, and $\xi_{DT}$ = 0.12 are comparable to @Barnacka [@et; @al.; @(2014)] and the jet Lorentz factor $\Gamma$ = 20, taken from @Aleksic [@et; @al.; @(2014)]. Using equation 4, the BLR energy density in the jet comoving frame is estimated as $U'{_{BLR}}$ = 6.41 erg cm$^{-3}$ and DT energy density as $U'{_{DT}}$ = 2$\times$10$^{-3}$ erg cm$^{-3}$. The temperature of the BLR is used from @Peterson [@(2006)], T$_{BLR}$ = 10$^{4}$K and the temperature of the DT region, T$_{DT}$ = 10$^{3}$K from @Ahnen [@et; @al.; @(2017)].
The Doppler factor ($\delta$) and Lorentz factor ($\Gamma$) for PKS 1510-089 have been chosen from an earlier study by @Aleksic [@et; @al.; @(2014)]. The sizes of the gamma-ray and X-ray emitting blobs are estimated by the relation R$<$ c$\tau_d$ $\delta$/(1+z), where $\tau_d$ denotes the doubling time in two different bands. The sizes of the emitting blobs are found to be 2.1$\times$10$^{16}$ cm and 1.2$\times$10$^{17}$ cm for gamma-ray and X-ray emission respectively.
The electron spectra for all the flares and the quiescent state evolve with time as the electrons lose energy radiatively by synchrotron and IC emission. The duration of each flare and the quiescent state are significantly longer than the cooling time scale of electrons, as a result the electron spectra become steady in a short time. The total time duration of flare A, B, C, and D are 50 days, 30 days, 28 days and 25 days respectively and the quiescent state Q2 lasted for 28 days. The synchrotron emission depends on the strength of the magnetic field and the luminosity of the relativistic electrons. The EC emission depends on the energy density and temperature of the external photons and also the luminosity of the relativistic electrons. The synchrotron self Compton (SSC) emission depends on the energy density of the synchrotron photons, which depends on the size of the blob, magnetic field and luminosity of the relativistic electrons. The SSC emission is found to be very low in our model compared to the external Compton emission. For the given magnetic field in the DT region the synchrotron emission is found to be below 10$^{-13}$ erg/cm$^{2}$/s, hence not visible in Figure 3.
The optical depth correction due to the absorption of gamma-rays by the EBL (extragalactic background light) is not important for the *Fermi*-LAT observed gamma-ray flux from PKS 1510-089. We have included the optical depth correction on the observed data points by MAGIC from @Ahnen [@et; @al.; @(2017)]. The de-absorbed data points are used in the SED modeling. To obtain the best model fit to the data points we have optimized the following parameters e.g. the magnetic field in the blob, luminosity and spectral index of injected relativistic electrons, and also their minimum and maximum energies ($\gamma_{min}$, $\gamma_{max}$).
We have assumed the ratio of pairs (electrons and positrons) to cold protons in the emission regions is $10:1$. The jet power in the relativistic electrons and positrons, or the magnetic field, or the cold protons is calculated with this expression $\rm P_{e,B,p}=\pi \, r_{blob}^2 \, \Gamma^2 \, c \,U_{e,B,p}$, where $U_{e,B,p}$ denotes the energy density in electrons and positrons, or magnetic field, or cold protons. The total jet power is always found to be lower than the Eddington’s luminosity of the source $6.86\times 10^{46}$ erg s$^{-1}$, calculated with the black hole mass given in @Abdo [@et; @al.; @(2010)]. The parameter values which can explain the SEDs of the flares A, B, C, D, and the quiescent state Q2 are listed in Table 2.
Discussion
==========
Below we discuss about our results and compare them with those of previous work.
Multi-wavelength Studies with SED Modeling
------------------------------------------
The flaring states identified as A, B, C, D and the quiescent states Q1, Q2 and Q3 are shown in the gamma-ray light curve in Figure 1 along with the light curves in other wavelengths. The flux doubling times in different wavelengths are found to be different, which motivated us to fit the SED with two-zone model. The values of the parameters used in two zone modeling are displayed in Table 2. The magnetic field in the first zone required to fit the optical and UV data points is in the range of 2.8 to 5.1 Gauss. This emission zone is located near the outer boundary of the BLR region. The magnetic field in the second zone located in the DT region is not constrained by optical or UV data in our model. It is assumed to be very low to minimize the jet power. However, in principle, it could be higher. The X-ray flux constrains the jet power in electrons and positrons in the second zone. In the first zone the magnetic field has more jet power than that in electrons and positrons. In this zone the electrons and positrons carry more energy during the flaring states. MAGIC detected very high energy gamma-rays [@Ahnen; @et; @al.; @(2017)] during Flare B. The maximum energy of the relativistic electrons and positrons in our model is the highest during the flare B. In the second zone also this jet power is expected to be higher during the flaring states if the X-ray flux is higher than that in the quiescent states.
@Abdo [@et; @al.; @(2010)] noted complex correlation between fluxes in different wavelengths during the flaring activity of PKS 1510-089 between September 2008 and June 2009. The high state of PKS1510-089 in 2009 was also studied by @Marscher [@et; @al.; @(2010)]. They found that the gamma-ray peaks were simultaneous with maxima in optical flux.
The 2009 GeV flares of PKS 1510-089 have been studied by @Dotson [@et; @al.; @(2015)]. They have discussed about the location of these flares. For two flares they have suggested that the emission region is at the DT region and for the other two at the vicinity of VLBI radio core. @Barnacka [@et; @al.; @(2014)] modeled the high energy flares detected in March 2009 from PKS 1510-089. They have used the photons in the BLR and DT regions for EC emission to model the flares. SSC emission is insignificant in their model. In their model the emission zone is located at a distance of $7\times 10^{17}$ cm from the black hole.
The low states of this source between 2012 to 2017 have been studied in detail recently using MAGIC data (MAGIC Collaboration; @Acciari [@et; @al.; @(2018)]). Their analysis shows the location of the gamma-ray emission region is close to the outer edge of the BLR region. They have chosen two scenarios with the emission regions located at $7\times10^{17}$ cm and $3\times10^{18}$ cm away from the black hole respectively. For the high state in 2015 @Ahnen [@et; @al.; @(2017)] located the emission region at $6\times10^{17}$ cm away from the black hole. These estimates are comparable to our results. In their work, the gamma-ray emission region has a size of $2.8\times10^{16}$ cm, which is also comparable to the size of the gamma-ray emission region found in our study $2.1\times10^{16}$ cm. Our Lorentz factor and Doppler factor values are similar to @Aleksic [@et; @al.; @(2014)].
The light curves of PKS 1510-089 and 3C 454.3 were studied by @Tavecchio [@et; @al.; @(2010)] for the period from August 4, 2008 to January 31, 2010 to constrain the location of emission region through rapid variability in gamma-rays in the Fermi-LAT data. From hour scale variability in gamma-ray flux they constrained the size of the emission region to be less than $4.8\times 10^{15}$ cm and $3.5\times 10^{15}$ cm for PKS 1510-089 and 3C 454.3 respectively for Doppler factor $\delta=10$. Extreme value of Doppler factor $\delta=50$ constrains the size of the emission region to less than 0.01 pc. They suggested such small emission regions are likely to be located near the black hole. They concluded that the far dissipation scenario, where the gamma-ray emission region is located 10-20 pc away from the black hole is disfavoured.
A time dependent modeling of gamma-ray flares of PKS 1510-089 has been carried out by @Saito [@et; @al.; @(2015)] within the framework of the internal shock scenario. They have shown that the emission region is located between 0.3 pc to 3 pc from the black hole depending on whether the jet is freely expanding or collimated. They have discussed about non-uniformity of Doppler factor across the jet due to the radial expansion of the outflow. This may result in time distortion in the observed gamma-ray light curve, in particular, asymmetric flux profiles with extended decay time.
The most variable blazar 3C 454.3 has been well studied and modeled with multiwavelength observations (@Finke [@(2016)]). Multiwavelength temporal variability in 3C 454.3 during its active state in 2014 has been studied by @Kushwaha [@et; @al.; @(2017)]. They found in some of the epochs IR/optical and gamma-ray fluxes show nearly simultaneous variation. Correlation in Optical and gamma-ray frequencies was observed in June, 2016 outburst of 3C 454.3 (@Weaver [@et; @al.; @(2019)]). Recently, @Rajput [@et; @al.; @(2019)] have analysed quasi-simultaneous data at optical, UV, X-ray and gamma-ray energies collected over a period of 9 years, August 2008 to February 2017. They identified four epochs when the source showed large optical flares. The optical and gamma-ray flares are correlated in two epochs. In two other epochs the flares in gamma-rays are weak or absent.
A correlation in optical and gamma-ray photons from flares of PKS 1510-089 during Jan 2009 to Jan 2010 has been suggested by @Castignani [@et; @al.; @(2017)], which could be a common feature among these blazars. This inference has also been used in our analysis to model the SEDs. We also note that in some FSRQs like 3C 279 the time lag between optical and gamma-ray emission could be due to the variations in the ratio of energy densities in external photon field and magnetic field with distance across the length of the jet (@Janiak [@et; @al.; @(2012)]).
Gamma-Radio Correlation
-----------------------
An interesting feature of these flares is the gradual increase in the radio flux over a long period of time. In the bottom panel of our Figure 1, the light curves at 15 GHz from OVRO observations and at 230 GHz from SMA observations are shown. DCF estimated between these two light curves does not show any clear peak or lag in Figure 2. Even at the end of the high state when the gamma-ray flux reached the quiescent state Q3 the radio flux continued to increase. The OVRO flux reached the maximum level in October, 2016 and subsequently, decreased slowly.
@Ahnen [@et; @al.; @(2017)] also reported gradual increase in radio flux in the second half of 2015. They have shown the light curve of the radio core at 43 GHz. A bright and slow radio knot K15 was ejected on MJD 57230$\pm$52. They associated the increase in radio flux with the ejection of the radio knot K15. Due to the large uncertainty in the ejection time of K15 it could not be associated with any particular GeV flare.
A similar feature was also observed with the gamma-ray high state in Feb-April 2012 when a radio knot K12 emerged from the core (@Aleksic [@et; @al.; @(2014)]). In the second half of 2011 PKS 1510-089 had a major outburst in radio flux. The outburst first peaked at higher frequency. The peak at 37 GHz was reached around 21 Oct, 2011 and later at 15 GHz around 15 Dec, 2011. After attaining the peaks the light curves gradually decayed. Small outbursts continued to happen after this. VLBA 43 GHz images show a new component (knot K11) in December 2011. This was also observed at 15 GHz in MOJAVE as reported by @Orienti [@et; @al.; @(2013)]. The temporal evolution of gamma-ray and radio flux suggests they are produced by different populations of electrons, located at different regions along the length of the jet.
Conclusions
===========
In this work, the high state of PKS 1510-089 in 2015 has been studied using the gamma-ray data from Fermi-LAT, Swift XRT/UVOT and radio data from OVRO and SMA observatory. Four flares are identified as A (MJD 57100 to MJD 57150), B (MJD 57150 to MJD 57180), C (MJD 57208 to MJD 57235) and D (MJD 57235 to MJD 57260) between quiescent states Q1 and Q3. Between flare B and C a quiescent state Q2 (MJD 57180 to MJD 57208) has also been identified. The epochs of MAGIC observations of flares in 2015 are within the duration of our flare B. We have also included MAGIC data for this flare from @Ahnen [@et; @al.; @(2017)]. We have inferred about the locations of the emission regions in different wavelengths from the flux doubling time scales. It is found that the source is less variable in X-rays and the flaring states cannot be clearly identified. In our work the optical and gamma-ray emission is assumed to be co-spatial. This region of emission is located within the edge of the BLR region and the X-ray emission could be from the DT region. The modeling has been done with the publicly available time dependent code GAMERA considering two emission zones. A log parabola distribution of injected electrons is propagated using the code GAMERA and subsequently, the synchrotron, EC and SSC emission has been calculated to fit the observed data. The parameter values used in our two zone model are displayed in Table 2. The data could be adequately fitted by adjusting the injected electron spectrum and the magnetic field. The jet power required in this scenario is below the Eddington’s luminosity of PKS 1510-089.\
**Acknowledgements :** The authors thanks the referee for fruitful comments to improve the paper. This work has made use of publicly available *Fermi*-LAT data from FSSC and XRT data analysis software (XRTDAS) developed by ASI science data center, Italy. Archival data from the SMA observatory and from the OVRO 40-m monitoring programme [@Richards; @et; @al.; @(2011)] have also been used in this research. N. Gupta acknowledges the hospitality of the Nicolaus Copernicus Astronomical Center. This work was partially supported by the Polish National Science Centre grant 2015/18/E/ST9/00580.
\[Table:TB\]
    
Abdo, A. A., Ackermann, M., et al. 2010, ApJ, 721, 1425 Acciari, V. A. et al., 2018, A & A, 619, 159 Abramowski, A., et al., 2013, A & A, 550, 4 Aleksi[ć]{}, J., Ansoldi, S., Antonelli, L. A. et al. 2014, A&A, 569, A46 Atwood, W. B., Abdo, A. A., et al., 2009, ApJ, 697, 1071 Acero, F., Ackermann, M., et al., 2015, ApJS, 218, 23 Ahnen, M. L., Ansoldi, S., Antonelli, L. A. et al. 2017, A & A, 603, A29 Barnacka, A., Moderski, R., Behera, B., Brun, P., Wagner, S., 2014, 567, A113 Basumallick, P. P. & Gupta, N., 2016, APh, 88, 1 Blumenthal, R. & Gould, G., 1970, Rev. Mod. Phys., 42, 237 Bonning, E, W., Bailyn, C., Urry, C. M., et al. 2009, ApJL, 697, L81 Böttcher M., Reimer A., Sweeney K. & Prakash A. 2013, ApJ, 768, 54 Breeveld, A. A., Landsman, W., et al., 2011, AIPC, 1358, 373-376 Brown, A., 2013, MNRAS, 431, 824-835 Castignani, G., Guetta, D., Pian, E., et al. 2014, A&A, 565, A60 Castignani, G., Pian, E., Belloni, T M., et al. 2017, A&A, 601, A30 Celotti, A., Padovani, P., & Ghisellini, G. 1997, MNRAS, 286, 415 Dotson, A., Georganopoulos, M., et al., 2015, ApJ, 809, 164 Edelson, R. A., & Krolik, J. H., 1988, ApJ, 333, 646 Finke, J. D., 2016, ApJ, 830, 94 Fossati,G., Celotti, A., Chiaberge, M., et al. 2000, ApJ, 541, 153-165 Fuhrmann, L., Larsson, S., et al. 2014, MNRAS, 441, 1899 Ghisellini, G., & Tavecchio, F., 2009, MNRAS, 397,985 Gurwell, M. A., Peck, A. B., Hostler, S. R., Darrah, M. R., & Katz, C. A. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 375,From Z-Machines to ALMA: (Sub)Millimeter Spectroscopy of Galaxies, ed. A. J. Baker, J. Glenn, A. I. Harris, J. G. Mangum, & M. S. Yun, 234 Janiak, M., Sikora, M., et al., 2012, ApJ, 760, 129 Jorstad, S. G., Marscher, A. P., Lister, M. L., et al. 2005, AJ, 130, 1418 Kalberla, P. M. W., Burton, W. B., et al., 2005, A & A, 440, 775-782 Kataoka, J., et al., 2008, ApJ, 672, 787-799 Kaur, N., Baliyan, K. S., 2018, A & A, 617, 59 Kushwaha, P., Gupta, A. C., Misra, R., Singh, K. P., 2017, MNRAS, 464, 2046 Larionov, V. M., Villata, M., et al., 2016, MNRAS, 461, 3047-3056 Liodakis, I., Marchili, N., et al. 2017, MNRAS, 466, 4625-4632 Marscher, A. P., Jorstad, S., Larionov, V., et al. 2010, ApJ, 710, L126 Massaro, E. Perri, M., Giomi, P. & Nesci, R., 2004, A & A, 413, 489 Nalewajko, K., Sikora, M., Madejeski, G., et al. 2012, ApJ, 760, 69 Nalewajko, K., 2013, MNRAS, 430, 1324-1333 Orienti, M. D., Ammando, F., Giroletti, M., et al. 2013, MNRAS, 428, 241 Paliya, V. S., 2015, ApJL, 808, L48 Peterson B. M., 2006 Alloin D. Lecture Notes in Physics, Vol. 693, Physics of Active Galactic Nuclei at all Scales Springer Berlin 77 Prince, R., Majumdar P., Gupta N., 2017, ApJ, 844, 62 Prince, R., Raman, G., et al. 2018, ApJ, 866, 16 Prince, R., 2019, ApJ, 871, 101 Rajput, B., Stalin, C. S. et al., 2019, MNRAS, 486, 1781 Richards, J, L., Max-Moerbeck, W., et al. 2011, ApJS, 194, 29 Roming, P. W. A., Kennedy, T. E., et al., 2005, ssr, 120, 95-142 Schlafly, E. F., Finkbeiner, D. P., 2011, ApJ, 737, 103 Saito, S., Stawarz, L., Tanaka, Y. T., et al., 2013, ApJL, 766, L11 Saito, S., Stawarz, L., et al., 2015, ApJ, 809, 171 Tanner, A. M., Bechtold, J., Walker, C. E., Black, J. H., & Cutri, R. M., 1996, AJ, 112, 62 Tavecchio, F., Ghisellini, G., et al., 2010, MNRAS, 405, L94-L98 Vaughan, S., Edelson, R., Warwick, R. S., & Uttley, P., 2003, MNRAS, 345, 1271 Weaver, Z. R., Balonek, T. J., et al., 2019, ApJ, 875, 15 Zacharias, M., Böttcher, M., Chakraborty, N., et al. 2016, arXiv:1611.02098 Zhang, Y. H., Celotti, A., Treves, A., Chiappetti, L., et al. 1999, ApJ, 527, 719
[^1]: https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl
[^2]: http://www.astro.caltech.edu/ovroblazars/index.php?page=sourcelist
[^3]: http://sma1.sma.hawaii.edu/callist/callist.html
[^4]: http://libgamera.github.io/GAMERA/docs/main$\_$page.html
|
---
address:
- |
Australian Catholic University\
25A Barker Rd\
Strathfield NSW 2135\
Australia
- |
School of Mathematics and Statistics\
University of Sydney\
NSW 2006\
Australia
- |
Département de Mathématiques\
ULB C.P. 216\
Bd. du Triomphe\
1050 Bruxelles\
Belgium
author:
- 'W. N. Franzsen'
- 'R. B. Howlett'
- 'B. Mühlherr'
title: Reflections in abstract Coxeter groups
---
Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite sugroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.
Introduction
============
The dihedral group of order 12 can be considered as Coxeter group of type $I_2(6)$ or as Coxeter group of type $A_1 \times I_2(3)$. This example shows that, in general, the set of reflections in a Coxeter system is not determined by the abstract group $W$ alone, but does indeed depend on the choice of the Coxeter generating set $R$. However there are a lot of examples of Coxeter systems $(W,R)$ where the abstract group does determine the set of reflections or even the set $R$ up to $W$-conjugacy. The main motivation for the present paper is to show that the latter holds for infinite Coxeter groups having a finite, irreducible and 2-spherical Coxeter generating set, which is our Theorem \[thm2sph\] below.
In view of the main result of [@CM] it suffices to show that these Coxeter groups determine the set of reflections. In order to achieve this goal we provide a handy criterion for an involution in an abstract Coxeter group $W$ to be a reflection with respect to any Coxeter generating set of $W$. Our principal observation is the following. Let $(W,R)$ be a Coxeter system and let $w \in W$ be an involution. If $w \not \in R^W$, then the centralizer of $w$ in $W$ contains a finite normal subgroup properly containing $\langle w \rangle$. This is an immediate consequence of Richardson’s result in [@RR]. Thus, if $w \in W$ is an involution having the property that $\langle w \rangle$ is a maximal finite normal subgroup of its centralizer in $W$, then $w$ is a reflection with respect to any Coxeter generating set of $W$.
It turns out that it is more convenient to work with the *finite continuation* of an involution rather than to consider finite normal subgroups of its centralizer. The finite continuation of a finite order element $w$ in a Coxeter group is defined to be the intersection of all maximal finite subgroups containing it; we write $\operatorname{FC}(w)$ for the finite continuation of $w$. In this paper we restrict our attention to finitely generated Coxeter groups. For these it is a consequence of a result of Tits that every element of finite order is contained in some maximal finite subgroup; so $\operatorname{FC}(w)$ is a finite subgroup of $W$ (see Corollary \[FCexis\] below). The main result of the present paper is a complete description of the finite continuation of a simple reflection in a Coxeter system of finite rank. Its proof constitutes the bulk of this paper.
[**Main Result:**]{}
*Let $(W,R)$ be a Coxeter system of finite rank. Then the following holds.*
- For each $r \in R$ the finite continuation of $r$ can be described.
- Given an involution $w \in W$ such that $\operatorname{FC}(w) = \langle w \rangle$, then $w \in R^W$.
Part a) of our main result is Theorem \[main\]. Its precise statement requires some preparation. Part b) is Corollary \[FCcrit1\].
The main result of this paper is in fact the first of two steps to reduce the isomorphism problem for Coxeter groups to its ‘reflection-preserving’ version. The second step is given in [@HM]. We refer to [@toro] for further information about the applications to the general isomorphism problem.
A special instance of the isomorphism problem for Coxeter groups is the question about their rigidity (see [@BMMN] for further information). In combination with the main result of [@CM] a consequence of our main result is the following rigidity result.
\[thm2sph\] Let $(W,R)$ be an irreducible, non-spherical Coxeter system such that $R$ is finite and such that $rr'$ has finite order for all $r,r' \in R$. Then the following assertions hold.
- For each $r \in R$ we have $\operatorname{FC}(r) = \langle r \rangle$.
- If $S \subseteq W$ is such that $(W,S)$ is a Coxeter system, then there exists $w \in W$ such that $S^w = R$.
- All automorphisms of $W$ are inner-by-graph.
In the language of [@BMMN], Part b) of the previous theorem means that an infinite, irreducible, 2-spherical Coxeter system is strongly rigid. Part c), which is an immediate consequence of Part b), improves the result of [@HRT].
To conclude this introduction we remark that characterizations results for reflections in even Coxeter groups have been obtained in [@BM]. Some of the results there can be deduced as corollaries of our main result as well.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The authors thank Frédéric Haglund for a helpful discussion on the subject.
Precise Statement of the Main Result
====================================
Recall that a Coxeter group is a group with a presentation of the form $$\label{eq:pres}
W=\operatorname{gp}\langle\,\{\,r_a\mid a\in\Pi\,\}\mid(r_ar_b)^{m_{ab}}=1
\text{ for all }a,b\in\Pi\,\rangle$$ where $\Pi$ is some indexing set, whose cardinality is called the *rank* of $W$ (relative to this presentation), and the $m_{ab}$ satisfy the following conditions: $m_{ab}=m_{ba}$, each $m_{ab}$ lies in the set $\{\,m\in\mathbb{Z}\mid m\ge 1\,\}\cup\{\infty\}$, and $m_{ab}=1$ if and only if $a=b$. When $m_{ab}=\infty$ the relation $(r_ar_b)^{m_{ab}}=1$ is interpreted as vacuous. We shall restrict attention to Coxeter groups of finite rank.
A *reduced expression* for an element $w\in W$ is a minimal length word expressing $w$ as a product of elements of the distinguished generating set $\{\,r_a\mid a\in\Pi\,\}$. We define $\ell(w)$ to be the length of a reduced expression for $w$.
As is well known (and as we shall describe in Section 3 below), every Coxeter group W can be realized geometrically as a group generated by reflections. In this realization of $W$ the reflections in $W$ are the conjugates of the generators $r_a$.
The *Coxeter graph* associated with the presentation above is the graph with vertex set $\Pi$ and edge set consisting of those pairs of vertices $\{a,b\}$ for which $m_{ab}\ge 3$. The edge $\{a,b\}$ is given the label $m_{ab}$. The *components* of $\Pi$ are the connected components of the graph, and we say that $W$ is *irreducible* if the graph is connected.
For each $I\subseteq\Pi$ we define $W_I$ to be the subgroup of $W$ generated by the set $\{\,r_a\mid a\in I\,\}$; we call these subgroups the *visible* subgroups of $W$. A *parabolic subgroup* of $W$ is any conjugate of a visible subgroup. We say that $I\subseteq\Pi$ is *spherical* if $W_I$ is finite, and we say that $\Pi$ (or $W$) is *$k$-spherical* if all $k$-element subsets of $\Pi$ are spherical.
The defitions given so far are fairly standard. In order to facilitate the precise statement of the main result, we introduce some nonstandard notation and terminology (in Definitions \[FC\], \[EO\], \[c3neighbour\], \[focus\] and \[halffocus\] below).
\[FC\] [If $w\in W$ has finite order, define the *finite continuation* of $w$, written $\operatorname{FC}(w)$, to be the intersection of all maximal finite subgroups of $W$ containing $w$.]{}
\[EO\] [The *odd graph* of $W$ is the graph $\operatorname{\Omega}(\Pi)$ obtained from the Coxeter graph by deleting the edges whose labels are infinite or even. For each $a\in\Pi$ we define $\operatorname{Odd}(a)$ to be the connected component of $\operatorname{\Omega}(\Pi)$ containing $a$. For each connected component $M$ of $\operatorname{\Omega}(\Pi)$ we define $\operatorname{E}(M)$ to be the union of $M$ with the set of all $b\in\Pi$ such that $m_{cb}$ is even for some $c\in M$. We also abbreviate $\operatorname{E}(\operatorname{Odd}(a))$ to $\operatorname{EOdd}(a)$.]{}
In the discussions below, when we refer to the components of $\operatorname{E}(M)$ we regard $\operatorname{E}(M)$ as the full subgraph of the Coxeter graph spanned by the vertices in $\operatorname{E}(M)$. In other words, the edges with even and infinite labels, deleted when forming the odd graph, are restored in $\operatorname{E}(M)$.
Note that if $a\in L\subseteq\Pi$ and $W_L$ is finite then $m_{ab}<\infty$ for all $b\in L$. Whether $m_{ab}$ is odd or even it follows that $b\in\operatorname{EOdd}(a)$. Thus $L\subseteq\operatorname{EOdd}(a)$.
\[c3neighbour\]
Let $M\subseteq\Pi$ be a connected component of $\operatorname{\Omega}(\Pi)$. We call $b\in\Pi\setminus M$ a $C_3$-*neighbour* of $M$ if $m_{bc}\in\{2,4\}$ for all $c\in E(M)$, the case $m_{bc}=4$ occurring for at least one $c$, and for each $c\in E(M)$ with $m_{bc}=4$ there is an $a\in M$ such that the following conditions are satisfied:
1. $m_{ba}=2$ and $m_{ca}=3$, and $m_{cd}=\infty$ for all $d\in M\setminus\{a,c\}$;
2. for all $e\in\Pi\setminus(M\cup\{b\})$, either $m_{ce}=\infty$ or $m_{ae}=m_{ce}=m_{be}=2$.
\[focus\]
Let $M\subseteq\Pi$ be a connected component of $\operatorname{\Omega}(\Pi)$, and let $a\in M$ and $b\in\Pi\setminus M$. We call the pair $(a,b)$ a *focus* of $M$ in $\Pi$ if the following conditions all hold.
1. All the edge labels of $M$ are 3, and $M$ is a tree.
2. For each $c\in M$, the set $C[b..c]\subseteq\Pi$ consisting of $b$ and those elements of $M$ that form the path from $a$ to $c$ in $M$ constitutes a system of type $C_k$ (for some $k$ dependent on $c$).
3. If $c,\,d\in M\cup\{b\}$ with $c\notin C[b..d]$ and $d\notin C[b..c]$ then $m_{cd}=\infty$.
4. If $m_{ce}\ne\infty$ for some $c\in M$ and $e\in\Pi\setminus(M\cup\{b\})$, then $m_{ce}=2=m_{de}$ for all $d\in C[b..c]$.
5. The vertices of $M\cup\{b\}$ do not form a spherical component of $\operatorname{E}(M)$.
\[halffocus\]
Let $M\subseteq\Pi$ be a connected component of $\operatorname{\Omega}(\Pi)$, and let $a,\,b\in M$. We call the two-element set $\{a,b\}$ a *half focus* of $M$ in $\Pi$ if $m_{ab}=2$ and the following conditions all hold.
1. We have $m_{ac}=m_{bc}\in\{2,3\}$ for all $c\in M\setminus\{a,b\}$, and $m_{ac}=m_{bc}\in\{2,\infty\}$ for all $c\in\Pi\setminus M$.
2. All the edge labels of $M\setminus\{b\}$ are 3, and $M\setminus\{b\}$ is a tree.
3. For each $c\in M\setminus\{a,b\}$, the set $D[a,b..c]\subseteq\Pi$ consisting of $b$ and those and those elements of $M\setminus\{b\}$ that form the path from $a$ to $c$ constitutes a system of type $D_k$ (for some $k$ dependent on $c$).
4. If $c,\,d\in M\setminus\{a,b\}$ with $c\notin D[a,b..d]$ and $d\notin D[a,b..c]$ then $m_{cd}=\infty$.
5. If $m_{ce}\ne\infty$ for some $c\in M\setminus\{a,b\}$ and $e\in\Pi\setminus M$, then $m_{ce}=2=m_{de}$ for all $d\in D[a,b..c]$.
6. The vertices of $M$ do not form a spherical component of $\operatorname{E}(M)$.
We are now able to give a precise statement of Part a) of our main result.
\[main\] For each connected component $M$ of $\operatorname{\Omega}(\Pi)$ there is at least one $a\in M$ such that $\operatorname{FC}(r_a)$ is a visible subgroup of $W$. We have the following possibilities.
[Case A:]{} Suppose that the component of $\operatorname{E}(M)$ containing $M$ is spherical, and let $a\in M$ be arbitrary. Then $\operatorname{FC}(r_a)=W_J$, where $J$ is the union of the spherical components of $\operatorname{E}(M)$.
[Case B:]{} Suppose that the component of $\operatorname{E}(M)$ containing $M$ is not spherical, and $M$ does not have any focus or half-focus in $\Pi$, and let $J'$ be the union of the spherical components of $\operatorname{E}(M)$ and the set of $C_3$-neighbours of $M$. If $a\in M$ is not adjacent in $\Pi$ to any $C_3$-neighbour of $M$ then $\operatorname{FC}(r_a)=W_{J'\cup\{a\}}$, and if $a\in M$ is adjacent in $\Pi$ to a $C_3$-neighbour of $M$ then $\operatorname{FC}(r_a)$ is not visible.
[Case C:]{} Suppose that $(a,b)$ is a focus of $M$. Then $\operatorname{FC}(r_a)=W_J$ where $J$ is the union of $\{a,b\}$ and the spherical components of $\operatorname{E}(M)$, and $\operatorname{FC}(r_c)$ is not visible for any $c\in M\setminus\{a\}$.
[Case D:]{} Suppose that $\{a,b\}$ is a half-focus of $M$. Then $\operatorname{FC}(r_a)=\operatorname{FC}(r_b)=W_J$, where $J$ is the union of $\{a,b\}$ and the spherical components of $\operatorname{E}(M)$, and $\operatorname{FC}(r_c)$ is not visible for any $c\in M\setminus\{a,b\}$.
Reflections and root systems
============================
Let ${\mathbb{R}}$ be the real field, and $V$ the vector space over ${\mathbb{R}}$ with basis $\Pi$. Let $B$ the bilinear form on $V$ such that for all $a,b\in\Pi$, $$B(a,b)=-\cos(\pi/m_{ab}).$$ To make our notation more compact we define $u\cdot v=B(u,v)$ for all $u,v\in V$. Note that $a\cdot a=1$ for all $a\in\Pi$, since $m_{aa}=1$.
For each $a\in V$ such that $a\cdot a=1$, the *reflection along $a$* is the transformation of $V$ given by $v\mapsto v-2(a\cdot v)a$. It is well known (see, for example, Corollary 5.4 of [@JH]) that $W$ has a faithful representation on $V$ such that, for all $a\in\Pi$, the element $r_a$ acts as the reflection along $a$. We shall identify elements of $W$ with their images in this representation. We also use the notation $r_a$ for the reflection along $a$ whenever $a\in V$ satisfies $a\cdot a=1$. It is straightforward to show that each reflection $r_a$ preserves the form $B$; hence all elements of $W$ preserve $B$. Furthermore, the equation $gr_ag^{-1}=r_{ga}$ holds for all $a\in V$ such that $a\cdot a=1$ and all transformations $g$ that preserve $B$.
We write $\operatorname{Ref}(W)$ for the set of all reflections in $W$. It is immediate from the above comments that if $\Phi=\{\,w a\mid w\in W,\ a\in \Pi\,\}$ then $\{\,r_b\mid b\in\Phi\,\}\subseteq\operatorname{Ref}(W)$.
The set $\Phi$ is called the *root system* of $W$, and elements of $\Phi$ are called *roots*. Elements of the basis $\Pi$ are called *simple roots*, and the reflections $r_a$ for $a\in \Pi$ are called *simple reflections*. A root is said to be *positive* if it has the form $\sum_{a\in\Pi}\lambda_a a$ with $\lambda_a\ge 0$ for all $a\in\Pi$, and *negative* otherwise. We write $\Phi^+$ for the set of all positive roots and $\Phi^-$ for the set of all negative roots.
\[somebasicfacts\] With the notation as above, the following statements hold.
1. Every negative root has the form $\sum_{a\in \Pi}\lambda_a a$ with $\lambda_a\le 0$ for all $a\in\Pi$. Furthermore, $\Phi^-=\{\,-b\mid b\in\Phi^+\,\}$.
2. If $w\in W$ and $a\in\Pi$ then $$\ell(wr_a)=
\begin{cases}
\ell(w)+1&\text{if $wa\in\Phi^+$,}\\
\ell(w)-1&\text{if $wa\in\Phi^-$.}
\end{cases}$$
3. If $t\in\operatorname{Ref}(W)$ then $t=r_b$ for some $b\in \Phi$.
4. The group $W$ is finite if and only if the bilinear form $B$ is positive definite.
5. The root system $\Phi$ is finite if and only if the group $W$ is finite.
Proofs of (1) and (2) can be found in [@JH Section 5.4], Theorem 4.1 in [@VD] includes both (4) and (5), and (3) is [@HRT Lemma 2.2].
The following result is well known.
\[class\] Let $a\in\Pi$. Then $\operatorname{Odd}(a)=\Pi\cap Wa$.
For each $w\in W$ we define $N(w)=\{\,b\in\Phi^+\mid wb\in\Phi^-\,\}$. By Part (2) of Lemma \[somebasicfacts\], if $w\ne 1$ then $N(w)\cap\Pi\ne\emptyset$. An easy induction shows that $N(w)$ has exactly $\ell(w)$ elements. In particular, $N(w)$ is a finite set. It is also easily shown that if $\Phi$ is finite then there is a unique $w\in W$ such that $N(w)=\Phi^+$. This element, which we denote by $w_\Pi$, is also the unique element of maximal length in $W$ (which is a finite group). Furthermore, $w_\Pi\Pi=-\Pi$.
For each $\Gamma\subseteq\Phi$ the subgroup $W_\Gamma$ generated by the set $\{\,r_a\mid a\in\Gamma\,\}$ is called a *reflection subgroup* of $W$. The set $\Phi_\Gamma=\{\,a\in\Phi\mid r_a\in W_\Gamma\,\}$ is called the *root subsystem* generated by $\Gamma$. Let $\Phi_\Gamma^+=\Phi_\Gamma\cap\Phi^+$ and $\Phi_\Gamma^-=\Phi_\Gamma\cap\Phi^-$, and define $$\Pi_\Gamma=\{\,a\in\Phi_\Gamma^+\mid N(r_a)\cap\Phi_\Gamma=\{a\}\,\}.$$ The main theorem of Deodhar [@VD2] and Theorem (3.3) of Dyer [@MD] yield the following result.
\[base\] For each $\Gamma\subseteq\Phi$ the group $W_\Gamma$ is a Coxeter group on the generating set $\{\,r_a\mid a\in\Pi_\Gamma\,\}$. The set $\{\,a\cdot b\mid a,\,b\in\Pi_\Gamma\text{ and }a\ne b\,\}$ is a subset of $\mathscr C=\{\,-\cos(\pi/m)\mid 2\le m\in{\mathbb{Z}}\,\}\cup(-\infty,-1]$. Moreover, if $\Delta$ is any subset of $\Phi^+$ such that $\{\,a\cdot b\mid a,\,b\in\Delta\text{ and }a\ne b\,\}\subseteq \mathscr{C}$ then $W_\Delta$ is a Coxeter group on the generating set $\{\,r_a\mid a\in\Delta\,\}$.
Note that the notation $W_\Gamma$ introduced above is an extension of the notation for visible subgroups introduced in Section 2. However, if $\Gamma\nsubseteq\Pi$ then $W_\Gamma$ need not be visible.
It is clear that if $I\subseteq\Pi$ then $W_I$ preserves the subspace $V_I$ of $V$ spanned by $I$, and acts on this subspace as a Coxeter group with $I$ as its set of simple roots. In this case $\Phi_I=\Phi\cap V_I$ and $\Pi_I=I$.
The following simple facts are well known.
\[subrootsys\] In the above situation, $\Phi_I=\Phi\cap V_I$. Furthermore, $w\in W$ normalizes $W_I$ if and only if $w\Phi_I=\Phi_I$. In particular, for all $a\in\Phi$, the reflection $r_a$ normalizes $W_I$ if and only if $a\in\Phi_I$ or $a\cdot b=0$ for all $b\in I$.
Suppose that $I\subseteq\Pi$ and $a\in\Pi\setminus I$, and let $L$ be the component of (the Coxeter graph of) $I\cup\{a\}$ to which $a$ belongs. If $W_L$ is finite we define $v[a,I]=w_Lw_{L\setminus\{a\}}$. It is easily seen that $v[a,I]I\subseteq I\cup\{a\}$, and that $v[a,I]b=b$ for all $b\in I\setminus L$. In particular, $v[a,I]I\in\mathscr{I}=\{\,J\subseteq\Pi\mid J=wI
\text{ for some $w\in W$ }\}$. It was proved in [@H] (for finite Coxeter groups) and in [@VD] (in the general case) that every element $w\in W$ satisfying $wI\subseteq\Pi$ can be expressed as a product of elements of the form $v[a,I']$, with $I'\in\mathscr{I}$ and $a\in\Pi\setminus I'$. That is, $$\label{canonexp}
w=v[a_1,I_1]v[a_2,I_2]\cdots v[a_n,I_n]$$ for some $I_i,\,a_i$ such that (for each $i$) the component of $I_i\cup\{a_i\}$ containing $a_i$ corresponds to a finite visible subgroup, $v[a_i,I_i]I_i=I_{i-1}$ for $1<i\le n$, and $I_n=I$. Furthermore, the following result holds.
\[normalizers\] Let $I,\,J\subseteq\Pi$. Then $\{\,w\in W\mid wW_Iw^{-1}=W_J\,\}=N(J,I)W_I$, where $N(J,I)=\{\,w\in W\mid wI=J\,\}$. Furthermore, for each $w\in N(J,I)$ and each $a\in\Pi\cap N(w)$ there is an expression for $w$ of the form (\[canonexp\]) above, with $(a_n,I_n)=(a,I)$ and $\ell(w)=\sum_{i=1}^n\ell(v[a_i,I_i])$.
The following lemma, which appears in [@NB Exercise 2d, p. 130], is fundamental to all of our arguments.
\[lemtits\] If $W$ is a Coxeter group and $H\le W$ is finite, then $H$ is contained in a finite parabolic subgroup of $W$.
One immediate consequence of Lemma \[lemtits\] is that every maximal finite subgroup of a Coxeter group is parabolic. Another consequence of the previous lemma is that each finite subgroup of $W$ is contained in a maximal finite parabolic subgroup. (Remember that we always assume that $W$ is finitely generated.) Thus the set of maximal finite subgroups of $W$ containing a given finite order element of $W$ is not empty, and hence we have the following fact.
\[FCexis\] If $w \in W$ has finite order, then $\operatorname{FC}(w)$ is a well-defined finite subgroup of $W$.
\[kilmoyerlem\] Let $I,J\subseteq \Pi$. Then every $(W_I,W_J)$ double coset in $W$ contains a unique element of minimal length; moreover, if $d$ is the minimal length element of $W_I d W_J$ then $W_I\cap dW_Jd^{-1}=W_K$, where $K=I\cap d J$.
See [@RC Theorem 2.7.4].
\[intersectionofparasispara\] The intersection of a finite number of parabolic subgroups is a parabolic subgroup.
The following consequence of Lemmas \[lemtits\] and \[kilmoyerlem\] is proved in [@FH2 Lemma 11].
\[maxvismax\]If $J$ is a maximal spherical subset of $\Pi$ then $W_J$ is a maximal finite subgroup of $W$. Furthermore, $W_J$ is not conjugate to any other visible subgroup of $W$.
Another important tool in our analysis of automorphisms is the classification of involutions in Coxeter groups, due to Richardson [@RR].
\[invclass\] Suppose that $w\in W$ is an involution. Then there is a $t\in W$ and a spherical $I\subseteq \Pi$ such that $w=t^{-1}w_It$ with $\ell(w)=\ell(w_I)+2\ell(t)$, and $w_I$ is central in $W_I$.
See [@FH2 Proposition 5].
\[-1type\] [We say that $I\subseteq\Pi$ is *of ($-$1)-type* if $W_I$ is finite and $w_I$ is central in $W_I$.]{}
The reason for the terminology is that $I$ is of $(-1)$-type if and only if there is an element of $W_I$ that acts on $V_I$ as multiplication by $-1$.
We need the following lemma.
\[parabconj\] Suppose that $I,\,J\subset\Pi$ with $I$ of ($-$1)-type, and suppose that $t\in W$ has the property that $tw_It^{-1}\in W_J$. Then $tW_It^{-1}\subseteq W_J$.
Let $a\in I$. Then $w_I(a)=-a$, and so $(tw_It^{-1})(ta)=-ta$, whence it follows that either $ta$ or $-ta$ is in the set $N(tw_It^{-1})$. But $N(tw_It^{-1})\subseteq\Phi_J$; so $ta\in\Phi_J$, and therefore $tr_at^{-1}=r_{ta}\in W_J$. Since $W_I$ is generated by $\{\,r_a\mid a\in I\,\}$, the result follows.
In particular, it follows from Lemma \[parabconj\] that if $I,\,J$ are both of $(-1)$-type and $tw_It^{-1}=w_J$ then $tW_It^{-1}=W_J$. Conversely, suppose that $tW_It^{-1}=W_J$, so that in fact $dW_Id^{-1}=W_J$ for all $d$ in $W_JtW_I$ (which equals $tW_I$). Taking $d$ to be the shortest element in $tW_I$, Lemma \[kilmoyerlem\] yields that $dI=J$, and hence $x\mapsto dxd^{-1}$ is a length-preserving isomorphism $W_I\to W_J$; consequently $dw_Id^{-1}=w_J$. If $w_I,\,w_J$ are central in $W_I,\,W_J$ we deduce that $tw_It^{-1}=w_J$. So we have proved the following result.
\[WItoWJ\] Suppose that $I,\,J$ are subsets of $\Pi$ that are both of ($-$1)-type. Then $\{\,t\in W\mid tw_It^{-1}=w_J\,\}=
\{\,t\in W\mid tW_It^{-1}=W_J\,\}$.
\[Z(w)normal\] Let $I\subset\Pi$ be of ($-$1)-type. Then $W_I\subseteq \operatorname{FC}(w_I)$.
Let $F$ be a maximal finite subgroup of $W$ such that $w_I\in H$. By Lemma \[lemtits\] there exist $t\in W$ and $J\subseteq\Pi$ such that $tFt^{-1}=W_J$. By Lemma \[parabconj\] and the fact that $w_I\in H$ it follows that $tW_It^{-1}\subseteq W_J$. Hence $W_I\subseteq t^{-1}W_Jt=F$.
\[key\] Let $W,\,W'$ be Coxeter groups of finite rank and $\alpha\colon W\to W'$ an isomorphism. Let $\Pi$ be the set of simple roots corresponding to the distinguished generating set of $W$, and let $a\in\Pi$. If $r_a^\alpha$ is not a reflection in $W'$ then the intersection of all maximal finite subgroups of $W$ containing $r_a$ is a parabolic subgroup of order greater than 2.
Write $\Pi'$ for the set of simple roots of $W'$. Observe that Lemma \[lemtits\] and Corollary \[intersectionofparasispara\] trivially imply that $\operatorname{FC}(r_a)$ is a parabolic subgroup of $W$.
Since $r_a^\alpha$ is not a reflection it follows from Proposition \[invclass\] that $r_a^\alpha=tw_It^{-1}$ for some $t\in W'$ and $I\subseteq\Pi'$ of $(-1)$-type and of rank at least 2. Clearly $\operatorname{FC}(r_a)^\alpha=t\operatorname{FC}(w_I)t^{-1}$, and by Proposition \[Z(w)normal\] we know that $W_I\subseteq \operatorname{FC}(w_I)$. Hence $(tW_It^{-1})^{\alpha^{-1}}\subseteq \operatorname{FC}(r_a)$, so that $\operatorname{FC}(r_a)$ has order greater than 2, as required.
\[FCcrit1\] Let $w \in W$ be an involution such that $\operatorname{FC}(w) = \langle w \rangle$ and let $S \subseteq W$ be such that $(W,S)$ is a Coxeter system. Then $w \in S^W$.
The finite continuation of a reflection
=======================================
Let $r$ be a reflection in $W$. Replacing $r$ by $wrw^{-1}$ replaces $\operatorname{FC}(r)$ by $w\operatorname{FC}(r)w^{-1}$, and so choosing $w$ suitably enables us to assume that $\operatorname{FC}(r)=W_J$, a visible parabolic subgroup. Furthermore, replacing $r$ by $trt^{-1}$ for suitable $t\in W_J$ enables us to assume that $r=r_a$ for some $a\in J$. (Note that these observations yield the first assertion of Theorem \[main\].)
\[finclos\] Let $a\in J\subseteq\Pi$, and suppose that $W_J$ is the intersection of all maximal finite subgroups of $W$ containing $r_a$. Then $\{\,w\in W\mid wr_aw^{-1}\in W_J\,\}$ is a subset of the normalizer of $W_J$ in $W$. Thus each $W\!$-conjugate of $r_a$ in $W_J$ is $N_W(W_J)$-conjugate to $r_a$, and $C_W(r_a)\subseteq N_W(W_J)$. Moreover, if $b\in\Pi\setminus J$ is such that $W_{J\cup\{b\}}$ is infinite then $m_{bc}=\infty$ for all $c\in J$ such that $r_c$ is conjugate to $r_a$ in $W$.
Let $\mathscr{S}$ be the set of all maximal finite subgroups of $W$ containing $r_a$, so that $W_J=\operatorname{FC}(r_a)=\bigcap_{F\in\mathscr{S}}F$. Suppose that $w\in W$ satisfies $wr_aw^{-1}\in W_J$, and let $F\in\mathscr{S}$. Then $wr_aw^{-1}\in W_J\subseteq F$, and so $r_a\in w^{-1}Fw$. Thus $w^{-1}Fw$ is a maximal finite subgroup of $W$ containing $r_a$, whence $w^{-1}Fw\in\mathscr{S}$. So $$\bigcap_{F\in\mathscr{S}}F\subseteq\bigcap_{F\in\mathscr{S}}w^{-1}Fw$$ and so $W_J\subseteq w^{-1}W_Jw$. Since $W_J$ is finite it follows that $w\in N_W(W_J)$.
Suppose that $c\in J$ with $r_c=wr_aw^{-1}$ for some $w\in W$. Clearly $F\mapsto wFw^{-1}$ is a bijection from the set of maximal finite subgroups of $W$ containing $r_a$ to the set of maximal finite subgroups of $W$ containing $r_c$, and so $\operatorname{FC}(r_c)=w\operatorname{FC}(r_a)w^{-1}$. But $w\operatorname{FC}(r_a)w^{-1}=wW_Jw^{-1}=W_J$ by the first part of the proof, and so $\operatorname{FC}(r_c)=W_J$. Now suppose that $b\in\Pi\setminus J$ with $m_{cb}<\infty$. Then $W_{\{c,b\}}$ is finite, and so contained in a maximal finite subgroup $F$. Since $r_c\in F$ we must have $\operatorname{FC}(r_c)\subseteq F$. It follows that the finite group $F$ contains both $W_J$ and $r_b$, and therefore $W_{J\cup\{b\}}$ is finite.
Assume, as in Proposition \[finclos\], that $a\in J\subseteq\Pi$ and $W_J=\operatorname{FC}(r_a)$, and suppose now that $J\ne\{a\}$. Suppose that $L\subseteq\Pi$ is such that $J\subseteq L$ and $W_L$ is finite. Then $W_L$ is a finite Coxeter group possessing a visible parabolic subgroup $W_J$ of rank greater than 1 that is normalized by the centralizer of some simple reflection $r_a\in W_J$. Indeed, $W_J$ is normalized by all $w\in W_L$ such that $wr_aw^{-1}\in W_J$. Equivalently, by Lemma \[somebasicfacts\] (3), $\{\,w\in W_L\mid wa\in\Phi_J\,\}\subseteq N_W(W_J)$. This is a very restrictive condition, which we now proceed to examine with a case-by-case investigation of the different types of finite Coxeter groups. For the course of this investigation, we can (and shall) assume that $L=\Pi$.
So we assume for now that $W$ is a finite Coxeter group of rank $n$, and our aim is to find all examples of the following phenomenon: there exist $\{a\}\subsetneqq J\subseteq\Pi$ such that the set $Q=\{\,w\in W\mid wa\in\Phi_J\,\}$ is a subset of $N_W(W_J)$. We assume that $J\ne\Pi$, since the condition is trivially satisfied otherwise.
If $K\subseteq\Pi$ is a component of the Coxeter graph such that $J\cap K=\emptyset$ then $W_K$ is a direct factor of $N_W(W_J)$; moreover, $Q=(Q\cap W_{\Pi\setminus K})W_K$. So removing $K$ from the graph will have no bearing on whether or not the condition $Q\subseteq N_W(W_J)$ holds. So we assume that there are no such components of $\Pi$. Exactly the same comments apply for a component $K$ of $\Pi$ such that $K\subseteq J$. So we also assume that there are none of these.
Assume that $\{a\}\subsetneqq J\subsetneqq\Pi$ and $Q\subseteq N_W(W_J)$. Suppose that $K\subseteq\Pi$ is a component of the Coxeter graph such that $a\notin K$. Then $r_ba=a$ for all $b\in K$; so $r_b\in Q\subseteq N_W(W_J)$, and it follows that $r_bc\in\Phi_J$ whenever $c\in J$. If $b\cdot c\ne 0$ then $b$ is in the support of $r_bc$, and so $r_bc\in\Phi_J$ implies $b\in J$. Since $K$ is connected it follows that if $K$ contains any element of $J$ then $K\subseteq J$. So either $K\cap J=\emptyset$ or $K\subseteq J$. But we have assumed that there are no such components. So the component of $\Pi$ that contains $a$ is the only component; that is, $\Pi$ is irreducible.
Observe that the group $\operatorname{Stab}(a)=\{\,w\in W\mid wa=a\,\}$ is a subset of $Q$ and hence of $N_W(W_J)$. Note also that $N_W(W_J)=\{\,w\in W\mid w\Phi_J=\Phi_J\,\}$, which is also the stabilizer of the subspace $V_J$ (since $V_J$ is the subspace spanned by $\Phi_J$ and $\Phi_J=V_J\cap\Phi$). Now $\operatorname{Stab}(a)$ is a parabolic subgroup of $W$ whose root system is $\Phi\cap a^\perp$, and the following table gives the type of this root system in all cases. $$\vbox{\offinterlineskip
\halign{\quad$\hfil#\hfil$\quad&\vrule#&\quad$\hfil#\hfil$\cr
W& depth 4.5 pt&\operatorname{Stab}(a)\cr
\noalign{\hrule}
A_n&height 9.5 pt depth 3.5 pt&A_{n-2}\cr
C_n&height 8.5 pt depth 3.5 pt&C_{n-2}+A_1\cr
C_n&height 8.5 pt depth 3.5 pt&C_{n-1}\cr
D_n&height 8.5 pt depth 3.5 pt&D_{n-2}+A_1\cr
F_4&height 8.5 pt depth 3.5 pt&C_3\cr
E_6&height 8.5 pt depth 3.5 pt&A_5\cr}}
\qquad\qquad\qquad
\vbox{\offinterlineskip
\halign{$\hfil#\hfil$\quad&\vrule#&\quad$\hfil#\hfil$\cr
W& depth 4.5 pt&\operatorname{Stab}(a)\cr
\noalign{\hrule}
E_7&height 8.5 pt depth 3.5 pt&D_6\cr
E_8&height 8.5 pt depth 3.5 pt&E_7\cr
H_3&height 8.5 pt depth 3.5 pt&A_1+A_1\cr
H_4&height 8.5 pt depth 3.5 pt&H_3\cr
I_2(2k)&height 8.5 pt depth 3.5 pt&A_1\cr
I_2(2k+1)&height 8.5 pt depth 3.5 pt&\emptyset\cr}}$$ (For $C_n$ there are two $W$ orbits of roots, giving two possibilities for $\operatorname{Stab}(a)$. For $F_4$ and $I_2(2k)$ there are also two $W$-orbits of roots, but $\operatorname{Stab}(a)$ has the same type of root system whichever orbit $a$ belongs to.) Since each irreducible constituent of its root system spans an irreducible $\operatorname{Stab}(a)$-submodule of $V$, the table shows that as a $\operatorname{Stab}(a)$-module, $V$ has composition length two or three or (in one case only) four: $a$ itself spans a trivial $\operatorname{Stab}(a)$ submodule of dimension 1, and $a^\perp$ is either irreducible of dimension $n-1$ (for types $F_4$, $E_6$, $E_7$, $E_8$, $H_4$, $I_2(2k)$ and one of the $C_n$ possibilities), or the direct sum of irreducibles of dimensions 1 and $n-2$ (for types $A_n$, $C_n$, $D_n$ when $n>4$, $H_3$ and $I_2(2k+1)$), or the direct sum of three irreducibles of dimension 1 (for type $D_4$). Furthermore, the summands of $a^\perp$ are pairwise nonisomorphic as $\operatorname{Stab}(a)$ modules, since even if they are of the same type their centralizers in $\operatorname{Stab}(a)$ are different.
Since $\{a\}\subsetneqq J\subsetneqq\Pi$ and $V_J$ is $\operatorname{Stab}(a)$-invariant, we see that $a^\perp=(V_J\cap a^\perp)\oplus V_J^\perp$, with both summands nonzero $\operatorname{Stab}(a)$-modules. So $\Pi$ is of type $A_n$, $C_n$, $D_n$ or $H_3$. Furthermore, except in type $D_4$, the two direct summands of $a^\perp$ are irreducible and not isomorphic, and are therefore the only proper $\operatorname{Stab}(a)$-submodules of $a^\perp$. We conclude that $V_J$ is spanned by $a$ and one of the summands of $a^\perp$, while $V_J^\perp$ is the other summand. In type $D_4$ we similarly deduce that $V_J$ is spanned by $a$ and one or two of the three 1-dimensional summands of $a^\perp$, and, correspondingly, $V_J^\perp$ is of either of type $A_1+A_1$ or of type $A_1$.
If $\Pi$ is of type $A_n$ then one of the summands of $a^\perp$ is of type $A_{n-2}$ while the other is a trivial 1-dimensional $\operatorname{Stab}(a)$-module. If $V_J^\perp$ is of type $A_{n-2}$ then $V_J$ must be of type $A_1$, since the orthogonal complement of a subsystem of type $A_{n-2}$ in $A_n$ contains only a rank 1 root system. This contradicts the assumption that $\{a\}\subsetneqq J$. So $J$ is of type $A_1+A_{n-2}$. Since $W_J$ is visible, we deduce that $a$ is an end node of the $A_n$ diagram, and the node adjacent to $a$ is the unique simple root not in $J$. However, if $n>3$ then the maximal length element of $W$ is in $Q$ but not in the normalizer of $W_J$. So $n=3$ and $J=\{a,c\}$, where $c$ is the other end node. It is readily checked that $Q$ has 8 elements and coincides with $N_W(W_J)$ (which is generated by $W_J$ and an element that interchanges $a$ and $c$).
If $\Pi$ is of type $C_n$ then one summand of $a^\perp$ is of type $C_{n-2}$ and the other of type $A_1$. The roots in the $A_1$ summand are in the same $W$-orbit as $a$. If $V_J^\perp$ is the $A_1$ component of $a^\perp$ then $V_J=(V_J^\perp)^\perp$ is of type $C_{n-2}+A_1$. This determines $J$ uniquely, since $W_J$ is visible. If $n\ge 4$ and $w$ is the longest element of the visible parabolic subgroup of type $A_{n-1}$, then $-wa\in\Pi\setminus\{b\}=J$, but $w\notin N_W(W_J)$. This contradicts the fact that $Q\subseteq N_W(W_J)$. So $n=3$, and the elements of $J$ are the end nodes $a,\,c$ of the $C_3$ diagram, the middle node $b$ being in the same $W$-orbit as $a$. Since $\Phi_J=\{\pm a,\pm c\}$ and $c$ is not in the same $W$-orbit as $a$ and $b$ we deduce that $Q=\{\,w\in W\mid wa=\pm a\,\}$. Furthermore, of the 6 roots in the $W$-orbit of $c$, only $c$ and $-c$ are orthogonal to $a$. So if $wa=\pm a$ then $wc=\pm c$. Thus if $w\in Q$ then $w\Phi_J=\Phi_J$, as required.
Continuing the discussion of $C_n$, suppose now that $V_J^\perp$ is the $C_{n-2}$ component of $a^\perp$. Then $V_J=(V_J^\perp)^\perp$ is of type $C_2$. Writing $J=\{a,b\}$, the fact that $\operatorname{Stab}(a)$ is of type $A_1+C_{n-2}$ means that it is $b$ rather than $a$ that is the end node of the $C_n$ diagram. If we put $c=r_ba$ then $\{\pm c\}$ is the component of $\Phi\cap a^\perp$ of type $A_1$. It follows that $\{\pm a\}=\{\pm r_bc\}$ is the $A_1$-component of $\Phi\cap(r_ba)^\perp=\Phi\cap c^\perp$. We see that $\operatorname{Stab}(a)=\langle r_c\rangle\times W'$ and $\operatorname{Stab}(c)=\langle r_a\rangle\times W'$, where $W'$ is a parabolic (not visible) subgroup of $W$ of type $C_{n-2}$. Indeed, the root system of $W'$ is $\Phi\cap V_J^\perp$. The roots in $\Phi_J$ that are in the same $W$-orbit as $a$ are $\pm a$ and $\pm c$, and so $$Q=\{1,r_a,r_b,r_br_a\}\operatorname{Stab}(a)=\{1,r_a,r_b,r_br_a\}\{1,r_c\}W'
=W_JW'.$$ Hence our requirement that $Q$ stabilizes $\Phi_J=\{\pm a,\pm b\}$ is indeed satisfied.
If $\Pi$ is of type $D_n$ with $n>4$ then one summand of $a^\perp$ is of type $D_{n-2}$ and the other of type $A_1$. The roots orthogonal to a $D_{n-2}$ subsystem form a system of type $A_1+A_1$. There are in fact two $W$-orbits of parabolic $A_1+A_1$ subsystems, and the orthogonal complement of a $D_{n-2}$ is perhaps better thought of as type $D_2$, since the visible parabolic in this orbit corresponds to the two nodes of the diagram that form the fork. So if $V_J^\perp$ is the $D_{n-2}$ summand of $a^\perp$ then $J=\{a,b\}$ consists to the two nodes of valency 1 that are adjacent to $c$, the node of valency 3. A similar statement applies for $D_4$ in the case that $V_J^\perp$ is of type $A_1+A_1$. In both cases the element $w=r_cr_ar_br_c\in W$ satisfies $wa=b$ and $wb=a$, and since $\Phi_J=\{\pm a,\pm b\}$ we see that $Q=\{1,r_a,w,wr_a\}\operatorname{Stab}(a)$. But $\operatorname{Stab}(a)=\langle r_b\rangle\times W'$ and $\operatorname{Stab}(b)=\langle r_a\rangle\times W'$, where $W'$ is the parabolic subgroup corresponding to the subspace $V_J^\perp$, and it follows readily that $Q$ stabilizes $\Phi_J=\{\pm a,\pm b\}$, as required.
Continuing the discussion of $D_n$, where $n\ge 4$, suppose now that $V_J^\perp$ is an $A_1$ component of $a^\perp$. Then $V_J=(V_J^\perp)^\perp$ is of type $A_1+D_{n-2}$. But the maximal length element of a visible $A_{n-1}$ subsystem containing $a$ takes $a$ to an element of $\Phi_J$ but does not normalize $W_J$. So our requirement that $Q\subseteq N_W(W_J)$ is not met.
Finally, suppose that $\Pi$ is of type $H_3$, so that $\operatorname{Stab}(a)$ is of type $A_1+A_1$. Then $V_J^\perp$ is of type $A_1$, and hence $J$ is of type $A_1+A_1$. Let $J=\{a,c\}$, and note that $c=wa$ for some $w\in W$. Since $N_W(W_J)$ is generated by $W_J$ and the central involution of $W$, we see that $c$ is not in the $N_W(W_J)$-orbit of $a$. Hence the element $w$ above is in $Q$ but not in $N_W(W_J)$, and so our requirements are not met.
We have thus established the following result.
\[cases\] Let $\Pi$ be the set of simple roots for the finite irreducible Coxeter group $W$, and suppose that $a\in J\subseteq\Pi$. Then $\{\,w\in W\mid wa\in\Phi_J\,\}$ is a subset of $N_W(W_J)$ if and only if one of the following situations occurs
1. $J=\{a\}$
2. $J=\Pi$
3. $\Pi=\{a,b,c\}$ is of type $C_3$, with $m_{ab}=3$ and $m_{bc}=4$, and $J=\{a,c\}$ of type $A_1+A_1$
4. $\Pi$ is of type $D_n$ or $A_3$, and $J=\{a,b\}$, where $a$ and $b$ are end nodes that are both adjacent to some $c\in\Pi$
5. $\Pi$ is of type $C_n$ and $J=\{a,b\}$ is of type $C_2$, with $b$ an end node of $\Pi$.
We return now to investigation of an arbitrary finite rank Coxeter group $W$. The next proposition is an immediate consequence of Proposition \[cases\] and the discussion preceding it.
\[finclos2\] Let $a\in J\subseteq L\subseteq\Pi$, and suppose that the group $W_L$ is finite and that $W_J = \operatorname{FC}(r_a)$. Let $J_0$ be the component of $J$ containing $a$ and $L_0$ the component of $L$ containing $J_0$. Then every component of $J$ that is not contained in $L_0$ is a component of $L$. Furthermore, if $\{a\}\ne J\cap L_0\ne L_0$ then $J\cap L_0=\{a,b\}$ for some $b$, and one of the following alternatives occurs
1. $L_0=\{a,c,b\}$ is of type $C_3$, with $m_{ac}=3$ and $m_{cb}=4$
2. $L_0$ is of type $C_n$ for some $n\ge 3$, with $b$ an end node and $J_0=\{a,b\}$ of type $C_2$
3. $L_0$ is of type $A_3$ or type $D_n$ for some $n\ge 4$, the nodes $a$ and $b$ having valency 1 and sharing a common neighbour.
One of the ingredients of alternative (2) of Proposition \[finclos2\] is that the component of $\operatorname{FC}(r_a)$ containing $a$ is of type $C_2$. We shall see that when this situation arises, $\operatorname{Odd}(a)$ has a focus in $\Pi$.
\[alt2\]Suppose that $a\in J\subseteq\Pi$ with $W_J=\operatorname{FC}(r_a)$, and let $J_0$ be the component of $J$ containing $a$. Suppose that $J_0=\{a,b\}$ is of type $C_2$. Then either $\operatorname{Odd}(a)\cup\{b\}$ is a spherical component of $\operatorname{EOdd}(a)$, or else $(a,b)$ is a focus of $\operatorname{Odd}(a)$ in $\Pi$.
We use induction on $k$ to prove that for all $k\ge 2$, if $b=c_1,\,a=c_2,\,c_3,\,\ldots,\,c_k$ are simple roots satisfying
1. $2<m_{c_ic_{i+1}}<\infty$ for all $i\in\{1,2,\ldots,k-1\}$, and
2. $c_1,\,c_2,\,\ldots,\,c_k$ are distinct from each other,
then $\{c_1,c_2,\ldots,c_k\}$ forms a system of type $C_k$. The case $k=2$ is immediately true.
Suppose that $k>2$. The inductive hypothesis tells us that $\{c_1,c_2,\ldots,c_{k-1}\}$ is of type $C_{k-1}$. The element $w=v[c_{k-1},\{c_{k-2}\}]\cdots v[c_4,\{c_3\}]v[c_3,\{c_2\}]$ has the property that $wa=wc_2=c_{k-1}$, and so if we write $d=wb$ then $$r_d=wr_bw^{-1}\in w\operatorname{FC}(r_a)w^{-1}=\operatorname{FC}(c_{k-1}),$$ since it is given that $b\in\operatorname{FC}(r_a)$. But $W_{\{c_{k-1},c_k\}}$ is finite, and so it follows that $\{r_d,r_{c_{k-1}},r_{c_k}\}$ generates a finite group. Now $d\cdot c_{k-1}=b\cdot a=-\cos(\pi/4)$ and $c_{k-1}\cdot c_k=-\cos(\pi/m)$ for some $m>2$. If $m\ge 4$ then $$c_k\cdot d=c_k\cdot\Bigl(b+\sqrt 2\sum_{i=2}^{k-1}c_i\Bigr)
\le -\sqrt 2(c_k\cdot c_{k-1})\le-1,$$ whence the reflection subgroup $W_{\{r_d,r_{c_k}\}}$ is infinite (by Theorem \[base\]), a contradiction. So $m=3$. If $m_{c_ic_k}>2$ for any $i\in\{1,2,\ldots,k-2\}$ then $$c_k\cdot d\le \sqrt 2(c_k\cdot c_{k-1})+c_k\cdot c_i<-1,$$ again giving a contradiction. So $m_{c_kc_i}=2$ for all $i\in\{1,2,\ldots,k-2\}$ and $m_{c_kc_{k-1}}=3$, and since $\{c_1,c_2,\ldots,c_{k-1}\}$ is a system of type $C_{k-1}$ it follows that $\{c_1,c_2,\ldots,c_k\}$ is a system of type $C_k$, as claimed.
If there were $c,\,d\in\operatorname{Odd}(a)$ with $3<m_{cd}<\infty$ then $b$ together with a minimal length odd-labelled path from $a$ to $\{c,d\}$ would yield $c_1,\,c_2,\,\ldots,\,c_k\in\Pi$ satisfying (1) and (2) above and not forming a system of type $C_k$, contradicting the result proved above. The same argument yields a contradiction if $c\in\operatorname{Odd}(a)$ and $d\in\Pi\setminus\operatorname{Odd}(a)$ with $3<m_{cd}<\infty$, unless $\{c,d\}=\{a,b\}$. So all edge labels in $\operatorname{Odd}(a)$ are $3$, if $c,\,d\in\operatorname{Odd}(a)$ are not adjacent in $\operatorname{Odd}(a)$ then $m_{cd}\in\{2,\infty\}$, and if $c\in\operatorname{Odd}(a)$ and $d\in\Pi\setminus\operatorname{Odd}(a)$ then $m_{cd}\in\{2,\infty\}$ unless $\{c,d\}=\{a,b\}$. Furthermore, any circuit in $\operatorname{Odd}(a)$ would similarly yield a contradiction (by combining the circuit with a minimal finite-labelled path connecting it to $b$). So $\operatorname{Odd}(a)$ is tree.
For each $c\in\operatorname{Odd}(a)$ let $C[b..c]\subseteq\Pi$ consist of $b$ and the unique path from $a$ to $c$ in $\operatorname{Odd}(a)$. The discussion above shows that $C[b..c]$ is always of type $C$. Now suppose that $c\in\operatorname{Odd}(a)$ and $e\in\Pi\setminus C[b..c]$ with $m_{ce}=2$. Write $C[b..c]=\{c_1,c_2,\cdots,c_k\}$, with $c_1=b$ and $c_k=c$, and let $d=b+\sqrt 2\sum_{i=2}^kc_i$. An argument similar to one used above shows that $r_d\in\operatorname{FC}(c)$, and hence $W_{\{d,c,e\}}$ is finite. So $d\cdot e>-1$. If $c_i\cdot e\ne0$ then $c_i\cdot e\le-1/2$; so it follows that there is at most one $i$ with $c_i\cdot e\ne0$. Suppose, for a contradiction, that there is exactly one such $i$. If $i>1$ then $d\cdot e=\sqrt 2(c_i\cdot e)$, and so $c_i\cdot e>-1/\sqrt 2$. Hence $m_{c_ie}=3$, and $d\cdot e=-1/\sqrt 2$. But this means that the edges $\{c,d\}$ and $\{d,e\}$ of the Coxeter graph of $\{d,c,e\}$ are both labelled $4$, contradicting the fact that $W_{\{d,c,e\}}$ is finite. So we must have $i=1$, and finiteness of $W_{\{c,d,e\}}$ forces $b\cdot e=d\cdot e=-1/2$. But now if we put $L=\{e\}\cup J$ then, in the notation of Proposition \[finclos2\], we have that $L_0=\{e,b,a\}$ is of type $C_3$ with $J\cap L_0=\{b,a\}$ of type $C_2$, and Proposition \[finclos2\] shows that this is not possible. We conclude that if $e\in\Pi$ has the property that $m_{ce}=2$ for some $c\in\operatorname{Odd}(a)$ then $m_{de}=2$ for all $d\in C[b..c]$. In particular, if $e\in\Pi\setminus(\operatorname{Odd}(a)\cup\{b\})$ and $m_{ce}\ne\infty$ for some $c\in\operatorname{Odd}(a)$ then $m_{ce}=2$, as shown above, and so $m_{de}=2$ for all $d\in C[b..c]$.
All that remains to prove now is that if $c,\,d\in\operatorname{Odd}(a)$ with $c\notin C[b..d]$ and $d\notin C[b..c]$, then $m_{cd}=\infty$. Since $c$ and $d$ are not adjacent in $\operatorname{Odd}(a)$ the only alternative is that $m_{cd}=2$; so suppose, for a contradiction, that this holds. Choose the vertex $e\in\operatorname{Odd}(a)$ on the (unique) path from $c$ to $d$ such that the distance from $e$ to $a$ is minimal. Let $c',\,d'$ be the neighbours of $e$ in the path from $c$ to $d$, with $c'$ between $e$ and $c$ and $d'$ between $e$ and $d$. Then $c'\in C[b..c]$, and since $m_{cd}=2$ it follows that $m_{c'd}=2$. Now since $d'\in C[b..d]$ and $m_{c'd}=2$ it follows that $m_{c'd'}=2$. Thus the set $L\subseteq\Pi$ consisting of $c'$ and $d'$ and the vertices on the path from $a$ to $e$ form a system of type $D$ (or $A_3$ if $e=a$). So $L$ is spherical, and since $b\in\operatorname{FC}(r_a)$ it follows that $L\cup\{b\}$ is spherical also. But this is impossible since $L\cup\{b\}$ is connected, has an edge labelled 4 (namely, $\{b,a\}$), and has a vertex of valency 3 (namely $e$).
The situation of alternative (3) of Proposition \[finclos2\] is very similar to that of alternative (2), and in this case it turns out that $\operatorname{Odd}(a)$ has a half-focus in $\Pi$.
\[alt3\]Suppose that $a\in J\subseteq\Pi$ with $W_J=\operatorname{FC}(r_a)$ and $\{a\}$ a component of $J$, and suppose that $J\cap\operatorname{Odd}(a)\ne\{a\}$. Then either $\operatorname{Odd}(a)$ is a spherical component of $\operatorname{EOdd}(a)$, or else there exists an element $b\in\operatorname{Odd}(a)$ such that $\{a,b\}$ is a half focus of $\operatorname{Odd}(a)$ in $\Pi$.
Let $b\in(J\cap \operatorname{Odd}(a))\setminus\{a\}$, and let $w\in W$ with $wa=b$. Then $w\in N_W(W_J)$, by Proposition \[finclos\], and so $$\operatorname{FC}(r_b)=\operatorname{FC}(wr_aw^{-1})=w\operatorname{FC}(r_a)w^{-1}=wW_Jw^{-1}=W_J.$$ Moreover, $w\Phi_J=\Phi_J$, and since $a\cdot c=0$ for all $c\in\Phi_J\setminus\{a\}$, it follows that $wa\cdot d=0$ for all $d\in\Phi_J\setminus\{wa\}$. So $\{b\}$ is a component of $J$. Note that $m_{ab}=2$, since $a$ and $b$ are in different components of $J$.
Let $c\in\Pi\setminus\{a,b\}$, and suppose first of all that $2<m_{bc}<\infty$. Since $\{b,c\}$ is spherical and $W_J=\operatorname{FC}(r_b)$ it follows that $J\cup\{c\}$ is spherical. Let $L=J\cup\{c\}$ and let $L_0$ be the component of $L$ containing $a$. By Proposition \[finclos2\], every component of $J$ that is not contained in $L_0$ is a component of $L$. But $b$ is adjacent to $c$ in $L$; so $\{b\}$ is not a component of $L$, and it follows that $b\in L_0$. Now $\{a\}\ne J\cap L_0$, since $b\in J\cap L_0$, and $J\cap L_0\ne L_0$, since $c\in L_0$ and $c\notin J$ (since $\{b\}$ is a component of $J$). Furthermore, the conditions of alternative (2) of Proposition \[finclos2\] are not satisfied, since $a$ and $b$ are not adjacent in $J$. So either alternative (1) or alternative (3) must hold, and since $c$ is the only element of $L$ not in $J$ it follows that $L_0=\{a,c,b\}$, with $m_{ac}=3$. But a symmetrical argument, with the roles of $a$ and $b$ reversed, shows that every $d\in\Pi$ with $2<m_{ad}<\infty$ has the property that $m_{bd}=3$. So $m_{ac}=m_{bc}=3$, and $\{a,c,b\}$ is of type $A_3$.
Now suppose that $m_{bc}=2$. Again since $\{b,c\}$ is spherical it follows that $J\cup\{c\}$ is spherical, and so $m_{ac}<\infty$. If $m_{ac}>2$ then, as we have just observed, it follows that $m_{bc}=3$, contrary to our assumption that $m_{bc}=2$. So $m_{ac}=m_{bc}=2$, and we have now shown that whenever $m_{bc}<\infty$ we have $m_{ac}=m_{bc}\in\{2,3\}$. Since a symmetrical argument gives the same conclusion whenever $m_{ac}<\infty$, we conclude also that $m_{ac}=\infty$ if and only if $m_{bc}=\infty$.
We now use induction on $k$ to prove that for all $k\ge 3$, if $b=c_1,\,a=c_2,\,c_3,\,\ldots,\,c_k$ are simple roots satisfying
1. $2<m_{c_ic_{i+1}}<\infty$ for all $i\in\{2,3,\ldots,k-1\}$, and
2. $c_1,\,c_2,\,\ldots,\,c_k$ are distinct from each other,
then $\{c_1,c_2,\ldots,c_k\}$ forms a system of type $D_k$ or $A_3$. The case $k=3$ follows from what we have proved above.
Suppose that $k>3$. The inductive hypothesis tells us that $\{c_1,c_2,\ldots,c_{k-1}\}$ is of type $D_{k-1}$ (or $A_3$ if $k=4$). The element $w=v[c_{k-1},\{c_{k-2}\}]\cdots v[c_4,\{c_3\}]v[c_3,\{c_2\}]$ has the property that $wa=wc_2=c_{k-1}$, and so if we write $d=wb$ then $$r_d=wr_{c_1}w^{-1}\in w\operatorname{FC}(r_a)w^{-1}=\operatorname{FC}(c_{k-1}),$$ since it is given that $b\in\operatorname{FC}(r_a)$. But $W_{\{c_{k-1},c_k\}}$ is finite, and so it follows that $\{r_d,r_{c_{k-1}},r_{c_k}\}$ generates a finite group. Now $d\cdot c_{k-1}=b\cdot a=0$ and $c_{k-1}\cdot c_k=-\cos(\pi/m)$ for some $m>2$. If $c_k\cdot c_i\ne 0$ for some $i\in\{1,2,\ldots,k-2\}$ then $$c_k\cdot d=c_k\cdot\Bigl(c_1+c_2+c_{k-1}+2\sum_{i=3}^{k-2}c_i\Bigr)\le
-(\tfrac12+\cos\tfrac\pi m)\le-1,$$ whence the reflection subgroup $W_{\{r_d,r_{c_k}\}}$ is infinite (by Theorem \[base\]), a contradiction. So $c_k\cdot d=c_{k-1}\cdot c_k=-\cos\frac\pi m$. Since the reflection subgroup generated by $\{r_d,r_{c_{k-1}},r_{c_k}\}$ is finite it follows that $m=3$. So we have shown that $m_{c_kc_{k-1}}=3$ and $m_{c_kc_i}=2$ for $i<k-1$, and since $\{c_1,c_2,\ldots,c_{k-1}\}$ is a system of type $D_{k-1}$ it follows that $\{c_1,c_2,\ldots,c_k\}$ is a system of type $D_k$, as claimed.
Note that $\operatorname{Odd}(a)\setminus\{b\}$ and $\operatorname{Odd}(a)\setminus\{a\}$ are both connected, since each element $c\in\operatorname{Odd}(a)$ that is is adjacent to $a$ is also adjacent to $b$, and vice versa. If there were $c,\,d\in\operatorname{Odd}(a)\setminus\{b\}$ with $3<m_{cd}<\infty$ then $b$ together with a minimal length odd-labelled path from $a$ to $\{c,d\}$ would yield $c_1,\,c_2,\,\ldots,\,c_k\in\Pi$ satisfying (1) and (2) above and not forming a system of type $D_k$, contradicting the result proved above. The same argument yields a contradiction whenever $c\in\operatorname{Odd}(a)\setminus\{a,b\}$ and $d\in\Pi\setminus\operatorname{Odd}(a)$ with $3<m_{cd}<\infty$. So all edge labels in $\operatorname{Odd}(a)$ are $3$, if $c,\,d\in\operatorname{Odd}(a)$ are not adjacent in $\operatorname{Odd}(a)$ then $m_{cd}\in\{2,\infty\}$, and if $c\in\operatorname{Odd}(a)$ and $d\in\Pi\setminus\operatorname{Odd}(a)$ then $m_{cd}\in\{2,\infty\}$. Furthermore, any circuit in $\operatorname{Odd}(a)\setminus\{b\}$ would similarly yield a contradiction (by combining the circuit with a minimal finite-labelled path connecting it to $b$). So $\operatorname{Odd}(a)\setminus\{b\}$ is tree. Of course, $\operatorname{Odd}(b)\setminus\{a\}$ is also a tree, by the same argument.
For each $c\in\operatorname{Odd}(a)\setminus\{a,b\}$ let $D[a,b..c]\subseteq\Pi$ consist of $b$ and the unique path from $a$ to $c$ in $\operatorname{Odd}(a)\setminus\{b\}$. The discussion above shows that $D[a,b..c]$ is of type $D$. Now suppose that $c\in\operatorname{Odd}(a)\setminus\{a,b\}$ and $e\in\Pi\setminus D[a,b..c]$ with $m_{ce}=2$. Write $D[a,b..c]=\{c_1,c_2,\cdots,c_k\}$, with $c_1=b$, $c_2=a$ and $c_k=c$, and let $d=c_1+c_2+c_k+2\sum_{i=3}^{k-1}c_i$. An argument similar to one used above shows that $r_d\in\operatorname{FC}(c)$, and hence $W_{\{d,c,e\}}$ is finite. So $d\cdot e>-1$. If $c_i\cdot e\ne0$ then $c_i\cdot e\le-1/2$; so it follows that $\{\,i\mid c_i\cdot e\ne0\,\}$ is a subset of $\{1,2,k\}$ with at most one element. But $c_k\cdot e=0$ since $m_{ce}=2$, and $c_1\cdot e=c_2\cdot e$ since $m_{af}=m_{bf}$ for all $f\in\Pi$. So $c_i\cdot e=0$ for all $i\in\{1,2,\ldots,k\}$. In particular, if $e\in\Pi\setminus\operatorname{Odd}(a)$ and $m_{ce}\ne\infty$ for some $c\in\operatorname{Odd}(a)$ then $m_{ce}=2$, as shown above, and so $m_{de}=2$ for all $d\in D[a,b..c]$.
All that remains to prove now is that if $c,\,d\in\operatorname{Odd}(a)\setminus\{a,b\}$ with $c\notin D[a,b..d]$ and $d\notin D[a,b..c]$, then $m_{cd}=\infty$. Since $c$ and $d$ are not adjacent in $\operatorname{Odd}(a)$ the only alternative is that $m_{cd}=2$; so suppose, for a contradiction, that this holds. Choose the vertex $e\in\operatorname{Odd}(a)\setminus\{b\}$ on the (unique) path from $c$ to $d$ such that the distance from $e$ to $a$ is minimal. Let $c',\,d'$ be the neighbours of $e$ in the path from $c$ to $d$, with $c'$ between $e$ and $c$ and $d'$ between $e$ and $d$. Then $c'\in C[b..c]$, and since $m_{cd}=2$ it follows that $m_{c'd}=2$. Now since $d'\in C[b..d]$ and $m_{c'd}=2$ it follows that $m_{c'd'}=2$. Thus the set $L\subseteq\Pi$ consisting of $c'$ and $d'$ and the vertices on the path from $a$ to $e$ form a system of type $D_k$, or $A_3$ if $e=a$. So $L$ is spherical, and since $b\in\operatorname{FC}(r_a)$ it follows that $L\cup\{b\}$ is spherical also. If $L=A_3$ then $m_{ac'}=m_{ad'}=3$, and since $m_{bc'}=m_{ac'}$ and $m_{bd'}=m_{ad'}$ we see that $L\cup\{b\}$ is of type $\widetilde A_3$, contradicting the fact that $L\cup\{b\}$ is spherical. Similarly if $L$ is of type $D_k$ then $L\cup\{b\}$ is of type $\widetilde D_k$, again giving a contradiction.
We also need to obtain further information about the situation of alternative (1) of Proposition \[finclos2\]. So for the next three lemmas we assume that $a\in J\subseteq L\subseteq\Pi$ with $L$ spherical and $W_J=\operatorname{FC}(r_a)$, and there exist $b\in J$ and $c\in L\setminus J$ such that $L_0=\{a,c,b\}$ is a component of $L$ of type $C_3$, with $m_{ac}=3$ and $m_{cb}=4$.
\[alt1a\] For all $e\in\Pi\setminus\{a,c,b\}$, either $m_{ce}=m_{ae}=m_{be}=2$ or $m_{ce}=\infty$. Moreover, $J\cap\operatorname{Odd}(a)=\{a\}$.
If $J\cap\operatorname{Odd}(a)\ne\{a\}$ then, since $\{a\}$ is a component of $J$, Proposition \[alt3\] applies, and it follows in particular that no vertex in $\operatorname{Odd}(a)$ lies on an edge with finite label different from 3. This contradicts $m_{bc}=4$. So $J\cap\operatorname{Odd}(a)=\{a\}$.
Suppose that $e\in\Pi\setminus\{a,c,b\}$ with $m_{ce}<\infty$. The group $r_cr_aW_{\{c,e\}}r_ar_c$ is finite and contains $r_cr_aW_{\{c\}}r_ar_c=W_{\{a\}}$; so there exists a maximal finite subgroup $G$ of $W$ containing $r_a$ and the reflection along $(r_cr_a)e$. Since $r_b\in\operatorname{FC}(r_a)\subseteq G$ it follows that $W_{\{b,(r_cr_a)e\}}$ is finite, and hence so is $W_{\{(r_ar_c)b,e\}}=r_ar_cW_{\{b,(r_cr_a)e\}}r_cr_a$. Hence $$\label{ineq}
(b+\sqrt2 c+\sqrt 2 a)\cdot e=(r_ar_c)b\cdot e>-1.$$ Assume, for a contradiction, that $m_{ce}\ne 2$. Then $c\cdot e\le-1/2<1/2\sqrt 2$, and so $(b+\sqrt 2 a)\cdot e>-1/2$, giving a contradiction if either $m_{be}\ne 2$ or $m_{ae}\ne 2$. So $b\cdot e=a\cdot e=0$, and the inequality \[ineq\] above gives $c\cdot e>1/\sqrt 2$. So $m_{ce}=3$. But now $W_{\{a,c,e\}}$ is of type $A_3$, hence finite, and hence contained in a maximal finite subgroup that also contains $\operatorname{FC}(r_a)=W_J$. Since $b\in J$ it follows that $\{a,c,e,b\}$ is spherical, which is false since it is of type $\widetilde B_3$. So $m_{ce}=2$. and it remains to show that $m_{ae}=m_{be}=2$.
Since $c\cdot e=0$ we deduce from \[ineq\] that $(b+\sqrt 2 a)\cdot e>-1$, and in particular it follows that $m_{ae}$ is 2 or 3. In either case $\{e,a,c\}$ is spherical (of type $A_3$ or $A_1+A_2$), and so $\{e,a,c,b\}$ is also spherical (since $r_b\in\operatorname{FC}(r_a)$). If either $m_{ae}\ne 2$ or $m_{be}\ne 2$ then applying Proposition \[finclos2\] with $L''=J\cup\{e,c\}$ in place of $L$ yields a contradiction, since if $L_0''$ is the the component of $L''$ containing $a$ then $\{a,b\}\subseteq L_0''\cap J\ne L_0''$ (since $c\notin J$), but $L_0''$ is not of type $C_3$ or $D_n$ since it contains $\{a,c,b,e\}$. So $m_{ae}=m_{be}=2$, as required.
\[alt1b\] Let $J'=J\setminus\{a\}$ and let $d\in\operatorname{Odd}(a)$. Then $m_{cd}\ne 2$. If $m_{db'}\ne 2$ for some $b'\in J'$ then $m_{db'}=4$, and there is a unique $a'$ adjacent to $d$ in $\operatorname{Odd}(a)$; moreover, $\{a',d,b'\}$ is of type $C_3$, and $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$. On the other hand, if $m_{db'}=2$ for all $b'\in J'$ then $\operatorname{FC}(r_d)=W_{J'\cup\{d\}}$.
We use induction on the distance from $d$ to $a$ in $\operatorname{Odd}(a)$. Observe that if $d=a$ then $m_{db'}=2$ for all $b'\in J'$, since $\{a\}$ is a component of $J$, and we have $\operatorname{FC}(r_d)=W_J=W_{J'\cup\{d\}}$ and $m_{cd}=3\ne 2$, as required.
Suppose now that $d\ne a$, and let $a=d_1,\,d_2,\,\ldots,\,d_k=d$ be a minimal length path from $a$ to $d$ in $\operatorname{Odd}(a)$. If $2\le i\le k-1$ then $d_i$ does not have valency 1 in $\operatorname{Odd}(a)$, and so $m_{d_ib'}=2$ for all $b'\in J'$, by the inductive hypothesis. The same is true for $i=1$, since $\{a\}$ is a component of $J$.
We prove first that $m_{cd}\ne 2$. Assuming, for a contradiction, that $m_{cd}=2$, then clearly $d\notin\{a,c,b\}$, and Lemma \[alt1a\] tells us that $m_{ad}=2$ and $m_{bd}=2$. It follows that $m_{bf}=2$ for all $f$ in the set $M=\{d_1,d_2,\ldots,d_k\}$, since $\{d_i\}$ is a component of $J'\cup\{d_i\}$ when $1\le i\le k-1$. So $wb=b$ for all $w\in W_M$. Furthermore, since $d$ and $a$ lie in the same connected component of $\operatorname{\Omega}(\Pi)$, we can choose $w\in W_M$ such that $wd=a$. Now since $wc\cdot a=c\cdot d=2$ we see that the reflection $r_{wc}$ centralizes $r_a$, and hence normalizes $\operatorname{FC}(r_a)=W_J$. By Lemma \[subrootsys\] it follows that either $wc\in\Phi_J$ or $wc\cdot e=0$ for all $e\in J$. But $wc\cdot b=c\cdot w^{-1}b=c\cdot b\ne 0$; so we must have $wc\in\Phi_J$, and hence $c\in w^{-1}\Phi_J\subseteq\Phi_{J\cup M}$. So $c\in M$, contradicting $m_{bf}=2$ for all $f\in M$. So $m_{cd}\ne 2$.
Write $a'=d_{k-1}$ and $\widetilde J=J'\cup\{a'\}$. Note that $\{a'\}$ is a component of $\widetilde J$, and $\operatorname{FC}(r_{a'})=W_{\widetilde J}$ (by the inductive hypothesis). Now since $\{d,a'\}$ is spherical, $\widetilde L=\widetilde J\cup\{d\}$ is spherical also. Let $\widetilde L_0$ be the component of $\widetilde L$ containing $a'$.
Consider first the case that $m_{db'}=2$ for all $b'\in J'$. Since also $m_{a'b'}=2$ for all $b'\in J'$, it follows that $r_d$ and $r_{a'}$ both fix all elements of $J'$. Since $v=v[d,\{a'\}]\in W_{\{a',d\}}$ satisfies $va'=d$, we conclude that $$\operatorname{FC}(r_d)=v\operatorname{FC}(r_{a'})v^{-1}=vW_{J'\cup\{a'\}}v^{-1}=W_{vJ'\cup\{va'\}}
=W_{J'\cup\{d\}},$$ as required.
Now suppose that $m_{db'}\ne 2$ for some $b'\in J'$. Then $b'\in \widetilde J\cap\widetilde L_0$, and so $\{a'\}\ne\widetilde J\cap\widetilde L_0\ne\widetilde L_0$. Applying Proposition \[finclos2\], we see that the situation of alternative (1) must hold: alternative (2) is ruled out since $a'$ is a component of $\widetilde J$, and alternative (3) is ruled out since $J'\cap\operatorname{Odd}(a)=\emptyset$. Hence $\widetilde L_0=\{a',d,b'\}$ is of type $C_3$, with $m_{db'}=4$ and $m_{da'}=3$. Furthermore, Lemma \[alt1a\] tells us that $m_{de}\in\{2,\infty\}$ for all $e\in\Pi\setminus\{a',d,b'\}$; so $a'$ is the unique neighbour of $d$ in $\operatorname{Odd}(a)$, as required.
\[alt1c\] Let $e\in\operatorname{EOdd}(a)\setminus\operatorname{Odd}(a)$ with $e\ne b$. Then $m_{be}=2$.
If $e\in J$ then it is clear that $m_{be}=2$, since $b$ is a component of $J$. So we may assume that $e\notin J$.
As above, write $J'=J\setminus\{a\}$. Since $e\in\operatorname{EOdd}(a)$ there exists a $d\in\operatorname{Odd}(a)$ with $m_{de}$ even. If $d$ is adjacent to some $b'\in J'$ then, by Lemma \[alt1b\], there is a unique $a'\in\operatorname{Odd}(a)$ adjacent to $d$; furthermore, $\{a',d,b'\}$ is of type $C_3$, and $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$. By Lemma \[alt1a\], since $m_{de}\ne\infty$ it follows that $m_{a'e}=m_{de}=2$. On the other hand, if $d$ is not adjacent to any element of $J'$ then Lemma \[alt1b\] tells us that $\operatorname{FC}(r_d)=W_{J'\cup\{d\}}$. So in either case there is an $a'\in\operatorname{Odd}(a)$ with $m_{a'e}$ even and $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$.
Choose such an $a'$. Since $m_{a'e}$ is even, $v[e,\{a'\}]a'=a'$; moreover $v[e,\{a'\}]$ is the reflection along some root $f=\lambda e+\mu a'$. Note that $f\cdot a'=0$, and hence $\lambda\ne 0$. Since $e\notin J$ it follows that $f\notin\Phi_J$. But $r_f$ centralizes $r_{a'}$, and hence normalizes $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$. By Lemma \[subrootsys\] it follows that $f\cdot b=0$. But also $a'\cdot b=0$, since $\{a'\}$ and $\{b\}$ are distinct components of $J'\cup\{a'\}$; so it follows that $e\cdot b=0$. Thus $m_{be}=2$, as required.
Lemmas \[alt1a\], \[alt1b\] and \[alt1c\] combine to yield the following result.
\[alt1d\] Suppose that $a\in J\subseteq L\subseteq\Pi$, with $L$ spherical and $\operatorname{FC}(r_a)=W_J$, and let $L_0$ be the component of $L$ containing $a$. Suppose that $L_0=\{a,c,b\}$ is of type $C_3$, with $m_{ac}=3$ and $m_{bc}=4$, and $J\cap L_0=\{a,b\}$. Then $b$ is a $C_3$-neighbour of $\operatorname{Odd}(a)$. Furthermore, $J\cap\operatorname{Odd}(a)=\{a\}$, and if $a'\in\operatorname{Odd}(a)$ is not adjacent to any $C_3$-neighbour of $\operatorname{Odd}(a)$ then $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$, where $J'=J\setminus\{a\}$.
If $m_{bd}\ne 2$ for some $d\in\operatorname{Odd}(a)$, then $m_{db}=4$, by Lemma \[alt1b\]. There is at least one $d\in\operatorname{Odd}(a)$ such that $m_{bd}=4$, namely $d=c$. Lemma \[alt1b\] tells us that for each $d\in\operatorname{Odd}(a)$ with $m_{bd}=4$ there is an $a'\in\operatorname{Odd}(a)$ such that $\{a',d,b'\}$ is a system of type $C_3$. Moreover, by Lemma \[alt1a\], if $e\in\Pi\setminus(\operatorname{Odd}(a)\cup\{b\})$ then either $m_{de}=\infty$ or $m_{ae}=m_{be}=m_{de}=2$, while if $e\in\operatorname{Odd}(a)\setminus\{a,c\}$ then $m_{de}=\infty$, since $m_{de}\ne 2$ by Lemma \[alt1b\]. And if $e\in\operatorname{EOdd}(a)\setminus(\operatorname{Odd}(a)\cup\{b\})$ then $m_{be}=2$, by Lemma \[alt1c\]. So $b$ satisfies all the requirements of a $C_3$-neighbour of $M=\operatorname{Odd}(a)$, as specified in Definition \[c3neighbour\].
It now follows from Lemma \[alt1b\] that if $a'\in\operatorname{Odd}(a)$ is adjacent to some $b'\in J'$ then $b'$ is a $C_3$-neighbour of $\operatorname{Odd}(a)$, and if $a'$ is not adjacent to any such $b'$ then $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$. Finally, $J\cap\operatorname{Odd}(a)=\{a\}$, by Lemma \[alt1a\].
Let $a,\,a'\in\Pi$, and suppose that $w\in W$ has the property that $wa=a'$. By Proposition \[normalizers\] there exist $a_i\in\operatorname{Odd}(a)$ and $c_i\in\Pi$ such that
- $a_1=a$ and $a_{k+1}=a'$,
- $m_{c_ia_i}\ne\infty$ and $v[c_i,\{a_i\}]a_i=a_{i+1}$, for all $i\in\{1,2,\ldots,k\}$,
- $w=v[c_k,\{a_k\}]\cdots v[c_2,\{a_2\}]v[c_1,\{a_1\}]$.
Now let $b$ be a $C_3$-neighbour of $\operatorname{Odd}(a)$. For each $c\in\operatorname{Odd}(a)$ that is adjacent to $b$, define $X(c)=b+\sqrt 2c+\sqrt2\tilde a$, where $\tilde a$ is the unique neighbour of $c$ in $\operatorname{Odd}(a)$, and for each $c\in\operatorname{Odd}(a)$ that is not adjacent to $b$, define $X(c)=b$. We show that $v[c_i,\{a_i\}]X(a_i)=X(a_{i+1})$ for all $i\in\{1,2,\ldots k\}$.
Suppose first that neither $c_i$ nor $a_i$ is adjacent to $b$. Then $X(a_i)=b$, and since $a_{i+1}\in\{a_i,c_i\}$ we have that $X(a_{i+1})=b$ also. Since $r_{a_i}$ and $r_{c_i}$ both fix $b$, and $v[c_i,\{a_i\}]\in W_{\{a_i,c_i\}}$, it follows that $$v[c_i,\{a_i\}]X(a_i)=v[c_i,\{a_i\}]b=b=X(a_{i+1}),$$ as required.
Next, suppose that $c_i$ is adjacent to $b$, but $a_i$ is not adjacent to $b$. Since $m_{c_ia_i}\ne\infty$ and $a\in\operatorname{Odd}(a)$ it follows that $c_i\in\operatorname{EOdd}(a)$. Since $b$ is a $C_3$-neighbour of $\operatorname{Odd}(a)$, it is not adjacent to any element of $\operatorname{EOdd}(a)\setminus\operatorname{Odd}(a)$; so $c_i\in\operatorname{Odd}(a)$, and, moreover, $m_{c_ie}=\infty$ for all $e\in\operatorname{Odd}(a)\setminus\{c_i\}$ apart from the unique neighbour of $c_i$ in $\operatorname{Odd}(a)$. So $a_i$ is this unique neighbour, $m_{c_ia_i}=3$, and $a_{i+1}=v[c_i,\{a_i\}]a_i=c_i$. Moreover, $m_{c_ib}=4$ and $m_{a_ib}=2$. So $$v[c_i,\{a_i\}]X(a_i)=r_{a_i}r_{c_i}b=b+\sqrt2c_i+\sqrt2a_i=X(c_i)=X(a_{i+1})$$ as required.
Now suppose that $a_i$ is adjacent to $b$, and let $\tilde a$ be the unique neighbour of $a_i$ in $\operatorname{Odd}(a)$. Since $m_{a_ie}=\infty$ for all $e\in\operatorname{Odd}(a)\setminus\{a_i,\tilde a\}$, if $c_i\in\operatorname{Odd}(a)$ then $c_i=\tilde a$. In this case we see that $$v[c_i,\{a_i\}]X(a_i)=r_{a_i}r_{c_i}(b+\sqrt2c_i+\sqrt2a_i)=b=X(c_i)=X(a_{i+1}),$$ since $a_{i+1}=v[c_i,\{a_i\}]a_i=c_i$. If $c_i=b$ then $v[c_i,\{a_i\}]=r_br_{a_i}r_b$, which fixes both $a_i$ and $X(a_i)=b+\sqrt 2a_i+\sqrt 2\tilde a$. So $v[c_i,\{a_i\}]X(a_i)=X(a_{i+1})$ in this case too. Finally, suppose that $c_i\notin\operatorname{Odd}(a)\cup\{b\}$. Since $m_{c_ia_i}\ne\infty$ we must have $m_{c_i\tilde a}=m_{c_ia_i}=m_{c_ib}=2$, (by the definition of a $C_3$-neighbour). So $$v[c_i,\{a_i\}]X(a_i)=r_{c_i}(b+\sqrt2c_i+\sqrt2\tilde a)
=b+\sqrt2c_i+\sqrt2\tilde a=X(a_{i+1})$$ since $a_{i+1}=r_{c_i}a_i=a_i$.
We have now now covered all cases, and shown that $v[c_i,\{a_i\}]X(a_i)=X(a_{i+1})$ for all $i\in\{1,2,\ldots k\}$. By a trivial induction it follows that $X(a_{k+1})=wX(a_1)$.
Thus we have established the following result.
\[c3lem\] Let $a\in\Pi$ and $w\in W$ such that $wa\in\Pi$. Suppose that $b$ is a $C_3$-neighbour of $\operatorname{Odd}(a)$ that is not adjacent to $a$. Then $$wb=\begin{cases}
b&\text{if $wa$ is not adjacent to~$b$}\\
b+\sqrt2 wa+\sqrt2\tilde a&\text{if $wa$ is adjacent to~$b$}
\end{cases}$$ where $\tilde a$ is adjacent to $wa$ in $\operatorname{Odd}(a)$.
We are now able to give a detailed description of the components of $J$ whenever $W_J$ is the finite continuation of a simple reflection.
\[components\] Suppose that $a\in J\subseteq\Pi$ with $W_J=\operatorname{FC}(r_a)$, and suppose that $K$ is a component of $J$. Then one of the following alternatives holds.
- $K=\{a\}=J\cap\operatorname{Odd}(a)$.
- $K=\{a,b\}$ is of type $C_2$, and $J\cap\operatorname{Odd}(a)=\{a\}$.
- $K=\{a\}$ or $K=\{b\}$, where $\{a,b\}=J\cap\operatorname{Odd}(a)$ is of type $A_1+A_1$.
- $K=\{b\}\nsubseteq\operatorname{Odd}(a)$, and $b$ is a $C_3$-neighbour of $\operatorname{Odd}(a)$.
- $\operatorname{Odd}(a)\subseteq K$, and $K$ is a component of $\operatorname{EOdd}(a)$.
- $K\cap\operatorname{Odd}(a)=\emptyset$, and $K$ is a component of $\operatorname{EOdd}(a)$.
We consider first the case that $K\cap\operatorname{Odd}(a)\ne\emptyset$, and start by supposing that there exists a spherical $L\subseteq\Pi$ with $J\subseteq L$ and $K$ not a component of $L$.
Choose such an $L$, and let $L_0$ be the component of $L$ containing $a$. By Proposition \[finclos2\], since $K$ is not a component of $L$ we must have $K\subseteq L_0$. So either $K=\{a\}$, in which case (a) above holds, or else $\{a\}\subsetneqq \{a\}\cup K\subseteq J\cap L_0$. Furthermore, $J\cap L_0\ne L_0$, since $K\ne L_0$. So if (a) does not hold then $\{a\}\ne J\cap L_0\ne L_0$, and so one of the alternatives (1), (2) or (3) of Proposition \[finclos2\] must hold.
Suppose that alternative (2) holds, so that $K=\{a,b\}=J\cap L_0$ for some $b$, and $\{a,b\}$ is of type $C_2$. By Proposition \[alt2\] we see that each $c\in\operatorname{Odd}(a)\setminus\{a\}$ lies in a type $C$ spherical subset $L'$ of $\Pi$ containing $\{a,b\}$. Since $J\cap L'=\{a,b\}$ (by Proposition \[finclos2\]) it follows that $c\notin J$. So $J\cap\operatorname{Odd}(a)=\{a\}$, and (b) above is satisfied.
Suppose that alternative (3) of Proposition \[finclos2\] holds, so that $J\cap L_0=\{a,b\}$ is of type $A_1+A_1$, and $b\in\operatorname{Odd}(a)$. Proposition \[alt3\] immmediately yields that $J\cap\operatorname{Odd}(a)=\{a,b\}$, and so (c) above is satisfied.
Suppose that alternative (1) of Proposition \[finclos2\] holds, so that $L_0=\{a,c,b\}$ with $m_{ac}=3$ and $m_{cb}=4$, and $J\cap L_0=\{a,b\}$. By Lemma \[alt1a\] we know that $b\notin\operatorname{Odd}(a)$, and since we have assumed that $K\cap\operatorname{Odd}(a)\ne\emptyset$, it follows that $K=\{a\}=J\cap\operatorname{Odd}(a)$. So (a) holds.
We have now dealt with all cases that arise if there is a spherical $L\subseteq\Pi$ with $J\subseteq L$ and $K$ not a component of $L$. So assume that $K$ is a component of every spherical $L$ containing $J$. We show that in this case $\operatorname{Odd}(a)\subseteq K$, and $K$ is a component of $\operatorname{EOdd}(a)$; that is, (e) above holds.
To show that $\operatorname{Odd}(a)\subseteq K$ it is clearly sufficient to show that if $a'\in K\cap \operatorname{Odd}(a)$ and $b$ is adjacent to $a'$ in $\operatorname{Odd}(a)$ then $b\in K$. Note that since $a'\in\operatorname{Odd}(a)$ there exists $w\in W$ with $a'=wa$, and since Proposition \[finclos\] yields that $w\in N_W(W_J)$ it follows that $\operatorname{FC}(r_{a'})=W_J$. Now the assumption that $b$ and $a'$ are adjacent in $\operatorname{Odd}(a)$ implies that $\{a',b\}$ is spherical, and therefore $J\cup\{b\}$ is spherical. But $K$ is a component of every spherical subset of $\Pi$ containing $J$; so it is a component of $J\cup\{b\}$. But $a'\in K$ and $b$ is adjacent to $a'$; so $b\in K$, as required.
Since $K\subseteq J\subseteq\operatorname{EOdd}(a)$ and $K$ is connected, saying that $K$ is a component of $\operatorname{EOdd}(a)$ is equivalent to saying that $m_{bc}=2$ whenever $b\in K$ and $c\in\operatorname{EOdd}(a)\setminus K$. So suppose that $c\in \operatorname{EOdd}(a)\setminus K$. Then there exists an $a'\in\operatorname{Odd}(a)$ such that $m_{a'c}$ is even. Thus $\{a',c\}$ is spherical, and as above it follows that $J\cup\{c\}$ is spherical. So $K$ must be a component of $J\cup\{c\}$, and since $c\notin K$ it follows that $m_{bc}=2$ for all $b\in K$, as required.
It remains to consider the case that $K\cap\operatorname{Odd}(a)=\emptyset$; we must show that either (f) or (d) holds. We start by supposing that there exists a spherical $L\subseteq\Pi$ and a $w\in W$ with $wJ\subseteq L$ and $wK$ not a component of $L$.
Choose such $L$ and $w$, and let $L_0$ be the component of $L$ containing $wa$. By Proposition \[finclos2\], since $wK$ is not a component of $L$ we must have $wK\subseteq L_0$. Now $wJ\cap L_0\ne L_0$ since $wK\ne L_0$, and $\{wa\}\ne wJ\cap L_0$ since $wa\notin wK$. So one of the alternatives (1), (2) or (3) of Proposition \[finclos2\] must hold. Alternative (3) can be ruled out, since in that case $wJ\cap L_0\subseteq\operatorname{Odd}(wa)$, which is impossible since $K\cap\operatorname{Odd}(a)=\emptyset$. If alternative (2) holds then $wK=wJ\cap L_0$ contains $wa$ and is of type $C_2$, whence $K$ contains $a$ and is of type $C_2$, and (b) is satisfied. If alternative (1) holds then since $wK\ne\{wa\}$ it follows from Proposition \[alt1c\] that $wK=\{b\}$, with $b$ a $C_3$-neighbour of $\operatorname{Odd}(a)$. Since $wa$ is not adjacent to $b$, it follows from Lemma \[c3lem\] that $w^{-1}b=b$, unless $a$ is adjacent to $b$, in which case $w^{-1}b=b+\sqrt 2a+\sqrt 2\tilde a$ for some $\tilde a$ in $\operatorname{Odd}(a)$. But this latter case cannot occur, since $w^{-1}b\in K\subseteq\Pi$. So $K=wK=\{b\}$, with $b$ a $C_3$-neighbour of $\operatorname{Odd}(a)$, and (d) holds.
Finally, suppose that $wK$ is a component of every spherical $L\subseteq\Pi$ such that $wJ\subseteq L$ for some $w\in W$. For each $c\in\operatorname{EOdd}(a)\setminus K$ there is then a sequence $a=a_0,a_1,\ldots,a_k=c$ in $\Pi$ such that $m_{a_{i-1}a_i}$ finite for all $i\in\{1,2,\ldots,k\}$ and odd for all $i\in\{1,2,\ldots,k-1\}$. We shall show that, for every such sequence, $m_{ba_i}=2$ for all $b\in K$ and $i\in\{0,1,\ldots,k\}$; in particular, this will show that $m_{bc}=2$ whenever $b\in K$ and $c\in\operatorname{EOdd}(a)\setminus K$, enabling us to conclude that $K$ is a component of $\operatorname{EOdd}(a)$.
The case $k=0$ is clear, since $a\in J\setminus K$ and $K$ is a component of $J$. Proceeding by induction, we may assume that $k>0$ and $m_{ba_i}=2$ for all $i\in\{1,2,\ldots,k-1\}$ and all $b\in K$. We see that the element $u=v[a_{k-1},\{a_{k-2}\}]v[a_{k-2},\{a_{k-3}\}]\cdots v[a_1,\{a_0\}]$ centralizes $W_K$ and has the property that $ua=a_{k-1}$, since the labels in the path from $a$ to $a_{k-1}$ are all odd. The group $u^{-1}W_{\{a_{k-1},a_k\}}u$ is finite and contains $u^{-1}r_{a_{k-1}}u=r_a$, and so there is a maximal finite subgroup $G$ of $W$ containing this group and also containing $W_J$.
Note that $W_J\cup\{u^{-1}r_cu\}\subseteq G=w^{-1}W_Lw$, for some $w\in W$ and spherical $L\subseteq\Pi$, the element $u$ being in the centralizer of $W_K$. We may choose $w$ to be the minimal length element of $W_Lw=W_LwW_J$, and it then follows from Lemma \[kilmoyerlem\] that $wJ\subseteq L$. Hence $wK$ is a component of $L$. Furthermore, since $wu^{-1}r_cuw^{-1}\in W_L$ we see that the root $wu^{-1}c$ is in $\Phi_L$ and not in $\Phi_{wK}=wu^{-1}\Phi_K$ (since $c\notin\Phi_K$). So $wu^{-1}c\cdot wu^{-1}b=0$ for all $b\in K$. So $c\cdot b=0$, or (equivalently) $m_{bc}=2$ for all $b\in K$, as required.
To complement the results we have obtained so far, our next task is to find conditions that ensure that a visible subgroup $W_K$ is contained in $\operatorname{FC}(r_a)$.
\[lem1a\] Let $a\in\Pi$ and $K$ a component of $\operatorname{EOdd}(a)$ such that $W_K$ is finite. Then $W_K\subseteq \operatorname{FC}(r_a)$.
Let $F$ be a maximal finite subgroup of $W$ with $r_a\in F$, and choose $w\in W$ such that $wFw^{-1}=W_L$ for some $L\subseteq\Pi$. We may replace $w$ by the minimal length element in the double coset $W_LwW_{\{a\}}$, since this does not affect the condition $wFw^{-1}=W_L$. So we have that $w^{-1}L\subseteq\Phi^+$, and, moreover, $r_a\in w^{-1}Lw\cap W_{\{a\}}=W_{w^{-1}L\cap\{a\}}$ by Lemma \[kilmoyerlem\]. So $wa\in L\subseteq\Pi$, and by Lemma \[normalizers\] we see that $w$ is a product of factors of the form $v[d,\{c\}]$, with $c,\,d\in\operatorname{EOdd}(a)$. Since $K$ is a component of $\operatorname{EOdd}(a)$ it follows that each $v[d,\{c\}]$ normalizes $W_K$, and therefore $w$ normalizes $W_K$. Moreover, since $wa\in L$ and $L$ is spherical, it follows that $L\subseteq\operatorname{EOdd}(wa)=\operatorname{EOdd}(a)$. So $W_L$ normalizes $W_K$. But $W_K$ is finite, by hypothesis, and $W_L$ is a maximal finite subgroup of $W$. So $W_K\subseteq W_L$, and $W_K=w^{-1}W_Kw\subseteq w^{-1}W_Lw=F$. Thus $W_K$ is contained in all maximal finite subgroups of $W$ containing $r_a$, as required.
\[morec3\] Let $a\in\Pi$ and let $b$ be a $C_3$-neighbour of $\operatorname{Odd}(a)$. If $a$ and $b$ are not adjacent in $\Pi$ then $r_b\in\operatorname{FC}(r_a)$.
Let $F$ be a maximal finite subgroup of $W$ with $r_a\in F$. As in the proof of Lemma \[lem1a\] there exist a $w\in W$ and a maximal spherical $L\subseteq\Pi$ with $wa=a'\in L$ and $F=w^{-1}W_Lw$. Since $L$ is spherical, $L\subseteq\operatorname{EOdd}(a)$.
Suppose first that $a'$ is not adjacent to $b$. Then $m_{ca'}=\infty$ for every $c\in\operatorname{Odd}(a)$ that is adjacent to $b$, and since $a'\in L$ it follows that no such $c$ is in $L$. Thus $m_{be}=2$ for all $e\in L\cap\operatorname{Odd}(a)$. But since also $m_{be}=2$ for all $e\in\operatorname{EOdd}(a)\setminus(\operatorname{Odd}(a)\cup\{b\})$, it follows that $m_{be}=2$ for all $e\in L\setminus\{b\}$. Thus $\{b\}$ is a component of $L\cup\{b\}$, and since $L$ is spherical it follows that $L\cup\{b\}$ is spherical. Maximality of $L$ tells us that $b\in L$. Moreover, Lemma \[c3lem\] gives $wb=b$, and so $r_b=w^{-1}r_bw\in w^{-1}W_Lw=F$.
On the other hand, suppose that $a'$ is adjacent to $b$. In this case Lemma \[c3lem\] gives $wb=b+\sqrt 2a'+\sqrt 2\tilde a$, where $\tilde a$ is the unique neighbour of $a'$ in $\operatorname{Odd}(a)$. Furthermore, since $m_{a'e}\in\{2.\infty\}$ for all $e\in\Pi\{\tilde a,a',b\}$, we see that $m_{a'e}=2$ for all $e\in L\setminus\{\tilde a,a',b\}$ (since $L$ is spherical). But the definition of a $C_3$-vertex also requires that $m_{\tilde ae}=m_{be}=2$ whenever $m_{a'e}=2$; so it follows that $\{\tilde a,a',b\}$ is a component of $L\cup\{\tilde a,a',b\}$, which is therefore spherical since $L$ and $\{\tilde a,a',b\}$ are both spherical. Maximality of $L$ tells us that $\{\tilde a,a',b\}\subseteq L$; so $wb=b+\sqrt 2a'+\sqrt 2\tilde a\in\Phi_L$, and $r_b=w^{-1}r_{wb}w\in w^{-1}W_Lw=F$.
So $r_b\in F$ in all cases, and so $r_b$ is contained in all maximal finite subgroups of $W$ containing $r_a$, as required.
We now prove the converse to Proposition \[alt2\].
\[alt2+\] Let $a\in\Pi$ and $b\in\Pi\setminus\operatorname{Odd}(a)$, and suppose that $(a,b)$ is a focus of $\operatorname{Odd}(a)$ in $\Pi$. Then $\operatorname{FC}(r_a)=W_J$, where $J$ is the union of $\{a,b\}$ and the spherical components of $\operatorname{EOdd}(a)$. Moreover, $\operatorname{FC}(r_{a'})$ is not visible for any $a'\in\operatorname{Odd}(a)\setminus\{a\}$.
For each $c\in\operatorname{Odd}(a)$ let $X(c)=b+\sqrt 2\sum_{i=1}^m c_i$ and $Y(c)=b+\sqrt 2\sum_{i=1}^{m-1} c_i$, where $c_1=a,\,c_2,\,\ldots,\,c_m=c$ is the unique path from $a$ to $c$ in $\operatorname{Odd}(a)$, noting that $X(c)$ and $Y(c)$ are roots in $\Phi_{C[b..c]}$. We remark, for later use, that $X(c)$ and $Y(c)$ are fixed by the reflections $r_b,\,r_{c_1},\,\ldots,\,r_{c_{m-2}}$.
Let $F=w^{-1}W_Lw$ be a maximal finite subgroup of $W$ containing $r_a$, with $L\subseteq\Pi$ and $w$ of minimal length in $W_LwW_{a}$. Then $wa=a'\in L$, by Lemma \[kilmoyerlem\]. Put $L_0=L\cap\operatorname{Odd}(a)$.
Choose $c\in L_0$ with $C[b..c]$ of maximal cardinality. If $d\in L_0$ then $m_{cd}\ne\infty$ (since $L_0$ is spherical), whence $d\in C[b..c]$ by condition (3) of Definition \[focus\]. So $L_0\subseteq C[b..c]$. Now if $e\in L\setminus L_0$ is arbitrary then $e\notin\operatorname{Odd}(a)$ (since $e\notin L\cap\operatorname{Odd}(a)$) and $m_{ce}<\infty$ (since $c,\,e\in L$ and $L$ is spherical). By condition (4) of Definition \[focus\] it follows that $m_{de}=2$ for all $d\in C[b..c]$. Since this holds for all $e\in L\setminus L_0$, and $C[b..c]$ and $L\setminus L_0$ are both spherical, it follows that $C[b..c]\cup(L\setminus L_0)$ is spherical. But this set contains $L$ (since $L_0\subseteq C[b..c]$) and since $L$ is a maximal spherical subset of $\Pi$ we conclude that $L=C[b..c]\cup(L\setminus L_0)$.
By Proposition \[normalizers\] and Lemma \[class\] there exist simple roots $e_1,\,e_2,\,\ldots,\,e_k$ and $d_1=a,\,d_2,\,\ldots,\,d_{k+1}=a'\in\operatorname{Odd}(a)$ with $w=v[e_k,\{d_k\}]\cdots v[e_2,\{d_2\}]v[e_1,\{d_1\}]$ and $v[e_i,\{d_i\}]d_i=d_{i+1}$ for all $i\in\{1,2,\ldots,k\}$. Moreover, $m_{e_id_i}<\infty$ for all $i$. Let $w_0=1$ and $w_i=v[e_i,\{d_i\}]w_{i-1}$; we will show that $$\{w_ib,-w_ib,w_i(b+\sqrt 2a),-w_i(b+\sqrt 2a)\}
=\{X(d_{i+1}),-X(d_{i+1}),Y(d_{i+1}),-Y(d_{i+1})\}$$ for all $i\in\{0,1,\ldots,k\}$. The case $i=0$ is trivial.
Proceeding inductively, suppose that $i>1$ and $$\{\pm w_{i-1}b,\pm w_{i-1}(b+\sqrt2a)\}=\{\pm X(d_i),\pm Y(d_i)\}.$$ It will be sufficient to show that $v[e_i,\{d_i\}]X(d_i)$ and $v[e_i,\{d_i\}]Y(d_i)$ both lie in the set $\{\pm X(d_{i+1}),\pm Y(d_{i+1})\}$.
Suppose first that $d_i=a$. Then $X(d_i)=b+\sqrt2 a$ and $Y(d_i)=b$. If $e_i\notin\operatorname{Odd}(a)$ then $m_{e_id_i}$ is even, and $d_{i+1}=d_i=a$. Furthermore, by condition (4) of Definition \[focus\]we have either $\{e_i,d_i\}=\{b,a\}$ or $m_{e_ib}=m_{e_ia}=2$. In the former case $v[e_i,\{d_i\}]=v[b,\{a\}]=r_br_ar_b$, giving $v[e_i,\{d_i\}]b=-b-\sqrt 2a=-X(a)$ and $v[e_i,\{d_i\}](b+\sqrt2a)=-b=-Y(a)$; in the latter case $v[e_i,\{d_i\}]=v[e_i\{a\}]=r_{e_i}$, giving $v[e_i,\{d_i\}]b=b=Y(a)$ and $v[e_i,\{d_i\}](b+\sqrt 2a)=b+\sqrt2a=X(a)$. If $e_i\in\operatorname{Odd}(a)$ then $a\in C[b..e_i]$, and by condition (2) of Definition \[focus\] we have $m_{e_ib}=2$ and either $m_{e_ia}=2$ or $m_{e_ia}=3$. If $m_{e_ia}=2$ then $d_{i+1}=d_i=a$, while if $m_{e_ia}=3$ then $d_{i+1}=e_i$. Furthermore, in former case we find that $v[e_i,\{d_i\}]b=r_{e_i}b=b=Y(a)$ and $v[e_i,\{d_i\}](b+\sqrt2a)=b+\sqrt2a=X(a)$, while in the latter case we find that $v[e_i,\{d_i\}]b=r_ar_{e_1}b=b+\sqrt 2a=Y(e_i)$ and $v[e_i,\{d_i\}](b+\sqrt2a+\sqrt2e_i)=X(e_i)$.
Now suppose that $d_i\ne a$. If $e_i\notin\operatorname{Odd}(a)\cup\{b\}$ then $m_{e_id_i}=2$ and $d_{i+1}=d_i$. Moreover, $m_{e_id}=2$ for all $d\in C[b..d_i]$, and so $v[e_i,\{d_i\}]=r_{e_i}$ fixes all the roots in $\Phi_{C[b..d_i]}$, including $X(d_i)=X(d_{i+1})$ and $Y(d_i)=Y(d_{i+1})$. If $e_i\in\operatorname{Odd}(a)\cup\{b\}$ and $\{e_i,d_i\}$ is not an edge of $\operatorname{Odd}(a)$ then we again have $d_{i+1}=d_i$ and $v[e_i,\{d_i\}]=r_{e_i}$. By condition (3) of Definition \[focus\] we either have $d_i\in C[b..e_i]$ or $e_i\in C[b..d_i]$. In the former case we have $m_{e_id}=2$ for all $d\in C[b..d_i]$, and as above we see that $r_{e_i}$ fixes $X(d_i)$ and $Y(d_i)$. In the latter case the remark made at the start of the proof implies that it is still true that $r_{e_i}$ fixes $X(d_i)$ and $Y(d_i)$. So we have shown that when $m_{e_id_i}=2$ it is true that $v[e_i,\{d_i\}]X(d_i)$ and $v[e_i,\{d_i\}]Y(d_i)$ both lie in the set $\{\pm X(d_{i+1}),\pm Y(d_{i+1})\}$, and it remains only to consider the case that $e_i$ and $d_i$ are adjacent in $\operatorname{Odd}(a)$. Note that in this case $d_{i+1}=e_i$.
Let $C[b..d_i]=\{b,c_1,\ldots,c_m\}$ with $c_1=a$ and $c_m=d_i$, and suppose that $e_i=c_{m-1}$ is the vertex adjacent to $d_i$ in $C[b..d_i]$. Then $$\displaylines{
v[e_i,\{d_i\}]X(d_i)=r_{c_m}r_{c_{m-1}}\Bigl(b+\sqrt2\sum_{j=1}^mc_j\Bigr)
=r_{c_m}\Bigl(b+\sqrt2\sum_{j=1}^mc_j\Bigr)\cr
=b+\sqrt2\sum_{j=1}^{m-1}c_j=X(e_i),\cr}$$ and similarly $$\displaylines{
v[e_i,\{d_i\}]Y(d_i)=r_{c_m}r_{c_{m-1}}\Bigl(b+\sqrt2\sum_{j=1}^{m-1}c_j\Bigr)
=r_{c_m}\Bigl(b+\sqrt2\sum_{j=1}^{m-2}c_j\Bigr)\cr
=b+\sqrt2\sum_{j=1}^{m-2}c_j=Y(e_i).\cr}$$ The alternative possibility is that $d_i$ is adjacent to $e_i$ in $C[b..e_i]$. Exactly the same calculations show that $v[e_i,\{d_i\}]X(d_i)=X(e_i)$ and $v[e_i,\{d_i\}]Y(d_i)=Y(e_i)$ in this case also.
The induction is now complete, and it follows in particular that $wb=w_kb$ is one of $\pm X(a')$ or $\pm Y(a')$. Hence $$wb\in\Phi_{C[b..a']}\subseteq\Phi_{C[b..c]}\subseteq\Phi_L.$$ Thus $wr_bw^{-1}\in W_L$, and so $r_b\in w^{-1}W_Lw=F$. Since $F$ was an arbitrary maximal finite subgroup of $W$ containing $r_a$, this shows that $r_b\in\operatorname{FC}(r_a)$.
Let $\widetilde M$ be the component of $\operatorname{EOdd}(a)$ containing $\operatorname{Odd}(a)$, and suppose, for a contradiction, that $\widetilde M$ is spherical. Clearly $b\in\widetilde M$, since $m_{ba}=4$, but $\widetilde M=\operatorname{Odd}(a)\cup\{b\}$ is not permitted, in view of condition (5) of Definition \[focus\]. So $\widetilde M\setminus(\operatorname{Odd}(a)\cup\{b\}\ne\emptyset$. But for $e\in\widetilde M\setminus(\operatorname{Odd}(a)\cup\{b\})$ and $c\in\operatorname{Odd}(a)$ we have $m_{ce}\ne\infty$, since $\widetilde M$ is spherical, and by condition (4) of Definition \[focus\] it follows that $m_{be}=m_{ce}=2$ for all $c\in\operatorname{Odd}(a)$. This contradicts the fact that $\widetilde M$ is connected.
Now suppose that $a'\in\operatorname{Odd}(a)$ is such that $\operatorname{FC}(r_{a'})=W_J$ for some $J\subseteq\Pi$, and let $J_0$ be the component of $J$ containing $a'$. Since $J_0\ne\widetilde M$ it follows from Proposition \[components\] that $J_0$ has rank at most 2. Now since there exists $w\in W_{C[b..a']}$ such that $wa=-a'$ and $wb=X(a')$, and since $r_b\in\operatorname{FC}(r_a)$, it follows that $r_{wb}=wr_bw^{-1}\in\operatorname{FC}(wr_aw^{-1})=\operatorname{FC}(r_{a'})$. Thus $X(a')\in\Phi_J$, and so $C[b..a']\subseteq J$. Since $J_0$ has rank at most 2, this means that $a'=a$ and $J_0=\{a,b\}$.
It remains to prove that $J$ is the union of $J_0$ and the spherical components of $\operatorname{EOdd}(a)$. By Lemma \[lem1a\] we know that all these components are contained in $J$. But if $K$ is any other component of $J$ such that $K\cap\operatorname{Odd}(a)=\emptyset$, then by Proposition \[components\] we see that $K=\{b'\}$, with $b'$ a $C_3$-neighbour of $\operatorname{Odd}(a)$. Since $b$ is the only element of $\Pi$ such that $m_{bc}\in\{2,4\}$ for all $c\in\operatorname{Odd}(a)$, we must have $b'=b$, contradicting the fact that the component of $J$ containing $b$ is $J_0=\{a,b\}$.
Next, we have the converse to Proposition \[alt3\].
\[alt3+\] Let $a\in\Pi$ and suppose that there exists a $b\in\operatorname{Odd}(a)$ such that $\{a,b\}$ is a half-focus of $\operatorname{Odd}(a)$ in $\Pi$. Suppose also that the vertices $\operatorname{Odd}(a)$ do not comprise a spherical subset of $\Pi$. Then $\operatorname{FC}(r_a)=W_J$, where $J$ is the union of $\{a,b\}$ and the spherical components of $\operatorname{EOdd}(a)$. Moreover, $\operatorname{FC}(r_{a'})$ is not visible for any $a'\in\operatorname{Odd}(a)\setminus\{a,b\}$.
For each $c\in\operatorname{Odd}(a)\setminus\{a,b\}$, define $$X(c)=b+a+c+2\sum_{i=2}^{m-1}c_i$$ where $c_1=a,\,c_2,\,\ldots,\,c_m=c$ is the unique path from $a$ to $c$ in $\operatorname{Odd}(a)\setminus\{b\}$. Then $X(c)$ is a root in $\Phi_{D[a,b..c]}$ and is fixed by the reflections $r_b,\,r_{c_1},\,\ldots,\,r_{c_{m-2}}$ and $r_{c_m}$. Define also $X(a)=b$ and $X(b)=a$.
Let $F=w^{-1}W_Lw$ be a maximal finite subgroup of $W$ containing $r_a$, with $L\subseteq\Pi$ and $w$ of minimal length in $W_LwW_{a}$. Then $wa=a'\in L$, by Lemma \[kilmoyerlem\]. Put $L_0=L\cap\operatorname{Odd}(a)$.
Choose $c\in L_0$ with $D[a,b..c]$ of maximal cardinality. If $d\in L_0$ then $m_{cd}\ne\infty$ (since $L_0$ is spherical), whence $d\in D[a,b..c]$ by condition (4) of Definition \[halffocus\]. So $L_0\subseteq D[a,b..c]$. Now if $e\in L\setminus L_0$ is arbitrary then $e\notin\operatorname{Odd}(a)$ (since $e\notin L\cap\operatorname{Odd}(a)$) and $m_{ce}<\infty$ (since $c,\,e\in L$ and $L$ is spherical). By condition (5) of Definition \[halffocus\] it follows that $m_{de}=2$ for all $d\in D[a,b..c]$. Since this holds for all $e\in L\setminus L_0$, and $D[a,b..c]$ and $L\setminus L_0$ are both spherical, it follows that $D[a,b..c]\cup(L\setminus L_0)$ is spherical. But this set contains $L$ (since $L_0\subseteq D[a,b..c]$) and since $L$ is a maximal spherical subset of $\Pi$ we conclude that $L=D[a,b..c]\cup(L\setminus L_0)$.
By Proposition \[normalizers\] and Lemma \[class\] there exist simple roots $e_1,\,e_2,\,\ldots,\,e_k$ and $d_1=a,\,d_2,\,\ldots,\,d_{k+1}=a'\in\operatorname{Odd}(a)$ with $w=v[e_k,\{d_k\}]\cdots v[e_2,\{d_2\}]v[e_1,\{d_1\}]$ and $v[e_i,\{d_i\}]d_i=d_{i+1}$ for all $i\in\{1,2,\ldots,k\}$. Furthermore, we have $m_{e_id_i}<\infty$ for all $i$. Let $w_0=1$, and $w_i=v[e_i,\{d_i\}]w_{i-1}$ for $i\ge 1$. We will show that $$\{w_ib,-w_ib\}=\{X(d_{i+1}),-X(d_{i+1})\}$$ for all $i\in\{0,1,\ldots,k\}$.
The case $i=0$ is trivial. Proceeding inductively, suppose that $i>1$ and $w_{i-1}b=\pm X(d_i)$. It will be sufficient to show that $v[e_i,\{d_i\}]X(d_i)=\pm X(d_{i+1})$.
Suppose first that $d_i=a$, so that $X(d_i)=b$. If $e_i\ne b$ then $m_{e_ib}=m_{e_ia}\in\{2,3\}$, since $m_{e_ia}=m_{e_id_i}\ne\infty$. We also have $m_{e_ia}=2$ if $e_i=b$. In the case $m_{e_ia}=3$ we have $v[e_i,\{d_i\}]=r_ar_{e_i}$, and $d_{i+1}=r_ar_{e_i}a=e_i$. Furthermore, $$v[e_i,\{d_i\}]X(d_i)=r_ar_{e_i}b=a+b+e_i=X(e_i)=X(d_{i+1}),$$ as required. In the case $m_{e_ia}=2$ we have $v[e_i,\{d_i\}]=r_{e_i}$, giving $d_{i+1}=r_{e_i}a=a$, and $$v[e_i,\{d_i\}]X(d_i)=r_{e_i}b=\pm b=\pm X(d_{i+1}),$$ since either $e_i=b$ or $m_{e_ib}=2$.
The case $d_i=b$ is the same as the case $d_i=a$ with $a$ and $b$ interchanged; so suppose that $d_i\notin\{a,b\}$. If $e_i\notin\operatorname{Odd}(a)$ then $m_{e_id_i}=2$ and $d_{i+1}=d_i$. Moreover, $m_{e_id}=2$ for all $d\in D[a,b..d_i]$, and so $v[e_i,\{d_i\}]=r_{e_i}$ fixes all the roots in $\Phi_{D[a,b..d_i]}$, including $X(d_i)=X(d_{i+1})$. If $e_i\in\operatorname{Odd}(a)$ and $\{e_i,d_i\}$ is not an edge of $\operatorname{Odd}(a)$ then we again have $d_{i+1}=d_i$ and $v[e_i,\{d_i\}]=r_{e_i}$. By condition (4) of Definition \[halffocus\] we either have $d_i\in D[a,b..e_i]$ or $e_i\in D[a,b..d_i]$. In the former case we have $m_{e_id}=2$ for all $d\in C[b..d_i]$, and as above we see that $r_{e_i}$ fixes $X(d_i)$. In the latter case it is still true that $r_{e_i}$ fixes $X(d_i)$, since the only simple reflection of $D[a,b..d_i]$ that does not fix $X(d_i)$ is the one corresponding to the vertex adjacent to $d_i$. So we have shown that when $m_{e_id_i}=2$ it is true that $v[e_i,\{d_i\}]X(d_i)$ and $v[e_i,\{d_i\}]Y(d_i)$ both lie in the set $\{\pm X(d_{i+1}),\pm Y(d_{i+1})\}$, and it remains to consider the case that $e_i$ and $d_i$ are adjacent in $\operatorname{Odd}(a)$. Note that in this case $d_{i+1}=e_i$.
Let $D[a,b..d_i]=\{b,c_1,\ldots,c_m\}$ with $c_1=a$ and $c_m=d_i$. Suppose first that $m>2$, and suppose that $e_i=c_{m-1}$ is the vertex adjacent to $d_i$ in $D[a,b..d_i]$. Then $$\begin{aligned}
v[e_i,\{d_i\}]X(d_i)&=r_{c_m}r_{c_{m-1}}\Bigl(b+a+c_m+2\sum_{j=2}^{m-1}c_j\Bigr)\\
&=r_{c_m}\Bigl(b+a+c_m+c_{m-1}+2\sum_{j=1}^{m-2}c_j\Bigr)\\
&=b+a+c_{m-1}+2\sum_{j=1}^{m-2}c_j=X(e_i).\end{aligned}$$ If $m=2$ and $e_i=b$ then $$v[e_i,\{d_i\}]X(d_i)=r_br_{d_i}(a+b+d_i)=a=X(b)=X(e_i),$$ and if $e_i=a$ then similarly $$v[e_i,\{d_i\}]X(d_i)=r_ar_{d_i}(a+b+d_i)=b=X(a)=X(e_i).$$ The alternative possibility is that $d_i=c_{m-1}$ is the vertex adjacent to $e_i=c_m$ in $D[a,b..e_i]=\{b,c_1,\ldots,c_m\}$. We calculate that $$\begin{aligned}
v[e_i,\{d_i\}]X(d_i)
&=r_{c_{m-1}}r_{c_m}\Bigl(b+a+c_{m-1}+2\sum_{j=2}^{m-2}c_j\Bigr)\\
&=r_{c_{m-1}}\Bigl(b+a+c_m+c_{m-1}+2\sum_{j=1}^{m-2}c_j\Bigr)\\
&=b+a+c_m+2\sum_{j=1}^{m-1}c_j=X(e_i),\end{aligned}$$ as required.
The induction is now complete, and it follows that $wb=w_kb=\pm X(a')$. Hence $$wb\in\Phi_{D[a,b..a']}\subseteq\Phi_{D[a,b..c]}\subseteq\Phi_L.$$ Thus $wr_bw^{-1}\in W_L$, and so $r_b\in w^{-1}W_Lw=F$. Since $F$ was an arbitrary maximal finite subgroup of $W$ containing $r_a$, this shows that $r_b\in\operatorname{FC}(r_a)$.
Note that since $W$ has a graph automorphism that swaps $r_a$ and $r_b$ and fixes all the other simple reflections, it must also be true that $r_a\in\operatorname{FC}(r_b)$.
Let $\widetilde M$ be the component of $\operatorname{EOdd}(a)$ containing $\operatorname{Odd}(a)$, and suppose, for a contradiction, that $\widetilde M$ is spherical. Note that $\widetilde M\ne\operatorname{Odd}(a)$, in view of condition (5) of Definition \[focus\]. So $\widetilde M\setminus\operatorname{Odd}(a)\ne\emptyset$. But for all $e\in\widetilde M\setminus\operatorname{Odd}(a)$ and $c\in\operatorname{Odd}(a)$ we have $m_{ce}\ne\infty$, since $\widetilde M$ is spherical, and by conditions (1) and (5) of Definition \[focus\] it follows that $m_{ce}=2$ for all $c\in\operatorname{Odd}(a)$. This contradicts the fact that $\widetilde M$ is connected.
Suppose that $a'\in\operatorname{Odd}(a)$ is such that $\operatorname{FC}(r_{a'})=W_J$ for some $J\subseteq\Pi$, and let $J_0$ be the component of $J$ containing $a'$. Since $J_0\ne\widetilde M$ it follows from Proposition \[components\] that $J\cap\operatorname{Odd}(a)$ has rank at most 2. Now suppose, for a contradiction, that $a'\notin\{a,b\}$. Since there exists an element $w\in W_{D[a,b..a']}$ such that $wa=a'$ and $wb=X(a')$, it follows that $r_{wb}=wr_bw^{-1}\in\operatorname{FC}(wr_aw^{-1})=\operatorname{FC}(r_{a'})$. Thus $X(a')\in\Phi_J$, and so $D[a,b..a']\subseteq J$, contradicting the fact that the rank of $J\cap\operatorname{Odd}(a)$ is at most 2. So we deduce that $a'=b$ or $a'=a$. Moreover, in either case we know that $\{a,b\}\subseteq J\cap\operatorname{Odd}(a)$, and since $J\cap\operatorname{Odd}(a)$ has rank at most 2 it follows that $J\cap\operatorname{Odd}(a)=\{a,b\}$.
By Lemma \[lem1a\] we know that all spherical components of $\operatorname{EOdd}(a)$ are components of $J$, and by Proposition \[components\] all other components of $J$ that intersect $\operatorname{Odd}(a)$ trivially correspond to $C_3$-neighbours of $\operatorname{Odd}(a)$. But clearly the conditions of Definition \[halffocus\] imply that $\operatorname{Odd}(a)$ has no $C_3$-neighbours. So we we conclude that $J$ is the union of $\{a,b\}$ and the spherical components of $\operatorname{EOdd}(a)$, as required.
Proof of Theorem \[main\] {#proof-of-theoremmain .unnumbered}
-------------------------
.
Let $M$ be a connected component of $\operatorname{\Omega}(\Pi)$, and write $\widetilde M$ for the component of $\operatorname{E}(M)$ containing $M$.
Suppose first that $\widetilde M$ is spherical, so that the conditions of Case A of Theorem \[main\] are satisfied, and let $a\in M$ be arbitrary. Observe that all $C_3$-neighbours of $M$ are contained in $\widetilde M$. Choose $a'\in M$ such that $\operatorname{FC}(r_{a'})$ is visible, and let $\operatorname{FC}(r_{a'})=W_J$. By Lemma \[lem1a\] we know that $\widetilde M$ is contained in $J$, and hence $a\in J$. So by Proposition \[finclos\] it follows that $\operatorname{FC}(r_a)=W_J$ also. By Proposition \[components\] the only possible components of $J$ apart from $J_0$ are the other spherical components of $\operatorname{E}(M)$, and by Lemma \[lem1a\] all of these are indeed components of $J$. So $J$ consists of the spherical components of $\operatorname{E}(M)$, as required.
Now suppose that $\widetilde M$ is not spherical. If there exists a $b\in\Pi\setminus M$ such that $(a,b)$ is a focus of $M$ then it follows from Proposition \[alt2+\] that $\operatorname{FC}(r_a)=W_J$, where $J$ is composed of $\{a,b\}$ and the spherical components of $\operatorname{E}(M)$, and $\operatorname{FC}(r_{a'})$ is not visible for any $a'\in\operatorname{Odd}(a)\setminus\{a\}$. Similarly, if there exists a $b\in M$ such that $\{a,b\}$ is a half-focus of $M$, then it follows from Proposition \[alt3+\] that $\operatorname{FC}(r_a)=\operatorname{FC}(r_b)=W_J$, where $J$ is composed of $\{a,b\}$ and the spherical components of $\operatorname{E}(M)$, and $\operatorname{FC}(r_{a'})$ is not visible for any $a'\in\operatorname{Odd}(a)\setminus\{a,b\}$.
Finally, suppose that $\widetilde M$ is not spherical and $M$ does not have a focus or a half focus. Suppose that $a\in M$ is such that $\operatorname{FC}(r_a)=W_J$ for some $J\subseteq\Pi$, and let $K$ be the component of $J$ containing $a$.
Suppose first that alternative (b) of Proposition \[components\] holds, so that $K=\{a,b\}$ is of type $C_2$, and $J\cap M=\{a\}$. Since $M$ does not have any focus in $\Pi$, it follows from Proposition \[alt2\] that $M\cup\{b\}$ is a spherical component of $\operatorname{E}(M)$. But the component of $\operatorname{E}(M)$ containing $M$ is $\widetilde M$, which, by our assumptions, is not spherical. So this case does not arise.
Alternative (c) of Proposition \[components\] is similarly impossible, by Proposition \[alt3\], and alternative (e) is also incompatible with the assumption that $\widetilde M$ is not spherical. So we conclude that alternative (a) holds: $K=\{a\}=J\cap\operatorname{Odd}(a)$. Note that all spherical components of $\operatorname{E}(M)$ are components of $J$, and by Proposition \[components\] the only other possible components are the sets $\{b\}$ such that $b$ is a $C_3$-neighbour of $M$.
Suppose that $b$ is a $C_3$-neighbour of $M$ that is adjacent to $a$. Let $\tilde a$ be the unique neighbour of $a$ in $M$. By Lemma \[morec3\] we know that $r_b\in\operatorname{FC}(r_{\tilde a})$, and so it follows that $\operatorname{FC}(r_{a})=r_{\tilde a}r_{a}\operatorname{FC}(r_{\tilde a})r_{a}r_{\tilde a}$ contains the reflection along the root $r_{\tilde a}r_ab=b+\sqrt 2a+\sqrt 2\tilde a$. Since $\operatorname{FC}(r_{a})=W_J$, it follows that both $b$ and $\tilde a$ are in $J$. But this contradicts the fact that the component of $J$ containing $a$ is just $\{a\}$.
This reasoning has shown that $a\in M$ is adjacent in $\Pi$ to a $C_3$-neighbour of $M$ then $\operatorname{FC}(r_a)$ is not visible. On the other hand, we know that there is at least one $a\in M$ such that $\operatorname{FC}(r_a)$ is visible. So we may choose an $a\in M$ such that $\operatorname{FC}(r_a)=W_J$ for some $J\subseteq\Pi$. Since $a$ is not adjacent to any $C_3$-neighbour of $M$ it follows by Lemma \[morec3\] that all $C_3$-neighbours of $M$ are in $J$. So we conclude that $J=J'\cup\{a\}$, where $J'$ is the union of the spherical components of $\operatorname{E}(M)$ and the $C_3$-neighbours of $M$.
It remains to prove that is $a'$ is any other element of $M$ that is not adjacent to any $C_3$-neighbour of $M$ then $\operatorname{FC}(r_{a'})=W_{J'\cup\{a'\}}$. Given such an $a'$, since $a'$ lies in $M=\operatorname{Odd}(a)$, we may choose $w\in W$ such that $wa=a'$. By Proposition \[normalizers\] we see that $w\in W_{\widetilde M}$, and so $w$ fixes all other components of $\operatorname{E}(M)$. And $w$ fixes all $C_3$-neighbours of $M$, by Lemma \[c3lem\]. So $w$ fixes $J'$, and it follows that $$\operatorname{FC}(r_{a'})=w\operatorname{FC}(r_a)w^{-1}=wW_{J'\cup\{a\}}w^{-1}=W_{wJ'\cup\{wa\}}=
W_{J'\cup\{a'\}},$$ as required. This completes the proof of Theorem \[main\].
Proof of Theorem \[thm2sph\] {#proof-of-theorem-thm2sph .unnumbered}
----------------------------
Let $a \in \Pi$ and $M = \operatorname{Odd}(a)$. As $\Pi$ is 2-spherical it follows that $\Pi = E(M)$, and, as $\Pi$ is non-spherical, it follows that Case A of Theorem \[main\] does not hold for $M$. As there are no $\infty$-labels in the Coxeter graph of $\Pi$, Cases C and D do not hold either. Hence we are in Case B. Since there are no $\infty$-labels in the Coxeter graph of $\Pi$, there are no $C_3$ neighbors of $M$. As $E(M) = \Pi$ is irreducible, there are no spherical components of $E(M)$. It follows now from Theorem \[main\] that there is an $a' \in \operatorname{Odd}(a)$ such that $\operatorname{FC}(r_{a'}) = \langle r_{a'} \rangle$. As $r_a$ and $r_{a'}$ are $W$-conjugate we have $\operatorname{FC}(r_a) = \langle r_a \rangle$ as well, and this completes the proof of Part a) of Theorem \[thm2sph\].
Let $S \subseteq W$ be such that $(W,S)$ is a Coxeter system. It follows from Part a) and Corollary \[FCcrit1\] that $r_a \in S^W$ for each $a \in \Pi$, and hence $\{ r_a \mid a \in \Pi \} \subseteq S^W$. As $\Pi$ is assumed to be non-spherical, irreducible and 2-spherical, it follows now from the main result of [@CM] that there is an element $w \in W$ such that $\{ r_a \mid a \in \Pi \} = S^w$. This completes the proof of Part b) of Theorem \[thm2sph\]. As Part c) is an immediate consequence of Part b) we are done.
[10]{}
P. Bahls and M. Mihalik, Reflection Independence in even Coxeter Groups, Preprint (2002) to appear in [*Geometriae Dedicata*]{}.
N. Bourbaki, , Chapitres 4, 5 et 6, Hermann, Paris, 1968.
N. Brady, J. McCammond, B. Mühlherr and W. Neumann, Rigidity of Coxeter Groups and Artin Groups, (2002), 91–109.
Brigitte Brink, On centralizers of reflections in Coxeter groups, (1996), 465–470. P. E. Caprace and B. Mühlherr, Reflection rigidity of 2-spherical Coxeter groups, Preprint (2003), 24p. to appear in [*Proc. London Math. Soc.*]{}.
Roger W. Carter, , J. Wiley & Sons, 1985.
V. V. Deodhar, On the root system of a Coxeter group, (1982), 611–630.
V. V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, (1989), 543–546.
Matthew Dyer, Reflection subgroups of Coxeter systems, (1990), 57–73.
W. N. Franzsen and R. B. Howlett, Automorphisms of nearly finite Coxeter groups, (to appear).
R. B. Howlett, Normalizers of parabolic subgroups of reflection groups, (1980), 62–80.
R. B. Howlett and B. Mühlherr, Isomorphisms of Coxeter groups which do not preserve reflections, Preprint (2004), 18p.
R. B. Howlett, P. J. Rowley and D. E. Taylor, On outer automorphism groups of Coxeter groups, (1997), 499–513.
James E. Humphreys, , Cambridge University Press, 1990 (Cambridge Studies in Advanced Mathematics, Vol. 29).
B. Mühlherr, The isomorphism problem for Coxeter groups, Preprint (2005), 15p, to appear Proceedings of the [*Coxeter Legacy Conference*]{}, Toronto, May 2004.
R. W. Richardson, Conjugacy classes of involutions in Coxeter groups, (1982), 1–15.
|
---
abstract: 'The quantum Hall effect is necessarily accompanied by low-energy excitations localized at the edge of a two-dimensional electron system. For the case of electrons interacting via the long-range Coulomb interaction, these excitations are edge magnetoplasmons. We address the time evolution of localized edge-magnetoplasmon wave packets. On short times the wave packets move along the edge with classical $E$ cross $B$ drift. We show that on longer times the wave packets can have properties similar to those of the Rydberg wave packets that are produced in atoms using short-pulsed lasers. In particular, we show that edge-magnetoplasmon wave packets can exhibit periodic revivals in which a dispersed wave packet reassembles into a localized one. We propose the study of edge-magnetoplasmon wave packets as a tool to investigate dynamical properties of integer and fractional quantum-Hall edges. Various scenarios are discussed for preparing the initial wave packet and for detecting it at a later time. We comment on the importance of magnetoplasmon-phonon coupling and on quantum and thermal fluctuations.'
address: |
$^1$Department of Physics, Indiana University, Bloomington, IN 47405\
$^2$Department of Physics, Colby College, Waterville, ME 04901
author:
- 'Ulrich Zülicke,$^1$ Robert Bluhm,$^2$ V. Alan Kostelecký,$^1$ and A. H. MacDonald$^1$'
title: 'Edge-Magnetoplasmon Wave-Packet Revivals in the Quantum Hall Effect'
---
Introduction
============
The quantum Hall (QH) effect occurs in two-dimensional (2D) electron systems (ES) whenever the chemical potential has a discontinuity which occurs at a magnetic-field-dependent density.[@ahmintro] For isotropic 2D ES, charge gaps occur at integer and certain non-integer but rational values of the filling factor $\nu :=
\frac{n}{B}\,\Phi_0$, where $\Phi_0 = hc/e $ is the magnetic-flux quantum. For the most part, we will assume in this paper that the 2D ES has a filling factor that is the inverse of an odd integer: $\nu
= 1/m$, with $m$ odd. When the QH effect occurs, the bulk of the system is incompressible in the absence of disorder, and the only gapless excitations are localized near the boundary of the finite QH sample.[@ahm:prl:90] For $\nu =1/m$ and the case of a sharp edge, [*i.e.*]{}, a confining potential that varies rapidly on a length scale of the order of the magnetic length $\ell=\sqrt{\hbar c /|e B|}$, the edge excitations are expected[@ahm:braz:96] to be well-described by a [*chiral Luttinger liquid*]{} (CLL) theory[@wen:prb:90; @wen:prb:91a; @wen:int:92] with a single branch of chiral bosons. In this theory, the edge of a two-dimensional electron system is thought of as a one-dimensional electron gas,[@emery; @sol:adv:79; @fdmh:jpc:81] which can be studied using bosonization techniques. The only low-lying excitations are collective bosonic density waves. For sharp $\nu =1/m$ edges in the quantum-Hall regime, it turns out[@ahm:prl:90; @wen:prb:90] that there is a single branch of bosons and that these are chiral, [*i.e.*]{}, they occur with only one sign of wave vector. If a long-range Coulomb interaction is present, the bosons are called edge magnetoplasmons[@hel1:85; @emplcaveat; @vav-sam:jetp:88] (EMP). It is possible to derive an expression for the energy dispersion relation of edge magnetoplasmons starting from the microscopic Hamiltonian for electrons moving in a strong magnetic field and interacting via a 3D Coulomb interaction.[@uz-ahm] The result, which is valid for both the disk and strip geometries, is $$\label{disperse}
\varepsilon^{\text{C}}_{M} = - \frac{M}{R} \ln{\left[\alpha
\frac{M}{R} \right]} \quad .$$ Expressions differing from Eq. (\[disperse\]) only in the constant $\alpha$ had been obtained earlier using a semiclassical approach.[@vav-sam:jetp:88; @wass:prb:90; @bla:phyb:92] Here, the constant $\alpha$ is of order unity and depends weakly on the geometry of the QH sample, $M$ is a positive integer, and $R$ is related to the perimeter $L$ of the QH sample: $R={L}/{2\pi}$. For the disk geometry, $R$ is the radius of the disk. In Eq. (\[disperse\]), as well as in all expressions to follow in this paper, we measure physical quantities in [*quantum-Hall units*]{} to simplify expressions. These units are defined in Table \[table\_nsu\].
We can think of the edge of a QH sample as an excitable one-dimensional medium, much like a string. The edge magnetoplasmons are the eigenmodes of this medium. For the case of a short-range interaction between the electrons, their dispersion is linear in the wave number: $\varepsilon^{\text{sr}}_{M} = v_{\text{F}} \, M / R$. If a long-range Coulomb interaction is present, the dispersion relation is nonlinear and is given by Eq. (\[disperse\]).
------------------------------------------------------------------------------------------
Quantity QH unit (in cgs) Values (for GaAs)
------------- ----------------------------------------- ----------------------------------
length $\ell = \sqrt{\hbar c / |e B|}$ $25.7 \times
B[\rm T]^{- 1/2} \,$ nm
energy $(\nu e^2) / (\pi \epsilon \ell)$ $1.39 \times \nu \,
B[\rm T]^{1/2}$ meV
time $(\hbar \pi \epsilon \ell) / (\nu e^2)$ $4.74 \times
10^{-13} / \nu B[\rm T]^{1/2}$ s
temperature $(\nu e^2) / (\pi \epsilon \ell k_B)$ $16 \times
\nu B[\rm T]^{1/2}$ K
------------------------------------------------------------------------------------------
: Quantum-Hall units. Throughout this paper, physical quantities are measured in these units. Besides the formal expression for each quantum-Hall unit, explicit values are given (as a function of magnetic field $B$ and filling factor $\nu$) which apply to 2D ES in GaAlAs/GaAs-heterostructures. The symbols $e$, $\hbar$, and $k_B$ denote the electron charge, Planck constant and Boltzmann constant, respectively.
\[table\_nsu\]
Using a time-dependent external potential, it is possible to excite a superposition of these eigenmodes. In this way, a wave packet can be created for the electron number density along the edge. We refer to these as edge-magnetoplasmon wave packets (EMP WP).
In this paper, we examine the formation and evolution of EMP WP in two-dimensional electron systems. Experimental studies[@hls:prb:83; @tal:jetp:89; @heit:prl:90; @heit:prl:91; @ray:prb:92; @zhi:prl:93; @hel2:85; @hel:91; @hel1:85; @wass:prb:90] of EMP have employed different excitation processes, including capacitive coupling[@ray:prb:92] or ohmic contacts[@zhi:prl:93] between the QH edge and an exciting voltage pulse. This paper is motivated most strongly by the capacitive coupling[@ray:prb:92] approach and we comment later on the interpretation of these experiments. The physical observables involved in these probes can all be expressed in terms of the operators involving the creation and annihilation operators for bosonic density fluctuations (EMP): $b_{M}^{\dagger}$ and $b_{M}$. In particular, the one-dimensional number density of electrons at the edge of a sample with filling factor $\nu$, defined by integrating perpendicular to the edge and comparing with the ground-state density, is related to boson creation and annihilation operators by[@wen:int:92] $$\label{density}
\varrho(\theta) = \sum_{M>0} \frac{(\nu M)^{1/2}}{2\pi R} \, \left[
b_{M} \, e^{i M \theta} + b^{\dagger}_{M} \, e^{- i M \theta} \right]
\quad .$$ In this expression, we have fixed the direction of the magnetic field. Reversing the field direction corresponds to interchanging $b_{M}^{\dagger}$ and $b_{M}$. In our convention, the EMP travel counterclockwise. The experimental configuration is shown in Fig. \[geometry\].
Localized electron wave packets have been produced and studied in atoms using short-pulsed lasers.[@ps; @az; @tenWolde; @yeazell; @meacher] A superposition of highly excited or Rydberg states is formed when a short laser pulse coherently excites a single electron far from the ground state of an atom. The resulting Rydberg wave packet is localized spatially, and its initial motion mimics the classical periodic motion of a charged particle in a Coulomb potential. The wave packet will eventually disperse and lose its classical character. However, the wave packet reassembles at later times in a sequence of fractional and full revivals and superrevivals.[@ps; @az; @ap; @nau; @sr] The revivals result from quantum interference between the different eigenstates in the superposition. The appearance of revivals is quite general and can occur in quantum systems other than Rydberg atoms,[@bkp] including systems with eigenenergies depending on more than one quantum number.[@revs23]
In the present work, we perform an analysis of the revival structure of EMP WP in two-dimensional electron systems that on short times exhibit classical $E$ cross $B$ drift motion in the electric field that confines the electronic system to a finite area. At longer times these wave packets disperse but we show that they can also have fractional and full revivals and superrevivals.
=8.5cm
We begin in Section \[sec2\] with a brief review of Rydberg wave-packet behavior, including a discussion of fractional and full revival of dispersed wave packets. Section \[sec3\] describes the dynamics of EMP in QH samples with sharp edges. The exact solution of the time-evolved edge-magnetoplasmon wave packet is obtained. We demonstrate that EMP WP can exhibit full and fractional revivals that we illustrate for one particular set of experimental parameters. The similarities between Rydberg wave packets and EMP WP are examined. In Section \[sec4\], we discuss experimental issues, taking into account the decay of EMP WP due to scattering processes occurring in the semiconductor host material. Our conclusions are presented in Section \[sec5\]. Some details of the derivations are given separately in appendices.
Atomic Rydberg Wave Packets {#sec2}
===========================
In this section, we present some background on atomic Rydberg wave packets. These wave packets form when a short-pulsed laser excites an atomic electron into a coherent superposition of large-quantum-number Rydberg states. Several theoretical approaches have been used to investigate Rydberg wave packets. They have been studied both numerically[@ps] and perturbatively.[@az] A description in terms of squeezed states has also been given.[@squeezed] Rydberg wave packets offer the possibility of approaching the classical limit of motion for electrons in atoms. Initially, the motion of a Rydberg wave packet is semiclassical, exhibiting the classical periodic motion of a charged particle in a Coulomb field. The period of the motion is the classical keplerian period $T_{\text{cl}}$. This semiclassical motion typically lasts for only a few cycles, after which the wave packet disperses and collapses. However, quantum interference effects subsequently cause the wave packet to undergo a sequence of fractional and full revivals.
The full revivals are characterized by the recombination of the collapsed wave packet into a form that resembles the initial wave packet. This first occurs at a time $t_{\text{rev}}$. At the full revival, the wave packet again oscillates with the classical period $T_{\text{cl}}$. The fractional revivals occur at earlier times equal to irreducible rational fractions of $t_{\text{rev}}$. At the fractional revivals, the wave packet separates into a set of equally weighted subsidiary wave packets. The motion of the subsidiary wave packets is periodic with period equal to a rational fraction of the classical period $T_{\text{cl}}$.[@ap] Eventually the periodic revivals fail, but on a still longer time scale they can reappear and a higher level of revival structure commences.[@sr] This occurs on a longer time scale called the superrevival time $t_{\text{sr}}$. At times equal to certain rational fractions of $t_{\text{sr}}$, distinct subsidiary waves form again, but with a period equal to a rational fraction of $t_{\text{rev}}$. These long-term fractional revivals culminate with the formation of a superrevival at the time $t_{\text{sr}}$. At the superrevival, the wave packet can resemble the initial wave packet more closely than at the full revival time $t_{\text{rev}}$. An analysis including the superrevival time scale has been performed[@sr] both for hydrogenic models and using supersymmetry-based quantum-defect theory,[@sqdt] to model the Rydberg alkali-metal atoms typically used in experiments. For times typically a few orders of magnitude greater than $t_{\text{sr}}$, atomic Rydberg wave packets spontaneously decay into lower-energy states by emitting photons.
Experiments[@tenWolde; @yeazell; @meacher; @wals] have studied the revival structure of Rydberg wave packets through times $\sim t_{\text{rev}}$. These experiments use a pump-probe method of detection involving either photoionization[@az] or phase-sensitive Ramsey interference and electric-field ionization.[@noord; @broers; @christian] In both these procedures, the wave packet is excited initially by a short laser pulse that creates a superposition of energy eigenstates with principal quantum number centered on a value $\bar n$. The wave packet initially forms near the nuclear core of the atom. After the pump pulse has passed, the wave packet evolves under the influence of the Coulomb potential, oscillating between inner and outer apsidal points of the keplerian ellipse corresponding to $\bar n$.
In pump-probe experiments using photoionization, a second laser pulse called the probe pulse ionizes the atom. The photoionization signal is measured as a function of the delay time $t$ between the pump and probe signals. The transition probability for absorbing the second photon is greatest when the wave packet is near the core and falls to zero as the wave packet moves away from the nucleus. As a result, the periodicity in the photoionization signal corresponds to the periodicity in the probability for the wave packet to return to its initial configuration. Experiments using this method have detected the initial periodic motion of the wave packet with period $T_{\text{cl}}$, as well as fractional revivals at delay times equal to fractions of $t_{\text{rev}}$, and a full revival at $t_{\text{rev}}$. The fractional revivals are characterized by periodicities in the photoionization signals equal to rational fractions of $T_{\text{cl}}$, as expected.
A second type of pump-probe experiment is based on Ramsey’s method of separated oscillating fields.[@norm] In this method, an initial laser pulse creates a wave packet, which is then excited by a second identical laser pulse. Depending on the relative phase between the two time-separated optical pulses, the upper-state population in the wave packet can be either increased or reduced by the second pulse. This population transfer between the excited levels and the ground state falls to zero as the wave packet moves towards its outer turning point. The population of the excited levels as a function of the delay time thus appears as a rapidly oscillating function, due to the Ramsey interference, that is modulated by an envelope dependent on the wave-packet motion. By monitoring the population of the excited levels as a function of time using electric-field ionization, the motion of the wave packet can be detected via the periodicities in the envelope function. Since electric-field ionization is more efficient than photoionization, the Ramsey method is better able to resolve fractional revivals. Indeed, using this method, fractional revivals consisting of as many as seven subsidiary wave packets have been detected.[@wals]
The wave function for a Rydberg wave packet can be written as a superposition of energy eigenstates $$\label{psi}
\Psi(\vec r,t) = \sum_n c_n \psi_n (\vec r) e^{-i E_n t / \hbar}
\quad ,$$ where $c_n$ are weighting coefficients and $\psi_n$ are energy eigenstates. For excitations with a short laser pulse, the coefficients $c_n$ are strongly peaked around a central value of $n$ equal to $\bar n$. This permits a Taylor expansion of the energy around the value $\bar n$. The first three derivative terms in this expansion define the time scales $T_{\text{cl}}$, $t_{\text{rev}}$, and $t_{\text{sr}}$ as follows:
\[gentimes\] $$\begin{aligned}
T_{\text{cl}} &=& \frac {2 \pi} {\vert (E_{\bar n})^\prime \vert}
\quad , \\
t_{\text{rev}} &=& \frac {2 \pi} {\frac 1 2 \vert (E_{\bar n}
)^{\prime\prime} \vert} \quad , \\
t_{\text{sr}} &=& \frac {2 \pi} {\frac 1 6 \vert (E_{\bar n}
)^{\prime\prime\prime} \vert} \quad .\end{aligned}$$
The primes denote derivatives with respect to $n$. The evolution of the wave packet is controlled by the interplay between these three time scales in the time-dependent phase.
For short times $t \ll t_{\text{rev}}$, only the first-derivative term in the Taylor expansion of $E_n$ contributes significantly, and we can write $\psi(\vec r,t) \approx \psi_{\text{cl}}(\vec r,t)$, where $$\label{psicl}
\psi_{\text{cl}}(\vec r,t) = \sum_n c_n \psi_n (\vec r)
e^{-i 2 \pi (n - \bar n)t / T_{\text{cl}} } \quad ,$$ and we have omitted an overall phase. The function $\psi_{\text{cl}}$ is periodic with period $T_{\text{cl}}$. This expression is valid only for $ t \ll t_{\text{rev}}$, however. Eventually the second-derivative term in the phase becomes appreciable, and the initial periodic motion is destroyed. The fractional revivals occur at times $t = m \,
t_{\text{rev}}/n$, where $m$ and $n$ are relatively prime integers ($m\le n$). At these times, it has been shown[@ap] that, with third and higher derivatives neglected, the wave function can be written as a sum of subsidiary wave functions: $$\label{fracrevs}
\Psi(\vec r,t) \approx \sum_{s=0}^{l-1} a_s \psi_{\text{cl}}(\vec r,t
+ \frac s l T_{\text{cl}}) \quad .$$ The coefficients $a_s$ and the integers $l$ depend on $m$ and $n$ and are defined in Ref. . The moduli of the $a_s$ are equal for all $l$, which means that the terms in the sum are equally weighted. The form of this equation shows that at the fractional revivals, the wave function equals a sum of subsidiary waves, each of which has the form of the initial wave function but is shifted in its orbit by a fraction of $T_{\text{cl}}$.
The periodicities in the motion of the wave packet are most easily studied using the autocorrelation function[@nau] $A(t) = \left< \Psi(\vec r,0) \vert \Psi(\vec r,t) \right>$. The absolute square of the autocorrelation function as a function of time gives the probability for the wave packet to return to its initial configuration. In a pump-probe experiment using photoionization, the periodicities in the photoionization signal should match those in $\vert A(t) \vert^2$, since both measure the probability for the wave packet to return to the core. Alternatively, in pump-probe experiments using the Ramsey method, the ionization signal depends on the real part of the autocorrelation function $\Re e \{ A(t) \} =
\Re e \left\{ \left< \Psi(\vec r,0) \vert \Psi(\vec r,t) \right>
\right \}$. This highly oscillatory function is modulated by an envelope that depends on the wave-packet motion.
In the following section, an analysis of the revival structure of edge-magnetoplasmon wave packets is presented and experimental detection methods are discussed.
Dynamics of EMP: the Edge-Magnetoplasmon Wave Packet {#sec3}
====================================================
The calculations in this section are based on the microscopic theory of QH samples with sharp edges and $\nu =1/m$. In this case, the edges have a single branch of chiral bosons. The calculations could readily be generalized to cases where the microscopic physics of the edge requires more elaborate boson models, if motivated by future experimental advances. For a schematic picture of the experimental geometry we have in mind, see Fig. \[geometry\]. We show that the edge of quantum-Hall systems can be prepared in a state which is a superposition of EMP narrowly distributed around a peak mode number. An exact expression for the time evolution of these wave packets can be obtained and these show an intricate revival structure. Numerical calculations for a particular set of model parameters are used to illustrate the latter.
Preparation and Evolution of an EMP WP {#sec3A}
--------------------------------------
We consider a QH sample subject to a time-dependent external potential which couples to the charge density of the system. We are motivated in large part by an experiment performed by Ashoori [*et al.*]{},[@ray:prb:92] which used capacitive coupling between the edge charge density and gates close to the edge of the QH sample to create EMP excitations and to detect their presence. The Hamiltonian describing this coupling is $$H = H_0 + V^{\text{ext}}(t) \quad ,$$ with the unperturbed time-independent Hamiltonian for free bosons (EMP) given by $$\label{freeplasm}
H_0 = \sum_{M>0} \varepsilon^{\text{C}}_{M} \, b_{M}^{\dagger} b_{M}
\quad ,$$ and the time-dependent external potential
\[potstart\] $$\begin{aligned}
V^{\text{ext}}(t) & = & u(t) \, R \int_{0}^{2\pi} d\theta \,\,
V^{\text{ext}}(\theta) \, \varrho(\theta) \\
& = & u(t) \sum_{M>0} (\nu M)^{1/2} \left[ V^{\text{ext}}_{-M} \,
b_{M} + V^{\text{ext}}_{M} \, b_{M}^{\dagger} \right] \,\, .
\label{potend}\end{aligned}$$
Here, $u(t)$ is the time-dependent voltage pulse applied to the exciting gate; see Fig. \[geometry\]. The angle $\theta$ parameterizes the coordinate along the edge. The quantity $\varrho(\theta)$ is the 1D electron density along the edge and $\varrho_{M}$ is its Fourier transform, while $V^{\text{ext}}(\theta)$ ($V^{\text{ext}}_{M}$) models (the Fourier transform of) the coupling between the gate and the 1D density along the edge, and $\nu$ is the filling factor. Equation (\[density\]) was used to obtain Eq. (\[potend\]). We have in mind the situation where the vertical separation between the plane containing the exciting gate and the plane containing the 2D electron layer is much smaller than the transverse size of the gate so that $V^{\text{ext}}(\theta) \approx 1$ for the portion of the edge under the gate and smoothly falls to zero outside this region. The time evolution of the system is calculated most straightforwardly in the Heisenberg picture where the operators carry all the time-dependence and the states are time-independent: $$b^{\pm}_{M}(t) := e^{i H t} \,\, b^{\pm}_{M} \,\, e^{- i H t}.$$ (Factors of $\hbar$ are absorbed in our quantum-Hall units.) The explicit form of these operators may be obtained by solving the Heisenberg equation of motion. In order to compress the notation, we write simultaneous equations for creation and annihilation operators; note that $b^{+}_{M} \equiv b^{\dagger}_{M}$ and $b^{-}_{M} \equiv b_{M}$. The Heisenberg equation of motion then reads
$$\begin{aligned}
i \partial_t \,\, b^{\pm}_{M}(t) &=& \left[ \,\, b^{\pm}_{M}(t) \, ,
\, H \,\, \right] \\
&=& \mp \varepsilon^{\text{C}}_{M} \, b^{\pm}_{M}(t) \mp (\nu M)^{1/2}
\, u(t) \, V^{\text{ext}}_{\mp M} \quad ,\end{aligned}$$
where the second line follows from the first using the commutation relations for bosonic operators with the Hamiltonian. The solution is $$\label{pertbos}
b^{\pm}_{M}(t) = \left( b^{\pm}_{M} + B_{\mp M}(t) \right) \,
\exp{\left[ \pm i \varepsilon^{\text{C}}_{M} t \right]} \quad ,$$ where $B_{\pm M}(t)$ are complex numbers: $$\label{field}
B_{\pm M}(t) = (\nu M)^{1/2} \, V^{\text{ext}}_{\pm M} \, (\mp i)
\int_{-\infty}^{t} d\tau \,\, u(\tau) \, \exp{\left[ \pm i
\varepsilon^{\text{C}}_{M} \tau \right]} .$$ The undisturbed edge is a collection of independent harmonic oscillators. When the edge is disturbed by an external potential which couples to the edge charge density, each harmonic oscillator is subject to a different time-dependent external force. Equation (\[pertbos\]) implies that when the edge is initially in its ground state, the effect of the external force is to put each oscillator in a coherent state described by the complex field $B_{\pm M}(t)$.
Inserting Eq. (\[pertbos\]) into the expression for the electron number density along the edge \[Eq. (\[density\])\], we obtain the Heisenberg-picture expression for the edge density operator:
$$\label{rhoexact}
\varrho(\theta, t) = \varrho_0(\theta, t) + \sigma(\theta, t) \quad ,$$
where $$\begin{aligned}
\varrho_0(\theta, t)
& =& \sum_{M>0} \frac{(\nu M)^{1/2}}{2\pi R} \, \left[ b_{M} \,\,
\exp{\left[ i (M \theta - \varepsilon^{\text{C}}_{M} t) \right] }
\,\, + \,\, b^{\dagger}_{M} \, \, \exp{\left[ -i (M \theta -
\varepsilon^{\text{C}}_{M} t ) \right] } \right] \quad , \\
\label{sigma} \sigma(\theta, t) &=& \sum_{M>0} \frac{(\nu M)^{1/2}}{2
\pi R} \, \left[ B_{M}(t) \exp{\left[ i (M \theta -
\varepsilon^{\text{C}}_{M} t) \right] } + B_{- M}(t) \exp{\left[ -i
(M \theta - \varepsilon^{\text{C}}_{M} t ) \right] } \right] \quad .\end{aligned}$$
Equation (\[rhoexact\]) is an [*exact*]{} expression capturing the impact of the external time-dependent potential $V^{\text{ext}}(t)$ on the edge charge density. Note that the effect of $V^{\text{ext}}(t)$ shows up only in Eq. (\[sigma\]) for $\sigma(\theta, t)$, and that $\varrho_0(\theta, t)$ does not contribute to the expectation value of $\varrho(\theta,t)$ since it does not conserve boson occupation numbers. As a result the density response to an external potential is temperature independent.
In the non-equilibrium state created by the excitation pulse, the time-dependent charge density, given by Eq. (\[rhoexact\]) \[$\left<
\varrho( \theta, t)\right> = \sigma(\theta, t)$\], is identical to the time-dependent charge density of a linear combination of classical normal modes. We will henceforth refer to this quantum state of the edge as an edge-magnetoplasmon wave-packet (EMP WP) state. The time-evolution of the wave packet is given by Eq. (\[sigma\]). As was discussed above, the spatial structure of the prepared wave packet as well as its evolution after switching off the potential is the same at any temperature.
It is shown in Appendix \[excite\] that it is possible to create a wave packet that has mode numbers strongly peaked around a central value $\tilde M$. One possible scenario uses a sequence of short voltage pulses on the gate to excite a superposition of eigenmodes, in analogy to the use of a short laser pulse to excite a superposition of Rydberg states. Using this method of excitation, it should be possible to produce wave packets with mean mode number $\tilde M
\approx 10$ – $100$.
To detect the wave packet, a second (detecting) gate can be located at coordinate $\theta_0$ relative to the exciting gate. (Alternately, the exciting gate could also serve as the detecting gate.) The charge signal at the detecting gate can be computed as a function of the delay time $t$: $$\label{Qt}
Q(t) = R\int_{0}^{2\pi} d\theta\,\, V^{\text{det}}(\theta - \theta_0)
\, \left< \varrho(\theta, t) \right> \quad ,$$ where $V^{\text{det}}(\theta) \approx 1 $ under the gate and smoothly goes to zero outside the gates because of fringe fields. We can loosely think of $Q(t)$ as the charge under the gate. The voltage signal on the gate should then be determined by the effective gate capacitance. (Note that we are not accounting for changes in the effective interaction between electrons in the 2D ES due to screening charges on the gates.) In the experiments of Ref. the voltage signal is approximately $1
{\rm \mu V}$ per electron under the gate. \[If the same gate were used for excitation and detection we would have $V^{\text{det}}(\theta) =
V^{\text{ext}}(\theta)$.\] Inserting Eq. (\[rhoexact\]) into Eq. (\[Qt\]), we obtain for $Q(t)$ $$\label{observe}
Q(t) = 2 \Re e \left\{ \frac{1}{2\pi R} \sum_{M>0} Q_{M}(t) \,
\exp{\left[ i \left( M \theta_0 - \varepsilon^{\text{C}}_{M} t \right)
\right] } \right\}$$ with the Fourier components given by
\[envelope\] $$\begin{aligned}
Q_{M}(t) &=& (\nu M)^{1/2} \, B_{M}(t) \, V^{\text{det}}_{-M} \\
&=& \nu M \, V^{\text{ext}}_{M} \, V^{\text{det}}_{-M} \, (- i)
\int_{-\infty}^{t} d\tau \,\, u(\tau) \, \exp{\left[ i
\varepsilon^{\text{C}}_{M} \tau \right]} \,\, . \label{envelope2}\end{aligned}$$
This method of creating an EMP WP using a time-dependent gate voltage and detecting the voltage pulse induced at a second gate by the time-dependent charge density of the propagating EMP WP is partially analogous to the phase-sensitive Ramsey method of detection for Rydberg wave packets.[@analogy]
We will refer to the picture of the gate-characteristic functions $V^{\text{ext}}(\theta)$ and $V^{\text{det}}(\theta)$ explained above, in which the gate is most sensitive to charges that are located in its immediate vicinity, as the [*local-capacitor model*]{}. For the calculations reported below we take $V^{\text{ext}}(\theta) =
V^{\text{det}}(\theta) = \exp{[-(\theta \, R / \zeta_{\text{G}})^2]}$ where $\zeta_{\text{G}}$ is the size of the gate. For the Fourier transforms we find that $V^{\text{ext}}_{M} = V^{\text{det}}_{M} \sim
(\zeta_{\text{G}} / R ) \, \exp{[- (M \, \zeta_{\text{G}} / R )^2]}$. The factor $V^{\text{ext}}_{M} V^{\text{det}}_{-M}$ in Eq. (\[envelope2\]) then precludes the observation of EMP modes with wave number $M > R/ \zeta_{\text{G}}$. This fact leads to an important constraint on the observability of EMP WP. If the excitation scheme (as discussed in detail in Appendix \[excite\]) creates an initial wave packet that is a superposition of EMP modes with dominant contribution from the mode with wave number $\tilde M \gg 1$, then optimal observability of this wave packet using the charge signal $Q(t)$ requires $\tilde M \zeta_{\text{G}} / R < 1$.
Revival Structure {#revistruc}
-----------------
To examine the revival structure of an EMP WP, we first consider the expectation value for the electron number density along the edge. We assume that the excitation pulse is turned off at time $\tau$. Then for $t > \tau$ $$\label{rho}
\left< \varrho(\theta,t) \right> = 2 \Re e \left\{ \sum_{M>0} c_M \,
e^{iM \theta} e^{-i \varepsilon^{\text{C}}_{M} t} \right\} \quad ,$$ where $$\label{cM}
c_M = \frac{\nu M}{2\pi R} \, V^{\text{ext}}_{M} \, \left( -i
\int_{-\infty}^{\tau} d\tau^\prime \,\, u(\tau^\prime) \, \exp{\left[
i \varepsilon^{\text{C}}_{M} \tau^\prime \right]} \right) \quad .$$ The coefficients $c_M$ act as weighting functions for the different modes $M$. Figure \[lasexc\] illustrates some possible distributions for the weightings in $M$ resulting from excitation sequences detailed in Appendix \[excite\]. As a result of the weighting distribution, only those energies $\varepsilon^{\text{C}}_{M}$ in Eq. (\[rho\]) near $\varepsilon^{\text{C}}_{\tilde M}$ will contribute to the sum. This permits a Taylor expansion of $\varepsilon^{\text{C}}_{M}$ around the central value $\varepsilon^{\text{C}}_{\tilde M}$.
The derivative terms in this expansion define the time scales that control the evolution and revival structure of the wave packet, as discussed above in the section on Rydberg wave packets. In this case the expressions for the time scales are \[cf. Eqs. (\[gentimes\])\]:
\[times\] $$\begin{aligned}
T_{\text{cl}} &=& \frac {2 \pi} {\vert (
\varepsilon^{\text{C}}_{\tilde M})^\prime \vert} = \frac {2 \pi R}
{(\ln \frac R {\alpha \tilde M} - 1)} \quad , \\
t_{\text{rev}} &=& \frac {2 \pi} {\frac 1 2 \vert (
\varepsilon^{\text{C}}_{\tilde M})^{\prime\prime} \vert} = 4 \pi R
\tilde M \quad , \\ t_{\text{sr}} &=& \frac {2 \pi} {\frac 1 6 \vert
(\varepsilon^{\text{C}}_{\tilde M})^{\prime\prime\prime} \vert} =
12 \pi R \tilde M^2 \quad .\end{aligned}$$
The primes denote derivatives with respect to $M$. Equations (\[times\]) define the classical orbit period, the revival time, and the superrevival time, respectively, for EMP WP.
For large values of $\tilde M \approx 10$ – $100$ and typical values of $R \approx 10^4$, we find $t_{\text{sr}} \gg t_{\text{rev}} \gg
T_{\text{cl}} \gg T_{\text{ph}} := 2\pi/
\varepsilon^{\text{C}}_{\tilde M}$. Therefore, for times $t \ll t_{\text{rev}}$, we can approximate $\left< \varrho(\theta,t) \right> \approx 2 \Re e \left\{ {\tilde
\varrho}_{\text{cl}} (\theta,t) \right\}$, where $$\begin{aligned}
\tilde \varrho_{\text{cl}}(\theta,t) &=& e^{i (\tilde M \theta - 2
\pi t / T_{\text{ph}} )} \, \sum_{M>0} c_M \, e^{i (M-\tilde M) (
\theta - 2 \pi t / T_{\text{cl}} ) } \nonumber \\
&=& e^{i (\tilde M \theta - 2 \pi t / T_{\text{ph}} )} \cdot
\varrho_{\text{cl}}(\theta,t) \quad.\end{aligned}$$ Here $T_{\text{ph}} = 2 \pi R / (\tilde M \, \ln[R/(\alpha \tilde M)
])$. Thus $\tilde \varrho_{\text{cl}}(\theta,t)$ is the product of a rapidly oscillating[@wiggles] (both in space and time) phase factor and a more slowly varying periodic envelope function. The period of the rapid spatial oscillations is $2\pi/\tilde M$, whereas $T_{\text{ph}}$ is the period of the rapid temporal oscillations. If the additional phase factor were not present, the charge density in this approximation would circulate around the edge without distortion and with period $T_{\text{cl}}$. This motion resembles the classical drift motion[@jackson] of the cyclotron orbit of a charged particle in a strong magnetic field; it is perpendicular to both magnetic and electric fields and has speed $v_{\text{dr}} = c E / B$. In the present case,the electric field is perpendicular to the edge of the 2D ES so that the drift is along the edge and the classical period is[@classcaveat] $T_{\text{cl}} \approx 2 \pi R /
v_{\text{dr}}$. The electric field which yields our classical orbit period is that from the neutralizing background required to stabilize a macroscopic system of charged particles.[@classcaveat] Because of the additional phase factor this classical motion appears as the envelope of a more rapid oscillation of edge charge density. In what follows, we focus on the evolution of this classical envelope function at longer times.
For times greater than $T_{\text{cl}}$, the second-derivative term will eventually contribute to the time-dependent phase, and we expect a revival structure analogous to the fractional and full revivals of Rydberg wave packets. Indeed, at the times $t = m\, t_{\text{rev}}/n$, with $m$ and $n$ relatively prime ($m\le n$), we can write the sum over $M$ \[in Eq. (\[rho\])\] as a sum of subsidiary functions, $$\label{rhofrac}
\left< \varrho(\theta,t) \right> \approx 2 \Re e \left\{ e^{i (\tilde
M \theta - 2 \pi t / T_{\text{ph}} )} \,\, \sum_{s=0}^{l-1} a_s \,
\varrho_{\text{cl}}(\theta,t + \frac s l T_{\text{cl}}) \right\}
\, .$$ The coefficients $a_s$ are given by $$\label{as}
a_s = \frac 1 l \sum_{M^\prime = 0}^{l-1} \exp[2 \pi i \frac m n
(M^\prime)^2] \, \exp [2 \pi i M^\prime \frac s l ] \quad ,$$ with $$\label{lcases}
l =\cases{\frac n 2&if~~$n = 0~~ ({\rm mod}~4)~~$,\cr
n&if~~$n \ne 0~~ ({\rm mod}~4)~~$.\cr }$$ The moduli $\vert a_s \vert$ are equal for all $l$. Therefore, the sum over $s$ in Eq. (\[rhofrac\]) consists of an equally weighted sum of subsidiary functions that are shifted in time by fractions of the classical period $T_{\text{cl}}$. This demonstrates that at the fractional revivals the EMP WP can be written as a sum of subsidiary waves. The motion of the wave packet is periodic with a period equal to a fraction of $T_{\text{cl}}$. The full revival occurs when $m=n=1$. In this case $l=1$, $a_0 = 1$ is the only nonzero coefficient in Eq. (\[rhofrac\]), and the wave-packet sum consists of a single term which has the same form as the initial wave packet. This analysis for $t \leq t_{\text{rev}}$ ignores contributions from the higher-order terms in the phase. As a result of these higher-order terms, the full revival is not a perfect revival in that it does not exactly equal the initial wave packet. Including the third-order term in the time-dependent phase, which depends on the time scale $t_{\text{sr}}$, would lead to an analysis of the superrevival structure analogous to that performed previously for Rydberg wave packets.
The experimentally measured quantity is not $\varrho (\theta,t)$ but the charge signal $Q(t)$ at the detecting gate defined in Eq. (\[observe\]). In this expression the weighting coefficients $Q_M(t)$ are constant after the excitation pulse has been turned off. Expanding $\varepsilon^{\text{C}}_{M}$ in a Taylor series in $M$ around the value $\tilde M$ shows that $Q(t)$ has a time dependence similar to $\left< \varrho (\theta,t) \right>$. The full and fractional periodicities in $T_{\text{cl}}$ should therefore be exhibited in the time dependence of the charge signal $Q(t)$.
Example
-------
As an example, we consider an EMP WP with $\tilde M = 50$. In the next section, we discuss the experimental feasibility of creating and observing a wave packet with mode numbers in this range. In realistic samples, such a wave packet will be damped and lose phase coherence due to electron-phonon interactions, and possibly also due to interactions with electrons in the bulk of the two-dimensional system when it has localized low-energy excitations. In this section, we ignore for the moment damping and loss of phase coherence of the wave packet and examine the resulting idealized revival structure for times up to $t_{\text{rev}}$.
As shown in Appendix \[excite\], there are many possible weighting distributions for the coefficients $c_M$, depending on the experimental configuration. The analysis of the revivals given above requires only that the coefficients $c_M$ be strongly peaked around a central value $\tilde M$. For the sake of definiteness we take a particular example in this subsection, modelling the coefficients $c_M$ by a Gaussian distribution centered on $\tilde M = 50$ with width $\sigma_M = 2$. We also set $\alpha = 1$ and $R = 10^4$. From Table \[table\_nsu\] for typical magnetic field strengths this corresponds to a sample with linear dimension $\sim 100$ $\rm \mu m$ and time scales starting from the nanosecond range. To be specific, we use $\nu = 1$ and $B = 10$ T, which fixes our time scales to $T_{\text{cl}} \approx 2.2$ ns, $t_{\text{rev}} \approx 942$ ns, and $t_{\text{sr}} \approx 141$ $\rm \mu s$. In the chiral-boson model the time evolution is independent of the overall strength of the signal so we plot results for $\left< \varrho(\theta,t) \right>$ calculated from Eq. (\[rho\]) as a function of the angle $\theta$ at various times in arbitrary units.
Figure \[rhoperiodic\]a shows the initial wave packet for the Gaussian model considered in this section, which sums to a smooth envelope on an oscillating background. Figures \[rhoperiodic\]b and \[rhoperiodic\]c show the wave packet at times $T_{\text{cl}}/2$ and $T_{\text{cl}}$, respectively. It is seen that initially the wave packet maintains its shape and exhibits the classical periodicity. However, for times beyond $T_{\text{cl}}$ quantum interference effects commence. Figure \[rhoperiodic\]d shows the wave packet at $50 \, T_{\text{cl}}$. After 50 classical periods, the wave packet has spread and is no longer localized.
Figure \[rhorevivals\] shows the fractional and full revivals. At $t = t_{\text{rev}}/6$, there are three nonzero terms in the sum in Eq. (\[rhofrac\]). The wave packet splits into three equally weighted subsidiary waves as illustrated in Fig. \[rhorevivals\]a. The motion of the wave packet is periodic with period $T_{\text{cl}}/
3$. Figure \[rhorevivals\]b shows the wave packet at $t_{\text{rev}}/4$, at which time it consists of a sum of two distinct subsidiary waves. Here, the motion is periodic with period $T_{\text{cl}}/2$. A full revival first occurs at $t_{\text{rev}}/2$. Figure \[rhorevivals\]c shows that a single wave packet has formed. This revival has the quantum-mechanical characteristic of being one-half cycle out of phase with the classical motion.[@ap] Only at the time $t_{\text{rev}}$ does a single wave packet form that is in step with the corresponding classical periodic motion. The full revival at $t_{\text{rev}}$ is shown in Fig. \[rhorevivals\]d. It evidently does not exactly match the initial wave packet. A smaller subsidiary wave packet has also formed.
The revival structure as a function of time can also be observed by computing the charge signal $Q(t)$. This is the observable that would be measured in an experiment. We evaluate Eq. (\[observe\]), using the same Gaussian weighting as above and ignoring the overall normalization.
Figures \[Qbig\]a and \[Qbig\]b show $Q(t)$ as a function of time for times up to and just beyond $t_{\text{rev}}/2$. Here, the revival structure is revealed through the periodicity of $Q(t)$. In Fig. \[Qbig\]a, the initial motion is clearly periodic with period $T_{\text{cl}} \approx 2.2$ ns. However, as the wave packet spreads and collapses, the peaks in $Q(t)$ disappear and fractional revivals start to emerge. The peaks at $t_{\text{rev}}/4 \approx 235$ ns have half the amplitude and periodicity of the initial peaks. This is because the wave packet has reformed into two distinct parts. The peaks at $t_{\text{rev}}/6 \approx 157$ ns have period $T_{\text{cl}}/3$ and even smaller amplitude, corresponding to the formation of three subsidiary wave packets. Figure \[Qbig\]b shows the full revival at $t_{\text{rev}}/2 \approx 471$ ns, which is one-half cycle out of phase with the classical motion. Here, the amplitude of the peaks matches that of the initial peaks, and the period is again $T_{\text{cl}}$.
Figure \[Qenlarged\] enlarges some of the regions in Fig. \[Qbig\]. Figure \[Qenlarged\]a shows the first few classical cycles for times up to 5 ns. Evidently, the individual peaks with period $T_{\text{cl}}$ actually consist of envelopes of highly oscillatory signals.[@wiggles; @analogy] These rapid oscillations cannot be resolved in Fig. \[Qbig\]. The period of the rapid oscillations is $T_{\text{ph}} \approx 0.036$ ns. Figure \[Qenlarged\]b shows an enlargement of $Q(t)$ near $t_{\text{rev}}/6$. The period of the envelope peaks is $T_{\text{cl}}/3 \approx 0.73$ ns. Figure \[Qenlarged\]c shows the charge signal near $t_{\text{rev}}/4$, where two wave packets are moving with period $T_{\text{cl}}/2 \approx 1.1$ ns. The charge signal near $t_{\text{rev}}/2$ is shown in Fig. \[Qenlarged\]d. The period is again $T_{\text{cl}}$, but some distortion of the signal is evident in comparison to the initial peaks in Fig. \[Qenlarged\]a.
This example demonstrates explicitly the formation of full and fractional revivals of EMP WP. The generic features are the same as for Rydberg wave packets. At the fractional revivals, the EMP WP is equal to a sum of subsidiary wave packets that move with a periodicity equal to a fraction of $T_{\text{cl}}$. The semiclassical behavior as well as the revivial structure of EMP WP can be detected experimentally by measuring periodicities in the envelope that modulates rapid oscillations in the charge signal $Q(t)$. These oscillations are the analog of the Ramsey fringes in Rydberg-wave-packet experiments which use the phase-sensitive method of detection.
Fluctuations: quantum and thermal
---------------------------------
The EMP WP is a many-mode coherent state: each plasmon mode represents a one-dimensional quantum harmonic oscillator, and the creation scheme described above results in each oscillator mode being excited into a coherent state. The above-mentioned detection scheme yields an average charge signal $Q(t)$ which reflects purely classical behavior of the EMP modes. Quantum fluctuation effects appear only in the noise spectrum of the charge signal.
In our detection scheme, we measure the expectation value of the charge operator $\hat Q(t)$:
$$\begin{aligned}
\hat Q(t) &=& R \int_{o}^{2\pi} d\theta \, V^{\text{det}} (\theta -
\theta_0) \, \varrho(\theta, t) \\
&=& \sum_{M>0} \left[ \nu M \right]^{1/2} \, \left\{
V^{\text{det}}_{-M} \, b_{M}(t) \, e^{i M \theta_0} + \rm h.c.
\right\} \,\,\, .\end{aligned}$$
It is possible to rewrite $\hat Q(t)$ as the sum of a term \[$=: \delta
\hat Q(t)$\] that has vanishing expectation value and contributes only to fluctuations, and a term \[$\equiv Q(t)$\] which does not fluctuate at all and equals the expectation value of the charge operator: $\hat
Q(t) = \delta \hat Q(t) + Q(t)$. Explicitly, we find $$\delta \hat Q(t) = \sum_{M>0} \left[ \nu M \right]^{1/2} \, \left\{
V^{\text{det}}_{-M} \, b_{M} \, e^{i [ M \theta_0 -
\varepsilon^{\text{C}}_{M} t ]} + \rm h.c. \right\} \quad ,$$ and $Q(t)$ was defined in Eq. (\[observe\]). The variance of the charge signal is readily evaluated:
\[noise\] $$\begin{aligned}
\big< \big[ \Delta \hat Q(t) \big]^2 \big> &=& \big< \big[ \Delta
\hat Q(t) \big]^2 \big>_{\text{qu}} + \big< \big[ \Delta \hat Q(t)
\big]^2 \big>_{\text{th}} \,\, , \\ \label{zeronoise}
\big< \big[ \Delta \hat Q(t) \big]^2 \big>_{\text{qu}} &=&
\sum_{M>0} \nu M \, | V^{\text{det}}_{M} |^2 \quad , \\
\big< \big[ \Delta \hat Q(t) \big]^2 \big>_{\text{th}} &=&
2 \sum_{M>0} \nu M \, | V^{\text{det}}_{M} |^2 \, n_{M}^{(0)}
\quad , \label{thermnoise}\end{aligned}$$
with $n_{M}^{(0)}$ being the thermal-equilibrium occupation number of the plasmon mode labelled by quantum number $M$. Equations (\[zeronoise\]) and (\[thermnoise\]) represent the noise due to quantum and thermal fluctuations, respectively.
We assume the detector characteristics to be determined primarily by geometrical properties, such as the dimension $\zeta_{\text{G}}$ of the capacitor plate. Within the local-capacitor model, we have $V^{\text{det}}_{M} \sim (\zeta_{\text{G}}/ R) \, \exp{
\{ - [M \, \zeta_{\text{G}} / R ] ^2 \} }$ which yields the result $$\big< \big[ \Delta \hat Q(t) \big]^2 \big> \approx \nu \, \times \,\,
\left\{ \begin{tabular}{cl}
1 & if $\zeta_{\text{T}} > \zeta_{\text{G}}$ \\
$\zeta_{\text{G}}/ \zeta_{\text{T}}$ & otherwise
\end{tabular}
\right.$$ where the thermal length $\zeta_{\text{T}} := v_{\text{dr}} / (
\sqrt{4\pi} k_B T)$. The regime where $\zeta_{\text{T}} >
\zeta_{\text{G}}$ is dominated by quantum fluctuations, whereas thermal fluctuations are more important if $\zeta_{\text{T}} <
\zeta_{\text{G}}$.
In order to judge the importance of the quantum and thermal fluctuations for experiments detecting the semiclassical behavior and revival structure of EMP WP, we compare the magnitude of the fluctuations to the amplitude of the charge signal $Q(t)$. As an order-of-magnitude estimate, we find for the case of an EMP WP which was created using the multiple-short-pulse technique (see Appendix \[laser\]): $$|Q(t)|^{\text{max}} \sim \nu \,\, \frac{\tilde M \,
\zeta_{\text{G}}}{R} \,\, \frac{u_0 \,
\zeta_{\text{G}}}{v_{\text{dr}}} \quad .$$ Typical drift velocities $v_{\text{dr}}$ are of the order of $5 \times
10^5$ m/s, so that we get a numerical estimate $$|Q(t)|^{\text{max}} \sim 3 \, u_0[\rm mV] \, \zeta_{\text{G}}[\mu m]
\,\, \times \,\, \nu \,\, \frac{\tilde M \,\zeta_{\text{G}}}{R}
\quad .$$ Here, $u_0$ denotes the amplitude of the voltage pulse which created the EMP WP. Remember that $\tilde M \zeta_{\text{G}} / R < 1$ is required to enable the detection of the EMP WP using the charge signal. For a signal-to-noise ratio greater than unity, the amplitude of the voltage pulse has to satisfy $$u_0 \gg \frac{R}{\tilde M \zeta_{\text{G}}} \,\,
\frac{v_{\text{dr}}}{\zeta_{\text{G}}} \,\,\, \rm max \left\{ 1, (
\zeta_{\text{G}} / \zeta_{\text{T}} )^{1/2} \right\}
\quad ,$$ or based on the numerical estimate above $$u_0[\rm mV] \gg \frac{0.3}{\zeta_{\text{G}}[\mu m]} \,\,\times \,\,
\frac{R}{\tilde M \zeta_{\text{G}}} \quad .$$
Finite Lifetime of Edge-Magnetoplasmon Wave Packets {#sec4}
===================================================
The previous section analyzed the formation and revival structure of EMP WP, assuming an infinite life time for the EMP which form the wave packet. It was shown that thermal effects have no influence on the preparation and evolution of these wave packets apart from contributing to fluctuations. The revival structures we have discussed require coherent evolution of the EMP system. In realistic systems, electrons in the 2D ES will be coupled to the semiconductor host material via various physical processes. For experimentally realistic temperatures and parameter ranges in semiconductors, the most important process will typically be the coupling of electrons to 3D acoustic phonons. This coupling leads to a finite life time for EMP, which we calculate in this section. Comparing the life time to the relevant time scales for semiclassical behavior ($T_{\text{cl}}$) and for revivals ($t_{\text{rev}}$), we can determine the observability of these effects.
Plasmon-Phonon Coupling – General
---------------------------------
The electron-phonon interaction is specified by the following contribution to the Hamiltonian,[@mahan] $$\label{elecphon}
H^{\text{el-ph}} = \sum_{\vec q, \lambda} M^{\lambda}(\vec q) \left(
a^{\dagger}_{- \vec q, \lambda} + a_{\vec q, \lambda} \right)
\varrho^{\text{3D}}_{\vec q} \quad ,$$ where the operators $a^{\dagger}_{\vec q, \lambda}$ ($a_{\vec q, \lambda}$) create (annihilate) phonons with 3D wave vector $\vec q$, polarization label $\lambda$, and normal-mode frequency $\omega^{\lambda}_{\vec q}$. Here, $\varrho^{\text{3D}}_{\vec q}$ denotes the 3D electron density in the QH sample and $M^{\lambda}(\vec q)$ is a coupling constant whose numerical value is known in most materials of interest. Since EMP are collective fluctuations of the electron density at the edge of our sample, the electron-phonon interaction leads to an effective coupling between phonons and EMP. By identifying the contribution to the 3D electron density from EMP we are able to derive a Hamiltonian which describes the coupling between the EMP and phonon systems. For example, for the disk geometry described in Fig. \[geometry\], the Hamiltonian has the form
----------------------------- ----------------- ----------------------------
Mass density $\rho$ $5300\,\rm kg / m^{3}$
Longitudinal sound velocity $c^l$ $5140\,\rm m/s$
Transversal sound velocity $c^t$ $3040\,\rm m/s$
Deformation potential $D$ 9.3 eV
Piezoelectric constant $h_{\text{14}}$ $1.2\times 10^9\,\rm V/ m$
----------------------------- ----------------- ----------------------------
: Parameters of GaAs/ heterostructures, according to Ref. .
\[numbers\]
$$\label{coupli}
H^{\text{pl-ph}} = \sum_{M>0 \atop \vec q, \lambda}
C^{\lambda}_{M}(\vec q) \left( a_{\vec q, \lambda} +
a^{\dagger}_{- \vec q, \lambda} \right) \left( b_{M} \, e^{i M
\theta_{\vec q}} + b^{\dagger}_{M} \, e^{- i M \theta_{\vec q}}
\right) \quad ,$$ where $\theta_{\vec q}$ denotes the azimuthal angle of the wave vector $\vec q$ in the plane of the 2D ES. Details of the derivation of Eq. (\[coupli\]) and an analytic expression for the coupling coefficients $C^{\lambda}_{M}(\vec q)$ are given in Appendix \[plasphon\]. It is possible to derive an expression similar to Eq. (\[coupli\]) which is valid for the strip geometry; this case is also discussed in Appendix \[plasphon\].
To investigate the effect of plasmon-phonon coupling on the evolution of an EMP WP, we consider a single-plasmon Matsubara Green’s function defined by $$\label{greensfunc}
{\cal G}_{M}(\tau) = - \big< \mbox{T}_\tau \, b_{M}(\tau)
b_{M}^{\dagger} (0) \big> \quad ,$$ and the Fourier transform ${\cal G}_{M}(\omega)$ of the retarded Green’s function, which is obtained from the Fourier transform of ${\cal G}_{M}(\tau)$ by continuation to real frequencies. In the absence of EMP-phonon coupling, ${\cal G}_{M}(\omega)$ reduces to the well-known result[@mahan] for free bosons: ${\cal G}_{M}^{0}(
\omega) = [\omega - \varepsilon_{M}^{\text{C}} + i \delta ]^{-1}$ reflecting the fact that EMP are the well-defined excitations of the system. The presence of phonons causes damping (and an energy shift) of the EMP, leading to the modified result ${\cal G}_{M}(\omega) =
[\omega - \tilde \varepsilon_{M} + i \, \Gamma_{M}/2]^{-1}$ with $\Gamma_{M}^{-1}$ being the life time of the EMP with wave number $M$. We find $$\label{damprate}
\Gamma_{M} = 2 \pi \sum_{\vec q, \lambda} \left| C^{\lambda}_{M}
(\vec q) \right|^2 \,\, \delta \big( \varepsilon_{M}^{\text{C}} -
\omega_{\vec q}^{\lambda} \big) \quad .$$ See Appendix \[selfen\] for details of the calculation.
The meaningful quantity to assess the effect of EMP-phonon coupling on the propagation of EMP WP is $\Gamma_{\tilde M}^{-1}$, which we call the life time of the EMP WP.
At energy and wave-vector scales appropriate for the observation of EMP WP, acoustic phonons with dispersion $\omega^{\lambda}_{\vec q}
= q\, c^{\lambda}$ are most important. In a polar semiconductor, both scattering from a deformation potential and piezoelectric effects contribute to the electron-phonon coupling, whereas in a non-polar semiconductor the piezoelectric part is absent. In the following two sub-sections, we discuss these two cases separately.
Polar Semiconductors: GaAs
--------------------------
At the long wavelengths used to construct EMP WP, piezoelectric coupling dominates the deformation-potential scattering of electrons. We therefore neglect the contribution from the deformation potential in this sub-section. Evaluation of Eq. (\[damprate\]) for the disk case yields $$\Gamma_{M} \approx \left( \frac{h_{14}}{\tilde c} \right)^2
\frac{\epsilon}{\rho} \, \frac{M}{R} \quad ,$$ with $\epsilon$ being the semiconductor bulk dielectric constant, and the piezoelectric coupling constant denoted by $h_{14}$. Here, the quantity $\tilde c$ is of the order of the speed of sound. This estimate is based on the observation that the drift velocity of the EMP WP is typically $\sim 100$ times larger than the speed of sound so that typical projections of the phonon wave vector onto the 2D plane lead to large Bessel-function arguments. (Typical phonon wavelengths are very small compared to the size of the disk.) Note that modes with higher wave number decay faster.
The relevant quantities to examine when assessing the possibility to observe wave-packet revivals are:
\[phonlimit\] $$\begin{aligned}
\frac{\Gamma_{\tilde M}^{-1}}{T_{\text{cl}}} &\sim& \left(
\frac{\tilde c}{h_{\text{14}}} \right)^2 \frac{\rho}{\epsilon} \,\,
\frac{\ln [R/\tilde M]}{\tilde M} \sim 100 \,\, \frac{\ln [R / \tilde
M]}{\tilde M} \,\,\, , \\ \label{revobs}
\frac{\Gamma_{\tilde M}^{-1}}{t_{\text{rev}}} &\sim& \left(
\frac{\tilde c}{h_{\text{14}}} \right)^2 \frac{\rho}{\epsilon} \,\,
\frac{1}{\tilde M^2} \sim \left(\frac{7}{\tilde M} \right)^2 \quad .\end{aligned}$$
Equations (\[phonlimit\]) give respectively the number of classical periods and the number of revivals which occur in the mean free time of an EMP WP. We see that, unless $\tilde M$ is very large, the initial periodicity should be observable. However, the likelihood of seeing revivals appears to be quite remote. Equation (\[revobs\]) shows that the wave packet loses coherence before it can revive. The typical values of parameters for GaAlAs/GaAs-heterostructures used in the calculation of the EMP WP life time are taken from Ref. and are shown in Table \[numbers\].
Interpretation of Previous Experiment
-------------------------------------
It is interesting to reexamine the experiments described in Ref. in the light of these expressions. In that work measurements were made on a QH sample in GaAs with the geometry sketched in Fig. \[geometry\] at temperature $T=0.3$ K. The values of the relevant parameters were: filling factor $\nu=1$, magnetic-field strength $B=5.1$ T ([*i.e.*]{} magnetic length $\ell
\approx 11$ nm), $R = 270\,\rm \mu m \approx 2.4 \times 10^4 \, \ell$ and $\zeta_{\text{G}} \sim 10\,\rm \mu m$. A single voltage pulse with amplitude $u_0 = 50$ mV and duration $T_{\text{exc}} = 100$ ps was applied to create the initial wave packet. The latter was observed to move around the disk sample with period $T_{\text{cl}} \approx 4$ ns while spreading rapidly. (Fewer than ten cycles can be discerned before the signal vanishes in the noise.) The data do [*not*]{} seem to consist of a rapid oscillation that is modulated by an envelope.
We believe that the EMP WP excited in this experiment was composed primarily of modes with $M < 5$, with $M=1$ possibly having the largest amplitude. Our analysis of the time scales would give $T_{\text{ph}}\approx T_{\text{cl}} \approx t_{\text{rev}} / 20$. In our interpretation of this experiment, the period of the rapid oscillation in the charge signal and the classical period (which determines the periodicity of the [*envelope function*]{} that modulates the rapidly oscillating charge signal) are nearly equal, and the revival time is just one order of magnitude larger. The absence of a rapid oscillation in the charge signal results from the near-equality of $T_{\text{ph}}$ and $T_{\text{cl}}$. The decay of the charge signal in $\sim 10$ classical periods can be consistently explained as being due to the spreading of the wave packet due to the non-linear dispersion of EMP. However, because of the small value of $\tilde M$ in this experiment, interference between first- and higher-order terms in the expansion of the non-linear dispersion of the EMP around $\tilde M$ would be expected to (and apparently does) obscure the fractional and full revivals. Exciting the EMP WP with a single voltage pulse results in a rather broad distribution of wave numbers in the EMP WP, so that the analysis of the revival structure, which is based on a sharply-peaked distribution of wave numbers around $\tilde M$, is certainly invalid.
With the drift velocity deduced from $R$ and $T_{\text{cl}}$, we estimate the thermal length to be $\zeta_{\text{T}} \approx 3\rm \mu
m$ which is smaller than $\zeta_{\text{G}}$. The experiment therefore was in a regime where thermal noise dominates. For a good signal-to-noise ratio, the requirement $u_0 \gg 2$ mV had to be satisfied; this criterion was met in the experiment under consideration. The life time of the wave packet as deduced from a calculation outlined in this section is $\sim 1000 \, T_{\text{cl}}$. Therefore, the rapid decay of the signal cannot be attributed to dissipation into the phonon system.
Non-polar Semiconductors: Si and Ge
-----------------------------------
In non-polar semiconductors, piezoelectric coupling is absent, and the rate of phonon emission by the plasmons is suppressed for the long-wavelength plasmons typically involved. In this case we find that the ratio of the life time and revival time, $$\frac{\Gamma_{\tilde M}^{-1}}{t_{\text{rev}}} \sim \left[
\frac{R}{\ln R} \,\,\, \frac{1}{\tilde{M}^2} \right]^2 \quad ,$$ can be made much larger than unity by adjusting the size of the QH sample. For a millimeter-size sample at typical magnetic fields ($2\pi R \approx 1$ mm, $B=10$ T), the ratio $\Gamma_{\tilde M}^{-1}/ t_{\text{rev}} \sim 1$ ($25$) for $\tilde M
= 50$ ($20$). The revival structure of EMP WP disussed in Sec. \[sec3\] should therefore be observable for samples with non-polar semiconductor host materials, [*e.g.*]{}, in Si/Ge-heterostructures.[@sigehetero]
Summary and Conclusions {#sec5}
=======================
In this paper, we have examined the formation and evolution of edge-magnetoplasmon wave packets in nanostructures. These wave packets are formed as superpositions of edge magnetoplasmons that are the only low-lying excitations in finite quantum-Hall samples. By using a sequence of short pulses in the excitation process, it is possible to produce a superposition with mode numbers sharply peaked around a central value $\tilde M$. We have shown that for such wave packets the initial motion is periodic with a period $T_{\text{cl}}$. After several of these cycles, the wave packet collapses and a sequence of fractional and full revivals commences. This revival structure is analogous to that of Rydberg wave packets in atomic systems; its relevant time scale is the revival time $t_{\text{rev}}$.
We find that experiments that use capacitive coupling to the charge-density fluctuation that is associated with the EMP WP both for creation and detection of the EMP WP are analogous to Rydberg wave-packet experiments that use the phase-sensitive Ramsey method of detection. In both types of measurement, the semiclassical motion as well as the revival structure is seen in the time variation of the envelope function of a rapidly oscillating signal.
We have shown that thermal effects have no influence on the propagation of the wave packet. Examining possible scenarios for energy loss of EMP, we found that plasmon-phonon coupling due to piezoelectric effects (in polar semiconductors such as GaAs) causes the wave packet to decay with a life time that is typically less than the revival time. However, for 2D ES fabricated in non-polar semiconductors such as Si or Ge, piezoelectric coupling is absent and it is possible to produce wave packets with large values of $\tilde M$ that will evolve for times of order $t_{\text{rev}}$ without appreciable decay. In this way, it should be feasible to detect fractional revivals in experiments. The analysis given in this work of a previous experiment[@ray:prb:92] that examined the classical motion of EMP WP only can serve as a guideline for future experimental studies of the EMP WP revival structure.
Possible Excitation Scenarios for the Initial Wave Packet {#excite}
=========================================================
Single Short Pulse
------------------
In the experiments on $\mu$m–size quantum dots[@ray:prb:92] by Ashoori [*et al.*]{} a single short pulse was applied to prepare the initial wave packet. In our model, the corresponding pulse characteristics is $$u(t) = \left\{
\begin{tabular}{cc}
$u_0$ & \,\, \mbox{ for $0\le t \le T_{\text{exc}}$} \quad , \\
0 & \,\, \mbox{ otherwise} \quad .
\end{tabular}
\right.$$ It is straightforward to calculate the field $B_{\pm M}(t)$ for this case. We find $$B_{\pm M}(t) = \left\{
\begin{tabular}{cc}
0 & \, \, $t\le 0$ \quad , \\
$(\nu M)^{1/2} \, V^{\text{ext}}_{\pm M} \, \frac{\mp 2 i
u_0}{\varepsilon^{\text{C}}_{M}} \, \exp{\left[ \pm i
\frac{\varepsilon^{\text{C}}_{M} t}{2} \right] } \,\,
\sin{\left[ \frac{\varepsilon^{\text{C}}_{M} t}{2} \right] }$ & \, \,
$0 < t < T_{\text{exc}}$ \quad , \\
$(\nu M)^{1/2} \, V^{\text{ext}}_{\pm M} \, \frac{\mp 2 i
u_0}{\varepsilon^{\text{C}}_{M}} \, \exp{\left[ \pm i
\frac{\varepsilon^{\text{C}}_{M} T_{\text{exc}}}{2} \right] } \,\,
\sin{\left[ \frac{\varepsilon^{\text{C}}_{M} T_{\text{exc}}}{2}
\right] }$ & \, \, $T_{\text{exc}} \le t$
\quad .
\end{tabular} \right.$$ This type of excitation cannot lead to an EMP WP state with a sharply peaked mode distribution and is unlikely to produce well-resolved revivals.
Multiple Short Pulses {#laser}
---------------------
In analogy with the excitation of Rydberg wave-packet states in atoms by laser pulses, we propose exciting the edge of a QH system using a series of $N$ short pulses each of duration $T_{\text{exc}}$. For the specific case of sinusoidal individual pulses this would give $$u(t) = \left\{
\begin{tabular}{cc}
$u_0 \, \sin\left[ \frac{2\pi}{T_{\text{exc}}} t \right]$ & \,\, for
$0\le t \le N \, T_{\text{exc}}$ \quad , \\ 0 & \,\, otherwise \quad .
\end{tabular} \right.$$ In this case we find that for $t \ge N T_{\text{exc}}$ $$B_{\pm M}(t) = (\nu M)^{1/2} \, V^{\text{ext}}_{\pm M} \, \frac{4\pi
u_0}{T_{\text{exc}}} \, \exp \left[ \pm i \frac{
\varepsilon^{\text{C}}_{M} N T_{\text{exc}}}{2} \right] \,\,
\frac{\sin \left[\varepsilon^{\text{C}}_{M} N T_{\text{exc}} /2
\right]}{\left( \varepsilon^{\text{C}}_{M}\right)^2 - \left(\frac{2
\pi}{T_{\text{exc}}} \right)^2 } \quad ,$$ which is sharply peaked around a value $\tilde M$ satisfying $\varepsilon^{\text{C}}_{\tilde M} = 2 \pi / T_{\text{exc}}$. Using our notation from Sec \[revistruc\], the EMP WP created with the multiple-short-pulse technique satisfies $T_{\text{ph}}\equiv
T_{\text{exc}}$. The duration of the short pulses determines the value of $\tilde M$, whereas the number of pulses $N$ determines the width of the peak in $B_{\pm M}(t)$ at $\tilde M$. Figure \[lasexc\] illustrates some possible distributions in $M$ that could be produced using this method.
Adiabatic Limit
---------------
An important limit of our general results is the case of an adiabatically varying potential $V^{\text{ext}}(t)$. In our formalism, this corresponds to a pulse characteristics $u(t)$ which varies on a time scale longer than the time scale set by the lowest EMP energy. The time integral in expression Eq. (\[field\]) is then dominated by the exponential, and $u(\tau)$ can be treated as a constant within the range of integration. We find $$B_{\pm M}(t) = (\nu M)^{1/2} \, V^{\text{ext}}_{\pm M} \, \frac{-
u(t)}{ \varepsilon^{\text{C}}_{M}} \, \exp{\left[ \pm i
\varepsilon^{\text{C}}_{M} t \right]} \quad ,$$ and the total density response $\tilde\varrho_{M}(t)$, derived from Eq. (\[sigma\]), is $$\label{adiabat}
\tilde\varrho_{M}(t) = - u(t) \, V^{\text{ext}}_{M} \,\, \frac{\nu
M}{\varepsilon^{\text{C}}_{M}} \quad .$$
The induced density in Eq. (\[adiabat\]) is the instantaneous ground-state density that minimizes the energy in the presence of the slowly-varying external potential. The energy in the presence of the external potential can be expressed in terms of the charge density as follows: $$E[\varrho] = \int d\theta \,\, \int d\theta' \,\, V^{\text{int}}(
\theta, \theta' ) \tilde\varrho(\theta) \, \tilde\varrho(\theta') +
u(t) \int d\theta \,\, V^{\text{ext}}(\theta)\, \tilde\varrho(\theta)
\quad .$$ The configuration that minimizes the energy functional is $$\label{variate}
\tilde\varrho_{M}(t) = - u(t)
\frac{V^{\text{ext}}_{M}}{V^{\text{int}}_{M}} = - u(t) \,
V^{\text{ext}}_{M} \,\, \frac{\nu M}{\varepsilon^{\text{C}}_{M}}
\quad ,$$ consistent with Eq. (\[adiabat\]).
Derivation of the Effective EMP–Phonon Coupling {#plasphon}
===============================================
In this section, the derivation of the effective coupling between the EMP and 3D phonons is given. We start with the full Hamiltonian \[Eq. (\[elecphon\])\] describing the interaction between 3D electrons with the 3D lattice in the sample. It is convenient to study the Fourier components of the 3D electron density $$\label{fourier}
\varrho^{\text{3D}}_{\vec q} = \int d^3 r \,\, e^{i \vec q \cdot
\vec r} \, \varrho^{\text{3D}}(\vec r) \quad .$$ As we are dealing with a 2D ES which is confined, say, to the $xy$-plane, we introduce the notation $\vec r = z \hat z +
\underbar r $ where $\hat z \perp \,\, \underbar r $ (in reciprocal space: $\vec q = q_z \hat z + \underbar Q $, $\hat z \perp \,\,
\underbar Q $) and assume the electron density to be peaked strongly at $z=z_0$: $\varrho^{\text{3D}}(\vec r) = \chi(z) \varrho^{\text{2D}}
(\underbar r )$. Then, the $z$-integration in Eq. (\[fourier\]) decouples from the rest of the 3D integral, and merely leads to a form factor $F^{\perp}(q_z) = \int dz \, e^{i z \cdot q_z} \chi(z)$. We are left with the 2D Fourier transform of the 2D electron density $\varrho^{\text{2D}}(\underbar r )$.
Disk Geometry
-------------
Specializing to the case of a QH sample in the disk geometry (see Fig. \[geometry\]), we can write approximately $$\label{densapprox}
\varrho^{\text{2D}}(\underbar r ) = \varrho_0 \,\, \Theta(R -
|\underbar r |) + \varrho^{\text{1D}}(\theta) \,\, \delta(R -
|\underbar r | ) \quad ,$$ where $\varrho_0 = \nu / 2\pi$, $\Theta(x)$ is the Heaviside step function, and $\theta$ is the coordinate along the edge (see Fig. \[geometry\]). The remaining integrals can be performed. The result is
$$\label{3Ddens}
\varrho^{\text{3D}}_{\vec q} = \varrho^{\text{bulk}}_{\vec q} +
\varrho^{\text{edge}}_{\vec q} \quad ,$$
$$\varrho^{\text{bulk}}_{\vec q} = F^{\perp}(q_z) \, \nu R^2 \, J_1(Q R)
\quad ,$$
$$\varrho^{\text{edge}}_{\vec q} = F^{\perp}(q_z) \, \sum_{M>0} i^M \,
J_{M}(Q R)\, [\nu M]^{1/2} \,\, \left( b_{M}^{\dagger} \, e^{- i M
\theta_{\vec q}} + b_{M} \, e^{i M \theta_{\vec q}} \right) \quad ,
\label{edgedens}$$
where we write $\theta_{\vec q}$ for the polar angle of the vector $\underbar Q $ in the $xy$-plane, [*i.e.*]{}, the azimuthal angle of $\vec q$ in 3D. We remind the reader that $\underbar Q $ is the projection of the wave vector $\vec q$ onto the plane where the 2D ES is located. To get Eq. (\[edgedens\]), we inserted Eq. (\[density\]) for $\varrho^{\text{1D}}(\theta)$. Using expression Eq. (\[3Ddens\]) for the 3D electron density $\varrho^{\text{3D}}_{\vec q}$ in Eq. (\[elecphon\]), we find the Hamiltonian Eq. (\[coupli\]) for the coupling between the EMP and phonon modes, with $$\label{diskres}
C^{\lambda}_{M}(\vec q) = [\nu M]^{1/2} \, M^{\lambda}(\vec q)
\,\, i^M \, J_{M}(Q R) \,\, F^{\perp}(q_z)$$ as the coupling strength. Note that $J_{M}(x)$ denotes the $M^{\text{th}}$-order Bessel function of the first kind.
Strip Geometry
--------------
For the sake of completeness, we give the corresponding results for the case of a QH bar (strip geometry). By QH bar, we mean a sample with periodic boundary conditions applied in the $\hat x$-direction, and open boundary conditions in the $\hat y$-direction. This configuration space corresponds to the surface of a cylinder with axis in the $\hat y$-direction. Although this geometry is not appropriate for the observation of EMP WP revivals, it can be useful in analyzing experiments in which edge disturbances travel along the edge of a Hall bar.[@zhi:prl:93]
In analogy with Eq. (\[densapprox\]), we write for the case of the strip geometry $$\varrho^{\text{2D}}(\underbar r ) = \varrho_0 \,\, \Theta(W - y) +
\varrho^{\text{1D}}(\theta) \,\, \delta(y - W ) \quad ,$$ with $W$ denoting the width of the strip. We again end up with an expression like Eq. (\[3Ddens\]), and find for the Hamiltonian describing the plasmon-phonon coupling: $$\label{stripcoup}
H^{\text{pl-ph}} = \sum_{M>0 \atop \vec q, \lambda} C(\vec q, \lambda)
\left( a_{\vec q, \lambda} + a^{\dagger}_{- \vec q, \lambda} \right)
\left( b_{M} \, \delta_{M, -R \cdot q_x} + b^{\dagger}_{M} \,
\delta_{M, R \cdot q_x} \right)$$ The coupling strength is $$C(\vec q, \lambda) = [\nu M]^{1/2} \, M^{\lambda}(\vec q) \,\,
F^{\perp}(q_z) \,\, e^{i W \cdot q_y} \quad ,$$ which is different from the result Eq. (\[diskres\]) we found for the disk case. Note that the factor $e^{i W \cdot q_y}$ is simply a form factor describing the profile of the charge density in $\hat y$-direction. Here, we have assumed a sharp confining potential and therefore used a delta-function. In general, this is not the experimentally realistic situation, and we will have to replace $e^{i W \cdot q_y}$ by a form factor $F^{\parallel}(q_y)$. A similar form factor should in principle be included in the analysis for the disk geometry as well, but would not be important for small $M$. Note the differences between the final expression for the coupling of EMP to phonons, Eq. (\[coupli\]) for the disk geometry, and Eq. (\[stripcoup\]) the for strip geometry.
Specialization: Acoustic Phonons in Semiconductors
--------------------------------------------------
For the physical situation we are concerned with in this paper, acoustic phonons play the dominant rôle. In polar semiconductors, as for instance in AlGaAs/GaAs heterostructures, phonon coupling occurs due to both deformation-potential and piezoelectric scattering. The bare 3D electron-phonon coupling reads $$M^{\lambda}(\vec q)
= \left( \frac{\hbar}{2\rho \Omega [\epsilon(Q)]^2
\omega^{\lambda}_{\vec q} }\right)^{\frac{1}{2}} \left[ D \, q \, \,
\delta_{\lambda, l} + i \,\, e h_{\text{14}} \,\,
{\cal M}_{\lambda}(\hat q) \right] \, ,$$ where $\rho$ and $\Omega$ denote the 3D bulk density and the sample volume, respectively. We have also introduced the strengths of the deformation potential ($D$) and the piezoelectric coupling ($h_{\text{14}}$). For more details on phonons in AlGaAs/GaAs heterostructures as well as numerical values of the parameters, see Ref. . A general reference on scattering mechanisms in metals and semiconductors is Ref. . The dielectric function $\epsilon(Q)$ incorporates screening of the original electron-phonon interaction due to many-electron effects. Here, the 2D ES is in the QH regime, and there is no screening, therefore we set $\epsilon(Q) \rightarrow
1$. Finally, the functions ${\cal M}_{\lambda}(\hat q)$ model the directional dependence of the piezoelectric phonon coupling. For a 2D ES which lies in the (100) plane of GaAs, we have[@lyo:prb:88]
$$\begin{aligned}
[{\cal M}_l(\hat q)]^2 &=& \frac{9}{2} \frac{Q^4 q_{z}^{2}}{q^6}
\quad , \\
\protect[{\cal M}_t(\hat q)]^2 &=& 2 \frac{Q^2 q_{z}^{4}}{q^6} +
\frac{1}{4} \frac{Q^6}{q^6} \quad ,\end{aligned}$$
for longitudinal and transverse modes respectively.
=8.5cm
Plasmon Self-Energy due to Phonon Coupling {#selfen}
==========================================
Due to the plasmon-phonon coupling, the plasmons acquire a non-vanishing imaginary part of the self-energy that is related to the rate of phonon emission/absorption by the plasmons. Here, we calculate the plasmon self-energy using diagrammatic perturbation theory. Due to azimuthal symmetry, the self-energy is diagonal in angular-momentum indices. As the Hamiltonian of the coupled EMP-phonon system is quadratic, the leading-order diagram gives the exact result for the self-energy. Note that we consider here the coupling of chiral 1D plasmons to 3D phonons; the problem of chiral 1D plasmons coupled to 1D phonons with implications for quantum-Hall edges has been discussed previously.[@hei:prl:96]
The full Hamiltonian of the coupled EMP-phonon system (without the external potential forming the initial wave packet) is $$H' = H^{\text{pl}}_{0} + H^{\text{ph}}_{0} + H^{\text{pl-ph}} \quad ,$$ where $H^{\text{pl}}_{0}$ is given by Eq. (\[freeplasm\]), the expression Eq. (\[coupli\]) for $H^{\text{pl-ph}}$ in the disk geometry has been derived in Appendix \[plasphon\], and $H^{\text{ph}}_{0}$ describes a system of free 3D phonons with dispersion relation $\omega^{\lambda}_{\vec q}$: $$H^{\text{ph}}_{0} = \sum_{\vec q, \lambda} \omega^{\lambda}_{\vec q}
\,\, a^{\dagger}_{\vec q, \lambda} \, a_{\vec q, \lambda} \quad .$$ We want to calculate the single-plasmon Matsubara Green’s function defined in Eq. (\[greensfunc\]), which can be written as a sum over all distinct connected diagrams. Three of the diagrams appearing in ${\cal G}_{M}(\tau)$ are shown in Fig. \[diagrams\]. All diagrams of higher than second order are reducible. The sums over phonon wave vectors that are implicit in the diagrams enforce the conservation of the plasmon wave number in each higher-order diagram. Using the standard definition[@mahan] for the phonon propagator $${\cal D}_\lambda^0 (\vec q, \tau_1 - \tau_2) = - \left< \mbox{T}_\tau
\left[ a^{\dagger}_{- \vec q, \lambda}(\tau_1) + a_{\vec q, \lambda}
(\tau_1) \right] \left[ a^{\dagger}_{\vec q, \lambda}(\tau_2) +
a_{-\vec q, \lambda} (\tau_2) \right] \right>_0$$ we find that the full plasmon propagator is given exactly by $${\cal G}_{M}(i \omega_n) = \big[ i \omega_n - \varepsilon^{\text{C}}_M
- \Sigma_{M}(i \omega_n) \big]^{-1}$$ with the plasmon self-energy $$\label{selfres}
\Sigma_{M}(i \omega_n) = \sum_{\vec q, \lambda} \left|
C^{\lambda}_{M}(\vec q)\right|^2 \, {\cal D}_\lambda^0 (\vec q, i
\omega_n) \quad ,$$ which is the contribution from second-order perturbation theory, expressed diagrammatically in Fig. \[diagrams\](d). The same result for the plasmon propagator is obtained when integrating out the phonon degrees of freedom in a path-integral expression for the partition function of the coupled EMP-phonon system.
After continuation to real frequencies, it is possible to find the imaginary part of the self-energy for the retarded Green’s function $$\Im m \, \Sigma_{M}(\omega) = \pi \sum_{\vec q, \lambda} \left|
C^{\lambda}_{M}(\vec q)\right|^2 \, \left[ \delta \big(\omega +
\omega_{\vec q} \big) - \delta \big( \omega - \omega_{\vec q} \big)
\right] \, .$$ From this expression, we read off the damping rate for the mode with wave number $M$ as expressed in Eq. (\[damprate\]).
[10]{}
A. H. MacDonald, in [*Proceedings of the 1994 Les Houches Summer School on Mesoscopic Quantum Physics*]{}, edited by E. Akkermans [*et al.*]{} (Elsevier Science, Amsterdam, 1995), pp. 659–720.
A. H. MacDonald, Phys. Rev. Lett. [**64**]{}, 220 (1990).
A. H. MacDonald, Brazilian J. Phys. [**26**]{}, 43 (1996).
X. G. Wen, Phys. Rev. B [**41**]{}, 12838 (1990).
X. G. Wen, Phys. Rev. B [**44**]{}, 5708 (1991).
X. G. Wen, Int. J. Mod. Phys. B [**6**]{}, 1711 (1992).
V. J. Emery, in [*Highly Conducting One-Dimensional Solids*]{}, edited by J. T. Devreese [*et al.*]{} (Plenum Press, New York, 1979), pp. 247–303.
J. S’olyom, Adv. Phys. [**28**]{}, 201 (1979).
F. D. M. Haldane, J. Phys. C [**14**]{}, 2585 (1981).
D. B. Mast, A. J. Dahm, and A. L. Fetter, Phys. Rev. Lett [**54**]{}, 1706 (1985).
Edge-magnetoplasmon collective excitations of 2D electron systems occur quite generally but, outside of the quantum-Hall regime, they have a finite life time, and the excitation spectrum does not completely bosonize. In macroscopic systems they are generally strongly overdamped in strong magnetic fields, except near the filling factors at which the quantum-Hall effect occurs. For early theoretical studies, see A. L. Fetter, Phys. Rev. B [**32**]{}, 7676 (1985); J.-W. Wu, P. Hawrylak, and J. J. Quinn, Phys. Rev. Lett. [**55**]{}, 879 (1985); and Ref. . For a review and additional references, see V. A. Volkov and S. A. Mikhailov, in [*Landau Level Spectroscopy*]{}, edited by G. Landwehr and E. I. Rashba (Elsevier Science, Amsterdam, 1991), pp. 855–907.
V. A. Volkov and S. A. Mikhailov, Pis’ma Zh. Eksp. Teor. Fiz. [**42**]{}, 450 (1985) \[JETP Lett. [**42**]{}, 556 (1985)\]; Zh. Eksp. Teor. Fiz. [**94**]{}, 217 (1988) \[Sov. Phys. JETP [**67**]{}, 1639 (1988)\].
U. Zülicke and A. H. MacDonald, Phys. Rev. B [**54**]{}, December 15 issue 1996 (to appear).
M. Wassermeier [*et al.*]{}, Phys. Rev. B [**41**]{}, 10287 (1990).
Ya. M. Blanter and Yu. E. Lozovik, Physica B [**182**]{}, 254 (1992).
S. J. Allen, H. L. Störmer, and J. C. M. Hwang, Phys. Rev. B [**28**]{}, 4875 (1983).
V. I. Tal’yanski[ĭ]{} [*et al.*]{}, Pis’ma Zh. Eksp. Teor. Fiz. [**50**]{}, 196 (1989) \[JETP Lett. [**50**]{}, 221 (1989)\].
T. Demel, D. Heitmann, P. Grambow, and K. Ploog, Phys. Rev. Lett. [**64**]{}, 788 (1990).
I. Grodnensky, D. Heitmann, and K. von Klitzing, Phys. Rev. Lett. [**67**]{}, 1091 (1991).
R. C. Ashoori [*et al.*]{}, Phys. Rev. B [**45**]{}, 3894 (1992).
N. B. Zhitenev, R. J. Haug, K. v. Klitzing, and K. Eberl, Phys. Rev. Lett. [**71**]{}, 2292 (1993); G. Ernst, R. J. Haug, J. Kuhl, K. v. Klitzing, and K. Eberl, Phys. Rev. Lett. [**77**]{}, 4245 (1996).
D. C. Glattli [*et al.*]{}, Phys. Rev. Lett [**54**]{}, 1710 (1985).
P. J. M. Peters [*et al.*]{}, Phys. Rev. Lett. [**67**]{}, 2199 (1991).
J. Parker and C. R. Stroud, Phys. Rev. Lett. [**56**]{}, 716 (1986).
G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A [**34**]{}, 1058 (1986); G. Alber and P. Zoller, Phys. Rep. [**199**]{}, 231 (1991).
A. ten Wolde, L. D. Noordam, A. Lagendijk, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. [**61**]{}, 2099 (1988).
J. A. Yeazell, M. Mallalieu, J. Parker, and C. R. Stroud, Phys. Rev. A [**40**]{}, 5040 (1989); J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Phys. Rev. Lett. [**64**]{}, 2007 (1990); J. A. Yeazell and C. R. Stroud, Phys. Rev. A [**43**]{}, 5153 (1991).
D. R. Meacher, P. E. Meyler, I. G. Hughes, and P. Ewart, J. Phys. B [**24**]{}, L63 (1991).
I. Sh. Averbukh and N. F. Perelman, Phys. Lett. [**139A**]{}, 449 (1989); C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, Phys. Rev. Lett. [**77**]{}, 3999 (1996).
M. Nauenberg, J. Phys. B [**23**]{}, L385 (1990).
R. Bluhm and V. A. Kostelecký, Phys. Rev. A [**50**]{}, R4445 (1994); Phys. Lett. A [**200**]{}, 308 (1995); Phys. Rev. A [**51**]{}, 4767 (1995).
See for example, R. Bluhm, V. A. Kostelecký, and J. Porter, Am. J. Phys. [**64**]{}, 944 (1996).
R. Bluhm, V. A. Kostelecký, and B. Tudose, Phys. Lett. A [**222**]{}, 220 (1996); Phys. Rev. A [**55**]{}, January 1 issue (1997) (to appear).
R. Bluhm and V. A. Kostelecký, Phys. Rev. A [**48**]{}, R4047 (1993); Phys. Rev. A [**49**]{}, 4628 (1994); R. Bluhm, V. A. Kostelecký, and B. Tudose, Phys. Rev. A [**52**]{}, 2234 (1995); Phys. Rev. A [**53**]{}, 937 (1996).
V. A. Kostelecký and M. M. Nieto, Phys. Rev. Lett. [**53**]{}, 2285 (1984); Phys. Rev. A [**32**]{}, 1293, 3243 (1985); R. Bluhm and V. A. Kostelecký, *ibid. [**47**]{}, 794 (1993).*
J. Wals, H. H. Fielding, J. F. Christian, L. C. Snoek, W. J. van der Zande, and H. B. van Linden van den Heuvell, Phys. Rev. Lett. [**72**]{}, 3783 (1994).
L. D. Noordam, D. I. Duncan, and T. F. Gallagher, Phys. Rev. A [**45**]{}, 4735 (1992).
B. Broers, J. F. Christian, J. H. Hoogenraad, W. J. van der Zande, H. B. van Linden van den Heuvell, and L. D. Noordam, Phys. Rev. Lett. [**71**]{}, 344 (1993).
J. F. Christian, B. Broers, J. H. Hoogenraad, W. J. van der Zande, and L. D. Noordam, Opt. Commun. [**103**]{}, 79 (1993).
N. F. Ramsey, [*Molecular Beams*]{} (Oxford University Press, Oxford, 1956).
There exists an analogy between the charge signal $Q(t)$ that is induced by the charge density of a moving EMP WP at a detecting gate and the observable that is employed to measure Rydberg wave packets using Ramsey’s method ([*i.e.*]{} the ionization signal). Both signals consist of a rapidly oscillating part which is modulated by an envelope function, and it is the latter that shows classical behavior on short times and revival structure on longer times.
Note that the existence of the rapid oscillations necessarily follows from the observability criterion for revivals: $t_{\text{rev}}\gg
T_{\text{cl}}$ implies $T_{\text{cl}}\gg T_{\text{ph}}$ (which is equivalent to $\tilde M\gg 1$). The charge-density configuration of the EMP WP consists of a finite classical wave train which has a large dominant wave number $\tilde M$. In this it differs from a Rydberg wave packet which is a complex wave.
See for example, J. D. Jackson, [*Classical Electrodynamics*]{}, Second Edition (Wiley, New York, 1975), p. 582.
Strictly speaking, the classical period $T_{\text{cl}}$ is not [*entirely*]{} determined by classical physics. This is a difference between the physics of EMP WP and Rydberg wave packets. As the EMP WP state is a many-body quantum state which is a superposition of many-electron collective modes (EMP), its time evolution is influenced by exchange-correlation effects among the electrons in the 2D ES. For the case of long-range interaction, these effects become more important at larger wave numbers $M$. However, for macroscopic QH samples, because of long-range interactions the influence of exchange-correlation effects on the EMP dispersion is relatively small,[@uz-ahm] and we recover approximately the classical drift period.
G. D. Mahan, [*Many-Particle Physics*]{} (Plenum, New York, 1990).
S. K. Lyo, Phys. Rev. B [**38**]{}, 6345 (1988), and references therein.
For measurements of the quantum Hall effect in Si/Ge-heterostructures, see, [*e.g.*]{}, D. Többen, F. Schäffer, A. Zrenner, and G. Abstreiter, Phys. Rev. B [**46**]{}, 4344 (1992); R. B. Dunford [*et al.*]{}, Surf. Sci. [**361/362**]{}, 550 (1996); K. Ismail, Physica B [**227**]{}, 310 (1996). A brief review of Si/Ge-heterostructures, including more references, can be found in G. Abstreiter, Solid State Comm. [**92**]{}, 5 (1994).
V. F. Gantmakher and Y. B. Levinson, [*Carrier Scattering in Metals and Semiconductors*]{} (North Holland, Amsterdam, 1987).
O. Heinonen and S. Eggert, Phys. Rev. Lett. [**77**]{}, 358 (1996).
|
---
abstract: 'We study a real, massive Klein-Gordon field in the Poincaré fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of *dynamical* boundary conditions of generalized Wentzell type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincaré fundamental domain of AdS. We completely solve the equations for the bulk and boundary fields and investigate the existence of bound state solutions, motivated by the analogous problem with Robin boundary conditions, which are recovered as a limiting case. Finally, we argue that both Robin and generalized Wentzell boundary conditions are distinguished in the sense that they are invariant under the action of the isometry group of the AdS conformal boundary, a condition which ensures in addition that the total flux of energy across the boundary vanishes.'
author:
- Claudio Dappiaggi
- 'Hugo R. C. Ferreira'
- 'Benito A. Juárez-Aubry'
title: |
Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime\
with dynamical boundary conditions of Wentzell type
---
Introduction
============
Classical and quantum field theory on asymptotically anti-de Sitter (AdS) spacetimes, and generally other spacetimes with boundaries, has been the target of significant attention in the last two decades, mainly inspired by the remarkable AdS/CFT correspondence [@Maldacena:1997re; @Witten:1998qj], see [@Ammon:2015] for a modern overview. The importance of this correspondence has gone well beyond its initial connection with the quantum gravity formulation in string theory and has become relevant in many low energy physics applications, ranging from nuclear to condensed matter physics [@Hartnoll:2009sz].
From a geometric standpoint, in contrast with their asymptotically flat or asymptotically de Sitter counterparts, asymptotically AdS spacetimes are not globally hyperbolic; the conformal asymptotic boundary at infinity is timelike. As a consequence, on an asymptotically AdS background, one cannot expect to find global solutions for hyperbolic equations, such as the Klein-Gordon one, only by assigning suitable initial data. These must be supplemented with appropriate boundary conditions imposed at the conformal boundary.
In previous work [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj], two of the authors analyzed the classical and quantum field theory of a massive scalar field propagating in anti-de Sitter (AdS) spacetime in $d+1$ spacetime dimensions, subject to homogeneous Robin boundary conditions, which include the familiar Dirichlet and Neumann boundary conditions as particular cases, see also [@Bussola:2017wki] for an analysis on BTZ spacetime. In that work, by means of a Fourier transform, the Klein-Gordon equation has been reduced to a Sturm-Liouville problem, which naturally provides all the admissible boundary conditions of Robin type for a specific range of the mass parameter of the field. In this context, studying all admissible Robin boundary conditions at once is a good strategy for finding the parameter space in mass and curvature coupling for which there exist bound state solutions, which decay exponentially away from the AdS boundary. These modes not only lead to instabilities in the classical linear theory but also pose an obstruction to the existence of a ground state for the underlying quantum theory.
These results call for two natural generalizations, the first by allowing the background to be a generic asymptotically AdS spacetimes, the second by fixing more general boundary conditions. In particular, in the AdS case, the second avenue implies in particular that one should treat boundary value problems outside of the realm of Sturm-Liouville theory.
From a structural standpoint, since there exist infinite choices of boundary conditions, the first step consists of identifying a natural subclass which is distinguished for its physical properties. To this end, it seems that one bit of information that could be used is the existence of a large group of isometries at the conformal boundary. Heuristically one expects that boundary conditions should be chosen in such a way to be compatible with the action of such group. The problem of translating such idea in a concrete mathematical tool can be addressed in the specific case of an AdS spacetime by adapting and reinterpreting the recent results of [@Ibort:2014sua; @Ibort2; @Pardo]. In particular one can realize that each boundary condition is nothing but an operator acting between (a suitable generalization of) the field restricted to the boundary and its normal derivative. From this perspective it is natural to require that such operator commutes with the scalar representation of the isometry group of the conformal boundary. By using this paradigm one restricts considerably the class of possible boundary conditions, while, at the same time, making clear that it is possible to go beyond those of Robin type.
In this paper, we study a massive scalar field in AdS in $d+1$ spacetime dimensions, AdS$_{d+1}$, subject to [*generalized Wentzell boundary conditions*]{} (WBCs). Specifically, we focus on the so-called Poincaré fundamental domain PAdS$_{d+1}$, which covers only a portion of the full AdS$_{d+1}$ and it has the advantage of being conformal to the half-Minkowski spacetime in $d+1$ dimensions, $\bHo^{d+1}$. As we shall see below, these boundary conditions have boundary data determined by a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincaré fundamental domain of AdS, a feature which is clearly reminiscent of the AdS/CFT framework, though here we limit ourselves to considering non interacting models.
In our investigation, although we limit ourselves to considering classical features of the underlying model, the existence of bound states in particular, our ultimate goal is to provide a full-fledged quantum system. This is in particular one of the key reasons why we shall not only focus on finding smooth solutions for the underlying dynamics, but we will also be interested in the square-integrability of the relevant functions.
As we show in this paper, the WBCs are *distinguished* in the sense that they are dynamical boundary conditions which are invariant under the action of the isometry group of the AdS boundary. Furthermore, as is the case with the simpler, nondynamical Robin boundary conditions, the total fluxes of symplectic and energy currents across the AdS conformal boundary vanish, as required for a closed system. Moreover, the Robin boundary condition eigenstate solutions to the Klein-Gordon operator can be recovered as suitable limits of the WBC ones.
The treatement of WBC boundary conditions in the classical and quantum field theoretic literature has appeared in the work of one of us, together with Barbero, Margalef-Bentabol and Villaseñor, [@G.:2015yxa; @Barbero:2017kvi], where the simple $(1+1)$-dimensional mechanical model of a finite string with point masses subject to harmonic potentials in the extrema has been studied in detail. In that work, the classical system is solved and the Fock quantization is performed, with the ultimate goal of constructing [*boundary Hilbert spaces*]{} where the dynamics of the extremal masses takes place, with the aid of the PDE [*Lions trace*]{} operators. It is further shown that the quantum mechanical dynamics in the boundary Hilbert spaces is non-unitary.
These boundary conditions have also been considered in [@Zahn:2015due] in $(d+1)$-dimensional Minkowski spacetime with one or two timelike boundaries. There, it was investigated the Fock space quantization of the underlying system and, in addition it was shown that the WBC ensure that the short-distance singularities of the two-point function for the boundary field has the form expected of a field living in a $d$-dimensional spacetime, contrary to other boundary conditions, for which the two-point function inherits the short-singularity of the $(d+1)$-dimensional bulk. This seems to be a very desired feature for holographic purposes. Previous explorations of the WBC in mathematical literature appear in *e.g.* [@Ueno:1973; @Favini:2002; @Coclite:2014].
While the main inspiration of this work comes from high energy physics, namely the AdS/CFT conjecture, and the connection to holographic renormalization [@Skenderis:2002wp] is a central motivation for us, we note that the boundary conditions that we consider, as well as related dynamical boundary conditions, are suitable for studying systems in a broad sprectrum of physical problems. We point out that dynamical boundary conditions are generally interesting from the point of view of modelling open systems in condensed matter physics. They are also relevant for lower-dimensional Chern-Simons theories coupled to electrodynamics, which can model e.g. effective topological insulators [@Martin-Ruiz:2015skg]. From a gravitational perspective, dynamical boundary conditions are interesting in the study of isolated horizons, providing an avenue for associating degrees of freedom at a horizon surface. This is attractive from a quantum gravity perspective. In loop quantum gravity, for example, a procedure for counting horizon degrees of freedom yields the Bekenstein-Hawking entropy [@Ashtekar:1997yu; @Ashtekar:1999wa].
The contents of this paper are as follows. In Sec. \[sec:AdS\] we review the basic geometric properties of AdS$_{d+1}$ and the closely related PAdS$_{d+1}$, the Poincaré fundamental domain of AdS$_{d+1}$ as a spacetime in its own right. In addition we show how the Klein-Gordon equation on PAdS$_{d+1}$ can be treated as a (generally singular) Klein-Gordon equation on half-Minkowski. In Sec. \[sec:WBCs\] we introduce WBCs. We motivate them in the form of an action principle for the Klein-Gordon field with boundary dynamical contributions in half-Minkowski spacetime. We then deal with the problem in PAdS$_{d+1}$ with the aid of the aforementioned conformal techniques, by solving both the bulk and boundary field equations in full generality. We consider separately two cases: in Sec. \[sec:regular\] the regular case, corresponding to the massless, conformally coupled (conformally transformed) scalar field (in half-Minkowski), and in Sec. \[sec:singular\] the singular case, corresponding to the general, massive scalar field. Additionally, in both cases, we investigate the existence of bound state mode solutions, which decay exponentially away from the boundary. Finally, in Sec. \[sec:maths\], we explicitly show the vanishing of the fluxes of symplectic and energy currents across the AdS boundary when WBCs are imposed, and explain how these boundary conditions are invariant under the boundary isometry group, making them distinguished. Throughout the paper we employ natural units in which $c = G_{\rm N} = 1$ and a metric with signature $({-}{+}{+}\cdots)$.
Anti-de Sitter spacetime and Klein-Gordon field {#sec:AdS}
===============================================
In this section, we briefly review the basic geometric properties of anti-de Sitter spacetime AdS$_{d+1}$ and introduce the Poincaré fundamental domain PAdS$_{d+1}$, on which we consider a classical scalar field satisfying the Klein-Gordon equation.
Geometry of anti-de Sitter spacetime
------------------------------------
The maximally symmmetric solution of the Einstein field equations with negative cosmological constant, $\Lambda$, is the anti-de Sitter spacetime, which we denote by AdS$_{d+1}$ in $d+1$ Lorentzian dimensions. As a manifold it is diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^d$ and it can be realized as an embedded submanifold in the ambient space $\mathbb{R}^{d+2}$ equipped with the metric $$g_{\mathbb{R}^{d+2}} = - \dd X_0^2 - \dd X_1^2 + \sum_{i,j = 2}^{d+1} \delta^{ij} \, \dd X_i \dd X_j \ ,
\label{Rd2metric}$$ via the equation $$-X_0^2 - X_1^2 + \sum_{i = 2}^{d+1} X_i^2 = \frac{d(d-1)}{\Lambda}\,$$ where each $X_i$, $i=0,...,d+2$ is a Cartesian coordinate, while $\delta^{ij}$ stands for the Kronecker delta. As a consequence AdS$_{d+1}$ comes equipped with the induced (Lorentzian, non-degenerate) metric.
We can give an explicit representation of these geometric structures by considering the [*Poincaré fundamental domain*]{} of AdS$_{d+1}$, denoted by PAdS$_{d+1}$. This is covered by the [*Poincaré coordinate patch*]{}, with $t \in \mathbb{R}$, $z \in \mathbb{R}^+$ and $x_i \in \mathbb{R}$, defined by $$\begin{aligned}
\left\{
\begin{array}{ll}
X_0 = \cfrac{\ell}{z} t, \\
\vspace*{-0.2cm} \\
X_1 = \cfrac{z}{2} \left(1 + \cfrac{1}{z^2} \left(- t^2 + \displaystyle\sum_{i = 1}^{d-1} x_i^2 + \ell^2 \right) \right), \\
\vspace*{-0.2cm} \\
X_i = \cfrac{\ell}{z} x_{i-1}, \quad i = 2, \ldots, d,\\
\vspace*{-0.2cm} \\
X_{d+1} = \cfrac{z}{2} \left(1 + \cfrac{1}{z^2} \left(- t^2 + \displaystyle\sum_{i = 1}^{d-1} x_i^2 - \ell^2 \right) \right),
\end{array}
\right.
\label{PAdScoods}\end{aligned}$$ where $\ell^2 = -d(d-1)/ \Lambda$. Thus, PAdS$_{d+1}$ is a Lorentzian spacetime with underlying manifold $\mathbb{R}^+ \times \mathbb{R}^d$ equipped with the metric $$g_{{\rm PAdS}_{d+1}} = \frac{\ell^2}{z^2} \left( - \dd t^2 + \dd z^2 + \sum_{i, j = 1}^{d-1} \delta^{ij} \dd x_i \dd x_j \right).
\label{gPAdS}$$ Eq. shows that (PAdS$_{d+1},g_{{\rm PAdS}_{d+1}}$) is conformal to the interior of the $(d+1)$-dimensional half-Minkowski spacetime, $(\bHo^{d+1},\eta_{d+1})$, with $\eta_{d+1} = \Omega^2 g_{{\rm PAdS}_{d+1}}$, where the conformal factor is $\Omega = z/\ell$. The conformal boundary of PAdS$_{d+1}$ can be attached at $z = 0$.
The Klein-Gordon field in PAdS$_{d+1}$
--------------------------------------
In this work, we consider a classical, real Klein-Gordon field $\phi: {\rm PAdS}_{d+1} \to \mathbb{R}$. Given initial data on a hypersurface of PAdS$_{d+1}$ for the Klein-Gordon wave equation, $$P \phi = \left( \Box_g^{(d+1)} - m_0^2 - \xi R \right) \phi = 0 \ ,
\label{KGPAdS}$$ where $\Box_g^{(d+1)}$ is the d’Alembert wave operator, $m_0$ is the mass, $\xi \in \mathbb{R}$ is the coupling to the scalar curvature, while $R = -d(d+1) / \ell^2$ is the Ricci scalar of the spacetime. From now on, we set $\ell = 1$. Although, for initial data, which are smooth and compactly supported in PAdS$_{d+1}$, a unique solution exists in its domain of dependence, in order to address the problem of global existence, one needs to impose boundary conditions at timelike infinity, which, in the Poincaré patch, corresponds to the conformal boundary.
In order to control such freedom, we follow the same strategy adopted in [@Dappiaggi:2016fwc] to switch from Eq. to the associated, conformally-transformed scalar field equation in $\mathbb{\mathring{\mathbb H}}^{d+1}$. Hence, defining $\Phi = \Omega^{\frac{1-d}{2}}\phi :\bHo^{d+1} \to \mathbb{R}$, the solutions of are in one to one correspondence with those of $$P_\eta \Phi \doteq \left( \Box_\eta^{(d+1)} - \frac{m^2}{z^2} \right) \Phi = 0 \ ,
\label{KGH}$$ where we have defined $m^2 \doteq m_0^2 + (\xi -\frac{d-1}{4d}) R$. A strategy for dealing with Eq. in the case of a minimally-coupled, real scalar field in $d = 3$ that does not rely on conformal transformations has appeared in [@Ayon-Beato:2018hxz]. Here, we consider problems for which $m^2 \geq -\frac{1}{4}$, which corresponds to the Breitenlohner-Freedman (BF) bound [@Breitenlohner:1982jf].
The boundary condition imposed at the conformal boundary of PAdS$_{d+1}$ that allows one to obtain global solutions for $\phi$ corresponds in to a boundary condition at $z = 0$, where the potential term becomes singular. In [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj] all possible homogeneous boundary conditions of Robin type have been analysed. It is the approach of this paper to consider more general, dynamical boundary conditions, which reduce to those of Robin type in a precise limiting sense. In particular, the boundary conditions that we consider are of [*generalized Wentzell type*]{}. As mentioned above, these have been considered in the mathematical physics literature in [@Zahn:2015due] for regular problems in the half-Minkowski spacetime, and also studied by one of the authors in [@G.:2015yxa; @Barbero:2017kvi] in the context of the quantization of a finite string coupled to point masses subject to harmonic restoring forces at the boundaries.
Wentzell boundary conditions {#sec:WBCs}
============================
In this section, we study the problem defined by Eq. in the bulk and Wentzell boundary conditions at $z = 0$. In Section \[sec:action\] we show that these boundary conditions can be obtained naturally starting from an action functional in the so-called regular case ($m^2 = 0$ in ). We study such case via a mode expansion in Section \[sec:regular\], finding the conditions under which there exists, together with the expected continuous spectrum, point spectrum contributions to the solutions. These indicate the existence of [*bound state mode solutions*]{} to the problem. Afterwards we show how to recover the solutions to the problem with Robin boundary conditions, obtaining full agreement with the results reported in [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj]. In Section \[sec:singular\] we repeat the analysis for the singular problem ($m^2 \in [-\frac{1}{4}, \frac{3}{4}) \setminus \{0\}$), also characterizing the spectrum and obtaining the Robin boundary problem solutions in a suitable limit in agreement with [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj].
Action {#sec:action}
-------
Let us motivate the introduction of the generalized Wentzell boundary conditions, by considering the usual action for a massless Klein-Gordon field in $\bHo^{d+1}$ together with a particular choice of boundary terms: $$\begin{aligned}
S & = - \frac{1}{2} \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^+} \!\!\!\! \dd z \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \ \partial_\mu \Phi \partial^\mu \Phi \nonumber \\
&\quad +\frac{c}{2} \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \left(-\dot{\Phi}^2 + \partial_i \Phi \partial^i \Phi + m^2_{\rm b} \Phi^2 \right) \, ,\end{aligned}$$ where $c$ and $m_{\rm b}^2$ are arbitrary real parameters at this stage and where repeated indexes are summed over with $\mu \in \{t, z, x_1, \ldots, x_{d-1}\}$ and $i \in \{x_1, \ldots, x_{d-1}\}$, and $\dd^{d-1} x = \prod_{i = 1}^{d-1} \dd x_i$. With a slight abuse of notation, we use the symbol $\Phi$ in the boundary integrand, in place of its restriction thereon. Implicitly we are also restricting our attention to kinematic configurations which are sufficiently regular at $z=0$, to make these operations meaningful.
The variation of the action yields $$\begin{aligned}
& dS(\Phi) \cdot \delta = \left. \frac{d}{d \lambda} S(\Phi + \lambda \delta) \right|_{\lambda = 0} \nonumber \\
& = - \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^+} \!\!\!\! \dd z \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \left( -\dot{\Phi} \dot{\delta} + \partial_z \Phi \partial_z\delta + \partial_i \Phi \partial^i \delta \right) \nonumber \\
&\quad +c \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \left(-\dot{\Phi} \dot{\delta} + \partial_i \Phi \partial^i \delta + m^2_{\rm b} \Phi \delta \right) \nonumber \\
& = \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^+} \!\!\!\! \dd z \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \, \delta \, \Box_\eta^{(d+1)} \Phi \nonumber \\
&\quad -c \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \, \delta \left(\Box_\eta^{(d+1)} \Phi - m^2_{\rm b} \Phi + \frac{1}{c} \partial_z \Phi \right) \, ,\end{aligned}$$ where, on the right hand side of the last equality above, the integration by parts in $z$ in the bulk action has contributed to the boundary term.
The extrema of the action, $dS(\Phi) = 0$, are $$\label{variation}
\begin{cases}
\Box_{\eta}^{(d+1)} \Phi = 0 \quad \text{in $\bHo^{d+1}$} \, , \\
\left(\Box_{\eta}^{(d)} - m^2_{\rm b} \right) F = - \dfrac{\rho}{c} \quad \text{in $\bR^d$} \, , \\
\Phi|_{z=0} = F \, , \quad \partial_z \Phi|_{z=0} = \rho \, .
\end{cases}$$ The boundary conditions of the problem are known as *generalized Wentzell boundary conditions* (WBCs), see [@Zahn:2015due] and references therein. These are dynamical boundary conditions, for which there is a boundary field $F$ required to coincide with the restriction of the bulk field at the boundary and to satisfy a Klein-Gordon equation with a source term, which is related to the derivative of the bulk field with respect to the direction orthogonal to the boundary. In the case in which the bulk field is massive, and the field equation is singular at the boundary, the explicit form of these boundary conditions need to be generalized, as the bulk field or its derivative may not be defined at the boundary. We discuss such generalization in Section \[sec:singular\].
Regular case {#sec:regular}
------------
A Klein-Gordon field, $\phi$, in PAdS$_{d+1}$, satisfying eq. with $m_0^2 = -(\xi - \frac{d-1}{4d})R$ can be mapped to the problem defined by Eq. with $m^2 = 0$. This defines, together with appropriate boundary conditions, a regular problem in $\bHo^{d+1}$. We choose the above-introduced WBCs, $$\label{eq:regsystem}
\begin{cases}
\Box_{\eta}^{(d+1)} \Phi = 0 \quad \text{in $\bHo^{d+1}$} \, , \\
\left(\Box_{\eta}^{(d)} - m^2_{\rm b} \right) F = - \dfrac{\rho}{c} \quad \text{in $\bR^d$} \, , \\
\Phi|_{z=0} = F \, , \quad \partial_z \Phi|_{z=0} = \rho \, .
\end{cases}$$ Here, the parameter $c$ is taken to be real and we restrict $m^2_{\rm b} \geq 0$, so that $m^2_{\rm b}$ is interpreted as a [*squared boundary field mass*]{} for the [*boundary field*]{} $F$. We further assume that the Fourier transforms for $\Phi$, $F$ and $\rho$ exist. It suffices for our purposes to consider these functions to identify tempered distributions.
### Bulk and boundary solutions
For the bulk field, we take the Fourier transform along the directions orthogonal to $z$, $$\label{eq:Fouriertransf}
\Phi(\underline{x},z) = \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{\Phi}(\underline{k},z) \, ,$$ where $\underline{x} \doteq (t, x_1, \ldots, x_{d-1})$, $\underline{k} \doteq (\omega, k_1, \ldots, k_{d-1})$ and $\widehat{\Phi}$ are solutions of $$\label{eq:regmodeeq}
- \frac{\dd^2}{\dd z^2} \widehat{\Phi}(\underline{k},z) = q^2 \, \widehat{\Phi}(\underline{k},z) \, , \quad
q^2 \doteq \omega^2 - \displaystyle\sum_{i=1}^{d-1} k_i^2 \, .$$ We note that the differential operator in the LHS of is of Sturm-Liouville type [@Zettl:2005]. Therefore we will work in this framework, whose first step calls for identifying those $\widehat{\Phi}(\underline{k},z)$ which are necessary to construct the fundamental solution associated to .
For the boundary field and source term, we also take the Fourier transform along all directions,
\[eq:FourierFrho\] $$\begin{aligned}
F(\underline{x}) &= \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{F}(\underline{k}) \, , \\
\rho(\underline{x}) &= \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{\rho}(\underline{k}) \, .\end{aligned}$$
In view of the theory for Sturm-Liouville equations should be treated as an eigenvalue equation on $L^2((0,\infty); \dd z)$ with spectral parameter $q^2$. Being $- \frac{\dd^2}{\dd z^2}$ the standard kinetic operator, its spectrum includes a continuous part, $q^2>0$, with a basis of eigensolutions $\big\{ \widehat{\Phi}_1, \, \widehat{\Phi}_2 \big\}$, with $$\begin{aligned}
\label{eq:fundbasisreg}
\widehat{\Phi}_1(\underline{k},z) = \frac{\sin(qz)}{q} \, , \qquad
\widehat{\Phi}_2(\underline{k},z) = - \cos(qz) \, .\end{aligned}$$ Observe that both solutions are square-integrable in a neighbourhood of the origin. We call $\widehat{\Phi}_1$ the *principal solution* at $z=0$, as it is the unique solution (up to scalar multiples) such that $\lim_{z \to 0} \widehat{\Phi}_1(\underline{k},z)/\widehat{\Psi}(\underline{k},z)=0$ for every solution $\widehat{\Psi}(\underline{k},z)$ which is not a scalar multiple of $\widehat{\Phi}_1$. The solution $\widehat{\Phi}_2$ is a nonprincipal solution and is not unique.
A general solution for $q^2>0$ may then be written as $$\label{eq:reglincomb}
\widehat{\Phi}(\underline{k},z) = A(\underline{k}) \widehat{\Phi}_1(\underline{k},z) + B(\underline{k}) \widehat{\Phi}_2(\underline{k},z) \, .$$ From and , one gets $$\label{eq:ABrhoFreg}
A(\underline{k}) = \widehat{\rho}(\underline{k}) \, , \qquad
B(\underline{k}) = - \widehat{F}(\underline{k}) \, .$$ These coefficients depend explicitly in $\underline{k}$, contrarily to the more common Robin boundary conditions. However, it is possible to recover the latter, as explained below.
It remains to obtain the boundary field in terms of the source term. From , it is easy to obtain $$\label{eq:Fitorho}
\widehat{F}(\underline{k}) = - \frac{\widehat{\rho}(\underline{k})}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \, .$$ Hence, the solution for $q^2>0$ may be written as $$\widehat{\Phi}(\underline{k},z) = \rho(\underline{k}) \left[ \widehat{\Phi}_1(\underline{k},z) + \frac{\widehat{\Phi}_2(\underline{k},z)}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \right] \, .$$ Observe that the term $q(\underline{k})^2 - m^2_{\rm b}$ contributes to a singular term which corresponds to two simple poles in the Fourier transform. These can be dealt with by means of standard, complex analysis techniques.
### Existence of bound states
Above, we analyzed the continuous part of the spectrum $q^2>0$ associated with the eigenvalue problem given by . Here, we investigate if there exist also negative eigenvalues, $q^2 < 0$, in the point spectrum with eigensolutions which satisfy the WBCs. Contrary to the continuous spectrum, in these case we must look for proper eigenfunctions, that is square-integrable solutions to .
For that let $\lambda = -q^2 > 0$ and consider $$\widehat{\Phi}_{\rm bs}(\underline{k},z) = - B(\underline{k}) \, e^{-\sqrt{\lambda}z} \, .$$ This is certainly a solution of the bulk field equation. If it additionally solves the WBCs for some choice of $\lambda$, it is a *bound state mode solution*, that is, a mode which exponentially decay with $z$.
The WBCs, together with , imply that $$\label{eq:lambdaeq}
\sqrt{\lambda} = c \left(q^2 - m^2_{\rm b} \right) = c \left(- \lambda - m^2_{\rm b} \right) \, .$$ It is clear that, with $m^2_{\rm b} \geq 0$, if $c\geq 0$ there is no positive $\lambda$ which solves the equation. For $c<0$, the solutions are $$\lambda = \frac{1}{2c^2} \left(1 - 2 m^2_{\rm b} c^2 \pm \sqrt{1 - 4 m^2_{\rm b} c^2}\right) \, .$$ If $m_{\rm b}=0$ there exists *one* strictly positive value of $\lambda$, which corresponds to one negative eigenvalue and, thus, one bound state. If $m_{\rm b}>0$, we have three cases:
- If $c<-1/(2m_{\rm b})$, then there is *no* strictly positive value of $\lambda$, and thus no bound states.
- If $c=-1/(2m_{\rm b})$, then there is *one* strictly positive value of $\lambda$, which corresponds to one negative eigenvalue and, thus, one bound state.
- If $-1/(2m_{\rm b})<c<0$, there are always *two* strictly positive values of $\lambda$, corresponding to two negative eigenvalues and, thus, two bound states.
### Robin boundary conditions
It is possible to recover Robin boundary conditions at $z=0$ from the WBCs through a specific choice of the boundary field mass term $m^2_{\rm b} $ and an appropriate limit of the constant $c$.
To see that, choose the boundary field mass such that $$m^2_{\rm b} = \frac{1}{c \kappa} \, ,$$ where $\kappa$ is a real number which is positive for $c>0$ and negative for $c<0$, [*i.e.*]{} keeping the squared boundary field mass positive. Then, Robin boundary conditions are recovered in the limit $c \to 0$: $$\widehat{\Phi}(\underline{k},z) = \rho(\underline{k}) \left[ \widehat{\Phi}_1(\underline{k},z) - \kappa \, \widehat{\Phi}_2(\underline{k},z) \right] \, .$$ The usual Dirichlet boundary conditions, for which $\widehat{\Phi}(\underline{k},0)=0$, correspond to $\kappa = 0$, whereas $\kappa \to \infty$ correspond to Neumann boundary conditions, for which $\partial_z\widehat{\Phi}(\underline{k},z)|_{z=0}=0$.
In [@Dappiaggi:2016fwc] it was found that one bound state mode solution exists when $\kappa < 0$, otherwise no such solution exists. From , if we set $m^2_{\rm b} = \frac{1}{c \kappa}$ and then take $c \to 0^-$, we obtain $$\sqrt{\lambda} = - \frac{1}{\kappa} \, .$$ Hence, if $\kappa < 0$ there is a strictly positive value of $\lambda$, which furthermore agrees with the result in [@Dappiaggi:2016fwc]. If $\kappa > 0$, there are no bound states in the limit $c \to 0^+$, also in agreement with the results of [@Dappiaggi:2016fwc].
Singular case {#sec:singular}
-------------
In the case of a massive scalar field in the Poincaré patch of AdS$_{d+1}$, the corresponding field equation for the conformally related field in $\bHo^{d+1}$ is singular at $z=0$ and the previous formulation of the WBCs is no longer valid, as the bulk field or its derivative with respect to $z$ may not be defined at $z=0$. In order to bypass this hurdle, we rewrite the underlying equation of motion in the following way: $$\label{eq:singsystem}
\begin{cases}
\left(\Box_{\eta}^{(d+1)} - \frac{m^2}{z^2} \right) \Phi = 0 \quad \text{in $\bHo^{d+1}$} \, , \\
\left(\Box_{\eta}^{(d)} - m^2_{\rm b} \right) F = - \dfrac{\rho}{c} \quad \text{in $\bR^d$} \, , \\
W_z \left[\Phi, \Phi_1\right] = F \, , \quad W_z \left[\Phi, \Phi_2\right] = \rho \, ,
\end{cases}$$ where $\big\{ \Phi_1, \, \Phi_2 \big\}$ is a basis of solutions, $W_z[u,v] \doteq u\frac{\partial v}{\partial z} - \frac{\partial u}{\partial z}v$ is the Wronskian betweens, and where $\Phi_1$ is chosen so that $\widehat{\Phi}_1$ is a principal solution at $z=0$. Since both $\widehat{\Phi}_1$ and $\widehat{\Phi}_2$ turn out to be solutions of an ODE with no first order derivative, see below, their Wronskian is constant in $z$. Hence we can normalize them so that $$\label{eq:condwronskian}
W_z \left[\widehat{\Phi}_1, \, \widehat{\Phi}_2\right] = 1 \, ,$$ and $\big\{ \widehat{\Phi}_1, \, \widehat{\Phi}_2 \big\}$ reduce to in the regular case.
### Bulk and boundary solutions
Using the same Fourier expansions as in in , $\widehat{\Phi}$ is now solution of $$\label{eq:singmodeeq}
\left(- \frac{\dd^2}{\dd z^2} + \frac{m^2}{z^2} \right)\widehat{\Phi}(\underline{k},z) = q^2 \, \widehat{\Phi}(\underline{k},z) \, .$$ Here, it is useful to remind ourselves that $m^2 = m_0^2 + (\xi - \frac{d-1}{4d})R$ and to introduce the convenient notation $$\nu \doteq \frac{1}{2} \sqrt{1+4m^2} \, .$$ The BF bound implies that $\nu \geq 0$.
A basis of solutions $\big\{ \widehat{\Phi}_1, \, \widehat{\Phi}_2 \big\}$ satisfying the required properties for the continuous part of the spectrum, $q^2 > 0$, is the following:
$$\begin{aligned}
\widehat{\Phi}_1(\underline{k},z) &= \sqrt{\frac{\pi}{2}} \, q^{-\nu} \sqrt{z} \, J_{\nu}(qz) \, , \label{eq:fundamentalsolutions1} \\
\widehat{\Phi}_2(\underline{k},z) &=
\begin{cases}
- \sqrt{\dfrac{\pi}{2}} \, q^{\nu} \sqrt{z} \, J_{-\nu}(qz) \, , & \nu > 0 \, , \\
- \sqrt{\dfrac{\pi}{2}} \sqrt{z} \left[ Y_{0}(qz) - \dfrac{2}{\pi} \log(q) \right] \, , & \nu = 0 \, .
\end{cases} \label{eq:fundamentalsolutions2}\end{aligned}$$
\[eq:fundamentalsolutions\]
The solution $\widehat{\Phi}_1$ is the principal solution at $z=0$ and is square-integrable near $z=0$ for all $\nu \geq 0$. The nonprincipal solution $\widehat{\Phi}_2$ is only square-integrable near $z=0$ if $\nu \in [0,1)$, hence, we only apply the WBCs for those values of the mass, namely $m^2 \in [-\frac{1}{4}, \frac{3}{4})$.
Hence, a general solution satisfying WBCs for $q^2 > 0$ and $\nu \in [0,1)$ is $$\label{eq:fundbasissing}
\widehat{\Phi}(\underline{k},z) = A(\underline{k}) \widehat{\Phi}_1(\underline{k},z) + B(\underline{k}) \widehat{\Phi}_2(\underline{k},z) \, ,$$ with $A(\underline{k}) = \widehat{\rho}(\underline{k})$ and $B(\underline{k}) = - \widehat{F}(\underline{k})$, where we used and . For $\nu \geq 1$, the only square-integrable solution is given by $\widehat{\Phi}_1$ and no boundary conditions need to be applied at $z=0$.
The boundary field $F$ is still given by the same expression of the regular case, $$\label{eq:Fitorho2}
\widehat{F}(\underline{k}) = - \frac{\widehat{\rho}(\underline{k})}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \, .$$ Hence, the bulk field may be written as $$\widehat{\Phi}(\underline{k},z) = \rho(\underline{k}) \left[ \widehat{\Phi}_1(\underline{k},z) + \frac{\widehat{\Phi}_2(\underline{k},z)}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \right] \, .$$
### Existence of bound states
Analogously to the regular case, we investigate if there exists negative eigenvalues, $q^2 < 0$, in the point spectrum of the singular eigenvalue problem given by with proper eigenfuctions which satisfy the WBCs.
Again letting $\lambda = -q^2 > 0$, consider $$\widehat{\Phi}_{\rm bs}(\underline{k},z) = \sqrt{z} \, K_{\nu}(\sqrt{\lambda}z) \, .$$ This is a solution of , and the WBCs, together with , imply that $$\label{eq:lambdanu}
\lambda^{\nu} = c \left(q^2 - m^2_{\rm b} \right) = c \left(- \lambda - m^2_{\rm b} \right) \, .$$ For $c \geq 0$ there is no positive $\lambda$ that solves the equation.
If $c<0$, we have several possibilities. If $m_{\rm b}=0$, then there is one strictly positive root, $$\lambda = (-c)^{\frac{1}{\nu-1}} \, ,$$ corresponding to one bound state. If $m_{\rm b}>0$ and $\nu=0$, there is a positive solution, $$\lambda = - \frac{1+cm^2_{\rm b}}{c} \, ,$$ when $- 1/m^2_{\rm b} < c < 0$, otherwise there is no positive solution, and hence no bound states. If $m_{\rm b}>0$ and $\nu \in (0,1)$, we cannot find analytical solutions of , but we can still obtain the number of positive roots. Let $$f(\lambda)=\lambda^{\nu}+c\lambda+c m^2_{\rm b} \, .$$ We want to know if $f$ has any positive roots for $c<0$ and $\nu \in (0,1)$. First, note that $f(0)=c m^2_{\rm b} < 0$ and that $\lim_{\lambda\to\infty}f(\lambda)=-\infty$ for $\nu \in (0,1)$ and $m_{\rm b} > 0$. Moreover, there is only one maximum at $\lambda_{\rm max} = (-c/\nu)^{1/(\nu-1)}$ with $$f(\lambda_{\rm max}) = (1-\nu) \left(\frac{-c}{\nu}\right)^{\frac{\nu}{\nu-1}} + c m^2_{\rm b} \, .$$ The maximum is positive, and hence there are two positive roots, if $-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}<c<0$. Otherwise, if $c=-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$, there is one positive root, and if $c<-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$ then there are no positive roots.
![\[fig:plot\]Plot of $f(\lambda)=\lambda^{\nu}+c\lambda+c m^2_{\rm b}$ for $\nu=1/3$ and $m_{\rm b}=1$ for different values of $c$.](WBC-plot)
We then conclude that, if $c<0$, $m_{\rm b} > 0$ and $\nu \in (0,1)$:
- If $c<-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$, then there is *no* strictly positive value of $\lambda$, and, thus, no bound states.
- If $c=-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$ (or $m_{\rm b}^2 = 0$), then there is *one* strictly positive value of $\lambda$, which corresponds to one negative eigenvalue and, thus, one bound state.
- If $-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}<c<0$, there are always *two* strictly positive values of $\lambda$, corresponding to two negative eigenvalues and, thus, two bound states.
These results are illustrated in Fig. \[fig:plot\]. Note that they are in agreement with the regular case, $\nu=\frac{1}{2}$.
Finally, if we consider the limit in which we recover Robin boundary conditions, by setting $m^2_{\rm b} = \frac{1}{c \kappa}$ and then taking $c \to 0$, we obtain from that $$\lambda^{\nu} = - \frac{1}{\kappa} \, .$$ Hence, if $\kappa < 0$ there is a strictly positive value of $\lambda$, and no positive values of $\lambda$ for $\kappa >0$, which agrees with the result in [@Dappiaggi:2016fwc].
Distinguishing structural properties of the generalized Wentzell boundary conditions {#sec:maths}
====================================================================================
In this section, we first show that imposing the generalized Wentzell boundary conditions (WBCs) at the PAdS boundary guarantees that the total fluxes of symplectic and energy currents across the boundary vanish, thus showing that the system is closed, as is the case with Robin (hence Dirichlet and Neumann) boundary conditions. We then provide a further explanation so as to why these and the Robin boundary conditions are *distinguished* on account of their interplay the scalar represetation of the isometry group at the conformal boundary.
Vanishing symplectic and energy flux across the boundary
--------------------------------------------------------
For the bulk field $\Phi$, now assumed to be complex-valued, we may define the *bulk symplectic current* as $$J_{\mu} \doteq -i \left(\overline{\Phi} \partial_{\mu} \Phi - \Phi \partial_{\mu} \overline{\Phi} \right) \, .$$ It is covariantly conserved, $\partial_{\mu}J^{\mu} = 0$, or equivalently $\dd \ast J = 0$. Using Stokes’ theorem, $$\begin{aligned}
0 = \int_{\bHo^{d+1}} \dd \ast J = \int_{\bR^d} \ast J \, ,\end{aligned}$$ which implies that, in combination of or , $$\begin{aligned}
\int_{\bR^d} \dd^d x \, \left(\rho \overline{F} - \overline{\rho} F \right) = 0 \, .\end{aligned}$$ This is a condition that both the source term $\rho$ and the boundary field $F$ must satisfy. One possibility is that the integrand itself vanishes, which, by , implies that the ratio $B/A$ in or must be real — that is, Robin boundary conditions must be imposed at the AdS boundary. But, more generally, the integrand does not need to vanish as long as its integral over the boundary is identically zero. Using or , one has $$\begin{aligned}
{}& \int_{\bR^d} \dd^d x \int_{\bR^d} \dd^d \underline{k}_1 \int_{\bR^d} \dd^d \underline{k}_2 \,
\left[ \frac{\widehat{\rho}(\underline{k}_1) \overline{\widehat{\rho}(\underline{k}_2)}}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} e^{i(\underline{k}_1-\underline{k}_2)\cdot \underline{x}} \right. \\
&\quad - \left. \frac{\overline{\widehat{\rho}(\underline{k}_1)} \widehat{\rho}(\underline{k}_2)}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} e^{-i(\underline{k}_1-\underline{k}_2)\cdot \underline{x}} \right] \\
&= \int_{\bR^d} \dd^d \underline{k}_1 \int_{\bR^d} \frac{\dd^d \underline{k}_2}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} \,
\left[ \widehat{\rho}(\underline{k}_1) \overline{\widehat{\rho}(\underline{k}_2)} - \overline{\widehat{\rho}(\underline{k}_1)} \widehat{\rho}(\underline{k}_2) \right] \\
&\quad \times \delta(\underline{k}_1-\underline{k}_2) \\
&= \int_{\bR^d} \frac{\dd^d \underline{k}}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} \,
\left[ \widehat{\rho}(\underline{k}) \overline{\widehat{\rho}(\underline{k})} - \overline{\widehat{\rho}(\underline{k})} \widehat{\rho}(\underline{k}) \right] = 0 \, .\end{aligned}$$ This shows that WBCs guarantees that the total symplectic flux across the boundary vanishes.
Again using or , we obtain $$\begin{aligned}
\int_{\bR^d} \dd^d x \, \eta^{\alpha\beta} \partial_{\alpha} \left(\overline{F} \partial_{\beta} F - F \partial_{\beta} \overline{F}\right) = 0 \, ,\end{aligned}$$ which suggests the definition of a *boundary symplectic current* $$\label{eq:boundarysymplecticcurrent}
J^{\partial}_{\alpha} \doteq -i \, c \left(\overline{F} \partial_{\alpha} F - F \partial_{\alpha} \overline{F}\right) \, .$$ Note, however, that it is not covariantly conserved, as $$\partial^{\alpha} J^{\partial}_{\alpha} = -i \left(\rho \overline{F} - \overline{\rho} F \right) \, ,$$ except in the particular case of Robin boundary conditions.
We note that the results presented above for the the symplectic current and for its flux across the boundary apply analogously to the energy density current for a real $\Phi$, defined by $J_{\mu}^E \doteq - T_{\mu\nu} k^{\nu}$, with $k = \partial_t$ and where the *bulk stress-energy tensor* is $$T_{\mu\nu} = \partial_{\mu} \Phi \partial_{\nu} \Phi - \frac{1}{2} g_{\mu\nu} \! \Big( \partial^{\lambda} \Phi \partial_{\lambda} \Phi + \tilde{m}^2 \Phi^2 \Big) \, ,$$ where $\tilde{m}^2 \doteq m_0^2 + \xi R$. Observe that, by using the bulk equations of motion, it holds $\partial^\mu T_{\mu\nu}=0$. We may also define a *boundary stress-energy tensor* as $$T_{\alpha\beta}^{\partial} = c \Big[\partial_{\alpha} F \partial_{\beta} F - \frac{1}{2} \eta_{\alpha\beta} \! \Big( \partial^{\lambda} F \partial_{\lambda} F + \tilde{m}^2 F^2 \Big) \Big] \, ,$$ which is however not covariantly conserved, $$\partial^{\alpha} T^{\partial}_{\alpha\beta} = \left[c (m_{\rm b}^2 - \tilde{m}^2) - \rho \right] \partial_{\beta} F \, ,$$ thus indicating that energy fluxes come from the bulk towards the boundary and leave the boundary into the bulk, as expected on physical grounds.
Interplay between the boundary conditions and the boundary isometry group
-------------------------------------------------------------------------
Considering boundary conditions of Robin type, but also those as in or in might appear at a first glance a mere academic exercise. Yet, as we already discussed in the introduction, the lessons learned from the study of the AdS/CFT correspondence indicate the importance of analyzing the possible interplays between bulk and boundary theories which are both dynamical. In this section, we discuss a different, structural property which indicates that both Robin and Wentzell boundary conditions are distinguished.
To this avail it is of paramount relevance that the underlying background is static. This allows us to shift from a purely hyperbolic equation such as the one in , ruled by the wave operator $\Box_\eta^{(d+1)}-\frac{m^2}{z^2}$, to an elliptic problem, governed by $$K_{\omega,m}\doteq-\nabla^2+\omega^2-\frac{m^2}{z^2} \, ,$$ where $\nabla^2=\sum_{i=1}^d \partial_i^2$ and $\omega$ is the Fourier parameter associated to the time coordinate.
Since we are interested in theories which can be coherently quantized, it is convenient to read the operator $K_{0,m}=-\nabla^2+\frac{m^2}{z^2}$ as the Hamiltonian of the underlying system. From this viewpoint, it is natural to interpret $K_{0,m}$ as a real, symmetric operator, acting on the Hilbert space $L^2(\mathring{\mathbb H}^d)$. Hence, in order for the underlying dynamics to describe a *closed* system, one needs to pick a self-adjoint extension of $K_{0,m}$, whose selection consists in turn on fixing suitable boundary conditions at $z=0$. At the classical level this guarantees that the total flux of symplectic and energy currents across the boundary vanishes, as shown explicitly in the previous section.
To better appreciate our freedom in this choice, we divide the analysis in two cases, $d=1$ and $d>1$. In the first case, and setting without loss of generality $m=0$, the Hamiltonian reduces to the kinetic operator on the half line. By using the theory of deficiency indices [@Moretti:2013cma Ch.5], the possible self-adjoint extensions are well-known: they are in one-to-one correspondence with boundary conditions of the form $\Phi|_{z=0}+\tan\alpha\,\partial_z\Phi|_{z=0}=0$, where $\alpha\in [0,\pi)$ can at most be made to be dependent on the spectral parameter, [*i.e.*]{} $\alpha=\alpha(\omega)$. This problem has been already investigated in [@Dappiaggi:2016fwc]
If $d>1$, the scenario is more intricate since $K_{0,m}$ is either essentially self-adjoint or the associated deficiency indices are infinite. The latter instance occurs for example when $m=0$. In other words there are infinite admissible choices for the ratio between the coefficients $A(\underline{k})$ and $B(\underline{k})$ appearing in or in . A physically motivated and mathematically sound selection criterion can be implemented by considering the interplay between the isometry group of the background and the operator $K_{0,m}$. Such problem has been studied only recently in a series of papers [@Ibort:2014sua; @Ibort2; @Pardo]. Another closely related analysis can be found in [@Asorey:2015lja; @Asorey:2017euv]. We will shortly review them and apply the procedure to the case at hand.
The starting point consists of investigating whether $K_{0,m}$ is an Hermitian operator on the Sobolev space $H^2(\mathbb H^d)$, where, for all $s>0$ and for all integer $d\geq 1$, $H^s(\mathbb{R}^d)=\{\psi\in L^2(M),\;|\;(\mathbb{I}-\nabla^2)^s\psi\in L^2(M)\}$, while $H^s(\mathbb{H}^d)=\{[\Psi]\;|\Psi\in H^s(\mathbb{R}^d)\;\Psi\sim\Psi^\prime,\;\textrm{iff}\; (\Psi-\Psi^\prime)|_{\mathbb{H}^d}=0\}$ — see [@Ibort2] or [@Adams] for a survey of the theory and of the key properties of Sobolev spaces. From now on, we will not write explicitly the symbol of equivalence classes since all our statements do not depend on the representative chosen in each of these classes.
To this avail, we observe that the following Green’s formula holds true for all $\Psi,\Psi^\prime\in H^2(\mathbb H^d)$: $$\label{Green_formula}
(\Psi,K_{\omega,m}\Psi^\prime)-(K_{\omega,m}\Psi,\Psi^\prime)=\widetilde{\Sigma}(\Psi,\Psi^\prime) \, ,$$ where $(,)$ stands for the inner product in $L^2(\mathbb H^d)$ while $$\label{boundary_form}
\widetilde{\Sigma}(\Psi,\Psi^\prime)=\langle\Gamma(\Psi),\Gamma(\partial_z\Psi^\prime)\rangle-\langle\Gamma(\partial_z\Psi),\Gamma(\Psi^\prime)\rangle \, .$$ where $\langle,\rangle$ is the $L^2$ inner product on the boundary $\mathbb{R}^{d-1}$. At the same time $\Gamma:H^s(\mathbb H^{d})\to H^{s-\frac{1}{2}}(\mathbb{R}^{d-1})$ is the so called *Lions trace* [@Adams Chap. 5]. For every $s>\frac{1}{2}$ this is a continuous and surjective operator which extends at the level of Hilbert spaces the standard restriction of smooth functions, namely, for every $\Psi\in C^\infty(\mathring{\mathbb H}^{d})\cap H^s(\mathbb H^{d})$, $\Gamma(\Psi)=\Psi|_{z=0}$. $\widetilde{\Sigma}$ is also known as [*Lagrange boundary form*]{} (in the case considered in this paper, corresponds to the boundary symplectic current introduced in ). A [*dense*]{} subspace $\mathcal{D}\subseteq\mathcal{H}_{\rm b}\doteq L^2(\mathbb{R}^{d-1})$ is called [*isotropic*]{} (with respect to $\widetilde{\Sigma}$) if $\widetilde{\Sigma}(\alpha,\beta)=0$ for all $\alpha,\beta\in\mathcal{D}$.
A direct inspection of unveils that, for $K_{\omega,m}$ to be a symmetric operator, it is mandatory that $\Sigma$ vanishes on its domain. While this is automatically true if one considers smooth and compactly supported functions on $\mathring{\mathbb H}^d$, from the viewpoint of the boundary Hilbert space, this choice is not informative since $\Gamma[C^\infty_0(\mathring{\mathbb H}^d)]=\{0\}$. Hence it is useful to consider the following relevant sets:
- For any $\mathcal{W}\subseteq\mathcal{H}_b\times\mathcal{H}_b$ its [*$\Sigma$-orthogonal subspace*]{} is $$\begin{aligned}
\mathcal{W}^\perp &\doteq \big\{(\varphi,\varphi^\prime)\in\mathcal{H}_b\times\mathcal{H}_b\;|\;\Sigma((\varphi,\varphi^\prime),(\psi,\psi^\prime))=0, \notag \\
&\qquad \;\forall (\psi,\psi^\prime)\in\mathcal{W}\times\mathcal{W}\big\} \, ,
\end{aligned}$$ where $\Sigma$ is the natural generalization of , [*i.e.*]{} $$\Sigma((\varphi,\varphi^\prime),(\psi,\psi^\prime)) = \langle\varphi,\psi^\prime\rangle-\langle\varphi^\prime,\psi\rangle \, .$$
- A subspace $\mathcal{W}$ is called [*$\Sigma$-isotropic*]{} if $\mathcal{W}\subseteq\mathcal{W}^\perp$ and [*maximally $\Sigma$-isotropic*]{} if $\mathcal{W}=\mathcal{W}^\perp$.
The advantage of considering those $\mathcal{W}$ which are maximally $\Sigma$-isotropic is two-fold. On the one hand, since $\Gamma$ is surjective, $\Gamma^{-1}[\mathcal{W}]$ identifies a natural domain of $K_{0,m}$ on which the right-hand side of vanishes automatically. On the other hand, it is possible to give an explicit characterization of these spaces. As a matter of fact, as proven in [@Pardo Lemma 3.1.4 & Prop. 3.1.5], letting $\mathcal{C}:\mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b}\to\mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b}$ be the [*unitary Cayley transform*]{} $$\label{eq:Cayley_transform}
\mathcal{C}(\varphi,\varphi^\prime) = \frac{1}{\sqrt{2}}\left(\varphi+i\varphi^\prime,\varphi-i\varphi^\prime\right) \, ,$$ it holds that
1. $\mathcal{W}$ is maximally $\Sigma$-isotropic if and only if $\mathcal{W}_c\doteq\mathcal{C}[\mathcal{W}]$ is maximally $\Sigma_c$-isotropic, where $$\label{Sigma_c}
\Sigma_c((\varphi,\varphi^\prime),(\psi,\psi^\prime)) = -i(\langle\varphi,\psi\rangle-\langle\varphi^\prime,\psi^\prime\rangle) \, .$$
It follows from the items above that, whenever $\mathcal{W}$ is maximally $\Sigma$-isotropic and for any unitary operator $U$, we can use to write $$\label{max_isotropic}
\mathcal{W} \doteq \{(\varphi,\varphi^\prime) \in \mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b} \;|\; \varphi-i\varphi^\prime = U(\varphi+i\varphi^\prime)\} \, .$$
If we recall that $\Sigma$ generalizes , we can identify $\varphi=\Gamma(\Psi)$ and $\varphi^\prime=\Gamma(\partial_z\Psi)$, $\Psi\in H^2(\mathbb H^d)$, which suggests that the choice of any $\mathcal{W}$ as in identifies a specific boundary condition. Formally this can be obtained inverting the identity in : $$\label{eq:inverse_formula}
\varphi^\prime=A_U\varphi,\quad A_U\doteq -i(\mathbb{I}-U)(\mathbb{I}+U)^{-1}.$$ As observed in [@Ibort:2014sua; @Ibort2], for to be both a well-defined mathematical expression and applicable to the case at hand, a sufficient requirement is that two conditions should be met. On the one hand, either $(\mathbb{I}+U)^{-1}$ exists or $-1$ is an element of the spectrum of $U$, which is not an accumulation point. It is noteworthy that, choosing Robin boundary conditions always falls in the first case. We stress that, if we recall the identification $\varphi^\prime=\Gamma(\partial_z\Psi)$, then we also need that $A_U$ is a continuous operator on $H^{\frac{1}{2}}(\mathbb{R}^d)$. Any unitary operator $U:\mathcal{H}_{\rm b}\to\mathcal{H}_{\rm b}$ meeting these requirements will be called [*admissible*]{}.
The next step consists of using the structures introduced above to characterize the self-adjoint extensions of the Hamiltonian operator $K_{0,m}$, whenever the deficiency indices are non-vanishing. Within this class, the prototypical case is the one in which we set $m=0$. Hence, from now on we focus our attention on $K\equiv K_{0,0}$, although all results apply also to the other scenarios.
The first step consists of translating $K$ into an Hermitian quadratic form. Following [@Ibort2], let $U$ be an admissible unitary operator so that $-1$ is not an element of its spectrum. Then we call $Q_U:D_{Q_U}\times D_{Q_U}\subset\mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b}\to\mathbb{C}$, $$\label{QF}
Q_U(\Phi,\Phi^\prime)\doteq\left( d\Phi,d\Phi^\prime\right)_{\Lambda^1}+\langle\Gamma(\Phi),A_U(\Gamma(\Phi^\prime))\rangle \, ,$$ where $(,)_{\Lambda^1}$ stands for the standard $L^2$-pairing between $1$-forms on a Riemannian manifold and where $\Phi,\Phi^\prime\in D_{Q_U}\equiv H^1(\mathbb H^d)$. In [@Ibort:2014sua; @Ibort2] it has been proven that $Q_U$ enjoys several properties, the most notable being that it is closable, namely there exists a domain $D^\prime_{Q_U}\supseteq D_{Q_U}$ on which $Q_U$ is closed with respect to the norm $$\|\Phi\|_Q^2=\|d\Phi\|_{\Lambda^1}^2+(1+C_U)\|\Phi\|^2_{H^1} \, , \quad \forall\Phi\in D^\prime_{Q_U} \, .$$
Hence, we can invoke [@Ibort:2014sua Th. 2.7 & 6.7] to conclude that the quadratic form $Q_U$ identifies a unique self-adjoint operator $K_U$ such that $D(K_U)= D^\prime_{Q_U}$ and there exists $\chi\in H^2(\mathbb H^d)$ for which $Q_U(\Phi,\Phi^\prime)=(\Phi,\chi)_{H^1}$ for all $\Phi\in D^\prime_{Q_U}$. In this case we set $K_U\Phi^\prime=\chi$ and $$Q_U(\Phi,\Phi^\prime)=(\Phi,K_U\Phi^\prime) \, , \quad \forall\Phi,\Phi^\prime\in D(K_U) \, .$$
In addition, it turns out that $K_U$ is a self-adjoint extension of $K$ uniquely and unambiguously identified by an admissible unitary operator $U:\mathcal{H}_{\rm b}\to\mathcal{H}_{\rm b}$.
The above digression serves us to recall that the choice of boundary conditions for $K_{\omega,0}$ is strongly tied to the identification of a maximally $\Sigma$-isotropic $\mathcal{W}\subset\mathcal{H}_{\rm b}$, which, in turn, corresponds to selecting a self-adjoint extension for $K$ via an admissible unitary operator $U$. Yet, since the number of the latter is infinite, one might wonder whether it is at least possible to identify a distinguished subclass.
To this end we observe that the Poincaré patch of AdS$_{d+1}$ has isometry group $Iso({\rm PAdS}_{d+1})=O(d-1,1)\ltimes\mathbb{R}^d$, that is the $d$-dimensional Poincaré group. On each constant time hypersurface, the relevant subgroup is $E(d-1)\doteq O(d-1)\ltimes\mathbb{R}^{d-1}$. Let $V:E(d-1)\to\mathcal{BL}(L^2(\mathring{\mathbb H}^d))$ be such that $$(V(g)\psi)(x)=\psi(g^{-1}x) \, ,\quad\forall \psi\in L^2(\mathring{\mathbb H}^d) \, ,$$ where $g^{-1}x$ stands for the geometric action of $g^{-1}$ on the point $x\in \mathring{\mathbb H}^d$. This is a unitary, strongly continuous representation of the Euclidean group. To analyze its interplay with , we start by considering $U=\mathbb{I}$. This choice identifies the so-called [*Neumann quadratic form*]{} $Q_N$ such that $$Q_N(\Phi)=\|d\Phi\|_{\Lambda^1}^2 \, , \quad D(Q_N)=H^1(\mathbb H^d) \, .$$ Observe that yields $\varphi^\prime=0$ if $U=\mathbb{I}$, which, in the case at hand, entails that we are considering Neumann boundary conditions. In addition, a direct calculation shows that $Q_N$ is invariant under the action of $V$, namely, for every $g\in E(d-1)$, it holds $Q_N(V(g)\Phi)=Q_N(\Phi)$.
Since $E(d-1)$ can also be read as a subgroup of the isometries of the boundary of PAdS$_{d+1}$, one can infer that $V$ has a trace along the boundary (see [@Ibort:2014sua Def. 6.9]), namely for every $\Phi\in H^1(\mathbb H^d)$, it holds $$\Gamma(V(g)\Phi)=v(g)\Gamma(\Phi) \, , \quad\forall g\in E(d-1) \, ,$$ where $\Gamma:H^1(\mathbb H^d)\to H^{\frac{1}{2}}(\mathbb{R}^{d-1})$ and $v:E(d-1)\to\mathcal{BL}(L^2(\mathbb R)^{d-1})$ is the strongly continuous, unitary representation implementing the geometric action $v(g)\varphi(y)=\varphi(g^{-1}y)$, $y$ being a point of $\mathbb{R}^{d-1}$. The most notable interplay between traceable representations and self-adjoint extensions of the operator $K$ are a consequence of [@Ibort:2014sua Th. 6.10], which entails that $K_U$ is an $E(d-1)-$invariant, self-adjoint extension of $K$ if and only if $[U,v(g)]=$ for all $g\in E(d-1)$.
From a physical point of view, invariance under the action of the underlying isometry group is a desired property and hence we call [*distinguished*]{} any self-adjoint extension of $K$ which is $E(d-1)$-invariant. Two examples are certainly of interest to our analysis. In the first we choose $A_U=\cot\alpha \, \mathbb{I}$, $\alpha\in [0,\pi)$, which via Cayley transform corresponds to $U=e^{i\alpha} \, \mathbb{I}$, see [@Moretti:2013cma Th. 5.34]. We observe that, on the one hand, a multiple of the identity operator $U$ commutes with every representation of $E(d-1)$, while on the other hand, , together with the identification of $\varphi^\prime=\Gamma(\partial_z\Phi)$ and of $\varphi=\Gamma(\Phi)$, yields the standard Robin boundary condition $\varphi^\prime = \cot\alpha \, \varphi$. Hence, Robin boundary conditions identify an $E(d-1)$-invariant self-adjoint extension of $K$. We should keep in mind that our interest towards the self-adjoint extensions of $K$ arises from having transformed the wave equation on $\PAdS_{d+1}$ (in the massless case) into an eigenvalue problem for the operator $K$. Hence, although $\alpha$ is a constant one might consider to make $\alpha$ dependent on the spectral parameter $\omega$. Yet this option should be discarded since, upon inverse Fourier transform, we would obtain a boundary condition which breaks manifestly Poincaré invariance.
A second, non trivial self-adjoint operator which commutes with the unitary representation $v$ for $E(d-1)$ is certainly $-\nabla^2_{d-1}$, the (unique self-adjoint extension of the Laplace-Beltrami operator on $\mathbb{R}^{d-1}$. In this case, although the rationale behind our selection criterion is fulfilled, the unitary operator built via Cayley transform from $-\nabla^2_{d-1}$ has a spectrum whose eigenvalues admit $-1$ has an accumulation point. In this case, one is still identifying a self-adjoint extension of $K$, but a more technical analysis is required, using the so-called quasi-boundary triples, see [@boundary_triple] for a short review.
In addition, in view of our need to reinstate the time coordinate via Fourier transform, a more natural choice consists of adding to $-\nabla^2_{d-1}$ a multiple of the identity operator, dependent on the spectral parameter, namely $\nabla^2_{d-1}+(\omega^2+m^2_{\rm b})\mathbb{I}$ where $m^2_{\rm b}> 0$ is a constant. By considering once again together with the identification of $\varphi^\prime=\Gamma(\partial_z\Phi)$ and of $\varphi=\Gamma(\Phi)$, we realize that this choice consists of considering the boundary condition $\varphi^\prime=(-\nabla^2_{d-1}+\omega^2+m^2_{\rm b})\varphi$, which, after an inverse Fourier transform with respect to $\omega$ yields exactly the Wentzell boundary condition and . In other words both Robin and Wentzell boundary conditions are distinguished in view of their interplay with the action of the boundary isometry group on the underlying spaces of functions.
Conclusions {#sec:conclusions}
===========
In this paper, we have considered a real, massive scalar field in the Poincaré fundamental domain of AdS in $d+1$ dimensions, subject to dynamical boundary conditions of generalized Wentzell type at the PAdS boundary. We solved the full system, for both the bulk and boundary fields, and verified that, depending on the values of the parameters of the theory, there might exist zero, one or at most two bound state mode solutions. Although we have not dwelt into the quantization of the underlying model, this result offers a clear indication concerning those values of the mass and of the curvature coupling parameter for which we can expect or rule out the existence of a ground state. Finally, we analyzed what makes this choice of dynamical boundary conditions distinguished, as they are invariant under the action of the isometry group of the PAdS boundary, and imply zero symplectic and energy density total flux accross the boundary.
As a perspective, we outline that in order to obtain the quantization of the theory, the first step will consist of constructing the bulk propagator / fundamental solutions, relating it to the one which stems from the boundary theory. It is of particular interest to obtain a map from a Hadamard state of the boundary theory to a Hadamard state of the bulk theory, which would constitute an AdS counterpart to the result in asymptotically flat spacetimes [@Dappiaggi:2017kka]. This is work in progress.
Finally, we note that the problem that we have studied in this paper belongs to a class of systems, those with dynamical boundary conditions, that are of relevance for a broad spectrum of physical models and theories. The techniques that we have employed in this work are applicable to different problems, ranging from condensed matter to gravitational physics, as well as quantum gravity and high energy physics.
The work of C. D. was supported by the University of Pavia. The work of H. F. was supported by the INFN postdoctoral fellowship “Geometrical Methods in Quantum Field Theories and Applications”, and in part by a fellowship of the “Progetto Giovani GNFM 2017 – Wave propagation on lorentzian manifolds with boundaries and applications to algebraic QFT” fostered by the National Group of Mathematical Physics (GNFM-INdAM). H. F. also acknowledges the hospitality of the ICN-UNAM and their support through a PAPIIT-UNAM grant IG100316. The work of B. A. J.-A. was supported by a Consejo Nacional de Ciencia y Tecnología (CONACYT, México) project 101712. B. A. J.-A. also acknowledges the hospitality of the INFN – Sezione di Pavia during the realization of part of this work, as well as the support of an International Mobility Award granted by the Red Temática de Física de Altas Energías (Red FAE-CONACYT).
[99]{}
J. M. Maldacena, Int. J. Theor. Phys. [**38**]{} (1999) 1113 \[Adv. Theor. Math. Phys. [**2**]{} (1998) 231\] \[hep-th/9711200\].
E. Witten, Adv. Theor. Math. Phys. [**2**]{} (1998) 253 \[hep-th/9802150\].
M. Ammon and J. Erdmenger, *Gauge/Gravity Duality: Foundations and Applications*, (Cambridge University Press, Cambridge, England, 2015).
S. A. Hartnoll, Class. Quant. Grav. [**26**]{}, 224002 (2009) \[arXiv:0903.3246 \[hep-th\]\].
C. Dappiaggi and H. R. C. Ferreira, Phys. Rev. D [**94**]{}, no. 12, 125016 (2016) \[arXiv:1610.01049 \[gr-qc\]\].
C. Dappiaggi and H. R. C. Ferreira, arXiv:1701.07215 \[math-ph\].
F. Bussola, C. Dappiaggi, H. R. C. Ferreira and I. Khavkine, Phys. Rev. D [**96**]{} (2017) no.10, 105016 \[arXiv:1708.00271 \[gr-qc\]\].
A. Ibort, F. Lledó and J. M. Pérez-Pardo, Annales Henri Poincaré [**16**]{} (2015) no.10, 2367.
A. Ibort, F. Lledó and J. M. Pérez-Pardo, J. Funct. Anal. [**268**]{} (2015) 634.
J. M. Pérez-Pardo, [*”On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry”*]{}, PhD thesis (2013) arXiv:1308.2158 \[math-ph\].
J. F. Barbero G., B. A. Juárez-Aubry, J. Margalef-Bentabol and E. J. S. Villaseñor, Class. Quant. Grav. [**32**]{}, no. 24, 245009 (2015) \[arXiv:1501.05114 \[math-ph\]\].
J. F. Barbero G., B. A. Juárez-Aubry, J. Margalef-Bentabol and E. J. S. Villaseñor, Class. Quant. Grav. [**34**]{} (2017) no.9, 095005 \[arXiv:1701.00735 \[gr-qc\]\].
J. Zahn, Annales Henri Poincaré [**19**]{}, no. 1, 163 (2018) \[arXiv:1512.05512 \[math-ph\]\].
T. Ueno, Proc. Japan Acad. 49 (1973), no. 9, 672-677.
A. Favini, et al, J. Evol. Equ.2, 1 (2002).
G. M. Coclite, et al, Commun. Pure Appl. Anal. 13, 419 (2014).
K. Skenderis, Class. Quant. Grav. [**19**]{} (2002) 5849 doi:10.1088/0264-9381/19/22/306 \[hep-th/0209067\]. A. Martín-Ruiz, M. Cambiaso and L. F. Urrutia, Phys. Rev. D [**92**]{} (2015) no.12, 125015 doi:10.1103/PhysRevD.92.125015 \[arXiv:1511.01170 \[cond-mat.other\]\].
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. [**80**]{} (1998) 904 doi:10.1103/PhysRevLett.80.904 \[gr-qc/9710007\].
A. Ashtekar, A. Corichi and K. Krasnov, Adv. Theor. Math. Phys. [**3**]{} (1999) 419 doi:10.4310/ATMP.1999.v3.n3.a1 \[gr-qc/9905089\].
E. Ayón-Beato, D. Higuita-Borja, J. A. Méndez-Zavaleta and G. Velázquez-Rodríguez, “Exact ghost-free bigravitational waves,” arXiv:1801.06764 \[hep-th\].
P. Breitenlohner and D. Z. Freedman, Annals Phys. [**144**]{} (1982) 249.
A. Zettl, [*Sturm-Liouville Theory,*]{} American Mathematical Society, (2005).
M. Asorey, D. García-Alvarez and J. M. Muñoz-Castañeda, Int. J. Geom. Meth. Mod. Phys. [**12**]{}, no. 06, 1560004 (2015) \[arXiv:1501.03752 \[hep-th\]\].
M. Asorey, A. Ibort and A. Spivak, Int. J. Geom. Meth. Mod. Phys. [**14**]{}, no. 08, 1740006 (2017).
V. Moretti, [*Spectral Theory and Quantum Mechanics: With an Introduction to the Algebraic Formulation*]{} (Springer, 2013).
R. A. Adams, J. J. F. Fournier, [*Sobolev Spaces*]{} (Academic Press, 2003)
J. Behrndt, T. Micheler, Proc. Appl. Math. Mech. [**11**]{}, (2011) 883
C. Dappiaggi, V. Moretti and N. Pinamonti, SpringerBriefs Math. Phys. [**25**]{} (2017) \[arXiv:1706.09666 \[math-ph\]\].
|
---
abstract: 'Dynamical instabilities due to spin-mixing collisions in a $^{87}$Rb $F=1$ spinor Bose-Einstein condensate are used as an amplifier of quantum spin fluctuations. We demonstrate the spectrum of this amplifier to be tunable, in quantitative agreement with mean-field calculations. We quantify the microscopic spin fluctuations of the initially paramagnetic condensate by applying this amplifier and measuring the resulting macroscopic magnetization. The magnitude of these fluctuations is consistent with predictions of a beyond-mean-field theory. The spinor-condensate-based spin amplifier is thus shown to be nearly quantum-limited at a gain as high as 30 dB.'
author:
- 'S. R. Leslie$^1$'
- 'J. Guzman$^{1,2}$, M. Vengalattore$^1$, J. D. Sau$^1$, M. L. Cohen$^{1,2}$, D. M. Stamper-Kurn$^{1,2}$'
title: Amplification of Fluctuations in a Spinor Bose Einstein Condensate
---
Accompanied by a precise theoretical framework and created in the lab in a highly controlled manner, ultracold atomic systems serve as a platform for studies of quantum dynamics and many-body quantum phases. Among these systems, gaseous spinor Bose Einstein condensates [@ho98; @ohmi98; @sten98spin; @schm04; @chan04], in which atoms may explore all sub-levels of a non-zero hyperfine spin $F$, provide a compelling opportunity to access the static and dynamical properties of a magnetic superfluid [@mies99meta; @stam99tunprl; @chan05nphys; @sadl06symm; @veng08helix].
We previously identified a quantum phase transition in an $F=1$ spinor Bose Einstein condensate between a paramagnetic and ferromagnetic phase [@sadl06symm]. This transition is crossed as the quadratic Zeeman energy term, of the form $q F_z^2$, is tuned through a critical value $q = q_0$; here, $F_z$ is the longitudinal ($\hat{z}$ axis) projection of the dimensionless vector spin operator $\bf{F}$. Accompanying this phase transition is the onset of a dynamical instability in a condensate prepared in the paramagnetic ground state, with macroscopic occupation of the $|m_z=
0\rangle$ magnetic sublevel [@lama07quench; @sait07quench; @zhan05instab]. This instability causes transverse spin perturbations to grow exponentially, producing atoms into the $|m_z = \pm 1\rangle$ sublevels. In contradiction with the mean-field prediction that the paramagnetic state should remain stationary because it lacks fluctuations by which to seed the instability, experiments revealed the spontaneous magnetization of such condensates after they were rapidly quenched across the phase transition.
In this Letter, we investigate the spin fluctuations that become amplified by the spin-mixing instability. In particular, we test whether these fluctuations correspond to quantum noise, i.e. to the zero-point fluctuations of quantized spin excitation modes that become unstable. For this, we use the spin-mixing instability as an amplifier, evolving microscopic quantum fluctuations into measurable macroscopic magnetization patterns. We present two main results. First, we characterize the spin-mixing amplifier and demonstrate its spectrum to be tunable by varying the quadratic Zeeman shift. This spectrum compares well with a theoretical model that accounts for the inhomogeneous condensate density and for magnetic dipole interactions. Second, we measure precisely the transverse magnetization produced by this amplifier at various stages of amplification, up to a gain of 30 dB in the magnetization variance. This magnetization signal corresponds to the amplification of initial fluctuations with a variance slightly greater than that expected for zero-point fluctuations.
Descriptions of the dynamics of initially paramagnetic spinor condensates [@lama07quench; @dams07quench; @uhlm07quench; @sait07quench; @sait05spont; @mias08] have focused on the effects of the quadratic Zeeman shift and of the spin-dependent contact interaction. The latter interaction has the mean-field energy density $c_2 n \langle \bf{F} \rangle^2$, and, with $c_2 = 4 \pi \hbar^2 \Delta a / 3m < 0$, favors a ferromagnetic state; here, $\Delta a=(a_{2}-a_{0})$ where $a_{{F_{\mbox{\scriptsize tot}}}}$ is the $s$-wave scattering length for collisions between particles of total spin ${{F_{\mbox{\scriptsize tot}}}}$ [@ho98; @ohmi98] and $m$ is the atomic mass. Excitations of the uniform condensate include both the scalar density excitations and also two polarizations of spin excitations with a dispersion relation given as $E_s^2({\bf{k}})=(\varepsilon_k+q)(\varepsilon_k+q-q_0)$, where $\varepsilon_k=\hbar^2 k^2/2 m$ and $q_0 = 2 c_2 n$. For $q>q_0$, spin excitations are gapped ($E_s^2
> 0$), and the paramagnetic condensate is stable. Below this critical value, the paramagnetic phase develops dynamical instabilities, defined by the condition $E_s^2<0$, that amplify transverse magnetization. The dispersion relation defines the spectrum of this amplification, yielding a wavevector-dependent time constant for exponential growth of the power in the unstable modes, $\tau = \hbar/2\sqrt{|E_s^2|}$.
The unstable regime is divided further into two regions. Near the critical point, reached by a “shallow” quench to $q_0/2 \leq q <
q_0$, the fastest-growing instability occurs at zero wavevector, favoring the “light-cone” evolution of magnetization correlations at ever-longer range [@lama07quench]. For a “deep” quench, with $q<q_0/2$, the instabilities reach a maximum growth rate of $1/\tau = q_0 / \hbar$. The non-zero wavevector of this dominant instability sets the size of magnetization domains produced following the quench.
Experimentally, we characterize the spectrum of this amplifier by seeding it with broadband noise and then measuring precisely the spectrum of its output. Similar to previous work [@sadl06symm], we produce condensates of $N_0 = 2.0 \times 10^{6}$ $^{87}$Rb atoms, with a peak density of $n = 2.6(1) \times 10^{14} \, \mbox{cm}^{-3}$ and a kinetic temperature of $\simeq 50 \, \mbox{nK}$, trapped in a linearly polarized optical dipole trap characterized by trap frequencies $(\omega_x,\omega_y,\omega_z)=2\pi\times(39,440,4.2)$ s$^{-1}$. Taking $\Delta a = -1.4(3) \, a_B$ [@vankemp02], with $a_B$ being the Bohr radius, the spin healing length $\xi_s=(8
\pi n |\Delta a|)^{-1/2} = 2.5 \, \mu\mbox{m}$ is larger than the condensate radius $r_y=1.6\,\mu\mbox{m}$ along the imaging axis ($\hat{y}$) . Thus, the condensate is effectively two-dimensional with respect to spin dynamics. For this sample, $q_0 = 2 c_2
\langle n \rangle = h \times 15$ Hz given the maximum $\hat{y}$-axis column-averaged condensate density $\tilde{n}$.
The quadratic Zeeman shift arises from the application of both static and modulated magnetic fields. A constant field of magnitude $B$, directed along the long axis of the condensate, leads to a quadratic shift of $q_{B}/h = (70 \, \mbox{Hz}/\mbox{G}^2) B^2$. In addition, a linearly polarized microwave field, detuned by $\delta/2\pi = \pm 35 \, \mbox{kHz}$ from the $|F=1, m_z = 0\rangle$ to $|F=2, m_z= 0\rangle$ hyperfine transition, induces a quadratic (AC) Zeeman shift of $q_{\mu}=- \hbar \Omega^{2}/4\delta$ where $\Omega$ is the Rabi frequency for the driving field.
The condensate is prepared in the $|m_z = 0 \rangle$ state using rf pulses followed by application of a $6 \, \mbox{G/cm}$ magnetic field gradient that expels atoms in the $|m_z = \pm 1\rangle$ states from the optical trap [@sadl06symm]. This preparation takes place in a static 4 G field and with no microwave irradiation, setting $q= q_{B} + q_\mu
> q_0$ so that the paramagnetic condensate is stable. Next, we increase the microwave field strength to a constant value, corresponding to a Rabi frequency in the range of $2 \pi \times (0$ – $1.5)\,
\mbox{kHz}$, to set $q_\mu$. To initiate the instability, we ramp the magnetic field over 5 ms to a value of $B = 230 \, \mbox{mG}$ (giving $q_B/h = 7.6 \,
\mbox{Hz}$). During separate repetitions of the experiment (for different values of $q_\mu$), the quadratic Zeeman shift at the end of the ramp was thus brought to final values $q_f/h$ between -2 and 16 Hz.
![Transverse magnetization produced near the condensate center after 87 ms (top) and 157 ms (bottom) of amplification at variable quadratic Zeeman shift $q_f$. Magnetization orientation is indicated by hue and amplitude by brightness (color wheel shown). The characteristic spin domain size grows as $q_f$ increases. The reduced signal strength for $q_f/h \geq 9$ Hz reveals the lower temporal gain of the spin-mixing amplifier. []{data-label="fig1"}](fig1.eps){width="40.00000%"}
Following the quench, the condensate spontaneously develops macroscopic transverse magnetization, saturating within about 110 ms to a pattern of spin domains, textures, vortices and domain walls [@sadl06symm]. Using a 2-ms-long sequence of phase-contrast images, we obtain a detailed map of the column-integrated magnetization ${\bf{\tilde{M}}}$ at a given time after the quench [@higb05larmor; @veng07mag]. The experiment is then repeated with a new sample.
The observed transverse magnetization profiles of spinor condensates (Fig. \[fig1\]) confirm the salient features predicted for the spin-mixing amplifier [^1]. The variation of the amplifier’s spatial spectrum with $q_f$ is reflected in the characteristic size $l_d$ of the observed spin domains, taken as the distance from the origin at which the magnetization correlation function, $$G(\delta\textbf{r})=\frac{\sum_\textbf{r}
\tilde{\textbf{M}}(\textbf{r}+\delta
\textbf{r})\cdot\tilde{\textbf{M}}(\textbf{r})}{(g_F \mu_B)2
\sum_\textbf{r} \tilde{n}(\textbf{r}+\delta
\textbf{r})\tilde{n}(\textbf{r})} , \label{corr}$$ acquires its first minimum; here $g_F \mu_B$ is the atomic magnetic moment. This characteristic size increases with increasing $q_f$ (Fig. 2). For $q_f / h \geq 9 \, \mbox{Hz}$, the magnetization features become long ranged and, thus, dominated by residual magnetic field inhomogeneities ($< 2 \, \mu$G).
![Characteristic domain size after 87 ms of amplification at variable quadratic Zeeman shift $q_f$. Data (circles) are averages over 5 experimental repetitions; error bars are statistical. Horizontal error bar reflects systematic uncertainty in $q_f$. Predictions based on numerical simulations for $|\Delta a| =
1.45\,a_B$ [@chan05nphys] (squares) and $1.07\,a_B$ [@wide06precision] (triangles) are shown, with error bar reflecting systematic uncertainty in the atomic density.[]{data-label="fig2"}](fig2_srl_v2.eps "fig:"){width="40.00000%"}\
The data also confirm the distinction between deep and shallow quenches. The spatially averaged magnetization strength during the amplification, quantified by $G(0)$ at $t = 87$ ms after the quench, is found to be constant for $0<q_f/h<6$ Hz, reflecting that the temporal gain of the amplifier is uniform over $0 \leq q
\leq q_0/2$ [@lama07quench]. For shallow quenches, with $q_{f}/h\geq7$ Hz, the measured magnetization decreases, reflecting a diminishing gain with increasing $q_f$ up to the transition point.
While the above observations are consistent with theoretical predictions, we note the unexpected outcome of quenches to negative values of $q_f$. Such quenches revealed a diminished amplifier gain, resulting in a complete suppression of the growth of magnetization for $q_f/h\leq -7$ Hz. We are unable to account for this behavior.
Having characterized the spin-mixing amplifier, let us consider the source of its input signal. For this, we develop a quantum field description of the spin-mixing instability [@lama07quench], working in the polar spin basis, where $\hat{\phi}_{n,k}$ is the annihilation operator for atoms of wavevector $k$ in the zero-eigenvalue states of $F \cdot {\textbf{n}}$. Treating a uniform condensate within the Bogoliubov approximation, one defines mode operators $\hat{b}_{n,k} = u \hat{\phi}_{n,k} + v
\hat{\phi}^\dagger_{n,-k}$ for the two polarizations of transverse ($n \in \{x,y\}$) spin excitations. The spin-dependent many-body Hamiltonian $H_{s}$ is then approximated as representing two independent parametric amplifiers: $$H_{s}= -\frac{i}{2} \sum_k \! |E_s(k)| \! \left(
b_{x,k}^2-b_{x,k}^{\dagger 2}+b_{y,k}^2-b_{y,k}^{\dagger 2}\right).$$ The parametric amplifiers serve to squeeze the initial state in each spin excitation mode, amplifying one quadrature of $b_{n,k}$ (its real part) and de-amplifying the other.
The above treatment may be recast in terms of spin fluctuations atop the paramagnetic state: fluctuations of the transverse spin, represented by the observables $F_x$ and $F_y$, and fluctuations in the alignment of the spinor, represented by the components $N_{yz}$ and $N_{xz}$ of the spin quadrupole tensor. We identify the Bogoliubov operators defined above as linear combinations of these observables. Based on this identification, we draw two conclusions. First, an ideally prepared paramagnetic condensate is characterized by quantum fluctuations of the Bogoliubov modes. In the linear regime, fluctuations in $b_{x,k}$ ($b_{y,k}$) correspond to projection noise for the conjugate observables $F_{x}$ ($F_y$) and $N_{yz}$ ($N_{xz}$). Second, the dynamical instabilities lead to a coherent amplification of these initial shot-noise fluctuations. While in the present work we observe only the magnetization, in future work both quadratures of the spin-mixing amplifier may be measured using optical probes of the condensate nematicity [@caru04imag] or by using quadratic Zeeman shifts to rotate the spin quadrature axes.
To test the validity of this quantum amplification theory, we determine $G(0)$ over the central region of the condensate after different intervals of amplification (Fig. \[fig3\]). We consider the linear-amplification theory to be applicable for $t
\leq 90$ ms, and, following Ref. [@lama07quench], perform a least-squares fit to a function of the form $$\left.G(0)\right|_t=G(0)|_{t_m} \times {\sqrt{t/t_m}} e^{(t-t_m)/\tau}$$ Here $\tau$ is the time constant characterizing the growth rate of the magnetization variance and $G(0)|_{t_m}$ is the value of $G(0)$ at time $t_m=77$ ms.
![Temporal evolution of the transverse magnetization variance $G(0)$, evaluated over the central $16 \times 124$ $\mu$m region of the condensate and averaging over 8 experimental repetitions; error bars are statistical. The contribution to $G(0)$ from imaging noise was subtracted from the data. Predictions from numerical calculations for $|\Delta a| = 1.45\,a_B$ and $1.07\,a_B$ are shown as black and gray lines, respectively.[]{data-label="fig3"}](fig3_srl_v2.eps "fig:"){width="40.00000%"}\
To compare our measurements to the amplifier theory outlined above, we performed numerical calculations of $\left.G(0)\right|_{t}$, taking into account the the inhomogeneous density profile; dipolar interactions; proper position-space, rather than momentum-space, spin excitation modes; and quantum fluctuations of the initial state [^2]. Such calculations, results of which are shown in Figs. \[fig3\] and \[fig4\], were performed for several values of the scattering length difference $\Delta a$ within the range of recent measurements [@chan05nphys; @wide06precision]. Our data are consistent with the quantum-limited amplification of zero-point quantum fluctuations in the case that $|\Delta a|$ lies in the upper range of its reported values. Taking $\tau$ to be determined instead solely on the basis of our measurements, the magnetization variance is measured to be between 1 and 50 times greater than that predicted by the zero-temperature quantum theory.
![The magnetization variance $G(0)_{t_m}$ at $t_m = 77$ ms and exponential time constant $\tau$ of the amplifier, obtained by fitting data in Fig. \[fig3\], are indicated by contours of the 1, 2, and 3 $\sigma$ confidence regions using a $\chi^2$ test. Predictions from numerical calculations of the zero-temperature quantum amplification theory, assuming different values of $|\Delta
a|$, are shown (circles and interpolating line). Error bars reflect systematic uncertainty in $q_f$ and in the condensate density. Time constants corresponding to reported values for $|\Delta a|$ are indicated at bottom.[]{data-label="fig4"}](fig4_srl_v2.eps "fig:"){width="40.00000%"}\
Several investigations were performed to identify technical or thermal contributions on top of the expected quantum spin fluctuations of our samples. We place an upper bound on thermal noise by performing a Stern-Gerlach analysis of populations $N_\pm$ in the $|m_z = \pm 1\rangle$ states just after the quench. Obtaining $N_\pm \leq 3 \times 10^2$ and assuming an incoherent admixture of Zeeman sublevels, the thermal contribution to $G(0)|_0$ is below $(2
N_\pm / N_0) = 3 \times 10^{-4}$. We confirmed that our results are insensitive to variations in the gradient strength, duration, and orientation used during the initial state preparation, and also to the delay (varied between 0 and 110 ms) between this preparation and the initiation of the spin amplifier. We checked against technical noise that would induce extrinsic Zeeman transitions during the experiment, finding that a condensate starting in the $|m_z =
-1\rangle$ state remained so for evolution times up to 400 ms following the quench. Altogether, these results suggest that paramagnetic samples were produced with a near-zero spin temperature, to which the quantum amplification theory may be expected to apply.
The microwave fields used to vary the quadratic Zeeman shift cause a slight mixing between the $F=1$ and $F=2$ hyperfine levels. We checked for the influence of this admixture by varying the relative contributions of the static and modulated field contributions to the final quadratic Zeeman shift $q_f = q_B + q_\mu$. As expected given the small value of $(\Omega / \delta)^2 < 10^{-3}$, no variation in the magnetization evolution for constant $q_f$ was observed.
It remains uncertain whether the zero-temperature amplifier theory should remain accurate, out to a gain in the magnetization variance as high as 30 dB, in a non-zero temperature gas subject to constant heating and evaporation from the finite-depth optical trap. Indeed, previous work showed a strong influence of the non-condensed gas on spin dynamics in a two-component gaseous mixture [@mcgu03normal]. We compared the amplification of magnetization at kinetic temperatures of 50 and 85 nK, obtained for different optical trap depths. We observed no variation, but note that the condensate fraction was not substantially varied in this comparison.
In this work, we have demonstrated near-quantum-limited amplification of magnetization fluctuations using dynamical instabilities in a spinor Bose gas. Just as the demonstration of matter-wave amplification in scalar condensates suggested a host of applications [@deng99fourwave; @kozu99amp; @inou00amp], the spin-mixing amplifier may serve as an important experimental tool in probing systems with multiple degrees of freedom or in spatially resolved magnetometry [@veng07mag].
We acknowledge insightful discussions with J. Moore and S.Mukerjee and experimental assistance from C. Smallwood. This work was supported by the NSF, the David and Lucile Packard Foundation, DARPA’s OLE Program, and the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Partial personnel and equipment support was provided by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences. S.R.L. acknowledges support from the NSERC.
[26]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
[^1]: We measure also the longitudinal magnetization of the condensate, but, consistent with prior observations, find that it remains small ($<15\%$ of the maximum magnetization)
[^2]: J.D. Sau et al., to be published
|
---
address: |
DESY CMS, Notkestra[ß]{}e 85,\
22607 Hamburg, Germany
author:
- 'K. Beernaert on behalf of the ATLAS and CMS collaborations'
title: 'Top-Quark Properties at the LHC'
---
Introduction {#sec:Introduction}
============
Due to its short lifetime, the top quark decays before it can form bound states and before its spin decorrelates. As a consequence we can study “bare” quark properties. The top quark has a mass of approximately $173$ GeV and may play a significant role in electro-weak symmetry breaking due to its large coupling to the Higgs boson. Measurements of the top-quark properties with increasing levels of precision test the SM and open the possibility to probe new physics. The data used for the studies presented here were collected in pp collisions in 2011 and 2012 at centre-of-mass energies of 7 and 8 TeV at the Large Hadron Collider (LHC) by the ATLAS [@1748-0221-3-08-S08003] and CMS [@Chatrchyan:2008aa] detectors.
Production Asymmetries {#sec:Production Asymmetries}
======================
The Tevatron forward-backward asymmetry measurements have initially shown some tension with the SM predictions [@D0; @CDF]. At the LHC, with a symmetric initial state, a charge asymmetry is measured, given by:
$$A_{C} = \frac{N(\Delta \left| y \right| > 0) - N(\Delta \left| y \right| < 0)}{N(\Delta \left| y \right| > 0) + N(\Delta \left| y \right| < 0\
)}
\label{eq.ChargeAsym}$$
The charge asymmetry can be set up using the rapidity $y$ of the top quarks ($\Delta \left| y \right|\,=\, \left| y_{t} \right| - \left| y_{\overline{t}} \right|$) or with the pseudo-rapidity $\eta$ of the leptons in the dilepton channel (replacing $\Delta \left| y \right|$ with $\Delta \left| \eta_{l} \right|\,=\,\left|\eta_{l^{+}}\right| - \left|\eta_{l^{-}} \right|$). In the SM, the top-quark production asymmetries are due to NLO QCD interference effects. Measurements of $A_{FB}$ at the Tevatron and $A_{C}$ at the LHC have complementary sensitivity to new physics models [@PhysRevD.84.115013; @AguilarSaavedra:2011ug]. ATLAS presents two results in the lepton+jets channel at $\sqrt{s}\,=$ 8 TeV. In one analysis [@Phys.Lett.B756], a minimum $t\overline{t}$ invariant mass of 0.75 TeV is imposed, and the final state is selected by looking for a resolved leptonic top-quark decay and a hadronic decay which is reconstructed as a large $R$-jet with substructure where $R = \sqrt{(\Delta\eta)^{2} +
(\Delta\phi)^{2}}$. In the other presented analysis [@Eur.Phys.J.C76], three signal regions are used based on the b-tag multiplicity. Full Bayesian unfolding is used to bring the distributions back to parton level. The inclusive charge asymmetry is measured as $A_C = [4.2\, \pm\, 3.2\,\mathrm{(stat.+syst.)}]\,\%$ [@Phys.Lett.B756] and $A_C = [0.9\, \pm\, 0.5\,\mathrm{(stat.+syst.)}]\,\%$ [@Eur.Phys.J.C76]. CMS also presents an analysis in the lepton+jets channel [@Phys.Rev.D93] at $\sqrt{s}\,=$ 8 TeV. The charge asymmetry is measured with a template fit using symmetric and asymmetric templates of $\Upsilon_{t\overline{t}} = \tanh{\Delta \left| y \right|_{t\overline{t}}}$. The fit parameter $\alpha$ represents the relative contribution of the symmetric and asymmetric templates. This results in $A_C = [0.33\, \pm\, 0.42\,\mathrm{(stat.+syst.)}]\,\%$. CMS presents a result in the dilepton channel [@arxiv:1603.06221], where the asymmetry is determined using the final-state leptons and reconstructed top quarks, leading to $A_C = [1.1\, \pm\, 1.3\,\mathrm{(stat.+syst.)}]\,\%$ based on the top quarks and $A^{\mathrm{lep}}_C = [0.3\, \pm\, 0.7\,\mathrm{(stat.+syst.)}]\,\%$ based on the leptons. The results are observed to be consistent with the SM, as seen from the summary in Fig. \[fig:Summary\_Asym\]. In addition, all analyses provide differential measurements of the charge asymmetry in a variety of variables, e.g. invariant mass $m(t\overline{t})$, transverse momentum $p_{T}(t\overline{t})$ and velocity $\beta(t\overline{t})$ of the top-quark pair.
Spin Correlations {#sec:Spin Correlations}
=================
In top-quark pair production, the SM predicts the spins of the top- and antitop-quark to be correlated. The spin information of the top quark can be accessed using the decay products. ATLAS presents an analysis [@Phys.Rev.D93012002] making use of the following double differential cross section:
$$\frac{1}{N}\frac{d^{2}N}{d\cos\theta_{1}dcos\theta_{2}} = \frac{1}{4}(1+B_{1}\cos\theta_{1} + B_{2}\cos\theta_{2} -C_{\mathrm{hel}}\cos\theta_{1}\cos\theta_{2})$$
where $\theta$ is the angle between the lepton direction in the top (anti-)quark parent rest frame and the top (anti-)quark parent in the $t\overline{t}$ rest frame. $B_{1,2}$ are proportional to the top-quark polarisation and are considered to be zero. Using this equation, the spin correlation strength $A_\mathrm{hel}$ can be directly extracted from $C_\mathrm{hel} = -A_\mathrm{hel}\alpha_1 \alpha_2$. ATLAS presents an analysis at $\sqrt{s}\,=$ 7 TeV [@Phys.Rev.D93012002] in the dilepton channel that results in $A_\mathrm{hel}\,=\,0.315\,\pm\,0.061\,\mathrm{(stat.)}\,\pm\,0.049\,\mathrm{(syst.)}$. CMS presents an analysis at $\sqrt{s}\,=$ 8 TeV [@Phys.Rev.D93052007] in the dilepton channel where several asymmetry variables are used to perform a direct measurement of the spin correlation strength and the top-quark polarization. A measurement of the spin correlation strength can be interpreted in terms of several BSM models. As an example, the CMS measurement has been used to set limits on top-quark chromomagnetic couplings [@Bernreuther] of $-0.053\,<\, \mathrm{Re}(\mu_t)\,<0.026$ for the chromomagnetic dipole moment (CMDM) and $-0.068\,<\,\mathrm{Im}(d_t)\,<\,0.067$ for the chromo-electric dipole moment (CEDM) both at the 95 % confidence level (CL). ATLAS presents an analysis at $\sqrt{s}\,=$ 8 TeV [@PRL114] where the spin correlation measurement is interpreted in terms of a Minimal Super-Symmetric Model (MSSM) where stop squarks decay 100 % into a top quark and a neutralino with the stop squark mass very close to the top-quark mass. Stop squark masses between the top-quark mass and 191 GeV are excluded at the 95 % CL. CMS presents an analysis at $\sqrt{s}\,=$ 8 TeV [@arxiv:1511.06170] in the lepton + jets channel where a matrix element method is used to set up a variable sensitive to spin correlation. Using matrix elements for $t\overline{t}$ production and decay using SM spin correlations and zero spin correlations, the likelihood ratio of these two hypotheses is used to perform a template fit and a hypothesis testing procedure. A SM fraction of $f^{SM}\,=\,0.72\,\pm\,0.08\,\mathrm{(stat.)}\,{}^{+0.15}_{-0.13}\,\mathrm{(syst.)}$ is measured.
Flavour-changing Neutral Currents {#sec:Flavour-changing Neutral Currents}
=================================
In the SM, FCNC are suppressed at tree-level due to the GIM mechanism. This leads to very small branching ratios of $t \rightarrow u/c + X$ with $X = g, \gamma, Z, H$ of $O(10^{-12} - 10^{-17})$. Several BSM models, such as MSSM, 2HDM, predict enhanced couplings for FCNC with branching ratios as high as $10^{-5}$. With the discovery of the Higgs boson, FCNC can now also be studied in $t\overline{t}$ where one of the top quarks decays as $t \rightarrow u/c + H$. A clear overview of the predictions can be found in the reviews [@rev1; @rev2]. CMS presents three analyses in this channel. One analysis [@CMS-PAS-TOP-14-020] makes use of the high branching fraction of $H \rightarrow b\overline{b}$ to look for $t\overline{t}$ production with decays of $t \rightarrow Hq \rightarrow b\overline{b}q$ in one leg and $t \rightarrow Wb \rightarrow l\nu b$ in the other, obtaining an observed limit of $B(t \rightarrow Hc) < 1.16 \%$ and $B(t \rightarrow Hu) < 1.92 \%$ at 95 % CL. Using the cleaner Higgs decay channel $H \rightarrow \gamma\gamma$ and looking for top-quark pairs with $t \rightarrow Hq \rightarrow \gamma\gamma q$ and $t \rightarrow Wb \rightarrow l\nu b\, \mathrm{or}\, q\overline{q}b$, observed limits are set of $B(t \rightarrow Hc) < 0.47 \%$ and $B(t \rightarrow Hu) < 0.42 \%$ at 95 % CL [@CMS-PAS-TOP-14-019]. Finally, using $t\overline{t}$ events where $t \rightarrow Hq \rightarrow ZZq\, \mathrm{or}\, WWq$ and $t \rightarrow Wq \rightarrow l\nu b$, CMS reports an observed limit of $B(t \rightarrow Hc) < 0.93 \%$ at 95 % CL [@CMS-PAS-TOP-13-017]. ATLAS presents an analysis at $\sqrt{s}\,=$ 8 TeV [@JHEP12_061] searching for FCNC in the channel $t \rightarrow Hq \rightarrow b\overline{b}q$ and $t \rightarrow Wb \rightarrow l\nu b$. Several signal categories are used based on jet and b-tag multiplicity. An observed limit of $B(t \rightarrow Hc) < 0.56 \%$ and $B(t \rightarrow Hu) < 0.61 \%$ is reported at the 95 % CL. In addition, a re-interpretation of previous $t\overline{t}H$ searches is performed in this analysis and combined limits are presented. Summaries of the observed limits on FCNC are shown in Fig. \[fig:SummaryFCNC\_CMS\]-\[fig:SummaryFCNC\_ATLAStuX\].
CP Violation {#sec:CP Violation}
============
A first search for CP violation in the $t\overline{t}$ sector has been pursued by CMS in the l+jets channel by inspecting T-odd observables [@CMS-PAS-TOP-16-001]. The observables are defined as $O_{2}\,\propto\,(\overrightarrow{p}_{b} + \overrightarrow{p}_{\overline{b}})\cdotp (\overrightarrow{p}_{l} \times \overrightarrow{p}_{j1})$, $O_{3}\,\propto\,Q_{l}\overrightarrow{p}_{b}\cdotp(\overrightarrow{p}_{l} \times \overrightarrow{p}_{j1})$, $O_{4}\,\propto\,Q_{l}(\overrightarrow{p}_{b} - \overrightarrow{p}_{\overline{b}})\cdotp(\overrightarrow{p}_{l} \times \overrightarrow{p}_{j1})$ and $O_{7}\,\propto\,(\overrightarrow{p}_{b} - \overrightarrow{p}_{\overline{b}})_{z}(\overrightarrow{p}_{b} \times \overrightarrow{p}_{\overline{b}})_{z}$. The statistically limited results are found to be in agreement with no CP violation in $t\overline{t}$ production and decay, and the following values have been measured for $A'_{CP}$: $O_2=+0.27\,\pm\,0.41\,\mathrm{(stat.)}\,\pm\,0.01\,\mathrm{(syst.)}$, $O_3\,=\,-0.71\,\pm0.41\,\mathrm{(stat.)}\,\pm\,0.03\,\mathrm{(syst.)}$, $O_4\,=\,-0.38\,\pm\,0.41\,\mathrm{(stat.)}\,\pm\,0.03\,\mathrm{(syst.)}$, $O_7\,=\,-0.06\,\pm\,0.41\,\mathrm{(stat.)}\,\pm\,0.01\,\mathrm{(syst.)}$.
Conclusions {#sec:Conclusions}
===========
Top-quark properties measurements at the LHC provide precision tests of the SM. Limits have been set on FCNC and precision measurements of spin correlations and the charge asymmetry are consistent with the SM.
References {#references .unnumbered}
==========
[99]{} ATLAS Collaboration, JINST [**3**]{}, S08003 (2008).
CMS Collaboration, JINST [**3**]{}, S08004 (2008).
D0 Collaboration, **100**, 142002 (2008).
CDF Collaboration, **101**, 202001 (2008).
Aguilar-Saavedra J. A. and Pérez-Victoria M., **84**, 115013 (2011).
Aguilar-Saavedra J. A. and Pérez-Victoria M., **09**, 097 (2011).
ATLAS Collaboration, B **756**, 52 (2016).
ATLAS Collaboration, C **76**, 87 (2016).
CMS Collaboration, D **93**, 034014 (2016).
CMS Collaboration, arXiv:1603.06221 (2016), submitted to PLB
ATLAS Collaboration, D **93**, 012002 (2016).
CMS Collaboration, D **93**, 052007 (2016).
W. Bernreuther and Z.-G. Si, B **725**, 115-122 (2013).
ATLAS Collaboration, **114**, 142001 (2015).
CMS Collaboration, arXiv:1511.06170 (2015), submitted to PLB
Aguilar-Saavedra J., B **35**, 2695 (2004).
Larios F., Martinez R. and Pérez M., A **21**, 3473-3494 (2006).
CMS Collaboration, CMS-PAS-TOP-14-020.
CMS Collaboration, CMS-PAS-TOP-14-019.
CMS Collaboration, CMS-PAS-TOP-13-017.
ATLAS Collaboration, **12**, 061 (2015).
CMS Collaboration, CMS-PAS-TOP-16-001.
|
---
abstract: 'We analyze the possibility of measuring the state of a movable mirror by using its interaction with a quantum field. We show that measuring the field quadratures allows to reconstruct the characteristic function corresponding to the mirror state.'
author:
- 'Blas M. Rodríguez and Héctor Moya-Cessa'
title: '**Relation between the field quadratures and the characteristic function of a mirror**'
---
The reconstruction of a quantum state is a central topic in quantum optics and related fields [@ris; @leo]. During the past years, several techniques have been developed, for instance the direct sampling of the density matrix of a signal mode in multiport optical homodyne tomography [@zuk], tomographic reconstruction by unbalanced homodyning [@wal], reconstruction via photocounting [@ban], cascaded homodyning [@kis] to cite some. There have also been proposals to measure electromagnetic fields inside cavities [@lut; @moy] and vibrational states in ion traps [@lut; @bar]. In fact the full reconstruction of nonclassical states of the electromagnetic field [@smi] and of (motional) states of an ion [@lei] have been experimentally accomplished. The quantum state reconstruction in cavities is usually achieved through a finite set of selective measurements of atomic states [@lut] that make it possible to construct quasiprobability distribution functions such as the Wigner function, that constitute an alternative representation of a quantum state of the field.
Recently there has been interest in the production of superposition states of macroscopic systems such as a moving mirror [@bouw]. It is therefore of interest to have schemes to measure the non-classical states that may be generated for the moving mirror. Here we will propose a method to relate the quadratures of the field to the characteristic function associated to the density matrix of the mirror.
The interaction between a quantum electromagnetic field and a movable mirror (treated quantum mechanically) has a relevant Hamiltonian given by [@manci]
$$H=\hbar(\omega a^{\dagger}a + \Omega b^{\dagger}b -g
a^{\dagger}a(b^{\dagger}+b)) \label{ham}$$
where $a$ and $a^{\dagger}$ are the annihilation and creation operators for the cavity field, respectively. The field frequency is $\omega$. $b$ and $b^{\dagger}$ are the annihilation and creation operators for the mirror oscillating at a frequency $\Omega$ and $$g = \frac{\omega}{L}\sqrt{\frac{\hbar}{2m\Omega}},$$ with $L$ and $m$ the lenght of the cavity and the mass of the movable mirror.
We can re-write the Hamiltonian (\[ham\])in the form [@bose] $$H= D_m(\eta a^{\dagger}a)\left(\omega a^{\dagger}a + \Omega
b^{\dagger}b -\epsilon (a^{\dagger}a)^2\right)D_m^{\dagger}(\eta
a^{\dagger}a)$$ where $ \epsilon=g\eta$ with $\eta=g/\Omega$ and the displacement operator is given by $$D_m(\beta)=e^{\beta b^{\dagger}-\beta^* b},$$ with $N=a^{\dagger}a$. Then the unitary evolution operator is simply $$U(t)= e^{\frac{-iHt}{\hbar}}D_m(\eta N)e^{-it\left(\omega N +
\Omega b^{\dagger}b -\epsilon N^2\right)}D_m^{\dagger}(\eta N)$$ We will consider the initial state of the field to be in a coherent state $$|\alpha\rangle =
e^{-\frac{|\alpha|^2}{2}}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle.$$ and the initial state of the mirror to be arbitrary and denoted by the density matrix $\rho_m$. We may calculate then $\langle
a\rangle$ in the form $$\langle a\rangle=\alpha e^{-i(\omega+\epsilon)t} Tr \left[\rho_m
D_m\left( \eta e^{i\Omega t}\right)D_m\left(- \eta\right)|\alpha
e^{2i(\epsilon t- \eta^2\sin\Omega t)}\rangle
\langle\alpha|\right]$$ where we have used several times the properties of permutation under the trace symbol. By using that $$D_m\left( \eta e^{i\Omega t}\right)D_m\left(- \eta
\right)e^{i\eta^2\sin\Omega t}= D_m\left( \eta(e^{i\Omega
t}-1)\right)$$ we may finally write $$\langle a\rangle=\alpha e^{-i(\omega+\epsilon)t}
e^{-i\eta^2\sin\Omega t} e^{-|\alpha|^2(\epsilon t-
\eta^2\sin\Omega t)} \chi_m\left( \eta(e^{i\Omega t}-1)\right)$$ where $\chi_m\left( \eta(e^{i\Omega t}-1)\right)$ is the characteristic function associated to the density matrix $\rho_m$. Therefore, by measuring the quadratures of the field (see for instance [@leo]) $\langle
X\rangle=\langle(a+a^{\dagger})\rangle/\sqrt{2}$ and $\langle Y
\rangle =-i\langle(a-a^{\dagger})\rangle/\sqrt{2}$ we may obtain the average value for the annihilation operator and hence, information about the state of the mirror through its characteristic function. The argument of the characteristic function may be changed in some range of parameters as $\omega\sim
10^{16} s^{1}$, $\Omega\sim 1$ kHz, $L\sim 1$ m and $m\sim 10 $ mg [@manci; @mey1; @mey2]. One could use the present method to reconstruct the quantum superpositions of a mirror state recently proposed by Marshall [*et al.*]{} around the origin to look for a negative Wigner function in this region.
What makes it possible to obtain information about the mirror state is the initial coherence of the field and the form of the Hamiltonian that has the term $b+b^{\dagger}$. Wilkens and Meystre [@wil] had shown that for the Jaynes-Cummings Model (JCM) (see for instance [@jcm]) it was possible to obtain information about the characteristic function of the field only if the system interacted with an extra (classical) field to allow several absorptions ($a^k$) or emissions \[$(a^{\dagger})^k$\]. The JCM by itself would allow one emission or absorption at a time because of the form of the interaction Hamiltonian $H_I=\lambda(a\sigma_++\sigma_-a^{\dagger})$ where the $\sigma$’s the usual spin operators and $\lambda$ the interaction constant.
However, if we do not make the rotating wave approximation in the atom field interaction it was shown that transforming the complete Hamiltonian by means of a unitary transformation gives [@moyac] $$H_T=\omega N + \omega_0
W\left(\frac{\lambda}{\omega}\sigma_z\right)$$ where $W\left(\frac{\lambda}{\omega}\sigma_z\right) =
D\left(\frac{\lambda}{\omega}\sigma_z\right) (-1)^N D^{\dagger}
\left(\frac{\lambda}{\omega}\sigma_z\right)$ is the Wigner operator [@vog]. This hints that keeping terms in the Hamiltonian proportional to the sum of annihilation and creation operators allows information about the system to be obtained.
In conclusion, we have shown that by measuring filed quadratures one may be able to reconstruct the characteristic function for the density matrix of the mirror.
We would like to thank CONACYT for support.
[aaaa]{} K. Vogel and H. Risken, Phys. Rev. A [**40**]{}, 2847 (1989). U. Leonhardt, [*Measuring the Quantum State of Light*]{}, (Cambridge, Cambridge University Press) 1997. A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, Phys. Rev. A [**54**]{}, 1678 (1996). S. Wallentowitz and W. Vogel, Phys. Rev. A [**53**]{}, 4528 (1996). K. Banaszek and K. Wodkiewcz, Phys. Rev. Lett. [**76**]{}, 4344 (1996). Z. Kis, T. Kiss, J. Janszky, P.Adam, S. Wallentowitz, and W. Vogel, Phys. Rev. A [**59**]{}, R39 (1999). L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. [**78**]{}, 2547 (1997). H. Moya-Cessa, S.M. Dutra, J.A. Roversi, and A. Vidiella-Barranco, J. of Mod. Optics [**46**]{}, 555 (1999); H. Moya-Cessa, J.A. Roversi, S.M. Dutra, and A. Vidiella-Barranco, Phys. Rev. A [**60**]{}, 4029 (1999). P.J. Bardroff, C. Leichtle, G. Scrhade, and W.P. Schleich, Phys. Rev. Lett. [**77**]{}, 2198 (1996). D.T. Smithey, M. Beck, M.G. Raimer, and A. Faradini, Phys. Rev. Lett. [**70**]{}, 1244 (1993); G. Breitenbach, S. Schiller, and J. Mlynek, Nature [**387**]{}, 471 (1997). D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. [**77**]{}, 4281 (1996). H. Moya-Cessa and P.L. Knight, Phys. Rev. A [**48**]{}, 2479 (1993). W. Marshall, C. Simon, R. Penrose and D. Bouwmeester, quant-ph/0210001. S. Mancini, V.I. Man’ko and P. Tombesi, Phys. Rev. A [**55**]{}, 3042 (1997). S. Bose, K. Jacobs and P.L. Knight, Phys. Rev. A [**56**]{}, 4175 (1997). A. Dorsel, J.D. McCullen, P. Meystre, E. Vignes, and H. Walther, Phys. Rev. Lett. [**51**]{}, 1550 (1983). P. Meystre, E.M. Wright, J.D. McCullen, and E. Vignes, J. Opt. Soc. Am. B[**2**]{}, 1830 (1985). M. Wilkens and P. Meystre, Phys. Rev. A[**43**]{}, 3832 (1991). P. Knight and B. Shore, Phys. Rev. A[**48**]{}, 642 (1993). H. Moya-Cessa, A. Vidiella-Barranco, J.A. Roversi and S.M. Dutra, J. Opt. B [**2**]{}, 21 (2000). W. Vogel and D.-G. Welsch *Lectures on Quantum Optics*, (Berlin, Akad. Verl., 1994).
|
---
author:
- 'S[ø]{}ren S. Larsen'
- Holger Baumgardt
- Nate Bastian
- Svea Hernandez
- Jean Brodie
bibliography:
- 'refs.bib'
date: 'Received 23 October 2018 / Accepted 29 January 2019'
title: Hubble Space Telescope photometry of multiple stellar populations in the inner parts of NGC 2419
---
Introduction
============
In the words of @Shapley1922, NGC 2419 is ‘a globular cluster of uncommon interest’. At a distance of 83 kpc [@Ripepi2007], it has long been recognised as one of the most remote globular clusters (GCs) in the Milky Way [@Baade1935; @Racine1975]. Despite the difficulties associated with the great distance, NGC 2419 has a number of unique characteristics that motivate the effort required to undertake detailed studies. It is among the most luminous and massive GCs in the Milky Way and it is also very extended; these properties together imply a very long relaxation time. @Baumgardt2018 found a total mass of $(9.8\pm1.4)\times10^5 M_\odot$ and a half-mass radius of $r_h = 24.2$ pc, corresponding to a half-mass relaxation time of $t_\mathrm{rh} = 55 \, \mathrm{Gyr}$. This is by far the longest among the GCs in the Milky Way. In many GCs, the present-day structural parameters have likely been significantly modified by dynamical effects such as mass segregation and orbital mixing [@Decressin2008; @Vesperini2013; @Dalessandro2014; @Dalessandro2018]. Because of the long relaxation time, such effects are expected to be minimal in NGC 2419 and it therefore offers one of the best opportunities to constrain the initial structural properties of a globular cluster.
The vast majority of GCs that have been studied in detail to date exhibit variations in the abundances of many of the light elements, with anti-correlated abundances of Na/O, C/N, and, less commonly, Al/Mg [@Carretta2009; @Cohen2002; @Shetrone1996; @Sneden2004]. NGC 2419 is no exception, and some of its abundance variations are in fact more extreme than in most other GCs: about half of the stars in NGC 2419 have extremely depleted Mg abundances, reaching $\mathrm{[Mg/Fe]}$ values as low as $-1$, and these stars are also enriched in potassium [@Cohen2012; @Mucciarelli2012a]. However, the overall metallicity spread appears to be very small (less than $\sim$0.1 dex), with a mean metallicity of $\mathrm{[Fe/H]} = -2.09$ [@Mucciarelli2012a; @Frank2015].
Photometric studies reveal an extremely extended blue tail of the horizontal branch in NGC 2419 [@Ripepi2007; @Dalessandro2008; @Sandquist2008], which suggests the presence of a population of strongly He-enriched stars [$\mathrm{Y}\approx0.36$; @DiCriscienzo2011; @DiCriscienzo2015]. @DiCriscienzo2011 also noted a significant spread in the F435W-F814W ($\sim B\!-\!I$) colours of red giant branch (RGB) stars, and estimated that the colour spread was consistent with about 30% of the stars being He-enriched. A spread in the colours of RGB stars was also found by @Beccari2013 [hereafter B2013] from ground-based $uVI$ photometry. According to B2013, stars with blue $u\!-\!V$ and $u\!-\!I$ colours tended to be more centrally concentrated within NGC 2419 than those with redder colours, and the authors argued that the blue colours were indicative of enhanced He. A more centrally concentrated distribution of stars with anomalous (i.e., non field-like) abundances of He and other elements would indeed be expected in some self-enrichment scenarios for the origin of multiple populations (MPs) in GCs [e.g. @DErcole2008; @Decressin2008; @Bastian2013a]. However, B2013 also found that stars with anomalous (low) Mg abundances tended to have redder $u\!-\!V$ colours than those with normal Mg abundances, in apparent conflict with the expectation that elevated He content would be coupled with depleted Mg abundances. A complication when interpreting the $uVI$ colours of red giants is that the SDSS $u$-band is also sensitive to N abundance variations. An increased amount of N will suppress the flux in the $u$-band and lead to redder $u\!-\!V$ colours compared to a N-normal population. The net effect on the colours of an ‘enriched’ population thus depends on the balance between the opposing effects of He- and N variations. This balance may change as a function of luminosity on the RGB. In this context, it is worth noting that the radial distributions were determined by B2013 for stars on the lower RGB, whereas the stars with spectroscopic Mg abundance measurements are found near the tip of the RGB. It is thus important to establish how the Mg abundance anomalies correlate with variations in other elemental abundances by independently constraining the He and N abundances of the different populations.
The most common spectroscopic tracers of multiple populations in GCs are the Na/O [@Carretta2009] and C/N [@Cohen2002] anti-correlations. Variations in CNO abundances are detectable photometrically through their effects on the OH, CN, NH, and CH molecular absorption bands in the blue part of the spectra of cool stars [e.g. @Sbordone2011; @Carretta2011], as demonstrated in spectacular fashion by the Hubble Space Telescope (HST) UV Legacy Survey of globular clusters [@Piotto2015; @Milone2017]. However, the distance of NGC 2419 makes UV observations of RGB stars relatively time consuming. From ground-based Str[ö]{}mgren $uvby$ photometry, @Frank2015 [hereafter F2015] found a significant spread in N abundance for RGB stars in the outer parts of NGC 2419, with an approximately equal mix of two populations with distinct N abundances being favoured over a single Gaussian distribution of N abundances. This is reminiscent of the bimodality observed in the $\mathrm{[Mg/Fe]}$ and $\mathrm{[K/Fe]}$ abundance ratios [@Mucciarelli2012a]. The Str[ö]{}mgren colours were found by F2015 to be consistent with the Mg-normal stars being N-normal, and Mg-poor stars being N-rich.
The inner regions of NGC 2419 are too crowded for accurate ground-based photometry of all but the brightest RGB stars, and the studies of B2013 and F2015 were both restricted to stars outside the central $\sim$50 (about one half-light radius) of the cluster. F2015 pointed out that the difference between the roughly equal fractions of N-normal and N-rich stars found by them in the outer regions of the cluster and the somewhat smaller fraction of He-rich stars in the centre [@DiCriscienzo2011] might suggest an inverse population gradient, in the sense that the enriched stars are somewhat less concentrated. While this might be at odds with theoretical expectations, it would not be completely unprecedented. In @Larsen2015, it was found that stars with enhanced N abundances were less concentrated than those with normal N abundances within the central regions of the GC M15, whereas an apparent reversal of this trend occurred at larger radii. However, this result has recently been challenged by @Nardiello2018, who found no significant difference in the radial distributions of the different populations in M15.
The correspondence between the stellar populations identified in the central regions of NGC 2419 (via the extended HB and the spread in optical colours on the RGB) and the constraints on N abundance variations in the outer parts remains unclear. What is still missing is a more robust way of establishing the contributions of N-abundance vs. He abundance variations to the colour variations in the central regions of the cluster. HST observations in the F275W filter used by @Piotto2015 would require impractically long integration times, but a viable alternative is offered by the narrow-band F343N filter, which is sensitive to the NH absorption band near 3400 Å [@Larsen2014a]. Here we present new HST observations of NGC 2419 in the F343N and F336W filters, which we combine with existing archival data in several optical filters to constrain the properties of multiple populations in the inner regions of the cluster.
Throughout this paper we assume a distance modulus of $(m-M)_0 = 19.60$ mag [@Ripepi2007] and a foreground extinction in the HST filters of $A_\mathrm{F336W} = 0.271$ mag, $A_\mathrm{F438W} = 0.220$ mag, $A_\mathrm{F555W} = 0.174$ mag, $A_\mathrm{F606W} = 0.151$ mag, $A_\mathrm{F814W} = 0.093$ mag, and $A_\mathrm{F850LP} = 0.073$ mag [@Schlafly2011 via the NASA/IPAC Extragalactic Database, NED]. Because of the low concentration of NGC 2419, the exact location of the centre is uncertain by several arcsec. The 2010 edition of the @Harris1996 catalogue gives the J2000.0 centre coordinates as (RA, Decl) = (07h38m08.47s, $+38^\circ52^\prime56\farcs8$) whereas NED lists the coordinates as (07h38m07.9s, $+38^\circ52^\prime48\arcsec$). @Dalessandro2008 used HST photometry to compute the location of the barycentre as (07h38m08.47s, $+38^\circ52^\prime55\arcsec$). From looking at our new HST images, the Harris coordinates seem a little too far north, and those given by the NED too far south. We thus adopted the coordinates from @Dalessandro2008.
Observations and data reduction
===============================
The WFC3 filters and multiple populations
-----------------------------------------
{width="16cm"}
The sensitivity of various photometric systems to multiple populations in GCs has been discussed in detail by previous studies [e.g. @Carretta2011; @Sbordone2011; @Piotto2015]. The photometric signatures can be grouped into two broad categories: 1) atmospheric effects, and 2) effects on stellar structure. In the first case, the observed spectral energy distribution (SED) is modified by variations in the strengths of strong molecular absorption features (CN, CH, NH, OH), which are linked to variations in the light-element abundances (C, N, O). As long as the C+N+O sum is constant, these abundance variations are not expected to have any significant effect on the structure of the star itself. In the second case, the abundance variations (typically an increased He content) do affect the structure of the star, and a He-enriched isochrone will generally be shifted to higher effective temperatures (bluer colours).
Figure \[fig:p1p2spec\] shows model spectra for two stars with properties similar to those found near the base of the RGB in NGC 2419 ($T_\mathrm{eff}=5254$ K, $\log g = 2.76$, and $\mathrm{[Fe/H]}=-2.0$). The model atmospheres and corresponding spectra were computed with the `ATLAS12` and `SYNTHE` codes [@Sbordone2004; @Kurucz2005] for normal ($\alpha$-enhanced) halo-like composition (P1) and the CNONa2 mixture of @Sbordone2011, which is typical of enriched stars (P2) in GCs ($\Delta$(C, N, O, Na) = $-0.6, +1.44, -0.8, +0.8$ dex). Also included are the transmission curves for the WFC3 filters used in this paper. We see that the F343N filter samples the NH feature near 3400 Å, which is much stronger in the N-rich P2 star. The broader F336W filter is also sensitive to variations in the OH bands bluewards of the NH feature, which are weaker in the P2 spectrum (due to the depleted O abundance). To the extent that stars in NGC 2419 follow the usual tendency for N and O to be anti-correlated, this enhances the ability of the F336W-F343N colour to distinguish between the different populations. We note also that the F438W filter includes the CH band near 4300 Å (the Fraunhofer G feature), which is weaker in the P2 spectrum because of the C depletion. The F555W, F606W, F814W, and F850LP filters include no strong molecular bands and are largely insensitive to the atmospheric effects of multiple populations, but they are, of course, sensitive to variations in effective temperature that may be caused by variations in He content.
Observations
------------
For the optical photometry we used archival data from observing programme GO-11903 (P.I.: J. Kalirai), which imaged the central parts of NGC 2419 with the Wide Field Camera 3 (WFC3) on board HST. These observations consist of pairs of un-dithered exposures in many filters. In this paper we use observations in F438W (exposure times of $2\times725$ s), F555W ($2\times580$ s), F606W (2$\times400$ s), F814W ($2\times650$ s), and F850LP ($2\times675$ s). The programme also includes exposures in several ultraviolet filters [which were used by @DiCriscienzo2015], but these are generally short and do not allow us to reach the required photometric accuracy for stars on the RGB.
Additional observations in F336W and F343N were obtained in cycle 25 under programme GO-15078 (P.I.: S. Larsen). This filter combination was chosen specifically to measure variations in the strength of the NH band near 3400 Å (Fig. \[fig:p1p2spec\]). The magnitude difference between the two filters is sensitive to the strength of the NH feature in a manner similar to, for example, the Str[ö]{}mgren $\beta$ index for the H$\beta$ line [@Stromgren1966].
The GO-15078 observations consist of three visits, of which two visits were allocated to F343N imaging and one visit to F336W. Each visit had a duration of three orbits and within each visit, the observations were dithered according to the C6 $3\times2$ dither pattern described in @Dahlen2010. Hence, NGC 2419 was observed in F343N for six orbits which yielded 12 exposures with a total exposure time of 17152 s. In F336W, the six exposures obtained during three orbits had a total exposure time of 8576 s. The roll angles, centre coordinates, and dither patterns were identical for all three visits. Given that the two filters have very similar central wavelengths, we expect that most systematic effects (reddening, sensitivity, etc.) will cancel out when calculating the difference between the F336W and F343N magnitudes.
Photometry
----------
For the analysis we used the ‘`*_flc`’ frames, which are corrected for charge-transfer inefficiencies by the instrument pipeline [@Ryan2016]. Photometry was carried out with `ALLFRAME` [@Stetson1994], following the procedure described in @Larsen2014a. After cleaning the individual pipeline-reduced frames of cosmic rays and multiplying them by the appropriate pixel area maps, `ALLFRAME` was set up to carry out photometry on each individual frame. Point-spread functions (PSFs) were determined from about 50 isolated, bright stars distributed evenly across each detector. We followed the standard approach of detecting stars with the `find` task in `daophot` and carrying out a first round of aperture- and PSF-fitting photometry, then detecting additional stars on the star-subtracted images generated by `ALLFRAME` in the first pass, and using the merged star catalogues as input for a second pass of PSF-fitting photometry [@Stetson1987]. The PSFs used in the second pass were redetermined from images in which all stars except the PSF stars had been subtracted.
The GO-15078 and GO-11903 data were reduced separately and the photometry catalogues were then merged. The photometry was calibrated to STMAG magnitudes using aperture photometry of the PSF stars and the 2017 photometric zero-points for an $r=10$ pixels aperture published on the WFC3 webpage[^1]. These are: $z_\mathrm{F343N} = 22.770$ mag, $z_\mathrm{F336W} = 23.517$ mag, $z_\mathrm{F438W} = 24.236$ mag, $z_\mathrm{F555W} = 25.651$ mag, $z_\mathrm{F606W} = 26.154$ mag, $z_\mathrm{F814W} = 25.861$ mag, and $z_\mathrm{F850LP} = 24.885$ mag.
Conveniently, the roll angles of the two datasets differ by less than 10 degrees (as indicated by the header keyword PA$\_$V3, which has a value of 276 deg for the GO-15078 observations and 269 deg for the GO-11903 observations) and the difference between the centre coordinates is only about $10\arcsec$. Hence, the overlap between the datasets is excellent. We used the `geomap` and `geoxytran` tasks in the `images.immatch` package in IRAF to define coordinate transformations between the GO-15078 and GO-11903 datasets. Because of the good overlap, most of the stars imaged on a given detector in GO-15078 were mapped onto the same detector in GO-11903, although a small fraction of CCD\#1 in GO-15078 was mapped onto CCD\#2 in GO-11903 and vice versa. We used about 100–150 stars on each CCD to define the transformations, which were fitted with 3rd order polynomials in the $x$ and $y$ coordinates. This yielded an r.m.s. scatter of about 0.05 pixels in the solutions. The pixel coordinates (measured in the GO-15078 frames) were further transformed to sky coordinates using the `wcs` package in `Astropy` [@AstropyCollaboration2018]. Small offsets to the HST coordinates (about 0216 in right ascension and 0061 in declination) were applied in order to match the Gaia astrometry, based on about 400 stars in common between our HST data and the Gaia DR2 catalogue [@GaiaCollaboration2016; @GaiaCollaboration2018]. The dispersions around the mean offsets were about $0\farcs026$ and $0\farcs015$ in right ascension and declination, respectively, from which we estimate the remaining systematic uncertainty on the astrometric calibration to be about $10^{-3}$ arcsec. The first few entries of the photometric catalogue are listed in Table \[tab:photometry\], and the full catalogue is available on-line.
![($m_\mathrm{F438W}-m_\mathrm{F850LP}, m_\mathrm{F438W}$) colour-magnitude diagram. Red symbols indicate stars that are saturated in the F850LP filter. \[fig:cmd\_bl\_b\] ](fig2.png){width="\columnwidth"}
![Spatial distribution of RGB stars relative to the centre of NGC 2419. \[fig:map\] ](fig3.pdf){width="\columnwidth"}
Figure \[fig:cmd\_bl\_b\] shows the ($m_\mathrm{F438W}$ vs. $m_\mathrm{F438W-F850LP}$) colour-magnitude diagram (CMD). Overall, the CMD is very similar to that shown in Fig. 1 of @DiCriscienzo2015, apart from an offset due to our use of STMAG (instead of VEGAMAG) zero-points. The CMD clearly shows the main features identified in previous studies, including the horizontal branch (HB) and its extended blue tail, a population of blue stragglers (BS), as well as the relatively narrow RGB. The photometry reaches a couple of magnitudes below the main sequence turn-off (MSTO), although we will concentrate exclusively on the RGB in this paper.
The red dashed line indicates the colour cut that we will use to separate RGB stars from potential HB and BS interlopers, which is given by: $m_\mathrm{F438W}-m_\mathrm{F850LP} > -0.12 - 0.45 \times (m_\mathrm{F438W} - 19.5)$. We use the F438W-F850LP colour combination for this purpose, partly because it offers a long colour baseline, which helps to separate the RGB from AGB stars in the range $19 \la m_\mathrm{F438W} \la 20$, and partly because the F850LP filter is less affected by saturation than the F555W and F814W filters. In F555W and F814W, saturation sets in at $m_\mathrm{F555W} \sim 18.9$ and $m_\mathrm{F814W} \sim 19.0$, respectively, whereas the saturation limit is about one magnitude brighter, at $m_\mathrm{F850LP} \sim 17.9$ in F850LP. Stars that are saturated in F850LP are indicated with red symbols in Fig. \[fig:cmd\_bl\_b\]. In the F336W, F343N, and F438W observations, even the brightest RGB stars remain unsaturated. While it is possible to recover the flux accurately even for stars that are several magnitude brighter than the saturation limit [@Gilliland2010], we have not attempted to do so here.
Figure \[fig:map\] shows the spatial distribution of RGB stars brighter than $M_\mathrm{F438W}=+2$ that are included in both the GO-15078 and GO-11903 datasets, relative to the adopted centre of NGC 2419. The coverage is spatially complete within a radius of $\approx70\arcsec$ (apart from the gap between the WFC3 detectors), and the outermost star is about 110 from the centre. In terms of the projected half-light radius [$r_{h,lp}=45\arcsec$; @Baumgardt2018], spatial coverage is thus complete to about $1.5 \, r_{h,lp}$.
Artificial star tests {#sec:artstar}
---------------------
![Input F438W magnitudes versus the difference between input and recovered F336W-F343N colour, $\delta_{o-i}$(F336W-F343N), for the artificial stars. The red lines show the 16% and 84% percentiles. \[fig:synt\_dunu\_b\] ](fig4.pdf){width="\columnwidth"}
![Here $\delta_{o-i}$(F336W-F343N) is plotted as a function of radial distance $R$ from the centre of NGC 2419 for artificial stars brighter than $M_\mathrm{F438W}=+2$. The red solid (dashed) lines show the 16% and 84% (2.5% and 97.5%) percentiles. \[fig:synt\_r\_dunu\] ](fig5.pdf){width="\columnwidth"}
The photometric accuracy and completeness were quantified by means of artificial star tests. Ten rounds of such tests were carried out, adding about 2050 stars to the HST images in each round with the `mksynth` task in the `BAOLAB` package [@Larsen1999]. The `mksynth` task models artificial stars by treating the PSF as a probability density function, from which events are picked at random and added to the image one by one until the desired number of counts has been reached. In this way, the images of simulated stars are subject to the same stochastic effects as those of real stars.
To ensure that the central regions of the cluster were well sampled, the artificial stars were grouped into a series of concentric annuli, centred on NGC 2419. These annuli had radii of $0 < r_1 < 100$ pixels (60 stars per round), $100 < r_2 < 200$ pixels (90 stars), $200 < r_3 < 400$ pixels (325 stars), $400 < r_4 < 600$ pixels (300 stars), $600 < r_5 < 800$ pixels (280 stars), $800 < r_6 < 1200$ pixels (540 stars), and $1200 < r_7 < 2000 $ pixels (480 stars). To avoid self-crowding among the artificial stars, we enforced a minimum separation of 20 pixels between any pair of artificial stars. A set of dedicated artificial PSF stars were also added.
The F814W magnitudes of the artificial stars were picked at random from the actual magnitude distribution of RGB stars in NGC 2419, with the faintest artificial stars having an absolute magnitude of $M_\mathrm{F814W} = +4$ ($M_\mathrm{F438W} \approx +3$). To determine the magnitudes in the other filters, we used `ATLAS12` and `SYNTHE` to compute synthetic spectra for 20 sampling points along the RGB of an $\alpha$-enhanced isochrone with $\mathrm{[Fe/H]}=-2$ and an age of 13 Gyr [@Dotter2007]. Magnitudes in the STMAG system were then determined by integrating the synthetic spectra over the filter transmission curves, and for a given $M_\mathrm{F814W}$ magnitude the magnitudes in other filters were then obtained by interpolation in the synthetic relations.
The `ALLFRAME` procedure was then repeated, the only modification with respect to the original photometry being that the PSFs were determined from the artificial PSF stars. This was done to ensure that inaccuracies in the PSF determination were propagated properly through the procedure, rather than fitting the artificial stars directly with the same PSFs that were used to generate them in the first place. The resulting photometry catalogues were matched against the input lists of artificial stars. An artificial star in the input catalogue was defined as recovered if a counterpart was found within a distance of 1 pixel in the `ALLFRAME` catalogue. The artificial star experiments were carried out separately for the GO-15078 and GO-11903 data, but using the same input catalogue of artificial stars. The photometry catalogues for the artificial stars could then subsequently be merged in the same way as was done for the science data.
Essentially all of the artificial stars added to the images were recovered by the `ALLFRAME` photometry procedure in both the GO-15078 and GO-11903 datasets. Even at the faintest magnitudes, about one magnitude below the limits adopted in our analysis, the recovery fraction remained at $>99\%$ at all radii. Hence, we can assume that the `ALLFRAME` photometry is unaffected by completeness effects over the magnitude range of interest here.
In Figure \[fig:synt\_dunu\_b\] we plot the input F438W magnitude vs. the difference between the input and recovered (output) F336W-F343N colour, $\delta_{o-i}$(F336W-F343N). The red curves indicate the 16% and 84% percentiles of $\delta_{o-i}$(F336W-F343N), computed in 0.25 mag bins. As expected, the scatter increases towards the faint end of the magnitude distribution. Taking half of the separation between the 16% and 84% percentiles as an estimate of the standard error, $\sigma_\mathrm{F336W-F343N}$, we find that this increases from $\sigma_\mathrm{F336W-F343N} = 0.008$ mag at the bright end to $\sigma_\mathrm{F336W-F343N} = 0.028$ mag at $M_\mathrm{F438W} = +2$, which will be the typical faint limit adopted in most of the subsequent analysis.
While crowding does not appear to affect our ability to detect RGB stars even at the centre of NGC 2419, it may still have an effect on the photometric errors. Figure \[fig:synt\_r\_dunu\] shows $\delta_{o-i}$(F336W-F343N) as a function of distance from the centre of NGC 2419 for stars brighter than $M_\mathrm{F438W} = +2$. Plots for other colours look very similar. We see that there is indeed a mild tendency for the scatter to increase towards the centre. Estimating the standard error in the same way as above, we find $\sigma_\mathrm{F336W-F343N} = 0.023$ mag in the innermost bin ($\langle R \rangle = 3\arcsec$), decreasing to $\sigma_\mathrm{F336W-F343N} = 0.016$ mag at radii $R>70\arcsec$.
When using the artificial star tests to estimate the error distributions for observed stars in NGC 2419, we need to correct for the fact that the radial distribution of the artificial stars does not exactly match that of the actual cluster stars. To this end, we assigned a radius-dependent weight $w_A(R)$ to each artificial star, where $w_A(R)$ is given by the ratio of the number of actual stars at radius $R$ to the number of artificial stars at the same radius. We computed these weights in radial bins of 100 pixels (4), and then assigned the corresponding $w_A(R)$ to all artificial stars within that radial bin. The weights ranged between $w_A(R)\approx0.15$ and $w_A(R)\approx0.35$. This relatively modest variation, combined with the weak radial dependence of the errors, means that the error distributions and corresponding estimates of the photometric errors change little when applying the weights.
![Histograms of the D$_\mathrm{F555W-F606W}$ distributions for observed RGB stars and artificial stars. \[fig:dvr\] ](fig6.pdf){width="\columnwidth"}
While artificial star tests such as those described above are widely used, it should be kept in mind that it is nearly impossible to include all effects that are present in the real data in such tests [see e.g. @Bellazzini2002; @Anderson2008]. The PSF of the artificial stars will typically not be an exact match to that of the real stars, and fitting the artificial stars with the same PSF used to generate them would fail to account fully for uncertainties in the PSF modelling. While we have attempted to account for this by redetermining the PSF from artificial stars, cases where artificial stars are blended with real stars may still be problematic. Furthermore, the real PSF varies across the WFC3 detectors, whereas our artificial star tests used a single PSF for all stars. Although spatial variations in the PSF are taken into account in the `ALLFRAME` photometry, the modelling of these variations is by necessity imperfect, and this is difficult to take into account in a fully realistic way in the artificial star tests. Even if the spatial variability of the PSF were fed back into the artificial star tests, it would be restricted to the parameterisation used by `ALLFRAME`.
To assess the fidelity of the artificial star tests, Figure \[fig:dvr\] shows a comparison of the observed spread in the F555W-F606W colours around the RGB with the corresponding spread for the artificial stars. The quantity D$_\mathrm{F555W-F606W}$ denotes the difference between the observed colours and a polynomial fit to the RGB for stars in the range $0 < M_\mathrm{F438W} < +2$. The F555W-F606W combination is expected to be insensitive to the abundance variations arising from the presence of multiple populations because of the small colour baseline and the lack of strong molecular features in these two bands, so that the spread in D$_\mathrm{F555W-F606W}$ should be due mainly to observational errors. As can be seen in the figure, the distributions for the observed and artificial stars are very similar indeed. To be more quantitative, we again estimated the dispersions from the 16th and 84th percentiles of the D$_\mathrm{F555W-F606W}$ colour distribution. For a Gaussian distribution, this will be similar to the dispersion computed in the usual way, but by using the percentiles we are less sensitive to extreme outliers. As indicated in the legend, the dispersion for the artificial stars ($\sigma_A = 0.0118$ mag) is very similar to that of the observations ($\sigma_\mathrm{Obs} = 0.0120$ mag). We note that including the correction for spatial coverage of the artificial stars makes virtually no difference; if this correction is omitted we find $\sigma_A = 0.0119$ mag. Very similar results were obtained from the F814W-F850LP combination, for which the same comparison yields $\sigma_A = 0.0153$ mag and $\sigma_\mathrm{Obs} = 0.0147$ mag.
We conclude that the artificial star tests, in spite of the potential concerns mentioned above, provide a fairly realistic estimate of the random uncertainties in our photometric analysis.
Results
=======
![$M_\mathrm{F438W}$ vs. $M_\mathrm{336W}-M_\mathrm{F343N}$ diagram. The filled coloured symbols are stars with $\mathrm{[Mg/Fe]}$ measurements [@Cohen2012; @Mucciarelli2012a]. \[fig:uuni\] ](fig7.pdf){width="\columnwidth"}
![(a): F438W vs. F336W-F343N colour-magnitude diagram with the 10% and 90% fiducial lines indicated. (b): verticalised CMD. \[fig:cmd\_unu\_b\] ](fig8a.pdf "fig:"){width="42mm"} ![(a): F438W vs. F336W-F343N colour-magnitude diagram with the 10% and 90% fiducial lines indicated. (b): verticalised CMD. \[fig:cmd\_unu\_b\] ](fig8b.pdf "fig:"){width="42mm"}
![Histogram of the $\Delta_\mathrm{F336W-F343N}$ colour distribution. The error distribution of the synthetic stars has been shifted by 0.015 mag to match the P2 peak. \[fig:hist\_dunu\] ](fig9.pdf){width="\columnwidth"}
Ultraviolet colours and N abundance variations {#sec:uvcol}
----------------------------------------------
In Fig. \[fig:uuni\] we show the $M_\mathrm{F438W}$ vs. $M_\mathrm{F336W}-M_\mathrm{F343N}$ diagram for RGB stars in NGC 2419. Here, and in the rest of the paper, we assume that the extinction in F343N is the same as in F336W. We use the F438W magnitude instead of F336W on the vertical axis, since the latter turns over somewhat below the tip of the RGB (i.e., the coolest and most luminous RGB stars are not the brightest in F336W). The loci of N-normal and N-rich stars are therefore not well separated at the bright end when plotting $M_\mathrm{F336W}$ vs. $M_\mathrm{F336W}-M_\mathrm{F343N}$. Filters centred at even longer wavelengths would potentially be even better than F438W, but here saturation becomes problematic.
The stars for which spectroscopic Mg abundance measurements are available in the literature [@Cohen2012; @Mucciarelli2012a] are highlighted with red ($\mathrm{[Mg/Fe]}>0$) and blue ($\mathrm{[Mg/Fe]}<0$) symbols. Also included in the figure are synthetic colours computed for standard field-like ($\alpha$-enhanced) and CNONa2 mixture, for He content of $Y=0.25$ (solid lines) and $Y=0.40$ (dashed lines). We used isochrones with $\mathrm{[Fe/H]}=-2$ and $t=13$ Gyr from the Dartmouth database [@Dotter2007] and colours were calculated with `ATLAS12`/`SYNTHE`, as described in Sect. \[sec:artstar\]. We also computed a set of model atmospheres and spectra in which the Mg abundance was decreased by 1 dex, but this was found to have a negligible effect on the colours. The model colours have been shifted by a small offset (0.02 mag) in the horizontal direction.
It is clear that the F336W-F343N colour index is a very effective discriminator of CNO content, specifically N abundance, while it is hardly affected by He content. The observed spread in F336W-F343N is roughly similar to the separation between the field-like and CNONa2 models, and much larger than the photometric errors. Henceforth we refer to stars appearing to the right in Fig. \[fig:uuni\] (i.e., those with field-like composition) as belonging to P1 and those to the left as P2. We will adhere to this nomenclature specifically in the context of CNO variations. It is clear that the Mg-poor stars are associated mostly with the N-rich population P2, and the Mg-normal stars mostly with the N-normal population P1, although there may not be an exact 1:1 correspondence.
Figure \[fig:uuni\] already hints at a bimodal distribution of the F336W-F343N colours. To further examine the properties of the colour distribution, we verticalised Fig. \[fig:uuni\], following a similar procedure to that described in @Milone2017. This entailed computing the offset in F336W-F343N for each star with respect to a fiducial line, and renormalising the offsets relative to those at some fixed magnitude to account for changes in the width of the colour distribution. While the model lines in Fig. \[fig:uuni\] roughly trace the extremes of the F336W-F343N colour distribution, it was found that a better result was obtained by determining the fiducial lines directly from the data. We computed the 10% and 90% percentiles of the F336W-F343N colours as a function of $M_\mathrm{F438W}$ in bins of 0.5 mag, and then fitted fourth-order polynomials to the percentile values vs. $M_\mathrm{F438W}$. In reality, the 10% and 90% lines were found to be nearly parallel (Fig. \[fig:cmd\_unu\_b\], panel (a)), with the separation varying between 0.101 mag at $M_\mathrm{F438W} = +2$ and 0.104 mag at $M_\mathrm{F438W} = +0.5$, and then narrowing slightly to 0.092 mag at $M_\mathrm{F438W} = -1.5$. We then defined the offset $\Delta_\mathrm{F336W-F343N}$ with respect to the 10% percentile line, scaled to the separation between the two lines at a reference magnitude of $M_\mathrm{F438W} = +1$.
Panel (b) of Fig. \[fig:cmd\_unu\_b\] shows the verticalised CMD, in which two vertical sequences are readily visible. For magnitudes fainter than $M_\mathrm{F438W} = +2$ the separation between the two sequences becomes less evident, presumably because of the increasing photometric errors (Fig. \[fig:synt\_dunu\_b\]). In the following, we therefore restrict the analysis to stars brighter than $M_\mathrm{F438W} = +2$, as indicated by the horizontal dashed line in Fig. \[fig:cmd\_unu\_b\]. This gives a sample of 1717 RGB stars.
Figure \[fig:hist\_dunu\] shows the distribution of $\Delta_\mathrm{F336W-F343N}$ for stars selected as described above. The histogram confirms that the distribution is clearly bimodal. The orange open histogram shows the error distribution, $\delta_{o-i}$(F336W-F343N) for artificial stars in the same magnitude range as the data included in the figure. The $\delta_{o-i}$(F336W-F343N) values denote the differences between the input and recovered F336W-F343N colours, which were scaled by the same magnitude-dependent factor as the $\Delta_\mathrm{F336W-F343N}$ offsets in order to facilitate comparison with the observed $\Delta_\mathrm{F336W-F343N}$ distribution. The error distribution has also been corrected for differences in the radial distributions of artificial and actual stars, using the weights $w_A(R)$ defined in Sect. \[sec:artstar\]. The error histogram appears only slightly narrower than the two peaks in the observed colour distribution, which suggests that much of the broadening of the peaks may be due to photometric errors.
To quantify the evidence for bimodality, we applied the KMM test [@Ashman1994] to the $\Delta_\mathrm{F336W-F343N}$ distribution. The KMM algorithm models the parent distribution of an observed sample as a sum of multiple Gaussians, and compares the likelihood obtained from the best fitting multi-Gaussian model with that obtained from a single Gaussian. The improvement of the multi-Gaussian fit with respect to a single Gaussian is expressed as a $p$-value. Here we used the KMM algorithm to carry out a double-Gaussian fit to the $\Delta_\mathrm{F336W-F343N}$ distribution, allowing the peaks ($\mu_1$ and $\mu_2$) and dispersions ($\sigma_1$ and $\sigma_2$) of both Gaussians to vary. The red and blue curves in Fig. \[fig:hist\_dunu\] represent the best-fitting double-Gaussian model estimated by the KMM algorithm. The KMM algorithm returned a $p$-value of $p<10^{-5}$, signifying that a double-Gaussian fit is a highly significant improvement over a single Gaussian (as was already evident from the histogram). The peak colours for P1 and P2 are $\mu_1 = 0.085$ mag and $\mu_2 = 0.015$ mag and the dispersions are $\sigma_1 = 0.024$ mag and $\sigma_2 = 0.021$ mag. The KMM algorithm assigned 939 stars (55%) and 778 stars (45%) to P1 and P2, respectively.
From the artificial star tests we find a dispersion of $\sigma_\mathrm{art} = 0.022$ mag based on the scaled $\delta_{o-i}$(F336W-F343N) values. This value is essentially independent of whether or not the weights $w_A(R)$ are applied ($\sigma_\mathrm{art} = 0.0227$ without the weights, 0.0224 when including them). This is very similar to the width of the P2 peak found by the KMM algorithm, and only slightly narrower than the P1 peak. Again, this is consistent with the visual impression from Fig. \[fig:hist\_dunu\], which shows the $\delta_{o-i}$(F336W-F343N) histogram to be very similar to the P2 Gaussian.
The KMM algorithm assigns stars with $\Delta_\mathrm{F336W-F343N}<0.047$ mag to P2 and those with $\Delta_\mathrm{F336W-F343N}>0.047$ mag to P1. In the remainder of this paper we associate stars with P1 and P2 based on this limit.
Optical colours and pseudo-chromosome maps
------------------------------------------
![F814W vs. F438W-F814W colour-magnitude diagram, showing the RGB for stars with $\Delta_\mathrm{F336W-F343N}>0.047$ (P1) and $\Delta_\mathrm{F336W-F343N}\le0.047$ (P2). The black curves are polynomial fits to the two sequences. Grey symbols indicate stars which are saturated in F814W. \[fig:rgb\_bi\_i\] ](fig10.pdf){width="\columnwidth"}
![Verticalised F438W vs. F438W-F814W colour-magnitude diagram. The horizontal dashed lines show the magnitude limits used in the construction of the pseudo-chromosome map. Grey symbols indicate stars which are saturated in F814W. \[fig:dbi\_b\] ](fig11.pdf){width="\columnwidth"}
![Pseudo-chromosome map and histograms of the $\Delta_\mathrm{F438W-F814W}$ colours for P1 and P2 stars. The thin orange histogram indicates the errors as determined from artificial star tests. The arrows show the effect of changing the He abundance by $\Delta \mathrm{Y}=0.15$, the overall metallicity by $\Delta \mathrm{[Z/H]} = +0.1$ dex, and other light element abundances by $\Delta$(\[(C, N, O, Na)/Fe\]) = $-0.6, +1.44, -0.8, +0.8$ dex [@Sbordone2011]. \[fig:chromo\] ](fig12.pdf){width="\columnwidth"}
The F336W-F343N colours provide an effective and clean way to assess nitrogen abundance variations. However, it is becoming increasingly clear that more than one parameter may be required to characterise the abundance variations in GCs, even in mono-metallic clusters. A clear demonstration is provided by the diversity observed among the chromosome maps of GCs [@Milone2017], which provide evidence of He abundance variations even among P1 stars in some clusters [@Lardo2018; @Milone2018].
While the chromosome maps defined by @Milone2017 make use of the F275W filter, we can use the colours available for NGC 2419 to construct pseudo-chromosome maps that have the same basic ability to separate variations in N- and He abundance. For the N abundance variations we continue to rely on the $\Delta_\mathrm{F336W-F343N}$ index. To trace He abundance variations, we can use optical colours, preferably with a long baseline. Here we opt for the F438W-F814W combination, noting that F438W-F850LP gives essentially identical results.
Figure \[fig:rgb\_bi\_i\] shows the $M_\mathrm{F814W}$ vs. $M_\mathrm{F438W} - M_\mathrm{F814W}$ CMD for RGB stars in NGC 2419. Stars associated with P1 and P2 based on their $\Delta_\mathrm{F336W-F343N}$ colours are plotted with red and blue symbols, respectively. It is evident that the P2 stars tend to have bluer F438W-F814W colours on average than the P1 stars, which is consistent with a higher He content for the P2 stars. We note that a difference in overall metallicity could potentially produce a similar effect on the RGB colours, in which case the bluer colours of the P2 stars would imply lower metallicities.
The two curves in Fig. \[fig:rgb\_bi\_i\] are polynomial fits to the P1 and P2 sequences. To verticalise the CMD we follow a slightly different procedure than for F336W-F343N, since the photometric errors contribute significantly to the scatter. Instead of using percentiles, we use the fits to the P1 and P2 sequences as fiducials, calculating the offsets $\Delta_\mathrm{F438W-F814W}$ with respect to the P1 fiducial line and normalising, as before, at $M_\mathrm{F438W} = +1$ ($M_\mathrm{F814W} = +1.3$). The resulting verticalised version of Fig. \[fig:rgb\_bi\_i\] is shown in Fig. \[fig:dbi\_b\]. Because of the saturation in F814W at bright magnitudes, we impose a bright magnitude limit of $M_\mathrm{F438W} = 0$, which leaves us with 1449 RGB stars.
Figure \[fig:chromo\] shows the pseudo-chromosome map obtained by plotting $\Delta_\mathrm{F336W-F343N}$ vs. $\Delta_\mathrm{F438W-F814W}$. The arrows indicate the effect of changing the He content by $\Delta \mathrm{Y} = 0.15$, the overall metallicity by $\Delta \mathrm{[Z/H]} = +0.1$ dex, and the CNO abundances according to the CNONa2 mixture. We have defined the axes such that the plot resembles the chromosome maps of @Milone2017 as closely as possible, i.e., with He content increasing towards the left and N increasing (and C and O decreasing) upwards. We note that the $\Delta$Z arrow is nearly anti-parallel to the $\Delta$Y arrow.
The mean $\Delta_\mathrm{F438W-F814W}$ colours of the P1 and P2 stars are $-0.007$ mag and $-0.050$ mag, respectively. Part of the colour difference may be caused by the differences in CNO abundances, since the $\Delta$CNO arrow is not exactly vertical. We will quantify this further below. The histograms in the upper panel show the colour distributions of the P1 and P2 stars together with the distribution of $\delta_{o-i}$(F438W-F814W) from the artificial star tests. As was done for $\delta_{o-i}$(F336W-F343N), a magnitude-dependent scaling was applied to the $\delta_{o-i}$(F438W-F814W) values for consistency with the verticalised colour distribution. For P1, the dispersion (as estimated from the 16th and 84th percentiles) is $\sigma_\mathrm{F438W-F814W} = 0.023$ mag and for P2 it is $\sigma_\mathrm{F438W-F814W} = 0.040$ mag. The standard deviation of the synthetic star colours is $\sigma=0.015$ mag. For the P2 stars, the $\Delta_\mathrm{F438W-F814W}$ distribution is thus significantly broader than the error distribution, and it appears that there may be some spread in $\Delta_\mathrm{F438W-F814W}$ also for the P1 stars, with a tail towards the blue that may indicate the presence of He-enriched stars.
Constraints on N and He abundance variations
--------------------------------------------
-------------------------------- ----------------------------------------------------- ------------------ ----------------- ---------------- ---------- ---------- ------------
Colour $\langle\Delta(\mathrm{P2-P1})\rangle_\mathrm{obs}$ $C_\mathrm{CNO}$ $C_\mathrm{He}$ $C_\mathrm{Z}$
CNO, He CNO, Z CNO, He, Z
(mag) (mag) (mag) (mag)
$\Delta_\mathrm{F336W-F343N}$ $-0.070$ $-0.093$ $-0.010$ $+0.003$ $-0.081$ $-0.087$ $-0.071$
$\Delta_\mathrm{F343N-F438W}$ $+0.113$ $+0.150$ $-0.032$ $+0.019$ $+0.114$ $+0.114$ $+0.112$
$\Delta_\mathrm{F336W-F438W}$ $+0.045$ $+0.056$ $-0.043$ $+0.022$ $+0.032$ $+0.026$ $+0.039$
$\Delta_\mathrm{F336W-F555W}$ $+0.013$ $+0.042$ $-0.075$ $+0.029$ $+0.009$ $+0.006$ $+0.017$
$\Delta_\mathrm{F336W-F814W}$ $-0.001$ $+0.042$ $-0.105$ $+0.032$ $-0.001$ $+0.003$ $+0.001$
$\Delta_\mathrm{F438W-F555W}$ $-0.026$ $-0.017$ $-0.039$ $+0.009$ $-0.027$ $-0.025$ $-0.027$
$\Delta_\mathrm{F438W-F814W}$ $-0.044$ $-0.017$ $-0.075$ $+0.013$ $-0.040$ $-0.029$ $-0.045$
$\Delta_\mathrm{F438W-F850LP}$ $-0.048$ $-0.017$ $-0.081$ $+0.013$ $-0.042$ $-0.029$ $-0.049$
$\Delta_\mathrm{F555W-F814W}$ $-0.019$ $+0.000$ $-0.038$ $+0.004$ $-0.013$ $-0.004$ $-0.020$
-------------------------------- ----------------------------------------------------- ------------------ ----------------- ---------------- ---------- ---------- ------------
CNO, He CNO, Z CNO, He, Z
---------------------------- --------- --------- ------------
$\mathscr{S}_\mathrm{CNO}$ 0.83 0.90 0.74
$\mathscr{S}_\mathrm{He}$ 0.34 … 0.64
$\mathscr{S}_\mathrm{Z}$ … $-1.08$ 1.17
r.m.s. (mag) 0.0065 0.0127 0.0024
: Scaling coefficients for the fits to the colour differences in Table \[tab:coldist\].[]{data-label="tab:coeff"}
![Colour distributions for P1 (red) and P2 stars (blue). The yellow histograms show the error distributions based on artificial star tests. The arrows indicate the effect of changing the He content by $\Delta$Y=$+0.15$ and the CNONa abundances by $\Delta$(\[(C, N, O, Na)/Fe\]) = $-0.6, +1.44, -0.8, +0.8$ dex. \[fig:hist\_dcol\] ](fig13.pdf){width="8cm"}
The pseudo-chromosome diagram utilises only two out of many possible colour combinations that can be formed from the available photometry. In Fig. \[fig:hist\_dcol\] we show the histograms of various other colour combinations for the P1 and P2 stars, together with the error distributions from the artificial star tests. Again, the error distributions have been corrected for spatial coverage and the magnitude-dependent verticalisation scaling. The error distributions have been scaled to the same total number as the P1 stars. Each panel also includes horizontal arrows indicating the effect of a change of 0.15 in the He abundance and a change in CNONa abundances from standard $\alpha$-enhanced abundances to the CNONa2 mixture.
In many of these colour combinations, the two populations appear separated to some degree. For colour combinations that use F438W as the blue filter instead of F336W or F343N, the P2 stars generally appear bluer than the P1 stars, consistent with the P2 stars being more He rich (or metal-poor) on average. In these cases, the He and CNO arrows are parallel (due to the effect of C variations on the G-band contained within the F438W filter), so that He and CNO abundance variations reinforce each other. For the combinations that do include F336W, the effect of modified CNO abundances tends to counterbalance the increase in He abundance for the P2 stars. For F336W-F814W these two effects appear to cancel out almost exactly on average, causing the colour distributions of P1 and P2 to be similar, whereas the smaller colour baseline for F336W-F555W implies that the CNONa variations win, making the P2 stars slightly redder on average than the P1 stars. This is reminiscent of the analysis by F2015, who found no difference between the Str[öm]{}gren $u-y$ colours of Mg-poor and Mg-normal RGB stars in NGC 2419, and attributed this to the opposite effects of He and CNO abundance variations on this particular colour. For F814W-F850LP (bottom panel), the two populations have essentially identical mean colours, which is consistent with the expectation that this colour combination should be insensitive to He- and CNO variations.
The dispersions of the P1, P2, and synthetic colour distributions are indicated in the legends of Fig. \[fig:hist\_dcol\]. In all cases (except F814W-F850LP), the colour spread for the P2 stars clearly exceeds that expected from the artificial star tests. In general, the colour spread may be due to a combination of CNO and He abundance variations. The F555W-F814W colour is expected to be a relatively clean tracer of He abundance variations (as indicated by the short CNONa2 arrow) and the significant spread in F555W-F814W for the P2 stars therefore corroborates the conclusion from the chromosome map that a He spread is likely present within P2. For P1, the case for a significant internal He spread is less strong, since the observed F555W-F814W distribution is only slightly broader than the error distribution, although there is again a hint of an asymmetric tail towards the blue. For F814W-F850LP we note that the observed dispersions of P1 and P2 are very similar to that of the artificial stars, which provides another verification that these tests give a realistic estimate of the photometric errors (cf. Sec. \[sec:artstar\]).
If we assume that the effects of He and CNO abundance variations, and potentially also of overall metallicity variations, can be combined linearly for each colour, then we can write $$\langle\Delta(\mathrm{P2-P1})_i \rangle = C_\mathrm{CNO,i} \mathscr{S}_\mathrm{CNO} + C_\mathrm{He,i} \mathscr{S}_\mathrm{He} + C_\mathrm{Z,i} \mathscr{S}_\mathrm{Z}
\label{eq:dcol}$$ where the coefficients $C_\mathrm{CNO,i}$, $C_\mathrm{He,i}$, and $C_\mathrm{Z,i}$ specify how the $i$th colour responds to variations in the CNO and He abundances and metallicity. Setting each of the scaling factors $\mathscr{S}_\mathrm{CNO}$, $\mathscr{S}_\mathrm{He}$, and $\mathscr{S}_\mathrm{Z}$ to unity corresponds to reference abundance changes of $\Delta[\mathrm{(C, N, O)/Fe}] = (-0.6, +1.44, -0.8)$ dex, $\Delta Y = 0.15$ (as indicated by the arrows in Fig. \[fig:hist\_dcol\]) and $\Delta Z = 0.1$ dex. We can then solve for the scaling factors that need to be applied to the reference abundance changes in order to best reproduce all of the observed colour differences $\langle\Delta(\mathrm{P2-P1})_i \rangle$.
In Table \[tab:coldist\] we list the mean observed colour differences between the P1 and P2 stars for each colour combination, with the exception of F814W-F850LP which contains little information and was included in Fig. \[fig:hist\_dcol\] only as a consistency check. The coefficients $C_\mathrm{CNO}$, $C_\mathrm{He}$, and $C_\mathrm{Z}$, were calculated from our synthetic photometry at a reference magnitude of $M_\mathrm{F438W} = +1$. The last three columns give the colour differences obtained by solving for variations in CNO and He (with Z fixed), in CNO and Z (with He fixed), and in all three parameters. In Table \[tab:coeff\] we give the corresponding best-fitting scaling factors $\mathscr{S}_\mathrm{CNO}$, $\mathscr{S}_\mathrm{He}$, and $\mathscr{S}_\mathrm{Z}$, which were found from a least-squares fit with the `lstsq` function in the `scipy.linalg` package in `Python`.
It is clear that, in all cases, a significant difference in mean CNO content between P1 and P2 is required to explain the colour differences, although the scaling factor $\mathscr{S}_\mathrm{CNO}$ is slightly less than unity for all fits. Hence, the implied average CNO difference between P1 and P2 is slightly smaller than assumed in the CNONa2 mixture. Table \[tab:coldist\] shows that a combination of variations in CNO and He content reproduces most of the P2-P1 colour differences to within about 0.01 mag, with an r.m.s. difference between the observed and modelled colour differences of 0.007 mag (Table \[tab:coeff\]). Assuming that $\Delta$Y scales linearly with $\mathscr{S}_\mathrm{He}$, the implied mean difference in He content is $\Delta \mathrm{Y} \simeq 0.05$. Keeping He fixed and allowing Z to vary instead produces a somewhat worse fit, with an r.m.s. difference of 0.013 mag between the observed and best-fit colours. In this case, the metallicity scaling factor is negative, $\mathscr{S}_\mathrm{Z}=-1.1$, implying that P2 would be on average 0.11 dex more metal-poor than P1. Allowing all three scaling factors to vary produces the smallest residuals (r.m.s. = 0.002 mag), with a larger variation in He abundance ($\Delta \mathrm{Y} \simeq 0.10$) and P2 now being more metal-rich than P1 by about 0.12 dex.
From the above, we conclude that small variations in mean metallicity may contribute to the observed colour differences, but these are largely degenerate with variations in He content and thus essentially unconstrained by our data. We can state that any differences in mean metallicity between P1 and P2 are likely less than about 0.1 dex, which is consistent with previous studies. The variations in CNO are slightly smaller than those corresponding to the CNONa2 mixture, so we may estimate that the mean difference in N abundance between P2 and P1 is $\Delta \mathrm{[N/Fe]} \approx 0.9 \times 1.44~\mathrm{dex} \approx 1.3~\mathrm{dex}$. The mean difference in He content is $\Delta \mathrm{Y} \approx 0.05$, or possibly slightly larger if P2 is also more metal-rich. The model-dependent nature of these estimates should, however, be emphasised [e.g. @Dotter2015].
Radial distributions {#sec:radist}
--------------------
Colour $R_h$(blue) $R_h$(red) $p_\mathrm{KS}$
-------------- ------------------------ ------------------------ -----------------
F336W-F343N $31\farcs3\pm1\farcs1$ $34\farcs6\pm0\farcs9$ 0.046
F343N-F438W $35\farcs4\pm0\farcs8$ $31\farcs4\pm1\farcs0$ 0.020
F336W-F438W $33\farcs7\pm1\farcs1$ $33\farcs5\pm1\farcs2$ 0.708
F336W-F555W $33\farcs7\pm1\farcs2$ $33\farcs5\pm1\farcs0$ 0.247
F336W-F814W $32\farcs6\pm1\farcs2$ $34\farcs0\pm1\farcs0$ 0.051
F438W-F555W $31\farcs8\pm1\farcs1$ $34\farcs8\pm0\farcs9$ 0.100
F438W-F814W $31\farcs8\pm1\farcs2$ $34\farcs8\pm0\farcs9$ 0.050
F438W-F850LP $31\farcs8\pm1\farcs1$ $35\farcs0\pm0\farcs9$ 0.038
F555W-F814W $31\farcs8\pm1\farcs1$ $35\farcs2\pm0\farcs9$ 0.014
F814W-F850LP $33\farcs2\pm1\farcs2$ $33\farcs7\pm1\farcs0$ 0.350
: Half-number radii for blue and red sub-populations, identified in various colours.[]{data-label="tab:rdist"}
![Cumulative radial distributions of P1 and P2 stars, selected based on their $\Delta_\mathrm{F336W-F343N}$ colours. P1E and P2E represent the “extreme” populations, which avoid the colour range between the peaks of P1 and P2. \[fig:rdist\_uun\] ](fig14.pdf){width="\columnwidth"}
As noted in the introduction, the long relaxation time of NGC 2419 means that the spatial distributions of sub-populations within the cluster are expected to be less affected by dynamical evolution than in other clusters. In this section we discuss the constraints on the radial distributions of subpopulations within NGC 2419.
In Fig. \[fig:rdist\_uun\] we show the cumulative radial distributions of P1 and P2 stars. The P2 stars are slightly more concentrated, but the difference between the P1 and P2 cumulative distributions does not appear dramatic and a Kolmogorov-Smirnov test returns $p_\mathrm{KS}=0.046$. From the cumulative distributions, the half-number radii for P1 and P2 are $34\farcs6\pm0\farcs9$ and $31\farcs3\pm1\farcs1$, respectively, with the $\sim10$% difference being significant at about 2.3$\sigma$ (the errors were estimated via bootstrapping). To get cleaner samples of P1 and P2 stars, we omitted stars with colours in the overlapping region between the peaks of P1 and P2, $\mu_2 < \Delta_\mathrm{F336W-F343N} < \mu_1$. Thus defining the extreme populations as P1E and P2E for $\Delta_\mathrm{F336W-F343N} > \mu_1$ and $\Delta_\mathrm{F336W-F343N} < \mu_2$, respectively, we get half-number radii of $34\farcs6\pm1\farcs1$ (P1E) and $32\farcs2\pm1\farcs7$ (P2E). The corresponding cumulative distributions, which are shown with dashed lines in Fig. \[fig:rdist\_uun\], are very similar to those of the full P1 and P2 samples, but due to the smaller numbers of stars the Kolmogorov-Smirnov test now gives $p_\mathrm{KS}=0.30$, indicating no significant difference. These half-number radii are all smaller than the half-light radii quoted in the literature, but this is as expected because the HST data do not include the outer parts of the cluster.
In general, analyses of radial trends can be affected by variations in differential reddening as well as instrumental effects such as variations in the PSF and flat-field errors across the field-of-view. Because of the very similar central wavelengths of the F336W and F343N filters and the small foreground extinction towards NGC 2419, it appears unlikely that reddening variations could produce significant spurious trends in the $\Delta_\mathrm{F336W-F343N}$ index.
To quantify the radial distributions in other colour combinations, we divided the RGB stars in NGC 2419 into blue and red samples in each colour, assigning (for simplicity) an equal number of stars to both groups. Table \[tab:rdist\] lists the half-number radii for the blue and red samples for each of the colour combinations shown in Fig. \[fig:hist\_dcol\]. In most of these colour combinations, differences in the radial distributions are only marginally significant. The most significant differences are found for F555W-F814W ($p_\mathrm{KS}=0.014$) and F343N-F438W ($p_\mathrm{KS}=0.020$). For F343N-F438W, the top panel in Fig. \[fig:hist\_dcol\] shows that this colour combination provides a fairly clear separation of the P1 and P2 stars, with the P2 stars having redder colours and being more concentrated. In the case of F555W-F814W, it is the blue stars that are more centrally concentrated. Blue F555W-F814W colours indicate enhanced He, which is characteristic of the P2 stars, so this is again consistent with the P2 stars being more centrally concentrated. These results are difficult to attribute to differential reddening variations, for which we would expect the difference $R_h$(blue) - $R_h$(red) to always have the same sign.
Instrumental effects are harder to rule out definitively. One check is provided by the F814W-F850LP colour, which is expected to be insensitive to multiple populations. For this colour, Table \[tab:rdist\] shows no significant difference between the radial distributions of blue and red samples. This colour distribution is also the narrowest, and thus more liable to be affected by instrumental effects, so the fact that no differences between red and blue subsamples are seen in this colour combination supports the notion that the differences seen in other colours are real.
Radial distributions: Helium or CNO as the main driver? {#sec:hecno}
-------------------------------------------------------
![Chromosome diagram divided according to He and CNO content. The half-number radii are indicated in each quadrant. The high quality measurements ($\sigma_\mathrm{F438W} < 0.02$ mag, $\chi_\nu^2 < 2$) are shown with open circles. \[fig:chromo\_xy\] ](fig15.pdf){width="\columnwidth"}
![Colour distributions in $\Delta_\mathrm{F555W-F850LP}$ of the four regions identified in Fig. \[fig:chromo\_xy\]. Histograms for the full samples are drawn with solid lines, whereas histograms for the stars remaining after error and $\chi^2$ cuts are shown with dotted lines. \[fig:hist\_dvl\_abcd\] ](fig16.pdf){width="\columnwidth"}
![Cumulative radial distributions of stars selected on their He and CNO content. Colour coding is the same as in Fig. \[fig:chromo\_xy\]. \[fig:rdist\_hecno\] ](fig17.pdf){width="\columnwidth"}
While the pseudo-chromosome diagram (Fig. \[fig:chromo\]) suggests some correlation between He and CNO abundance variations, a significant He abundance spread is implied for at least the P2 stars, and possibly there is also some spread for the P1 stars. In the previous sections, we have used the CNO abundance variations as our primary means to split the cluster stars into sub-populations, with the tacit assumption that the intra-population He spreads can be treated as a perturbation on top of the CNO variations. This view seems to be supported by the clear bimodality in the CNO-sensitive colours, whereas the separation into distinct He-rich and He-normal groups is much less clear. Nevertheless, it appears worthwhile to examine in more detail how the radial distributions are affected by CNO as well as He abundance variations.
To this end, Fig. \[fig:chromo\_xy\] again shows the pseudo-chromosome diagram, now divided into four regions. The stars with the highest quality photometry are shown with open circles (photometric errors in F438W less than $\sigma_\mathrm{F438W} = 0.02$ mag and `ALLFRAME` $\chi_\nu^2 < 2$) and other stars are shown with solid dots. The black lines that separate the four regions have the same slopes as the arrows in Fig. \[fig:chromo\] and thus separate the stars by their He- and CNO content.
Stars in the lower right-hand corner (region A) have the most field-like composition (normal He, normal CNO), whereas stars in the upper left-hand corner (region C) have the most GC-like composition (enhanced He and N). In each region we also indicate the half-number radius of the corresponding radial distribution and the number of stars. Fig. \[fig:hist\_dvl\_abcd\] shows the $\Delta_\mathrm{F555W-F850LP}$ distributions for the four regions defined in Fig. \[fig:chromo\_xy\]. Histograms drawn with solid lines represent the full sample, and those drawn with dotted lines are for the high quality sub-sample. Like $\Delta_\mathrm{F555W-F814W}$, the $\Delta_\mathrm{F555W-F850LP}$ combination is mainly sensitive to He abundance, and is used here because it is entirely independent of the colours used in Fig. \[fig:chromo\_xy\]. The legend in Fig. \[fig:hist\_dvl\_abcd\] indicates the mean colour offsets relative to the A stars. These confirm that the C stars are the most He enriched population, and that the D stars also have a significant He enhancement relative to the A stars. Comparing the histograms for the full set of D stars with the high-quality subsample, it can be seen that the quality cut mainly removes stars with colours more similar to those of the A stars, which may have scattered into the D region. This leaves a cleaner sample of pure D stars, so that the offset for the high-quality D stars is in fact slightly greater than for the full sample. The offset of $-0.041$ mag between A and C corresponds to $\Delta\mathrm{Y} = 0.13$, which would suggest a He fraction as high as $Y\simeq0.38$ for the C stars. The C and D stars together correspond to about 29% of the total number of RGB stars in the pseudo-chromosome map, which is comparable to the estimated fraction of RGB stars that are converted to extreme HB stars [@Sandquist2008]. It is thus tempting to associate the extreme HB stars with the He-enriched RGB stars in NGC 2419.
The cumulative radial distributions of the four populations are shown in Fig. \[fig:rdist\_hecno\]. The least concentrated stars are those belonging to the A group, while the D stars are formally the most concentrated, but none of the differences are highly significant. Grouping the stars by He content, we find half-number radii of $34\farcs2$ for the (He-normal) AB stars and $30\farcs5$ for the (He-rich) CD stars, respectively, with a K-S $p-$value of $p_\mathrm{KS} = 0.03$ for the comparison of the cumulative distributions. Grouping instead by CNO content, the corresponding half-number radii are $34\farcs0$ (AD) and $32\farcs5$ (BC) with $p_\mathrm{KS}=0.25$. In both cases, the stars with the most field-like composition have the most extended distribution, and there is a suggestion that the correlation with He abundance may be somewhat stronger than with CNO content, in agreement with the results in Sec. \[sec:radist\].
Kinematics
----------
![Kinematics. Top: AD (CNO-normal) stars. Bottom: BC (CNO-enriched) stars. \[fig:kinematics\] ](fig18a.pdf "fig:"){width="8cm"} ![Kinematics. Top: AD (CNO-normal) stars. Bottom: BC (CNO-enriched) stars. \[fig:kinematics\] ](fig18b.pdf "fig:"){width="8cm"}
To test for kinematic differences between the different populations, we cross correlated the positions of the stars that we determined from the HST images against the positions of stars with measured radial velocities from [@Baumgardt2018], accepting stars with a positional difference of less than $0\farcs5$ as a match. This resulted in 64 matches. We then subtracted the average velocity of NGC 2419, $v_r=-20.6 \pm 0.2$ km s$^{-1}$ found by [@Baumgardt2018] from the velocities of all stars and applied a maximum-likelihood test according to $$\ln \Lambda = \sum_i \frac{1}{\sqrt{ \sigma^2 + \epsilon^2_i}} e^{-\frac{1}{2}(v_i-v_{rot} \sin(\theta_i-\theta_0))^2/(\sigma^2 + \epsilon^2_i)}$$ to determine the best-fitting values of the velocity dispersion $\sigma$, the amount of rotation $v_{rot}$ and the position angle $\theta_0$ of rotation for the different components. In the above formula, $\theta_i$ is the direction from the centre of NGC 2419 to a star (measured anti-clockwise from north), and $v_i$ and $\epsilon_i$ are the velocity and the velocity error of each star after subtraction of the mean velocity of NGC 2419.
The stars with radial velocity measurements were assigned to P1 (standard CNO) and P2 (enriched CNO) using the $\Delta_\mathrm{F336W-F343N}$ index, as discussed in Sect. \[sec:uvcol\]. We obtained velocity dispersions of $\sigma = 4.76^{+0.66}_{-0.56}$ km s$^{-1}$ for the 38 stars assigned to P1, and $\sigma = 5.78^{+0.89}_{-0.72}$ km s$^{-1}$ for the remaining 26 P2 stars. Because these stars are mostly located within a few magnitudes of the tip of the RGB, assigning them to the He-normal and He-rich populations is more difficult. We made a rough assignment based on the F438W-F850LP colour. The corresponding dispersions for the standard and He-rich stars are $\sigma = 5.06^{+0.75}_{-0.62}$ km s$^{-1}$ and $\sigma = 5.67^{+0.76}_{-0.62}$ km s$^{-1}$ respectively. Hence there is some indication that the stars with field-like abundances have lower velocity dispersions, in agreement with their more extended spatial distribution. However the differences between the populations are still within the error bars.
The only group of stars for which we find significant rotation are the P1 stars, for which we find a rotation velocity of $v_r=2.4 \pm 1.1$ km s$^{-1}$ with a position angle of $\theta_0=337 \pm 25$ degrees. The CNO enriched (P2) stars, instead, do not show any significant rotation (Fig. \[fig:kinematics\]). We formally find a rotation velocity for the P2 stars of $1.02\pm1.47$ km s$^{-1}$, which is compatible with the same rotation as the P1 stars, as well as no rotation. These results could indicate that the P1 stars in NGC 2419 formed from an initially slowly rotating gas cloud which imprinted its rotation signature on these stars, while the P2 stars may have formed from kinematically more mixed gas.
To test the significance of the detection of rotation for the P1 stars, we randomly drew 38 velocities from a Gaussian distribution with a dispersion of 5 km s$^{-1}$ and errors randomly distributed between 0.5 and 2.5 km s$^{-1}$. We assigned random position angles to these velocities (i.e., no underlying rotation). We then measured how often a rotation signal with more than $2.2 \sigma$ was found. This happened in 43 out of 500 cases, corresponding to a false positive rate of about 8%.
Discussion
==========
Despite the unusual chemical abundance patterns found by spectroscopic studies, such as the presence of extremely Mg-depleted and K-enhanced stars [@Cohen2012; @Mucciarelli2012a], the photometric evidence indicates a relatively normal range of CNO abundance variations within NGC 2419. The N-sensitive $\Delta_\mathrm{F336W-F343N}$ index reveals a clearly bimodal distribution of N abundances, with a difference in average $\mathrm{[N/Fe]}$ value of $\sim1.3$ dex between N-normal P1 stars and N-rich P2 stars. Our analysis is consistent with that by @Frank2015, who found a range in $\mathrm{[N/Fe]}$ of less than 1.3 dex from Str[ö]{}mgren photometry for the outer parts of the cluster. While it should be kept in mind that the photometric estimates of the $\mathrm{[N/Fe]}$ variations are uncertain, and the difference in average N abundance between P1 and P2 may underestimate the full range somewhat, the estimated N abundance variations in NGC 2419 are not particularly extreme compared to those found in other GCs, where a range up to $\sim2$ dex in $\mathrm{[N/Fe]}$ has been found [e.g. @Yong2008].
Our photometry is consistent with previous evidence that a significant fraction of the stars in NGC 2419 have enhanced He abundances. Assuming that metallicity variations are negligible, we find a mean difference of $\Delta$Y$\simeq 0.05$ between P1 and P2, implying $\langle$Y$\rangle \simeq 0.30$ for the P2 stars if the P1 stars have a normal He fraction of Y$\simeq0.25$. However, the total range is likely greater, since both populations (especially P2) show evidence of an intrinsic He spread. NGC 2419 is yet another example of a cluster where significant He spreads are found within the sub-populations identified via CNO-sensitive colours [@Nardiello2018; @Milone2017; @Lardo2018; @Milone2018]. The difference between the mean He abundances of P1 and P2 is relatively large compared to those found by @Milone2018, who found mean differences between 0 and 0.05 in $\Delta$Y for Galactic GCs, but it is consistent with their result that the larger differences tend to be found in more massive clusters. Again, it is worth emphasising that the absolute values of the He abundance spreads derived from the photometry should not be taken too literally.
A double-Gaussian fit to the $\Delta_\mathrm{F336W-F343N}$ distribution assigns about 45% of the RGB stars to the N-rich P2. For a massive GC like NGC 2419, this is a relatively low fraction of enriched stars, with enriched fractions of $\sim70$% or more being common for clusters with masses approaching $10^6 M_\odot$ [@Milone2017; @Bastian2018]. The enriched fraction estimated from a simple double-Gaussian fit would increase if we consider that some of the P1 stars may not be truly field-like, but the stars in region A of Fig. \[fig:chromo\_xy\] still account for nearly 1/2 of all stars in the diagram. These estimates refer to the inner regions of NGC 2419 covered by our HST photometry, and the global fraction of enriched stars could be even lower if there are radial gradients. According to @Beccari2013, the fraction of stars with blue $u\!-\!I$ and $u\!-\!V$ colours decreases significantly outwards for radii between 100 and 400. If these blue colours indicate enhanced He abundance, as assumed by B2013, then this would indeed imply a decreasing enriched fraction outwards. However, the strong radial trend seen in the $u\!-\!V$ and $u\!-\!I$ colours is somewhat puzzling, given the very small separation between P1 and P2 in the (nearly) equivalent F336W-F555W and F336W-F814W colours, and the lack of strong radial trends in these colours in the inner regions of the cluster (Fig. \[fig:hist\_dcol\] and Table \[tab:rdist\]).
![Cumulative radial distributions of RGB stars in the outer regions of NGC 2419, using Str[ö]{}mgren photometry from @Frank2015 \[fig:frank\] ](fig19.pdf){width="\columnwidth"}
@Frank2015 found an enriched fraction of $53\pm5$% in the outer parts of NGC 2419 from a double-Gaussian fit to their measurements of the N-sensitive Str[ö]{}mgren $\delta_4$ index. This is formally slightly higher than our estimate for the inner regions. F2015 did not investigate radial trends in detail, except by noting that their inferred enriched fraction is higher than that found for the inner regions of the cluster by other studies. To see whether their photometry can constrain radial trends, we downloaded their photometric catalogue and defined the $\Delta_{\delta_4}$ and equivalent $\Delta_\mathrm{b-y}$ and $\Delta_\mathrm{u-y}$ parameters with respect to a ridge line for the ‘clean RGB sample’ in the same way that F2015 did.
The comparisons of the resulting cumulative radial distributions for sub-samples divided according to $\Delta_{\delta_4}$, $\Delta_\mathrm{u-y}$, and $\Delta_\mathrm{b-y}$ are shown in Fig. \[fig:frank\]. The F2015 sample of RGB stars is smaller than our HST sample for the central regions (177 stars), so the statistical significance of any results is inevitably lower. Nevertheless, there is no significant difference between the radial distributions of N-normal and N-rich subsamples as defined by $\Delta\delta_4$ ($p_\mathrm{KS}=0.33$). Likewise, when dividing the sample according to $\Delta_\mathrm{u-y}$, the two radial distributions are essentially identical ($p_\mathrm{KS}=0.94$). This is seemingly at odds with the strong trends found by B2013, since the Str[ö]{}mgren $u\!-\!y$ index is expected to behave very similarly to the $u\!-\!V$ colour. Indeed, F2015 found that the $u\!-\!y$ colours showed no correlation with Mg abundance, which they attributed to the opposite effects of CNO and He abundance variations on this colour, a similar conclusion to that reached from our analysis. When dividing according to $\Delta_\mathrm{b-y}$, there is a somewhat significant difference ($p_\mathrm{KS}=0.038$), with the blue stars being more centrally concentrated. While the Str[ö]{}mgren $b-y$ colour is insensitive to CNO abundance variations [@Carretta2011], it does depend on He abundance (through $T_\mathrm{eff}$), and the mild tendency for the stars with blue $\Delta_\mathrm{b-y}$ colours (i.e. enhanced He) to be more centrally concentrated would be consistent with our results for the central regions.
The spectroscopically identified Mg-poor stars appear to be associated primarily with P2, and the Mg-normal stars with P1. So far, we have not discussed Na, which is perhaps the best established spectroscopic tracer of multiple populations. The Na-O anticorrelation is present in nearly all GCs where it has been looked for [@Carretta2009], and it is thus natural to inquire about the behaviour of Na in NGC 2419. Unfortunately, the picture remains unclear in this regard. While @Cohen2012 found a significant spread in $\mathrm{[Na/Fe]}$ within the cluster ($\sim 1$ dex), they found no correlation between $\mathrm{[Na/Fe]}$ and $\mathrm{[Mg/Fe]}$, with a difference of only 0.04 dex between the mean Na abundances of (five) Mg-deficient and (eight) Mg-normal stars. Five of the stars measured by @Cohen2012 fall within our HST data; two of these happen to have low Na abundances (S810 and S1166) and also have normal Mg abundances. These two stars have F336W-F343N colours consistent with normal N abundances. The other three (S1004, S1065, S1131) are Mg-deficient and Na-rich, and their F336W-F343N colours are relatively blue, indicating enhanced N abundances (Fig. \[fig:uuni\]). In this sense, the behaviour of these five stars is as expected, but the broader implications for the behaviour of Na remain unclear, given that many of the Mg-normal stars measured by @Cohen2012 are, in fact, Na-rich.
Our data do not provide strong constraints on metallicity variations except that they are small, with a mean metallicity difference of $<0.1$ dex between P2 and P1. This is in agreement with the findings by most previous studies [@Mucciarelli2012a; @Frank2015], although there are also claims of a larger metallicity spread [@Lee2013].
Summary and conclusions
=======================
We have used new HST/WFC3 imaging in the F343N and F336W filters, combined with archival optical HST/WFC3 data, to study the multiple populations in the central regions of the remote globular cluster NGC 2419. The data are spatially complete within a radius of $R=70\arcsec$ (28 pc), or about 1.5 projected half-light radii. The combination of UV and optical filters allowed us to constrain variations in He and N abundances for red giants in the inner regions of the cluster. We combined the photometry with radial velocity measurements from the literature [@Baumgardt2018] to examine the kinematics of the different populations. Our main findings are as follows:
- The F336W-F343N colour distribution is clearly bimodal, as confirmed by a KMM test ($p<10^{-5}$). A double-Gaussian fit assigns 55% of the stars to a population with F336W-F343N colours indicative of field-like nitrogen abundances (P1), and the rest to a population with nitrogen-enhanced composition (P2).
- From a comparison of the mean optical colours of P1 and P2 stars with model calculations, we estimate a mean difference in the He content of $\Delta \mathrm{Y} \simeq 0.05$ between the two populations. Small metallicity differences ($<0.1$ dex) could also contribute to the colour differences.
- For the P2 stars, the observed spread in optical colours such as F555W-F814W and F438W-F814W is greater than the observational uncertainties. This most likely indicates a He spread at least within P2, with some stars possibly having He content as high as $Y\simeq0.38$. Analysis of the pseudo-chromosome map suggests that a small fraction (about 16%) of the P1 stars may also be significantly He-enriched.
- The P2 stars are somewhat more centrally concentrated within the cluster than the P1 stars, with some hint that the difference is driven primarily by differences in mean He content. The difference in the half-number radii is, at any rate, modest (about 10%) and only moderately significant.
- The P1 stars have a slightly lower velocity dispersion than the P2 stars, although the difference is not statistically significant. Nevertheless, the difference is in agreement with the more extended spatial distribution of the P1 stars. We find evidence of rotation for the P1 stars, whereas the data for the P2 stars are consistent with no rotation, as well as the same rotation as the P1 stars.
- Stars for which spectroscopic measurements indicate a significant Mg-deficiency ($\mathrm{[Mg/Fe]}<0$) are associated primarily with the nitrogen-rich population.
In terms of the main elements studied and discussed in this paper, the abundance patterns seen in NGC 2419 are relatively unsurprising. The P1 stars identified through their field-like N abundances also tend to have relatively field-like He and Mg abundances, while the N-enriched P2 stars tend to have enhanced He and (strongly) depleted Mg. Nevertheless, the correlations between N and He have real scatter, and the same may well be true for N vs. Mg. It is well to keep in mind, however, the apparent lack of any correlation between Na and Mg. Here we cannot directly address the relation between Na and N or He, and this certainly appears to be a problem worthy of further investigation.
The failure of current scenarios for the formation of GCs to provide a satisfactory account of the observed properties of multiple populations in general is well documented [@Bastian2018], as are the additional problems associated with the complex chemistry of NGC 2419 specifically [e.g. @DiCriscienzo2011; @Cohen2012; @Mucciarelli2012a; @Carretta2013]. The relative similarity of the radial distributions of the different populations found here poses yet another potential complication for many scenarios.
The possibility that NGC 2419 may be the nucleus of a disrupted dwarf galaxy has been discussed by many authors. Recent proper motion measurements, combined with the radial velocity of NGC 2419, appear to be consistent with membership of the Sagittarius dwarf spheroidal galaxy [@Massari2017; @Sohn2018], but since Sagittarius already has a nucleus [@Bellazzini2008] this would argue against NGC 2419 also being a nucleus. It is not clear, in any case, that this would help explain its peculiar abundance patterns, which are not observed in dwarf galaxies [@Salgado2019]. Other more plausible candidates for nuclei (such as $\omega$ Cen and M54) display a chemical inventory that is quite different from what is seen in NGC 2419, with significant metallicity spreads but no reported Mg-K anticorrelation [@Carretta2013]. While NGC 2419 appears to represent a relatively extreme manifestation of the multiple populations phenomenon, it should be noted that the GC NGC 2808 shares some of the features seen in NGC 2419, such as the Mg-K anticorrelation [@Mucciarelli2015], although other details differ.
Hence, it seems that NGC 2419 falls within the range of behaviours that a successful theory for GC formation must be able to explain.
We thank the anonymous referee for a careful reading of the manuscript and several helpful comments. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. N.B. gratefully acknowledges financial support from the Royal Society (University Research Fellowship) and the European Research Council (ERC-CoG-646928-Multi-Pop). J.B. acknowledges support for HST Program number GO-15078 from NASA through grant HST-GO-15078.02 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
Photometry
==========
[^1]: <http://www.stsci.edu/hst/wfc3/analysis/uvis_zpts/>
|
---
abstract: 'Generalized Einstein relation between the mobility and diffusion in conductors with a large built-in field near the thermodynamic equilibrium has been derived.'
author:
- 'Sergey A. Ktitorov'
title: 'Generalized Einstein relation in conductors with a large built-in field'
---
The mobility and diffusion coefficients depend on the particle energy. When the system under consideration is in the inequilibrium state and the current does not vanish, the electons are hot and their energy is determined by the balance of power, which electrons acquire from the electric field and transfere to the thermostat. Mobility is determined by the mean kinetic energy of electrons and within the framework of the quasi-hydrodynamic approximation can be considered as a function of the local electric field, if inhomogeneities are spread along the current lines [@shura] (this is the case for N-shape current-voltage characteristics). Attempts to generalize the Einstein relation are directed mainly to analysis of strongly inequilibrium state (see [@strong], for instance). In the opposite case of S-shape charaterisitics the mobility can be considered as a function of the local current. We encounter quite different situation, when the system is near the thermodynamic equilibrium, but the built-in electric field is strong enough so that the mobility depends mainly on the potential energy: $\mu=\mu\left(
\phi\right) .$ Therefore, the diffusion coefficient depends on the potential as well: $D=D\left( \phi\right) .$ We consider here a one-dimensional distribution of the electric field and the electronic density. The total current reads:$$j=-en\left( x\right) \mu\left[ \phi\left( x\right) \right] \frac{d\phi
}{dx}-e\frac{d}{dx}\left[ D\left( \phi\right) n\left( x\right) \right]
,\text{ \ \ , }E=-d\phi/dx,\text{ \ \ }\frac{dj}{dx}=0. \label{current}$$
We have in the thermodynamic equilibrium:$$j=0\text{, \ \ }n\left( x\right) =n_{0}e^{-\frac{e\phi}{T}}.
\label{equilibrium}$$
Carrying out differetiation in (\[current\]) we have$$j=-en\left( x\right) \mu\left[ \phi\left( x\right) \right] \frac{d\phi
}{dx}-e\frac{dD}{d\phi}n\left( x\right) \frac{d\phi}{dx}-eD\left(
\phi\right) \frac{dn}{d\phi}\frac{d\phi}{dx}, \label{current2}$$
where $\frac{dn}{d\phi}=-\frac{e}{T}n.$ Thus, we obtain the differential equation for the diffusion coefficient:$$\frac{dD}{d\phi}+\frac{d\log n}{d\phi}D+\mu\left( \phi\right) =0
\label{diffusion}$$
or $$\frac{dD}{d\phi}-\frac{e}{T}D+\mu\left( \phi\right) =0$$
Let us denote $\mu\left( \phi=0\right) \equiv\mu_{0}.$ Then we can introduce the dimensionless variables:$$\widetilde{\mu}=\frac{\mu}{\mu_{0}},\text{ \ \ }\psi=\frac{e\phi}{T},\text{
\ \ }\widetilde{D}=\frac{eD}{\mu_{0}T}. \label{dimensionless}$$
We obtain the following differential equation for the dimensionless diffusion coefficient:$$\frac{d\widetilde{D}}{d\psi}-\widetilde{D}+\widetilde{\mu}\left( \psi\right)
=0 \label{dimlessdiffus}$$
Formulae (\[dimensionless\]) show that the dimensional factor in the diffusion coefficient has the standart form $D/\widetilde{D}=\mu_{0}T/e.$ However, the relation between the diffusion coefficient and the mobility is non-local, as it can be seen from the equation (\[dimlessdiffus\]). The solution of this equation with the initial condition $\widetilde{D}\left(
0\right) =$ $\widetilde{D}_{0}$ reads:$$\widetilde{D}\left( \psi\right) =\exp\left( \psi\right) \left[
\widetilde{D}_{0}-\int_{0}^{\psi}dz\widetilde{\mu}\left( z\right)
\exp\left( -z\right) \right] \label{dimlesssolution}$$
Let us consider the trivial case of constant mobility: $\mu=\mu_{0},$ and, therefore, $\widetilde{\mu}=1,$ $\ \widetilde{D}_{0}=1.$ The solution (\[dimlesssolution\]) takes the form:$$\widetilde{D}\left( \psi\right) =\exp\left( \psi\right) \left[ 1-\int
_{0}^{\psi}dz\exp\left( -z\right) \right] =1. \label{constant}$$
This calculation confirms consistency of our description.
Now we consider the case of the exponential law mobility dependence $\widetilde{\mu}=\exp\left( a\psi\right) .$$$\widetilde{D}\left( \psi\right) =\exp\left( \psi\right) -\exp\left(
\psi\right) \int_{0}^{\psi}dz\exp(\left( a-1)z\right) =\frac{a\exp\left(
\psi\right) }{a-1}-\frac{\exp\left( a\psi\right) }{a-1} \label{exp}$$
This expression tends to unity, when $a\longrightarrow0$ or $\psi
\longrightarrow0.$ The diffusion coefficient vanishes identically at $a-1.$
Let us consider now the case of linear dependence of the mobility on the potential: $\widetilde{\mu}=1+a\psi.$ Then we have:$$\widetilde{D}\left( \psi\right) =\exp\left( \psi\right) \left[ 1-\int
_{0}^{\psi}dz\exp\left( -z\right) \right] =2+\psi-\exp\left( \psi\right)
. \label{linear}$$
Thus, we see that the diffusion vs mobility relation near the equilibrium is generically non-local. The conventional Einstein relation $D=\mu T/e$ is valid in the limit of low built-in potential.
[9]{}
A.F. Volkov, Sh.M. Kogan. Uspehi Fizicheskih Nauk, V. **96**, 633 (1968).
V.Blickle, T. Speck, C. Lutz, U. Seifert, C. Bechinger. Phys. Rev. Lett. **98**, 210601 (2007).
|
---
abstract: |
Given a $(m-2)$-form $\zw$ and a volume form $\zW$ on a $m$-manifold one defines a bi-vector $\zL$ by setting $\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$ for any $1$-forms $\za,\zb$. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, $(\zL,\zL_1 )$ is always represented by a couple of $(m-2)$-forms $\zw,\zw_1$ and a volume form $\zW$. Here one shows that, for $m\zmai 5$ and odd and $(\zL,\zL_1 )$ generic, $(\zL,\zL_1 )$ is flat if and only if there exists a $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$.
Moreover, we use this result for constructing several examples of linear or Lie Poisson pairs that are generic and non-flat.
MSC: 37K10, 53D17, 53A60
address: ' Geometr[í]{}a y Topolog[í]{}a, Facultad de Ciencias, Campus de Teatinos, s/n, 29071-M[á]{}laga, Spain'
author:
- 'Francisco-Javier Turiel'
title: Flatness of generic Poisson pairs in odd dimension
---
Introduction {#sec-1}
============
Henceforth differentiable means $C^{\zinf}$ in the real case and holomorphic in the complex one. Manifolds, real or complex, and objects on them are assumed to be differentiable unless another thing is stated.
Poisson pairs or bi-Hamiltonian structures, introduced by F. Magri [@MA1] and I.Gelfand and I. Dorfman [@GD1], are a powerful tool for integrating many equations from Physics. At the same time the study of their geometric properties, regardless other aspects, gives rise to several interesting problems, global [@RI1; @RI2] and mostly local, for instance the theory of Veronese webs, notion introduced by I.Gelfand and I. Zakharevich in codimension one and later on extended to any codimension by other authors [@GZ1; @GZ2; @PA1; @TU1; @TU2; @ZA1].
Often, for applications, it is important to determine [*in a practical way*]{} whether generic Poisson pairs are flat or not (see section \[sec-2\] for definitions.)
In even dimension the problem is solved since a local classification is known [@TUa], and flatness is equivalent to say that the Poisson pair defines a $G$-structure. On the contrary in odd dimension no general and practical criterion of flatness is known, except for dimension three where a simple explicit obstruction has been constructed by A. Izosimov [@IZ1]. In this work one gives a such criterion by making use of the differential forms.
More exactly, given a $(m-2)$-form $\zw$ and a volume form $\zW$ on a $m$-manifold one defines a bi-vector $\zL$ by setting $\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$ for any $1$-forms $\za,\zb$. In this way, locally, a Poisson pair $(\zL,\zL_1 )$ is always represented by a couple of $(m-2)$-forms $\zw,\zw_1$ and a volume form $\zW$. Here one shows that, for $m\zmai 5$ and odd and $(\zL,\zL_1 )$ generic, $(\zL,\zL_1 )$ is flat if and only if there exists a $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$.
In a second time we apply this result to generic, linear or Lie, Poisson pairs (see section \[sec-5\] for definitions.) It is well known the difficulty for constructing this kind of Poisson pairs (for a non-trivial example of linear Poisson pair in dimension five see [@KO1].) Our result allows to construct, in any dimension $\zmai 5$, examples of these pairs which are non flat; for instance on the dual space of some Lie algebra of truncated polynomial fields in one variable (examples \[eje-1\] and \[eje-8\].)
Moreover one shows that if the dual space of a non-unimodular Lie algebra of dimension $\zmai 5$ supports a generic linear Poisson pair, then it supports a non-flat generic linear Poisson pair too. This result applies to some semi-direct product of the affine algebra and an ideal of dimension one (example \[eje-7\].)
On the other hand from a Lie algebra one constructs a second one, called secondary, whose dual space support a generic non-flat linear Poisson pair provided that some minor conditions hold; just it is the case of the special affine algebra (example \[eje-6\].)
In section \[sec-8\] many examples of generic non-flat Lie Poisson pairs are given, one of them containing the affine algebra as subalgebra (see remark \[rem-6\].)
From our examples follows that for the linear Lie algebra, and therefore for any Lie algebra ${\mathcal A}$, there exist two other Lie algebras ${\mathcal B}$ and ${\mathcal B'}$, which contain ${\mathcal A}$ as subalgebra, such that ${\mathcal B}^*$ support a generic non-flat linear Poisson pair and ${\mathcal B'}^*$ a generic non-flat Lie Poisson pair. Thus a natural question arises: to determine the minimal dimension of such Lie algebras and describe the structural relation among ${\mathcal A}$, ${\mathcal B}$ and ${\mathcal B'}$.
Finally, sections \[sec-9\] and \[sec-10\] include a completely description of generic and non-flat linear Poisson pair or Lie Poisson pair, respectively, in dimension three.
Preliminaries {#sec-2}
=============
Let $V$ be a real or complex vector space of dimension $m$ and ${\mathbb K}={\mathbb R},{\mathbb C}$. Given a volume form $\zW$ on $V$, to each $\zw\zpe\zL^{m-2} V^*$ one may associated a bi-vector $\zL$ through the formula $$\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$$ where $\za,\zb\zpe V^*$ and the quotient means the scalar $a$ such that $\za\zex\zb\zex\zw =a {\zW}$; in this way an isomorphism between $\zL^{m-2} V^*$ and $\zL^{2} V$ is defined. Thus a bi-vector $\zL$ on $V$ can be represented by a couple of forms $(\zw,\zW)$ where $\zw\zpe\zL^{m-2} V^*$ and $\zW\zpe\zL^{m} V^*-\{0\}$. Of course $(\zw,\zW)$ and $(\zw',\zW')$ represent the same bi-vector if and only if $\zw'=b\zw$ and $\zW'=b\zW$ for some $b\zpe {\mathbb K}-\{0\}$.
Note that for any $\za\zpe V^*$ its $\zL$-Hamiltonian $\zL(\za,\quad)$ is just the vector $v_{\za}$ such that $i_{v_{\za}}\zW=-\za\zex\zw$.
\[lem-1\] Consider a bi-vector $\zL$ represented by $(\zw,\zW)$. If $\zW=ae_{1}^{*}\zex\zps\zex e_{m}^{*}$ and $\zL=\zsu_{_1\zmei i<j\zmei m}a_{ij}e_i \zex e_j$, where $\{e_{1},\zps,e_{m}\}$ is a basis of $V$, then $$\zw=\zsu_{_1\zmei i<j\zmei m}(-1)^{i+j-1}aa_{ij}e_{1}^{*}\zex\zps\zex
{\widehat e_{i}^{*}}\zps\zex{\widehat e_{j}^{*}}\zex\zps e_{m}^{*}$$ (as usual terms under hat are deleted.)
Recall that a bi-vector can be described by a $r$-form, whose kernel equals the image of the bi-vector, and a $2$-form , whose restriction to the kernel of the $r$-form is symplectic; indeed, identify the bi-vector to the dual bi-vector of the restriction of the $2$-form.
\[lem-2\] On $V$ consider $1$-forms $\za_1 ,\zps,\za_ r$ and a $2$-form $\zb$ such that $\za_1 \zex\zps\zex\za_ r \zex\zb^k \znoi 0$ where $m=2k+r$ . Then $\za_1 \zex\zps\zex\za_ r$, $\zb$ describe a bi-vector that is represented, as well, by $(k\za_1 \zex\zps\zex\za_ r \zex\zb^{k-1},\za_1 \zex\zps\zex\za_ r \zex\zb^{k})$.
Consider a couple of bi-vectors $(\zL,\zL_1 )$ on $V$. By definition [*the rank of*]{} $(\zL,\zL_1 )$ is the maximum of ranks of $(1-t)\zL+t\zL_1$, $t\zpe\mathbb K$. Note that $rank(\zL,\zL_1 )$ equals $rank((1-t)\zL+t\zL_1 )$ for any $t\zpe\mathbb K$ except a finite number of scalars. Therefore, by considering $\zL'=(1-a)\zL+a\zL_1 $ and $\zL'_{1}=(1-a_{1})\zL+a_{1}\zL'_{1}$ for suitable $a\znoi a_1$, one may assume $rank\zL=rank\zL_1 =rank(\zL,\zL_1 )$ if necessary.
The classification of couples $(\zL,\zL_1 )$ is due to Gelfand and Zakharevich [@GZ2; @TU2]; just pointing out that $(\zL,\zL_1 )$ is the product of $m-rank (\zL,\zL_1 )$ Kronecker blocks and, perhaps, a symplectic factor \[this last one only if there exists $b\zpe\mathbb C$ such that $rank((1-b)\zL+b\zL_1 )<rank(\zL,\zL_1 )$ or $rank(\zL-\zL_1 )<rank(\zL,\zL_1 )$.\]
Let $M$ be a real or complex manifold of dimension $m$ and $(\zL,\zL_1 )$ a couple of bi-vectors on it. One will say that [*at a point*]{} $p$ the couple $(\zL,\zL_1 )$ is:
1. [*flat*]{} if, in some coordinates around $p$, $\zL$ and $\zL_1$ can be simultaneously written with constant coefficients.
2. [*a $G$-structure*]{} if there exists an open neighborhood $A$ of $p$ such that, for any $q\zpe A$, the algebraic couples $(\zL,\zL_1 )(q)$ on $T_q M$ and $(\zL,\zL_1 )(p)$ on $T_p M$ are isomorphic.
3. [*a Poisson pair or a bi-Hamiltonian structure*]{} if, on some open neighborhood of $p$, $\zL$, $\zL_1$ and $\zL+\zL_1$ are Poisson (structures); in this case $a\zL+b\zL_1$ is Poisson for any $a,b\zpe\mathbb K$.
When the properties above hold at every point of $M$, $(\zL,\zL_1 )$ is called flat, a $G$-structure or a Poisson pair respectively.
Note that (1) implies (2) and (3).
One will say that $(\zw,\zw_1 ,\zW)$ [*represents*]{} $(\zL,\zL_1 )$, when $(\zw,\zW)(q)$ and $(\zw_1 ,\zW)(q)$ represent $\zL(q)$ and $\zL_1 (q)$ respectively for every $q\zpe M$. If $(\zw',\zw'_1 ,\zW')$ is another representative of $(\zL,\zL_1 )$ then $(\zw',\zw'_1 ,\zW')=f(\zw,\zw_1 ,\zW)$, that is $\zw'=f\zw$, $\zw'_{1}=f\zw_1$ and $\zW'=f\zW$, where $f$ is a function without zeros. The existence of a representative of $(\zL,\zL_1 )$ on $M$ only depends on the existence of a volume form. Since our problem is local, we may suppose that it is the case without loss of generality.
\[pro-1\] Let $(\zL,\zL_1 )$ be a Poisson pair on M represented by $(\zw,\zw_1 ,\zW)$. If $(\zL,\zL_1 )$ is flat at $p$ then, about this point, there is a closed $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_{1}=\zl\zex\zw_{1}$.
Moreover if a Poisson structure ${\tilde\zL}$ represented by $({\tilde\zw},{\tilde\zW})$ is flat on a neighborhood of $p$, then around this point there exists a function ${\tilde g}$ with no zeros such that ${\tilde g}{\tilde\zw}$ is closed.
By flatness, and always around $p$, there exists a representative $(\zw',\zw'_1 ,\zW')$ of $(\zL,\zL_1 )$ such that $d\zw'=d\zw'_{1}=0$. On the other hand $(\zw,\zw_1 ,\zW)=h(\zw',\zw'_1 ,\zW')$ for some function $h$ with no zeros. Moreover one can assume $h=\zmm e^{g}$. Then $d\zw=dg\zex\zw$ and $d\zw_1 =dg\zex\zw_1$.
The second part of proposition \[pro-1\] is obvious
The proposition above provides a necessary condition for flatness that in some cases, as one shows in the next two sections, is sufficient too.
The generic case in odd dimension {#sec-3}
=================================
In this section $m=2n-1\zmai 3$, so $n\zmai 2$. Let $(\zL,\zL_1 )$ a couple of Poisson structures. Suppose $(\zL,\zL_1 )$ [*generic*]{} (at each point.) This is equivalent to assume that $(a\zL+b\zL_1 )^{n-1}$ has no zeros for any $(a,b){\zpe\mathbb C}^{2}-\{0\}$; so its algebraic model has just a Kronecker block and no symplectic factor, and $(\zL,\zL_1 )$ defines a $G$-structure. Moreover if a $1$-form $\zt$ is Casimir for both $\zL$ and $\zL_1$, that is $\zL(\zt,\quad)=\zL_{1}(\zt,\quad)=0$, then $\zt=0$. Thus if $(\zw,\zw_1 ,\zW)$ represents $(\zL,\zL_1 )$ and $\zt\zex\zw=\zt\zex\zw_1 =0$ then $\zt=0$.
Therefore if there exists $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$, it is unique. Besides, if $(\zw',\zw'_1 ,\zW')$ represents $(\zL,\zL_1 )$ as well, then we can assume $(\zw',\zw'_1 ,\zW')=\zmm e^{g}(\zw,\zw_1 ,\zW)$. So $d\zw'=(\zl+dg)\zex\zw'$ and $d\zw'_1 =(\zl+dg)\zex\zw'_1$. This shows that the existence of this kind of $1$-form does not depend on the representative while its exterior derivative is intrinsic. Consequently $d\zl$ can be constructed even if representatives are local only.
In general there is no reason for the existence of a such $\zl$ except in dimension three. More exactly:
\[pro-2\] Let $(\zL,\zL_1 )$ be a couple of Poisson structures and $(\zw,\zw_1 ,\zW)$ a representative. Assume $m=3$ and $(\zL,\zL_1 )$ generic. Then there exists a $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1=\zl\zex\zw_1$ if and only if $(\zL,\zL_1 )$ is a Poisson pair; in this case $Im\zL\zin Im\zL_1 \zco Kerd\zl$.
Moreover $d\zl=0$ if and only if $(\zL,\zL_1 )$ is flat.
In dimension three a bi-vector $\zL'$ represented by $(\zw',\zW')$ is Poisson if and only if $\zw'$ is completely integrable, that is $\zw'\zex d\zw'=0$. Besides if $\zL'(q)\znoi 0$ then $Im\zL'(q)=Ker\zw'(q)$.
In our case as $\zw\zex d\zw=0$ and the problem is local, there exists a function $\zf$ without zeros such that $d(\zf\zw)=0$. Thus changing of representative allows us to suppose $\zw$ closed.
Since $Im\zL$ and $Im\zL_1$ are in general position about any point, one may choose coordinates $(x_1 ,x_2 ,x_3 )$ such that $\zw=dx_1$ and $\zw_1 =e^{h}dx_2$. Moreover by modifying the third coordinate if necessary one can suppose $\zW=dx_1 \zex dx_2 \zex dx_3$. The couple $(\zL,\zL_1 )$ is a Poisson pair if and only if $\zL+\zL_1 $ is Poisson, that is if and only if $(\zw+\zw_1 )\zex d(\zw+\zw_1 )=0$ (note that $(\zw+\zw_1, \zW)$ represents $\zL+\zL_1 $,) which is equivalent to say that $h=h(x_1, x_2 )$.
On the other hand as $d\zw=0$ the $1$-form $\zl$, if any, has to be equal $gdx_1$, which implies that $d\zw_1=e^{h}dh\zex dx_2 =ge^{h}dx_1 \zex dx_2$. In other words a such $\zl$ exists just when $h=h(x_1, x_2 )$; in this case $\zl=(\zpar h/\zpar x_1 )dx_1$. That proves the first part of proposition \[pro-2\].
Now suppose $d\zl=0$; then $\zpar^{2}h/\zpar x_2 \zpar x_1 =0$ and $h=h_1 (x_1 )+h_2 (x_2 )$ so $e^{-h_1}\zw$ and $e^{-h_1}\zw_1$ are closed. Therefore changing of representative allows to suppose $\zw$, $\zw_1$ closed. Consequently there exist coordinates $(y_1 ,y_2 ,y_3 )$ such that $\zw=dy_1$ and $\zw_1 =dy_2$. By modifying the third coordinate if necessary we may suppose $\zW=dy_1 \zex dy_2 \zex dy_3$; now is obvious that $(\zL,\zL_1 )$ is flat. The converse follows from proposition \[pro-1\].
\[rem-1\] [By proposition \[pro-2\] in dimension three $d\zl$ is an invariant of the Poisson pair $(\zL,\zL_1 )$, which vanishes just when $(\zL,\zL_1 )$ is flat. A straightforward computation shows that, up to multiplicative constant, this invariant equals the curvature form introduced by Izosimov [@IZ1]]{}
For odd dimension greater than or equal to five one has:
\[teo-1\] Consider a Poisson pair $(\zL,\zL_1 )$ on a manifold $M$ of dimension $m=2n-1\zmai 5$ represented by $(\zw,\zw_1 ,\zW)$. Assume that $(\zL,\zL_1 )$ is generic everywhere. Then $(\zL,\zL_1 )$ is flat if and only if there exists a $1$-form $\zl$ such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$.
The existence of $\zl$ when $(\zL,\zL_1 )$ is flat follows from proposition \[pro-1\]. Conversely suppose that $\zl$ exists; for proving the flatness, which is a local question, it will be enough to show that the Veronese web associated is flat [@GZ1; @GZ2; @TU1; @TU2].
One starts proving that $d\zl=0$. For every $a\zpe \mathbb K$ the Poisson structure $\zL+a\zL_1$ is represented by $(\zw+a\zw_1 ,\zW)$ so, by proposition \[pro-1\], (locally) there is a function $g_a$ with no zeros such that $g_a (\zw+a\zw_1 )$ is closed. Therefore $0=(\zl+(dg_a )/g_a )\zex g_a (\zw+a\zw_1 )$ and $\zl+(dg_a )/g_a $ has to be a Casimir of $\zL+a\zL_1$.
This implies that $d\zl=d(\zl+(dg_a )/g_a )$ is divisible by any ($1$-form) Casimir of $\zL+a\zL_1$. Thus $d\zl$ is divisible by every Casimir of $\zL+a\zL_1$, $a\zpe \mathbb K$. But at each point all these Casimirs span a vector space of dimension$\zmai 3$. Since $d\zl$ is a $2$-form necessarily vanishes.
As it is known, given any different and non-vanishing real numbers $a_1 ,\zps,a_n$, around each point $p\zpe M$, there exist coordinates $(x,y)=(x_1 ,\zps,x_n ,y_1 ,\zps,y_{n-1})$ and functions $f_1 ,\zps,f_n$ only depending on $x$ such that $\zL$ is given by $(a_1 \zpu\zpu\zpu a_n)\zsu_{j=1}^{n}a_{j}^{-1}f_j dx_j$ and $\zsu_{j=1}^{n-1}dx_j \zex dy_j$ while $\zL_1$ is given by $\za=\zsu_{j=1}^{n}f_j dx_j$ and $\zsu_{j=1}^{n-1}a_j dx_j \zex dy_j$, where $f_1 ,\zps,f_n$ have no zeros, $d\za=0$ and $\za\zex d(\za\zci J)=0$ when $\za$ is regarded as a $1$-form in variables $x=(x_1 ,\zps,x_n )$ and $J=\zsu_{j=1}^{n}a_{j}(\zpar/\zpar x_{j})\zte dx_j$; that is $\za\zci J=\zsu_{j=1}^{n}a_{j}f_j dx_j$ and $\za\zci J^{-1}=\zsu_{j=1}^{n}a_{j}^{-1}f_j dx_j$ (see page 893 of [@TU1] and section 3 of [@TU2]). For simplifying computations we choose $a_1 ,\zps,a_n$ in such a way that $a_1 \zpu\zpu\zpu a_n =1$.
By lemma \[lem-2\] the Poisson structure $\zL$ is represented by $$\zw'=(n-1)(\za\zci J^{-1})\zex(\zsu_{j=1}^{n-1}dx_j \zex dy_j )^{n-2}$$ $$\zW'=(\za\zci J^{-1})\zex(\zsu_{j=1}^{n-1}dx_j \zex dy_j )^{n-1}$$ and $\zL_1$ by $$\zw'_1 =(n-1)\za\zex(\zsu_{j=1}^{n-1}a_j dx_j \zex dy_j )^{n-2}$$ $$\zW'_1 =\za\zex(\zsu_{j=1}^{n-1}a_j dx_j \zex dy_j )^{n-1}\, .$$
Note that $\zW'=\zW'_1$ because $a_1 \zpu\zpu\zpu a_n =1$. On the other hand $d\zw'_1 =0$; therefore for the representative $(\zw',\zw'_1 ,\zW')$ there exists a $1$-form $\zl'=\zf\za$ such that $d\zw'=\zl'\zex\zw'$. But $d\zl'=d\zl=0$ so $\zl'=dg$ for some function $g=g(x)$ such that $\za\zex dg=0$, and $e^{-g}\zw'$, $e^{-g}\zw'_1$ will be closed.
Observe that ${\tilde\za}=e^{-g}\za$ is closed, $e^{-g}f_1 ,\zps,e^{-g}f_n$ have no zeros, ${\tilde\za}\zex d({\tilde\za}\zci J)=0$ while $\zL$, $\zL_1$ are given by ${\tilde\za}\zci J^{-1}$, $\zsu_{j=1}^{n-1}dx_j \zex dy_j$ and ${\tilde\za}$, $\zsu_{j=1}^{n-1}a_j dx_j \zex dy_j$ respectively. In other words, considering ${\tilde\za}$ instead $\za$ and calling it $\za$ again allows to assume, without loss of generality, $\zl'=0$ and $\zw'$ closed.
As $Ker(\za\zci J^{-1})=Im\zL$ is involutive $d(\za\zci J^{-1})=\zt\zex(\za\zci J^{-1})$ for some $1$-form $\zt$. Hence $0=d\zw'=\zt\zex\zw'$ since $(\zsu_{j=1}^{n-1}dx_j \zex dy_j )^{n-2}$ is closed. Therefore $\zt$ is a Casimir of $\zL$ and $\zt=h\za\zci J^{-1}$ for some function $h$. But in this case $\zt\zex(\za\zci J^{-1})=0$ and $\za\zci J^{-1}$ will be closed. In other words $$0=d(\za\zci J^{-1})= \zsu_{1\zmei k<j\zmei n}\zcizq
a_{j}^{-1}(\zpar f_{j}/\zpar x_{k})
-a_{k}^{-1}(\zpar f_{k}/\zpar x_{j})\zcder dx_k \zex dx_j$$ so $a_{j}^{-1}(\zpar f_{j}/\zpar x_{k})-a_{k}^{-1}(\zpar f_{k}/\zpar x_{j})=0$. Since $(\zpar f_{j}/\zpar x_{k})=(\zpar f_{k}/\zpar x_{j})$, necessarily $(\zpar f_{j}/\zpar x_{k})=0$ if $j\znoi k$. Thus $f_j =f_j (x_j )$, $j=1,\zps,n$.
Recall that the web associated to $(\zL,\zL_1 )$ is completely determined by $J$ and $\za$ regarded on the (local) quotient manifold $N$ of $M$ by the foliation $\zing_{t\zpe\mathbb K}Im(\zL+t\zL_1 )$ (see [@TU1; @TU2] again;) of course $(x_1 ,\zps,x_n )$ can be regarded too as coordinates on $N$ around $q$, where $q$ is the projection on $N$ of point $p$. Now consider functions ${\tilde x}_1 ,\zps,{\tilde x}_n$ such that $d{\tilde x}_j =f_j (x_j )dx_j$, $j=1,\zps,n$; then $({\tilde x}_1 ,\zps,{\tilde x}_n )$ are coordinates on $N$ about $q$, $\za=d{\tilde x}_1 +\zps+d{\tilde x}_n$ and $J=\zsu_{j=1}^{n}a_j (\zpar/\zpar {\tilde x}_{j})\zte d{\tilde x}_j$, which shows the flatness of the web associated to $(\zL,\zL_1 )$.
\[eje-A\]
On ${\mathbb K}^5$ with coordinates $x=(x_1 ,x_2 ,x_3 ,x_4 ,x_5 )$ consider the bi-vectors $$\zL=\zpizq x_2 {\frac {\zpar} {\zpar x_1}}+x_3 {\frac {\zpar} {\zpar x_3 }}\zpder
\zex{\frac {\zpar} {\zpar x_4}}
+\zpizq x_1 {\frac {\zpar} {\zpar x_1}}+x_2 {\frac {\zpar} {\zpar x_2}}
+x_3 {\frac {\zpar} {\zpar x_3 }}\zpder\zex{\frac {\zpar} {\zpar x_5}}$$
$$\zL_1 =x_1 {\frac {\zpar} {\zpar x_2 }}\zex{\frac {\zpar} {\zpar x_4}}
+\zpizq x_1 {\frac {\zpar} {\zpar x_1}}+x_2 {\frac {\zpar} {\zpar x_2}}
\zpder\zex{\frac {\zpar} {\zpar x_5}}\, .$$ .3truecm
First note that $\zL+t\zL_1 =X_t \zex (\zpar /\zpar x_ 4)+Y_t \zex (\zpar /\zpar x_ 5)$ where $X_t$, $Y_t$, $(\zpar /\zpar x_ 4)$ and $(\zpar /\zpar x_ 5)$ commute among them, which implies that every $\zL+t\zL_1$ is a Poisson structure and $(\zL,\zL_1 )$ a Poisson pair (in fact a Lie Poisson pair, see section \[sec-5\].)
Let $A$ be the complement of the union of hyperplanes $x_k =0$, $k=1,2,3$, and in addition to these $x_2 =(a_j +1)x_1$, $j=1,2$, where $a_1 ,a_2$ are the roots of $t^2 +t+1=0$, if ${\mathbb K}={\mathbb C}$. Then $(\zL,\zL_1 )$ is generic on $A$.
Set $\zW=dx_1 \zex dx_2 \zex dx_3 \zex dx_4 \zex dx_5$. Then $\zL$ is represented by $$\zw=(-x_3 dx_1 \zex dx_2 +x_2 dx_1 \zex dx_3 -x_1 dx_2 \zex dx_3)\zex dx_4
\hskip 2truecm$$ $$\hskip 2truecm+(x_3 dx_1 \zex dx_2 +x_2 dx_2 \zex dx_3 )\zex dx_5$$ and $\zL_1$ by $\zw_1 =(x_2 dx_1 -x_1 dx_2 )\zex dx_3 \zex dx_4 -x_1 dx_1 \zex dx_3 \zex dx_5\, .$
Moreover $$d\zw=-3dx_1 \zex dx_2 \zex dx_3 \zex dx_4+dx_1 \zex dx_2 \zex dx_3 \zex dx_5$$ and $$d\zw_1 =-2dx_1 \zex dx_2 \zex dx_3 \zex dx_4
=(2x_{1}^{-1}dx_1 )\zex\zw_1\, .$$
As $dx_3$ is a Casimir of $\zL_1$, if $d\zw_1 =\zl\zex\zw_1$ then $\zl=2x_{1}^{-1}dx_1 +fdx_3$.
On the other hand if we assume $d\zw=\zl\zex\zw$ then $\zL(\zl,\quad)=(\zpar /\zpar x_ 4)+3(\zpar /\zpar x_ 5)$ since the contraction of $-\zL(\zl,\quad)$ and $\zW$ equals $\zl\zex\zw=d\zw$. But at the same time $$\zL(\zl,\quad)={\frac {2x_2} {x_1}}{\frac {\zpar} {\zpar x_4 }}
+2{\frac {\zpar} {\zpar x_5}}+x_3 f\zpizq {\frac {\zpar} {\zpar x_4 }}+
{\frac {\zpar} {\zpar x_5}}\zpder$$ which implies that $(2x_{1}^{-1}x_2 +x_3 f)$ and $x_3 f$ have to be constant, [*contradiction*]{}. In short $(\zL,\zL_1 )$ is not flat at any point of $A$.
A practical criterion for local flatness is the following:
\[lem-3\] Let $M$, $(\zL,\zL_1 )$ and $(\zw,\zw_1 ,\zW)$ be as in theorem \[teo-1\]. On some neighborhood of a point $p$ of $M$ consider a vector field $X$ tangent to $Kerd\zw$ and a $1$-form $\za$ Casimir of $\zL_1$, both of them non-vanishing at $p$. Assume $d\zw(p)\znoi 0$ and $d\zw_1 =0$ on this neighborhood. Then $(\zL,\zL_1 )$ is flat at $p$ if and only if $\zL(\za,\quad)$, about $p$, is functionally proportional to $X$.
In this proof objects will be considered on a suitable neighborhood of $p$. First observe that $\zL(\za,\quad)$ does not vanishes anywhere because no common Casimir of $\zL$ and $\zL_1$ other that zero exists. Recall that $i_{\zL(\za,\quad)}\zW=-\za\zex\zw$. Thus there is a function $f$ such that $d\zw=f\za\zex\zw$ if and only if $\zL(\za,\quad)$ and $X$ are functionally proportional. When $\zL(\za,\quad)$ and $X$ are proportional it suffices setting $\zl=f\za$.
Conversely if $\zl$ exists necessarily $\zl=f\za$ since $d\zw_1 =0$, so $d\zw=\zl\zex\zw=f\za\zex\zw$ and $\zL(\za,\quad)$ and $X$ are proportional.
Other cases {#sec-4}
===========
The foregoing results extend to any analytic Poisson pair $(\zL,\zL_1 )$ on a $m$-manifold $M$ provided that $rank(\zL,\zL_1 )=m,m-1$. Indeed, if $m$ is even and $rank(\zL,\zL_1 )=m$, then $(\zL,\zL_1 )$ is flat at a point $p\zpe M$ if and only if it defines a $G$-structure. This is a straightforward consequence of the classification of pairs of compatible symplectic forms [@TUa] since, up to linear combination, one may suppose symplectic $\zL$ and $\zL_1$; in this case analyticity is not need. On the other hand:
\[teo-2\] On a real analytic or complex manifold $M$, of odd dimension $m$, consider a point $\zpe M$ and a Poisson pair $(\zL,\zL_1 )$, represented by $(\zw,\zw_1 ,\zW)$, whose rank equals $m-1$. Then $(\zL,\zL_1 )$is flat at $p$ if and only if $(\zL,\zL_1 )$ defines a $G$-structure at $p$ and there is a closed $1$-form $\zl$, about $p$, such that $d\zw=\zl\zex\zw$ and $d\zw_1 =\zl\zex\zw_1$.
As the problem is local, up to linear combination, one may suppose $rank\zL=rank\zL_1 =rank(\zL,\zL_1 )=m-1$. Obviously the conditions of the theorem are necessary (see proposition \[pro-1\].) Conversely, since $(\zL,\zL_1 )$ defines a $G$-structure at $p$, from the product theorem for Poisson pairs [@TU3] follows that, always about $p$, $(M,\zL,\zL_1 )$ splits into a product of two Poisson pairs $(M',\zL',\zL'_1 )\zpor (M'',\zL'',\zL''_1 )$, $p=(p',p'')$, the first one Kronecker with a single block, so generic, and symplectic the second one.
Moreover $(\zL'',\zL''_1 )$ defines a $G$-structure at $p''$ because $(\zL,\zL_1 )$ does at $p$; therefore it is flat at $p''$ and one can choose a representative $(\zw'',\zw''_1 ,\zW'')$ with $d\zw''=d\zw''_1 =0$.
If $dimM'=1$ the proof is finished since $\zL'=\zL'_1 =0$; therefore assume $dimM'\zmai 3$. Let $(\zw',\zw'_1 ,\zW')$ be a representative of $(\zL',\zL'_1 )$. Then with the obvious identification (by means of the pull-back by the canonical projections forms on $M'$ or $M''$ can be regarded as forms on $M$) $$(\zw'\zex\zW''+\zW'\zex\zw'',\zw'_1 \zex\zW''+\zW'\zex\zw''_1 ,\zW'\zex\zW'')$$ represents $(\zL,\zL_1 )$.
Note that the existence of a closed form as in theorem \[teo-2\] happens for any representative because all of them are functionally proportional. Let us denote by ${\tilde\zl}$ that corresponding to the representative above and $\zl'$ its restriction to $M'\zpor \{p''\}$, identified to $M'$ in the obvious way. Then $\zl'$ is closed, $d\zw'=\zl'\zex\zw'$ and $d\zw'_1 =\zl'\zex\zw'_1$. Therefore by proposition \[pro-2\] and theorem \[teo-1\] $(\zL',\zL'_1 )$ is flat at $p'$, which allows to conclude the flatness of $(\zL,\zL_1 )$ at $p$.
\[rem-2\]
In the real case the analyticity is needed for applying the product theorem; nevertheless an unpublished results by the author states that if $(\zL,\zL_1 )$ defines a $G$-structure and $rank(\zL,\zL_1 )=dimM-1$ then the product theorem holds in the $C^{\zinf}$-category too.
If $dimM'=3$ then there always exists $\zl$ because this fact essentially depends on $(M',\zL',\zL'_1 )$. By the same reason when $dimM'\zmai 5$ the condition $d\zl=0$ is unnecessary.
Pairs on the dual space of a Lie algebra {#sec-5}
========================================
The remainder of this work is essentially devoted to the generic case in odd dimension when $\zL$ is linear and $\zL_1$ linear or constant.
\[lem-4\] Consider a couple of analytic bi-vectors $(\zL,\zL_1 )$ on a connected non-empty open set $A\zco{\mathbb K}^m$, $m=2n-1\zmai 3$. If $(\zL,\zL_1 )(p)$ is generic for some $p\zpe A$, then the set of points $q\zpe A$ such that $(\zL,\zL_1 )(q)$ is generic is open and dense.
Fixed $s\zpe \mathbb K$ the equation $(\zL+s\zL_1 )(\zl,\quad)=0$ is an homogeneous linear system, on $({\mathbb K}^m )^*$, with analytic coefficients (like functions on $A$.) Since $(\zL+s\zL_1 )^{n-1}(p)\znoi 0$, at each point near $p$ the vector subspace of its solutions has dimension one. Thus around $p$ there is a solution $\zl_{s}=\zsu_{j=1}^{m}f_j dx_j$, with $\zl_{s}(p)\znoi 0$, where every $f_j$ is a rational function of the coefficients functions of $\zL$ and $\zL_1$. Multiplying by a suitable polynomial allows to assume, without loss of generality, that $f_1 ,\zps,f_n$ are polynomials in the coefficients functions of $\zL$ and $\zL_1$. Therefore $\zl_s$ is defined on $A$ and by analyticity $(\zL+s\zL_1 )(\zl_s ,\quad)=0$ everywhere.
Consider $n$ different scalars $s_1 ,\zps,s_n$. Then $(\zl_{s_1} \zex\zps\zex\zl_{s_n} )(p)\znoi 0$ and by analyticity $\zl_{s_1} \zex\zps\zex\zl_{s_n}$ does not vanish at any point of a dense open set $A'$.
By a similar reason each $(\zL+s_j \zL_1 )^{n-1}$ does not vanish at any point of a dense open set $A_j$. Take any $q\zpe A'\zin A_1 \zin\zps\zin A_n$; then $(\zL+s_j \zL_1 )^{n-1}(q)\znoi 0$, $j=1,\zps,n$, and $(\zl_{s_1} \zex\zps\zex\zl_{s_n} )(q)\znoi 0$, which only is possible if $(\zL,\zL_1 )(q)$ is generic.
\[rem-3\] [It is easily seen that lemma \[lem-4\] holds in analytic connected manifolds too.]{}
Let ${\mathcal A}$ be a Lie algebra of finite dimension and $G$ a connected Lie group of algebra ${\mathcal A}$. Then any element of $\zL^{r}{\mathcal A}$, respectively $\zL^{r}{\mathcal A}^*$, can be regarded like a left invariant $r$-field, respectively $r$-form, on $G$. Thus we may consider the Lie and exterior derivatives on ${\mathcal A}$ and from the formulas on $G$ deduce the corresponding formulas on ${\mathcal A}$. Here we adopt the differentiable point of view or the algebraic one depending on the convenience for working with. Recall that closed means cocycle and exact cobord. One say that $\zr\zpe {\mathcal A}^*$ is a [*contact form*]{} if $dim {\mathcal A}=2k+1$ and $\zr\zex(d\zr)^k$ is a volume form, while $\zb\zpe\zL^{2} {\mathcal A}^*$ is called [*symplectic*]{} when $d\zb=0$ and $rank\zb=dim {\mathcal A}$.
Remember that on ${\mathcal A}^*$ one defines the [*Lie-Poisson structure*]{} $\zL$ by considering the elements of ${\mathcal A}$ as linear functions and setting $\zL(a,b)=[a,b]$. In the coordinates associated to any basis of ${\mathcal A}^*$ the coefficients of $\zL$ are linear functions. Conversely any Poisson structure on a vector space with linear coefficients is obtained in this way.
On the other hand a $2$-form $\zb\zpe\zL^{2} {\mathcal A}^*$ can be seen as a Poisson structure $\zL$ on ${\mathcal A}^*$ with constant coefficients. Remember that $(\zL,\zL_1 )$ is compatible if and only if $d\zb=0$. In this case $(\zL,\zL_1 )$ will be named [*a linear Poisson pair or a linear bi-Hamiltonian structure*]{}.
Other option consists in considering a second Lie algebra structure $[\quad,\quad]_1$ on ${\mathcal A}$ and its associated Lie-Poisson structure $\zL_1$ on ${\mathcal A}^*$. Then $(\zL,\zL_1 )$ is compatible if and only if $[\quad,\quad]+[\quad,\quad]_1$ defines a Lie algebra structure. In this case $(\zL,\zL_1 )$ will be called [*a Lie Poisson pair or a Lie bi-Hamiltonian structure*]{}.
In both cases, because lemma \[lem-4\], when $dim{\mathcal A}=2n-1\zmai 3$ one will say that $(\zL,\zL_1 )$ is [*generic*]{} if it is generic at some point (so almost everywhere), and [*flat*]{} if it is flat at every generic point.
Set $I_0 =\{a\zpe{\mathcal A}\zbv tr[a,\quad]=0\} $ where $tr[a,\quad]$ is the trace of the adjoint endomorphism $[a,\quad]\zdp b\zpe{\mathcal A}\zfl [a,b]\zpe{\mathcal A}$. Note that $I_0$ is an ideal, which contains the derived ideal of ${\mathcal A}$, and that we will call the [*unimodular ideal of ${\mathcal A}$*]{}. Its codimension equals zero when ${\mathcal A}$ is unimodular and one otherwise. As Lie algebra $I_0$ is unimodular.
\[lem-5\] Let $\{e_1 ,\zps,e_m \}$ be a basis of a Lie algebra ${\mathcal A}$ and $(x_1 ,\zps,x_m )$ the coordinates on ${\mathcal A}^*$ associated to the dual basis. Set $\zW=dx_1 \zex\zps\zex dx_m$ and $X=\zsu_{j=1}^{m}(tr[e_j ,\quad ])(\zpar/\zpar x_j )$. Assume that $(\zw,\zW)$ represents the Lie-Poisson structure $\zL$ of ${\mathcal A}^*$. Then $d\zw=i_X \zW$.
Therefore $d\zw=0$ if and only if ${\mathcal A}$ is unimodular.
Suppose that $[e_i ,e_j ]=\zsu_{k=1}^{m}a_{ij}^{k}e_k$; then $$\zL=\zsu_{1\zmei i<j\zmei m}\zpizq\zsu_{k=1}^{m}a_{ij}^{k}x_k \zpder
{\frac {\zpar} {\zpar x_i}}\zex {\frac {\zpar} {\zpar x_j}}$$ and by lemma \[lem-1\] $$\zw=\zsu_{1\zmei i<j\zmei m}(-1)^{i+j-1}
\zpizq\zsu_{k=1}^{m}a_{ij}^{k}x_k \zpder
dx_1 \zex\zps\zex{\widehat dx_{i}}\zex\zps\zex{\widehat dx_{j}}\zex\zps\zex dx_m .$$
Finally a straightforward computation yields $$d\zw=\zsu_{j=1}^{m}(-1)^{j}\zpizq\zsu_{i=1}^{m}a_{ij}^{i}\zpder
dx_1 \zex\zps\zex{\widehat dx_{j}}\zex\zps\zex dx_m$$ $$=\zsu_{j=1}^{m}(-1)^{j-1}(tr[e_j ,\quad])
dx_1 \zex\zps\zex{\widehat dx_{j}}\zex\zps\zex dx_m.$$
\[rem-4\]
Assume $m\zmai 3$ and odd. First consider a constant Poisson structure $\zL_1$ on ${\mathcal A^*}$ such that $(\zL, \zL_1 )$ is compatible and generic. If ${\mathcal A}$ is unimodular, as $\zL_1$ is represented by a constant $(m-2)$-form $\zw_1$, by proposition \[pro-2\] and theorem \[teo-1\] the pair $(\zL, \zL_1 )$ is flat.
Now suppose $\zL_1$ given by a second Lie algebra structure $[\quad,\quad]_1$ and $(\zL, \zL_1 )$ compatible and generic. Again by proposition \[pro-2\] and theorem \[teo-1\], when $[\quad,\quad]$ and $[\quad,\quad]_1$ are unimodular, $(\zL, \zL_1 )$ is flat. Flatness happens as well if the derived ideal of $[\quad,\quad]$ equals ${\mathcal A}$; indeed, for $a\znoi 0$ small enough the derived ideal of $[\quad,\quad]+a[\quad,\quad]_1$ equals ${\mathcal A}$ too; therefore $[\quad,\quad]$ and $[\quad,\quad]+a[\quad,\quad]_1$ are unimodular and$(\zL, \zL+a\zL_1 )$ flat.
A couple of $2$-forms $\zb,\zb_1$ on a vector space $V$, of dimension $2n-1\zmai 3$, is named [*generic*]{} if it is generic as bi-vectors on $V^*$, that is if $(s\zb+t\zb_1 )^{n-1}\znoi 0$ for any $(s,t)\zpe {\mathbb C}^2 -\{0\}$; note that when $\zb_{1}^{n-1}\znoi 0$ it suffices to check that $(\zb+t\zb_1 )^{n-1}\znoi 0$ for any $t\zpe {\mathbb C}$.
Observe that if $\zL$ is the Lie-Poisson structure on the dual space ${\mathcal B}^*$ of a Lie algebra ${\mathcal B}$, then $\zL(\za)=-d\za$ for each $\za\zpe {\mathcal B}^*$, where $d\za$ is regarded as constant bi-vector on ${\mathcal B}^*$. Therefore, when $dim{\mathcal B}=2n-1\zmai 3$, a linear (respectively Lie) Poisson pair $(\zL,\zL_1 )$ on ${\mathcal B}^*$, where $\zL_1$ is associated to $\zb\zpe\zL^2 {\mathcal B}^*$ (respectively to a second bracket $[\quad,\quad]_1$), is generic at $\za\zpe {\mathcal B}^*$ if and only if $(d\za,\zb)$ (respectively $(d\za,d_1 \za)$) is generic.
\[eje-1\]
Let ${\mathcal P}$ be the Lie algebra of vector fields $f\zpu(\zpar/\zpar u)$, on $\mathbb K$, such that $f$ is a polynomial in $u$ and $f(0)=0$. Set ${\mathcal P}_m =u^{m}{\mathcal P}$, $m\zpe \mathbb N$, which is an ideal of ${\mathcal P}$ of codimension $m$. The quotient ${\mathcal A}={\mathcal P}/{\mathcal P}_m$ (we omit the subindex for sake of simplicity) is a Lie algebra of dimension $m$ that we called the truncated Lie algebra (of ${\mathcal P}$ at order $m$). Denote by $e_k$ the class of $u^k (\zpar/\zpar u)$; then $\{e_1 ,\zps,e_m \}$ is a basis of ${\mathcal A}$ and $[e_i ,e_j ]=(j-i)e_{i+j-1}$ if $i+j\zmei m+1$ and zero otherwise.
Throughout this example one will assume $m=2n-1\zmai 5$. Then $de_{m}^* =-\zsu_{j=1}^{n-1}2(n-j)e_j^* \zex e_{2n-j}^*$ and $de_{m-1}^* =-\zsu_{j=1}^{n-1}(2(n-j)-1)e_j^* \zex e_{2n-j-1}^*$. Moreover $(de_m^* ,de_{m-1}^* )$ is generic; this follows from the next lemma since $(de_m^* )^{n-1}$ and $(de_{m-1}^* )^{n-1}$ do not vanish.
\[lem-6\] For every $t\zpe{\mathbb C}-\{0\}$ one has $(e_m^* +te_{m-1}^* )\zex(de_m^* +tde_{m-1}^* )^{n-1}\znoi 0$.
Regard ${\mathcal A}$ like a vector subspace of the $2n$-dimensional vector space $V$ of basis $\{e_1 ,\zps,e_m ,{\tilde e}\}$ and ${\mathcal A}^*$ as a vector subspace of $V^*$ in the obvious way ($\za({\tilde e})=0$ for every $\za\zpe{\mathcal A}^*$). Set $\zt= -\zsu_{j=1}^{n-1}2(n-j)e_j^* \zex e_{2n-j}^*+e_{n}^* \zex{\tilde e}^*$, which is a symplectic form, and $\zt_1 =-\zsu_{j=1}^{n-1}(2(n-j)-1)e_j^* \zex e_{2n-j-1}^*$. Let $J$ be the endomorphism given by the formula $\zt_1 (v,w)=\zt(Jv,w)$. Then $J=\zsu_{j=1}^{n-2}a_j e_{j+1}\zte e_{j}^* +a{\tilde e}\zte e_{n-1}^*
+\zsu_{k=n}^{2(n-1)}b_k e_{k+1}\zte e_{k}^*$ for some non-vanishing scalars $a_j$, $a$, $b_k$, $j=1,\zps,n-2$, $k=n,\zps,2(n-1)$. Note that $J$ is nilpotent so every $I+tJ$, $t\zpe \mathbb C$, is invertible and each $\zt+t\zt_1$, $t\zpe \mathbb C$, symplectic.
Moreover the vector subspace spanned by $\{e_1 ,\zps,e_{n-1}, {\tilde e}\}$ is Lagrangian for all symplectic form $\zt+t\zt_1$, $t\zpe \mathbb C$. Thus there exists a vector $v(t)=\zsu_{j=1}^{n-1}c_j (t)e_j +c(t) {\tilde e}$ such that $i_{v(t)}(\zt+t\zt_1 )=e_{m}^* +te_{m-1}^*$, $t\zpe \mathbb C$. It is easily checked that $c(t)\znoi 0$ when $t\znoi 0$; in this case $v(t)\znope{\mathcal A}$ and the restriction of $i_{v(t)}(\zt+t\zt_1 )^n$ to ${\mathcal A}$ does not vanish. But this restriction equals $n(e_m^* +te_{m-1}^* )\zex(de_m^* +tde_{m-1}^* )^{n-1}$.
On ${\mathcal A}^*$ consider coordinates $(x_1 ,\zps,x_m )$ associates to the basis $\{e_{1}^* ,\zps,e_{m}^* \}$ and the Lie -Poisson structure $\zL$. Set $\zL_1 \zeq de_{m}^*$. As $(de_{m}^* ,de_{m-1}^* )$ is generic, $(\zL,\zL_1 )$ is generic at $e_{m-1}^* \zpe{\mathcal A}^*$. Besides [*it is not flat at this point*]{}. Indeed, $dx_n$ is a Casimir of $\zL_1$ and $$\zL(dx_n ,\quad)= (1-n)x_n {\frac {\zpar} {\zpar x_1 }}
+ (2-n)x_{n+1}{\frac {\zpar} {\zpar x_2 }}
+\zsu_{j=3}^{n-1}f_j {\frac {\zpar} {\zpar x_j }}\, .$$
In our case the vector field $X$ of lemma \[lem-5\] equals $a(\zpar/\zpar x_1 )$ with $a\znoi 0$. Since $\zL(dx_n ,\quad)$ and $\zpar/\zpar x_1$ are not proportional about $e_{m-1}^*$, the non-flatness of $(\zL,\zL_1 )$ at $e_{m-1}^*$ follows from lemma \[lem-3\] (see remark below).
\[rem-5\]
Recall that a linear vector field on a vector space $V$ is proportional, about some point $p\zpe V$, to a constant vector field if and only if the associated endomorphism has rank zero or one. Let ${\mathcal B}$ be a finite dimensional Lie algebra and $\zL'$ the Lie-Poisson structure of ${\mathcal B}^*$. Consider a constant $1$-form $\za$ on ${\mathcal B}^*$; then there exists just a $b_{\za}\zpe V$ such that $\za$ is the exterior derivative of $b_{\za}$ regarded as a linear function on ${\mathcal B}^*$. On the other hand, $\zL'(\za,\quad)$ is a linear vector field whose associated endomorphism is the dual map of $[b_{\za},\quad]\zdp{\mathcal B}\zfl{\mathcal B}$. Thus $\zL'(\za,\quad)$ is proportional, about some $p\zpe{\mathcal B}^*$, to a constant vector field if and only if $rank[b_{\za},\quad]\zmei 1$.
Therefore if $\zb\zpe\zL^{2}{\mathcal B}^*$ is a cocycle and $\za$ a Casimir of $\zL''\zeq\zb$, that is $\zb(b_{\za},\quad)=0$, such that $rank[b_{\za},\quad]\zmai 2$, then the linear vector field $\zL'(\za,\quad)$ is never proportional to a constant vector field about any element of ${\mathcal B}^*$. Just is the case of the foregoing example, where $\zL_1 \zeq de_{m}^*$, $\za= dx_n$ and $b_{\za}=e_n$.
Other results for the linear case {#sec-6}
=================================
For non-unimodular Lie algebras, from a flat Poisson pair one may construct another one that is not flat. More exactly:
\[pro-3\] Consider a linear Poisson pair $(\zL,\zL_1 )$ on the dual space ${\mathcal A}^*$ of a non-unimodular Lie algebra ${\mathcal A}$, of dimension $m=2n-1\zmai 5$, and an element $p$ of ${\mathcal A}^*$. If $(\zL,\zL_1 )$ is generic and flat at $p$, then every linear Poisson pair $(\zL,\zL_1 +a\zL(p))$, $a\zpe\mathbb K -\{0\}$, is generic and non-flat at $p$.
Therefore, given a non-unimodular Lie algebra ${\tilde{\mathcal A}}$ of odd dimension $\zmai 5$, if there exist $\tilde\za,\tilde\zb\zpe{\tilde{\mathcal A}}^*$ such that $(d\tilde\za,d\tilde\zb)$ is generic, then on ${\tilde{\mathcal A}}^*$ there is a generic and non-flat linear Poisson pair.
Clearly $(\zL,\zL_1 +a\zL(p))$ is generic at $p$. On the other hand, let $X$ be the vector field of lemma \[lem-5\] and $\za,\za_{a}$ $1$-forms on ${\mathcal A}^*$ Casimir of $\zL_1$ and $\zL_1 +a\zL(p)$ respectively, both of them non-vanishing at $p$. As $a\znoi 0$ vectors $\zL(\za,\quad)(p)$ and $\zL(\za_{a},\quad)(p)$ are linearly independent (that follows from being generic since $dim{\mathcal A}\zmai 5$.) By lemma \[lem-3\], $X(p)$ and $\zL(\za,\quad)(p)$ are linearly dependent because $(\zL,\zL_1 )$ is flat. Therefore $X$ and $\zL(\za_{a},\quad)$ are independent at $p$, so about $p$.
For the second assertion it is enough to remark that $(\tilde\zL,\tilde\zL_1 )$, where $\tilde\zL$ is the Lie-Poisson structure on ${\tilde{\mathcal A}}^*$ and $\tilde\zL_1\zeq d\tilde\zb$, is generic at $\tilde\za$; if $(\tilde\zL,\tilde\zL_1 )$ is flat at $\tilde\za$ apply the first statement.
\[eje-2\]
Let ${\mathcal A}$ be the Lie algebra of basis $\{e_1 ,\zps,e_5 \}$ given by $[e_1 ,e_5 ]=e_5$, $[e_2 ,e_3 ]=e_3$, $[e_2 ,e_4 ]=-e_4$ and $[e_i ,e_j ]=0$, $i<j$, otherwise (that corresponds on ${\mathbb K}^3$ to vector fields $e_1 =\zpar/\zpar z_1$, $e_2 =\zpar/\zpar z_2$, $e_3=exp(z_2 )(\zpar/\zpar z_3 )$, $e_4=exp(-z_2 )(\zpar/\zpar z_3 )$ and $e_5=exp(z_1 )(\zpar/\zpar z_1 )$). Then with respect to the coordinates associated to the dual basis $\{e_{1}^* ,\zps,e_{5}^* \}$ one has $$\zL=x_5 {\frac {\zpar} {\zpar x_1 }}\zex{\frac {\zpar} {\zpar x_5 }}
+x_3 {\frac {\zpar} {\zpar x_2 }}\zex{\frac {\zpar} {\zpar x_3 }}
-x_4 {\frac {\zpar} {\zpar x_2 }}\zex{\frac {\zpar} {\zpar x_4 }}.$$
On the other hand $e_{1}^* \zex e_{2}^* +e_{3}^* \zex e_{4}^*$ is a $2$-cocycle, so $(\zL,\zL_1 )$, where $\zL_1 =(\zpar/\zpar x_1 )\zex(\zpar/\zpar x_2 )
+ (\zpar/\zpar x_3 )\zex(\zpar/\zpar x_4 )$, is a Poisson pair. It is easily checked that $dx_5$ is a $\zL_1$-Casimir and $(\zL,\zL_1 )$ generic at $p=(0,0,1,0,1)$.
The vector field $X$ given by lemma \[lem-5\] equals $\zpar/\zpar x_1$ while $\zL(dx_5 ,\quad)=-x_5 (\zpar/\zpar x_1 )$, so $(\zL,\zL_1 )$ is flat at $p$ (lemma \[lem-3\].) Nevertheless $(\zL,\zL_1+a\zL(p))$, $a\zpe{\mathbb K}-\{0\}$, is not flat (proposition \[pro-3\].) For checking this fact directly, observe that $dx_5 -adx_2 -a^2 dx_4$ is a Casimir of $$\zL_1+a\zL(p)={\frac {\zpar} {\zpar x_1 }}\zex{\frac {\zpar} {\zpar x_2 }}
+{\frac {\zpar} {\zpar x_3 }}\zex{\frac {\zpar} {\zpar x_4 }}
+a\zpizq {\frac {\zpar} {\zpar x_1 }}\zex{\frac {\zpar} {\zpar x_5 }}
+{\frac {\zpar} {\zpar x_2 }}\zex{\frac {\zpar} {\zpar x_3 }}\zpder$$ and $$\zL(dx_5 -adx_2 -a^2 dx_4 ,\quad)(p)=-{\frac {\zpar} {\zpar x_1}}
-a{\frac {\zpar} {\zpar x_3}}\, ;$$ now apply lemma \[lem-3\].
For unimodular algebras a more sophisticated construction is needed. Until the end of this section ${\mathcal A}$ will denote a Lie algebra of dimension $m=2n-1\zmai 3$. To each element $\za\zpe{\mathcal A}^*$ such that $(d\za)^{n-1}\znoi 0$ one associates a Lie subalgebra ${\mathcal A}_{\za}$ as follows:
1. If $\za\zex(d\za)^{n-1}\znoi 0$ then ${\mathcal A}_{\za}=0$.
2. If $\za\zex(d\za)^{n-1}=0$ then there exists $v\zpe{\mathcal A}$ such that $i_v d\za=-\za$; moreover if $i_w d\za=-\za$ for another $w\zpe{\mathcal A}$ then $w-v\zpe Kerd\za$. By definition ${\mathcal A}_{\za}$ will be the $2$-dimensional vector space spanned by $v$ and $Kerd\za$, which is a Lie subalgebra because one has $L_v d\za=-d\za$ that implies $[v,Kerd\za]\zco Kerd\za$.
In this case $v$ will be named a [*Hamiltonian of*]{} $\za$.
\[lem-7\] Suppose ${\mathcal A}$ unimodular. Consider $\za\zpe{\mathcal A}^*$ such that $(d\za)^{n-1}\znoi 0$. Then ${\mathcal A}_{\za}$ is either non-abelian or zero.
Assume $dim{\mathcal A}_{\za}=2$. Let $v$ a Hamiltonian of $\za$ and $u$ a basis of $Kerd\za$. Take $\zt\zpe\zL^{m}{\mathcal A}^* -\{0\}$. Then $i_u \zt=c(d\za)^{n-1}$ with $c\znoi 0$.
Since ${\mathcal A}$ is unimodular, $L_v \zt=0$; now if $[u,v]=0$ one has $$0=i_u L_v \zt=L_v (i_u \zt)=cL_v ((d\za)^{n-1})=-c(n-1)(d\za)^{n-1}\znoi 0$$ [*contradiction*]{}.
For the purpose of this work, a couple $(\za,\zb)\zpe{\mathcal A}^* \zpor{\mathcal A}^*$ will be called [*generic*]{} if $(d\za,d\zb)$ is generic and ${\mathcal A}_{(s\za+t\zb)}$ is non-abelian or zero for all $(s,t)\zpe{\mathbb C}^2 -\{0\}$; note that if ${\mathcal A}_{\zb}$ is non-abelian or zero, it suffices to check the property for every ${\mathcal A}_{(\za+t\zb)}$, $t\zpe{\mathbb C}$. Even when ${\mathcal A}$ is a real Lie algebra, this definition is meaningful by complexifying it. On the other hand, by lemma \[lem-7\], when ${\mathcal A}$ is unimodular $(\za,\zb)$ is generic if $(d\za,d\zb)$ is generic.
The next step is to construct a new Lie algebra ${\mathcal B}_{\mathcal A}$, called [ *the secondary algebra*]{} (of ${\mathcal A}$). Set ${\mathcal B}_{\mathcal A}={\mathcal A}\zpor{\mathcal A}\zpor\mathbb K$ endowed with the bracket $$[(v,v',s),(w,w',t)]=([v,w],[v,w']-[w,v']+tv'-sw',0).$$
This new algebra is just the extension of ${\mathcal A}$ by ${\mathcal A}$, this second time regarded as an abelian ideal, by means of the adjoint representation plus more one dimension for taking into account $I\zdp{\mathcal A}\zfl{\mathcal A}$. Clearly ${\mathcal B}_{\mathcal A}$ is not unimodular and its dimension equals $2m+1$.
If $\{{\tilde e}_1 ,\zps, {\tilde e}_m\}$ is a basis of ${\mathcal A}$, $[{\tilde e}_i ,{\tilde e}_j ]=\zsu_{k=1}^{m}c_{ij}^{k}{\tilde e}_k$, $i,j=1,\zps,m$, and we set $e_r =({\tilde e}_r ,0,0)$, $f_r =(0,{\tilde e}_r ,0)$, $r=1,\zps,m$ and $e=(0,0,1)$, then $\{e_1 ,\zps,e_m ,f_1 ,\zps,f_m ,e\}$ is a basis of ${\mathcal B}_{\mathcal A}$ and $[e_i ,e_j ]=\zsu_{k=1}^{m}c_{ij}^{k}e_k$, $[e_i ,f_j ]=-[f_j ,e_i ]=\zsu_{k=1}^{m}c_{ij}^{k}f_k$, $[f_j ,e]=-[e,f_j ]=f_j$, $i,j=1,\zps,m$, while the other brackets vanish.
In coordinates $(x,y,z)=(x_1 ,\zps,x_m ,y_1 ,\zps,y_m ,z)$ associated to the dual basis $\{e_{1}^* ,\zps,e_{m}^* ,f_{1}^* ,\zps,f_{m}^* ,e^*\}$ the Lie-Poisson structure on ${\mathcal B}_{\mathcal A}^{*}$ writes: $$\zL=\zsu_{1\zmei i<j\zmei m}\zsu_{k=1}^{m}c_{ij}^{k}x_k
{\frac {\zpar} {\zpar x_i}}\zex{\frac {\zpar} {\zpar x_j}}\hskip 4 truecm$$ $$\hskip 3 truecm +\zsu_{i,j,k=1}^{m}c_{ij}^{k}y_k
{\frac {\zpar} {\zpar x_i}}\zex{\frac {\zpar} {\zpar y_j}}
+\zpizq\zsu_{k=1}^{m}y_k {\frac {\zpar} {\zpar y_k}}\zpder
\zex{\frac {\zpar} {\zpar z}}.$$
\[pro-4\] Let ${\mathcal A}$ be a Lie algebra of dimension $m=2n-1\zmai 3$. If there exists a generic couple $(\za,\zb)\zpe{\mathcal A}^* \zpor{\mathcal A}^*$ such that $\zb\zex(d\zb)^{n-1}\znoi 0$, then on ${\mathcal B}_{\mathcal A}^*$ there is some non-flat generic linear Poisson pair.
Observe that the the center of ${\mathcal A}$ is zero because it is included in $Kerd\za\zin Kerd\zb$, which is zero since $(d\za,d\zb)$ is generic. Regarding $(d\za,d\zb)$ as a couple of bi-vectors on ${\mathcal A}^*$ and taking into account that Casimirs of $d\za+td\zb$, $t\zpe \mathbb K$, correspond to elements of $Ker(d\za+td\zb)$ show the existence of a polynomial curve $\zg(t)=\zsu_{j=0}^{n}t^j a_j$ in ${\mathcal A}$, with $a_0 ,\zps,a_n$ linearly independent, such that $\zg(t)$, $t\zpe \mathbb K$, is a basis of $Ker(d\za+td\zb)$ (see [@TU1; @TU2].) Now choose $\zr\zpe {\mathcal A}^*$ such that $\zr(a_0 )=1$ and $\zr(a_k )=0$, $k=1,\zps,n$; then $\zr(\zg(t))\znoi 0$ for every $t\zpe \mathbb C$ (in the real case complexify ${\mathcal A}$.)
On the other hand we identify ${\mathcal B}_{\mathcal A}^*$ and ${\mathcal A}^* \zpor{\mathcal A}^* \zpor(\mathbb K )^*$ in the obvious way. Set $\zL_1 =\zL(0,\zb,0)$; we shall prove that $(\zL,\zL_1 )$ is generic and non-flat at $(\zr,\za,0)$.
First let us check that $rank\zL_1 =2m$. As $\zb\zex (d\zb)^{n-1}\znoi 0$, there exists a basis $\{{\tilde e}_1 ,\zps, {\tilde e}_m\}$ of ${\mathcal A}$ such that $\zb={\tilde e}_{1}^*$ and $d\zb=-\zsu_{j=1}^{n-1}{\tilde e}_{2j}^*
\zex{\tilde e}_{2j+1}^*$. Then in coordinates $(x,y,z)$ of ${\mathcal B}_{\mathcal A}^*$ one has $$\zL_1 =\zsu_{j=1}^{n-1}\zpizq
{\frac {\zpar} {\zpar x_{2j}}}\zex{\frac {\zpar} {\zpar y_{2j+1}}}
-{\frac {\zpar} {\zpar x_{2j+1}}}\zex{\frac {\zpar} {\zpar y_{2j}}}\zpder
+{\frac {\zpar} {\zpar y_{1}}}\zex{\frac {\zpar} {\zpar z}}$$ whose rank equals $2m$.
Observe that $dx_1$ is a Casimir of $\zL_1$, which corresponds to $e_1 \zpe {\mathcal B}_{\mathcal A}$. As $rank([e_1 ,\quad])\zmai 2$ because ${\mathcal A}$ has trivial center, from lemmas \[lem-3\] and \[lem-5\] and remark \[rem-5\] follows the non-flatness of $(\zL,\zL_1 )$ at $(\zr,\za,0)$ provided that it is generic.
Since $rank\zL_1 =2m$, for verifying that $(\zL,\zL_1 )$ is generic at $(\zr,\za,0)$ it suffices to show that every $(\zL+t\zL_1 )(\zr,\za,0)$, $t\zpe \mathbb C$, has rank $2m$, which is equivalent to see that $\zL(\zr,\za+t\zb,0)$, $t\zpe \mathbb C$, has rank $2m$. Set $\zt=\za+t\zb$; by complexifying ${\mathcal A}$ if necessary, we may suppose that it is a complex Lie algebra without loss of generality. If $\zt\zex(d\zt)^{n-1}\znoi 0$ by considering a second basis $\{{\tilde e}_1 ,\zps, {\tilde e}_m\}$ of ${\mathcal A}$ (denoted as the first one for sake of simplicity) such that $\zt={\tilde e}_{1}^*$ and $d\zt=-\zsu_{j=1}^{n-1}{\tilde e}_{2j}^* \zex{\tilde e}_{2j+1}^*$, in coordinates $(x,y,z)$ on ${\mathcal B}_{\mathcal A}^*$ one has $$\zL(\zr,\zt,0) =\zsu_{1\zmei i<j\zmei m}a_{ij}
{\frac {\zpar} {\zpar x_i}}\zex{\frac {\zpar} {\zpar x_j}}\hskip 6truecm$$ $$+\zsu_{j=1}^{n-1}\zpizq
{\frac {\zpar} {\zpar x_{2j}}}\zex{\frac {\zpar} {\zpar y_{2j+1}}}
-{\frac {\zpar} {\zpar x_{2j+1}}}\zex{\frac {\zpar} {\zpar y_{2j}}}\zpder
+{\frac {\zpar} {\zpar y_{1}}}\zex{\frac {\zpar} {\zpar z}}\hskip 1truecm$$ $$\hskip 1truecm =\zsu_{j=1}^{n-1}\zpizq
{\frac {\zpar} {\zpar x_{2j}}}\zex v_{2j+1}
-{\frac {\zpar} {\zpar x_{2j+1}}}\zex v_{2j}\zpder
+{\frac {\zpar} {\zpar y_{1}}}\zex{\frac {\zpar} {\zpar z}}\, ,$$ .3truecm where $v_k =(\zpar/\zpar y_{k})+\zsu_{r=1}^{m}b_{kr}(\zpar/\zpar x_{r})$, $k=2,\zps,m$ for suitable scalars $b_{kr}$, whose rank equals $2m$.
Now assume $\zt\zex(d\zt)^{n-1}=0$. As $(d\zt)^{n-1}\znoi 0$ since $(d\za,d\zb)$ is generic and $\zr_{\zbv Kerd\zt}\znoi 0$ because $\zr(\zg(t))\znoi 0$, there exists a basis $\{{\tilde e}_1 ,\zps, {\tilde e}_m\}$ of ${\mathcal A}$ (denoted as the first and second ones) such that $\zt={\tilde e}_{2}^*$, $d\zt=-\zsu_{j=1}^{n-1}{\tilde e}_{2j}^*
\zex{\tilde e}_{2j+1}^*$ and $\zr={\tilde e}_{1}^*$. Thus $\{{\tilde e}_{1},{\tilde e}_{3}\}$ is a basis of ${\mathcal A}_{\zt}$, $[{\tilde e}_{1},{\tilde e}_{3}]=c_{13}^{1}{\tilde e}_{1}$ with $c_{13}^{1}\znoi 0$ since ${\mathcal A}_{\zt}$ is non-abelian, and in coordinates $(x,y,z)$ on ${\mathcal B}_{\mathcal A}^*$ one has $x_1 (\zr,\zt,0)=y_2 (\zr,\zt,0)=1$ while the remainder coordinates of $(\zr,\zt,0)$ vanish. Therefore .3truecm $$\zL(\zr,\zt,0) =\zsu_{1\zmei i<j\zmei m}c_{ij}^{1}
{\frac {\zpar} {\zpar x_i}}\zex{\frac {\zpar} {\zpar x_j}}\hskip 6truecm$$ $$\hskip 1truecm +\zsu_{j=1}^{n-1}\zpizq
{\frac {\zpar} {\zpar x_{2j}}}\zex{\frac {\zpar} {\zpar y_{2j+1}}}
-{\frac {\zpar} {\zpar x_{2j+1}}}\zex{\frac {\zpar} {\zpar y_{2j}}}\zpder
+{\frac {\zpar} {\zpar y_{2}}}\zex{\frac {\zpar} {\zpar z}}$$ $$=c_{13}^{1}{\frac {\zpar} {\zpar x_1}}\zex{\frac {\zpar} {\zpar x_3}}
+{\frac {\zpar} {\zpar x_2}}\zex w_{3}
-\zpizq{\frac {\zpar} {\zpar x_{3}}}
+{\frac {\zpar} {\zpar z}}\zpder\zex{\frac {\zpar} {\zpar y_2}}\hskip 2truecm$$ $$\hskip 2truecm +\zsu_{j=2}^{n-1}\zpizq {\frac {\zpar} {\zpar x_{2j}}}\zex w_{2j+1}
-{\frac {\zpar} {\zpar x_{2j+1}}}\zex w_{2j}\zpder ,$$ .3truecm where $w_k =(\zpar/\zpar y_{k})+\zsu_{r=1}^{m}{\tilde b}_{kr}(\zpar/\zpar x_{r})$, $k=3,\zps,m$ for suitable scalars ${\tilde b}_{kr}$. Clearly its rank equals $2m$ since $c_{13}^{1}\znoi 0$.
\[pro-5\] Consider an unimodular Lie algebra ${\mathcal A}$ of dimension $m=2n-1\zmai 3$; assume that this algebra possesses some contact form. If there exist $\za,\zb\zpe{\mathcal A}^*$ such that $(d\za,d\zb)$ is generic then on ${\mathcal B}_{\mathcal A}^*$ there exists some non-flat generic linear Poisson pair.
The set of contact forms is open and dense in ${\mathcal A}^*$. As to be generic is an open property, there exists $\zb'\zpe{\mathcal A}^*$ such that $(d\za,d\zb')$ is generic and $\zb'\zex(d\zb')^{n-1}\znoi 0$. Now apply lemma \[lem-7\] and proposition \[pro-4\].
\[eje-3\]
Proposition \[pro-4\] (proposition \[pro-5\] is just a particular case of the foregoing one) can be applied to a $3$-dimensional Lie algebra ${\mathcal A}$ if and only if either it is simple or there exists a basis $\{e_1 ,e_2 ,e_3 \}$ such that $[e_1 ,e_2 ]=e_3$, $[e_1 ,e_3 ]=ae_2 +be_3$ with $a\znoi 0$ and $[e_2 ,e_3 ]=0$. Indeed, the simple case is obvious; therefore assume solvable ${\mathcal A}$.
If proposition \[pro-4\] applies, then ${\mathcal A}$ possess contact forms and its center is trivial (see the beginning of the proof of proposition \[pro-4\].) In this case a computation shows the existence of this basis (as its center equals zero ${\mathcal A}$ contains a $2$-dimensional abelian ideal ${\mathcal A}_0$ such that $[v,\quad]\zdp{\mathcal A}_0 \zfl{\mathcal A}_0$ is an isomorphism for any $v\znope{\mathcal A}_0$; the existence of contact forms implies that $[v,\quad]\zdp{\mathcal A}_0 \zfl{\mathcal A}_0$ is never multiple of identity.)
Conversely, when a such basis exists it suffices to set $\za=e_{2}^*$ and $\zb=e_{3}^*$.
\[eje-4\]
Consider the Lie algebra ${\mathcal A}$ of basis $\{e_1 ,\zps,e_{2n-1}\}$, $n\zmai 2$, given by $[e_{2j-1},e_{2j}]=-e_{2j}$ and $[e_{2j-1},e_{2n-1}]=-ae_{2n-1}$ with $a\znoi 0$, $j=1,\zps,n-1$, and $[e_{i},e_{r}]=0$, $i<r$, otherwise (this algebra corresponds to consider vector fields $e_{2j-1}=\zpar/\zpar x_j$ and $e_{2j}=exp(-x_j )(\zpar/\zpar x_n )$, $j=1,\zps,n-1$, and $e_{2n-1}=exp(-\zsu_{k=1}^{n-1}ax_{k})(\zpar/\zpar x_n )$ on ${\mathbb K}^n$.) Set $\za=\zsu_{j=1}^{n-1}e_{2j}^*$ and $\zb=\zsu_{j=1}^{n-1}a_j e_{2j}^* +e_{2n-1}^*$ where $a_1 ,\zps,a_{n-1}$ are distinct and non-vanishing scalars.
Then $$(d\za+td\zb)^{n-1}=\zpizq\zsu_{j=1}^{n-1}(a_j t+1)e_{2j-1}^{*}\zex e_{2j-1}^{*}
+at(\zsu_{j=1}^{n-1}e_{2j-1}^{*})\zex e_{2n-1}^{*} \zpder^{n-1}$$ $$=(n-1)!\zpizq\zpr_{j=1}^{n-1}(a_j t+1)\zpder e_{1}^{*}\zex\zps\zex e_{2n-2}^{*}$$ $$+(n-1)!at\zsu_{k=1}^{n-1}\zcizq\zpizq\zpr_{j=1;j\znoi k}^{n-1}(a_j t+1)\zpder
e_{1}^{*}\zex\zps\zex{\widehat e_{2k}^{*}}\zex\zps\zex e_{2n-1}^{*}\zcder$$ so non-zero for any $t\zpe\mathbb C$.
On the other hand $$(\za+t\zb)\zex(d\za+td\zb)^{n-1}=(n-1)!(a[1-n]+1)t\zpizq\zpr_{j=1}^{n-1}(a_j
t+1)\zpder e_{1}^{*}\zex\zps\zex e_{2n-1}^{*}$$ while $\zb\zex(d\zb)^{n-1}=(n-1)!(a[1-n]+1)a_1 \zpu\zpu\zpu a_{n-1}
e_{1}^{*}\zex\zps\zex e_{2n-1}^{*}$.
Let us suppose $a\znoi (n-1)^{-1}$. Then $\zb$ is a contact form and $(d\za,d\zb)$ is generic. Thus if $a=-1$, as ${\mathcal A}$ is unimodular, one may apply proposition \[pro-5\] to $(\za,\zb)$. In the general case we have to examine algebras ${\mathcal A}_{(\za+t\zb)}$, $t\zpe\mathbb C$. They are of dimension two just when $t=0, -a_{1}^{-1} ,\zps,-a_{n-1}^{-1}$.
A basis of ${\mathcal A}_{\za}$, which corresponds to $t=0$, is $\{\zsu_{j=1}^{n-1}e_{2j-1},e_{2n-1}\}$, so this algebra is not abelian. Now suppose $t=-a_{1}^{-1}$ (the remainder cases are similar); then a basis of ${\mathcal A}_{(\za-a_{1}^{-1}\zb)}$ is $\{e_2 ,(a^{-1}-n+2)e_1 +\zsu_{j=2}^{n-1}e_{2j-1} \}$ and it is non-abelian if and only if $a^{-1}-n+2\znoi 0$, that is $a\znoi (n-2)^{-1}$.
Summing up, proposition \[pro-4\] may be applied just when $a\znoi 0, (n-1)^{-1}, (n-2)^{-1}$.
\[eje-5\]
Let ${\mathcal A}$ be the truncated Lie algebra of dimension $m=2n-1\zmai 3$ of example \[eje-1\]. Recall that the couple $(de_{m}^* ,de_{m-1}^* )$ was generic, so $(d\za,d\zb)$ where $\za=e_{m}^*$ and $\zb=e_{m}^* +e_{m-1}^*$ is generic too. By lemma \[lem-6\] $\zb$ is a contact form and ${\mathcal A}_{(\za+t\zb)}=\{0\}$ unless $t=0,-1$. Therefore to see that $(\za,\zb)$ is generic one has to show that ${\mathcal A}_{e_{m}^*}$ and ${\mathcal A}_{e_{m-1}^*}$ are not abelian.
But $\{e_1 ,e_n \}$ is a basis of ${\mathcal A}_{e_{m}^*}$ and $[e_1 ,e_n]=(n-1)e_n$ while $\{e_1 ,e_m \}$ is a basis of ${\mathcal A}_{e_{m-1}^*}$ and $[e_1 ,e_m]=(m-1)e_m$, so $(\za,\zb)$ is generic and proposition \[pro-4\] may be applied to it.
The special affine algebra {#sec-7}
==========================
In this section we show that proposition \[pro-5\] may be applied to this Lie algebra. Let $V$ be a real or complex vector space of dimension $n\zmai 2$ and ${\mathcal Aff}(V)$ the affine algebra of $V$, which can be regarded too like the algebra of polynomial vector fields on $V$ of degree $\zmei 1$. Recall that ${\mathcal Aff}(V)= {\mathcal I}\zdi sl(V)\zdi V$ where ${\mathcal I}$ consists of linear vector fields multiples of identity, $sl(V)$ is the special linear algebra of $V$ and $V$ the ideal of constant vector fields. On the other hand, the dual space ${\mathcal Aff}(V)^*$ will be identify to ${\mathcal I}^* \zdi sl(V)^* \zdi V^*$ in the obvious way.
Denote by $kil$ the Killing form of $sl(V)$, which is non-degenerate; therefore if one sets $\za_g =kil(g,\quad)$, $g\zpe sl(V)$, then $g\zpe sl(V)\zfl\za_g \zpe sl(V)^*$ is an isomorphism of vector spaces. Moreover, given $g,h\zpe sl(V)$ then $(d\za_g )(h,\quad)=-\za_{[g,h]}$; thereby $(d\za_g )(h,\quad)=0$ if and only if $[g,h]=0$.
For any $g\zpe gl(V)$ and $\zt\zpe V^*$ the subspace spanned by $g,\zt$ means that spanned by $\zt$ and the dual endomorphism $g^*$. One will need the following results:
\[lem-8\] On a real or complex vector space $E=E_1 \zdi E_2$, of even dimension, consider a couple of $2$-forms $\zl,\zl_1$ such that $Ker\zl\zcco E_2$, $\zl_{1\zbv E_1}=0$ and $\zl_{1\zbv E_2}=0$. One has:
\(a) If $\zl_{1\zbv Ker\zl}$ is symplectic then $\zl+\zl_1$ is symplectic too.
\(b) If the corank of $\zl_{1\zbv Ker\zl}$ equals two and $dim (E_1 \zin Ker(\zl_{1\zbv Ker\zl}))\zmai 1$, then the corank of $\zl+\zl_1$ equals two.
\[lem-9\] For any $n\zmai 2$ one may find real numbers $a_1 ,\zps,a_n$, $b_1 ,\zps,b_n$, $c_1 ,\zps,c_n$ satisfying:
\(I) $a_i \znoi a_j$, $b_i \znoi b_j$ and $c_i \znoi c_j$ whenever $i\znoi j$; moreover no $c_i$, $i=1,\zps,n$, vanishes.
\(II) $\zsu_{i=1}^{n}a_i =\zsu_{i=1}^{n}b_i =0$.
\(III) For every $t\zpe{\mathbb C}-\{0\}$, at least $n-1$ elements of the family $a_1 +tb_1 ,\zps,a_n +tb_n$ are different. Moreover is a such family includes two equal elements then $1+tc_i \znoi 0$ for every $i=1,\zps,n$.
Consider a family $a'_1 ,\zps,a'_n$ of distinct rational numbers and a second one $b'_1 ,\zps,b'_n$ of rationally independent real numbers. Set $t_{ij}=(a'_i -a'_j )(b'_i -b'_j )^{-1}$, $i\znoi j$. Then $a'_i +tb'_i =a'_j +tb'_j$ if and only if $t=-t_{ij}$.
Suppose $a'_i +tb'_i =a'_j +tb'_j$ and $a'_k+tb'_k =a'_r +tb'_r$ for some $i<j$ and $k<r$ with $(i,j)\znoi (k,r)$. Then $t=-t_{ij}=-t_{kr}$, that is $t_{ij}=t_{kr}$, and an elementary computation shows that $b'_1 ,\zps,b'_n$ are not rationally independent; therefore (III) holds for these two families.
Observe that (III) holds too for $a'_1 +a,\zps,a'_n +a$ and $b'_1 +b,\zps,b'_n +b$ whatever $a,b$ are, because each $t_{ij}$ does not change. In other words, setting $a=-\zsu_{i=1}^{n}(a'_i /n)$ and $b=-\zsu_{i=1}^{n}(b'_i /n)$ shows the existence of two families $a_1 ,\zps,a_n$ and $b_1 ,\zps,b_n$ satisfying (I), (II) and (III).
Finally, choose $c_1 ,\zps,c_n \zpe{\mathbb R}-(\{0\}\zun\{t_{ij}^{-1} \zbv
1\zmei i<j\zmei n\})$ that are distinct among them.
\[pro-6\] Given a $n$-dimensional, $n\zmai 2$, real or complex vector space consider $g\zpe sl(V)$ and $\zt\zpe V^*$ and regard $\za_g +\zt$ like an element of ${\mathcal Aff}(V)^*$. Assume diagonalizable $g$. One has:
\(a) If the eigenvalues of $g$ are distinct and $g,\zt$ span $V^*$, then $d(\za_g +\zt)$ is symplectic.
\(b) If, at least, $n-1$ eigenvalues are different and $g,\zt$ span a vector subspace of dimension $n-1$, then $rank(d(\za_g +\zt))=n^2 +n-2=dim{\mathcal Aff}(V)-2$. Moreover $Ker(d(\za_g +\zt))$ is not included in $sl(V)\zdi V$.
Set $E= {\mathcal Aff}(V)$, $E_1 ={\mathcal I}\zdi sl(V)$, $E_2 =V$, $\zl=d\za_g$ and $\zl_1 =d\zt$.
\(a) In this case there is a basis $\{v_1 ,\zps,v_n \}$ of $V$ such that $g=\zsu_{j=1}^{n}a_j v_j \zte v_{j}^{*}$, where $a_i \znoi a_j$ if $i\znoi j$, and $\zt=\zsu_{j=1}^{n} v_{j}^{*}$. Then $\{v_1 \zte v_{1}^{*},\zps,v_n \zte v_{n}^{*},v_1 ,\zps,v_n \}$ is a a basis of $Kerd\za_g$ and $$d\zt_{\zbv Kerd\za_g}=\zmm\zpizq\zsu_{j=1}^{n}(v_j \zte
v_{j}^{*})^{*}\zex v_{j}^{*}\zpder_{\zbv Kerd\za_g}$$ where $\{\{v_i \zte v_{j}^{*}\},i,j=1,\zps,n,v_1 ,\zps, v_n \}$ is the basis of ${\mathcal Aff}(V)$ associated to $\{v_1 ,\zps,v_n \}$ and $\{\{(v_i \zte v_{j}^{*})^{*}\},i,j=1,\zps,n,v_{1}^* ,\zps, v_{n}^* \}$ its dual basis. Now apply (a) of lemma \[lem-8\].
\(b) This time there are two possible cases. First assume that all eigenvalues of $g$ are distinct; then there exists a basis $\{v_1 ,\zps,v_n \}$ of $V$ such that $g=\zsu_{j=1}^{n}a_j v_j \zte v_{j}^{*}$, with $a_i \znoi a_j$ if $i\znoi j$, and $\zt=\zsu_{j=1}^{n-1} v_{j}^{*}$.
On the other hand $Kerd\za_g$ is the same as before while $$d\zt_{\zbv Kerd\za_g}=\zmm\zpizq\zsu_{j=1}^{n-1}(v_j \zte
v_{j}^{*})^{*}\zex v_{j}^{*}\zpder_{\zbv Kerd\za_g}$$ and it suffices applying (b) of lemma \[lem-8\] for computing the rank.
Now suppose that two eigenvalues are equal; in this case there exists a basis $\{v_1 ,\zps,v_n \}$ of $V$ such that $g=\zsu_{j=1}^{n-2}a_j v_j \zte v_{j}^{*}
+a_{n-1}(v_{n-1} \zte v_{n-1}^{*}+v_{n} \zte v_{n}^{*})$, with $a_i \znoi a_j$ if $i\znoi j$, and $\zt=\zsu_{j=1}^{n-1} v_{j}^{*}$. Then $\{v_1 \zte v_{1}^{*},\zps,v_{n-2} \zte v_{n-2}^{*},\{v_{k} \zte v_{r}^{*}\},
k,r=n-1,n,v_1 ,\zps,v_n \}$ is a basis of $Kerd\za_g$ and $d\zt_{\zbv Kerd\za_g}$ equals $$\zmm\zpizq\zsu_{j=1}^{n-2}(v_j \zte
v_{j}^{*})^{*}\zex v_{j}^{*}+(v_{n-1} \zte v_{n-1}^{*})^{*}\zex v_{n-1}^{*}
+(v_{n-1} \zte v_{n}^{*})^{*}\zex v_{n}^{*}\zpder_{\zbv Kerd\za_g}$$ and it is enough to apply (b) of lemma \[lem-8\] for computing the rank.
Finally note that in both cases $v_{n} \zte v_{n}^{*}$ belongs to $Ker(d\za_g +\zt)$.
\[eje-6\]
Let ${\mathcal Aff}_{0}(V)$ be the special affine algebra of $V$, that is ${\mathcal Aff}_{0}(V)=sl(V)\zdi V$. Consider a basis $\{v_1 ,\zps,v_n \}$ of $V$ and scalars $a_1 ,\zps,a_n$, $b_1 ,\zps,b_n$, $c_1 ,\zps,c_n$ as in lemma \[lem-9\]. Set $g=\zsu_{j=1}^{n}a_j v_j \zte v_{j}^{*}$, $h=\zsu_{j=1}^{n}b_j v_j \zte v_{j}^{*}$, $\zt=\zsu_{j=1}^{n} v_{j}^{*}$ and $\zm=\zsu_{j=1}^{n} c_j v_{j}^{*}$. Let ${\tilde\za}=\za_g +\zt$ and ${\tilde\zb}=\za_h +\zm$ that are $1$-forms on ${\mathcal Aff}(V)$; then $\za={\tilde\za}_{\zbv {\mathcal Aff}_{0}(V)}$ and $\zb={\tilde\zb}_{\zbv {\mathcal Aff}_{0}(V)}$ are contact forms on ${\mathcal Aff}_{0}(V)$. Indeed, we prove it for $\za$ the other case is analogous. By (a) of proposition \[pro-6\] $d\tilde\za$ is symplectic, so there is $z\zpe{\mathcal Aff}(V)$ such that $i_z d\tilde\za=\tilde\za$, which implies that $L_z ((d\tilde\za)^{n(n+1)/2})\znoi 0$ that is to say $z\znope{\mathcal Aff}_{0}(V)$. But $z$ is a basis of the kernel of ${\tilde\za}\zex(d{\tilde\za})^{(n(n+1)/2)-1}$, hence its restriction to ${\mathcal Aff}_{0}(V)$, which equals ${\za}\zex(d{\za})^{(n(n+1)/2)-1}$, is a volume form.
Moreover $(d\za,d\zb)$ is generic. Let us see it. Clearly $rank(d\za)$ and $rank(d\zb)$ equal $dim{\mathcal Aff}_{0}(V)-1$, so it suffices to show that the rank of $d\za+td\zb$, $t\zpe{\mathbb C}-\{0\}$, is maximal. If $d(\tilde\za+t\tilde\zb)$ is symplectic reason as before. If not, taking into account that $\tilde\za+t\tilde\zb=\za_{g+th}+(\zt+t\zm)$, $g+th=\zsu_{j=1}^{n}(a_j +tb_j) v_j \zte v_{j}^{*}$ and $\zt+t\zm=\zsu_{j=1}^{n}(1+t c_j) v_{j}^{*}$, by proposition \[pro-6\] and lemma \[lem-9\] we have two cases:
1. The family $a_1 +tb_1 ,\zps,a_n +tb_n$ just includes two equal elements but no $1+tc_i$, $i=1,\zps,n$, vanishes.
2. All $a_1 +tb_1 ,\zps,a_n +tb_n$ are distinct but one element of the family $1+tc_1 ,\zps,1+tc_n$ vanishes.
In both cases, by (b) of proposition \[pro-6\], $rank(d(\tilde\za+t\tilde\zb))=n^2 +n-2$ and $Kerd(\tilde\za+t\tilde\zb)\znoco{\mathcal Aff}_{0}(V)$, so $rank(d(\za+t\zb))=rank(d(\tilde\za+t\tilde\zb))=dim{\mathcal Aff}_{0}(V)-1$.
Summing up, one may apply proposition \[pro-5\] to ${\mathcal Aff}_{0}(V)$ and $(\za,\zb)$.
\[eje-7\]
Consider $\{v_ ,\zps,v_n \}$, $a_1 ,\zps,a_n$, $b_1 ,\zps,b_n$, $c_1 ,\zps,c_n$ as in the foregoing example. Let $a$ be any scalar. Set ${\mathcal A}(V,a)={\mathcal Aff}(V)\zdi \mathbb K$ endowed with the bracket defined below. Regard ${\mathcal Aff}(V)$, $\mathbb K$ as subsets of ${\mathcal A}(V,a)$ and ${\mathcal Aff}(V)^*$, ${\mathbb K}^*$ like subsets of ${\mathcal A}(V,a)^*$ in the obvious way. Let $e$ be the unit of $\mathbb K$ seen in ${\mathcal A}(V,a)$ and $e^*$ the element of ${\mathcal A}(V,a)^*$ given by $e^* (e)=1$, $e^* ({\mathcal Aff}(V))=0$. On the other hand let $id\zpe{\mathcal Aff}(V)$ be the morphism identity of $V$, that is $id=\zsu_{j=1}^{n} v_j \zte v_{j}^{*}$. Now we define a structure of Lie algebra on ${\mathcal A}(V,a)$, for which ${\mathcal Aff}(V)$ is a subalgebra, by putting $[id,e]=-ae$ and $[sl(V)\zdi V,e]=0$.
Set $\za_1 =\tilde\za$ and $\zb_1 =\tilde\zb +e^*$ where $\tilde\za,\tilde\zb$, defined in the preceding example, are now regarded as elements of ${\mathcal A}(V,a)^*$. Note that $de^* =(a/n)\zsu_{j=1}^{n}(v_j \zte v_{j}^{*})^* \zex e^*$, so $$(d\zb_1 )^{n(n+1)/2}=(d{\tilde\zb})^{n(n+1)/2}\hskip6truecm$$ $$+aC\zpizq\zsu_{j=1}^{n}(v_j \zte v_{j}^{*})^*\zpder
\zex(d{\tilde\zb})^{(n(n+1)/2)-1}\zex e^* \znoi 0$$ where $C$ is non-zero constant, while $$\zb_1 \zex(d\zb_1 )^{n(n+1)/2}=(aC'+1)(d{\tilde\zb})^{n(n+1)/2}\zex e^*$$ where $C'$ is another constant (perhaps zero). As $d\tilde\zb$ is symplectic on ${\mathcal Aff}(V)$ it follows that $\zb_1$ is a contact form on ${\mathcal A}(V,a)$ if and only if $aC'+1\znoi 0$.
On the other hand $(d\za_1 ,d\zb_1 )$ is generic if $a\znoi 0$. Indeed, since $d\tilde\za$ and $d\tilde\zb$ are symplectic on ${\mathcal Aff}(V)$ it is enough to show that $d\za_1 +td\zb_1$, $t\zpe{\mathbb C}-\{0\}$, has maximal rank. If $d\tilde\za+td\tilde\zb$ is symplectic on ${\mathcal Aff}(V)$ it is clear. Otherwise by proposition \[pro-6\] the rank of $d\tilde\za+td\tilde\zb$ on ${\mathcal Aff}(V)$ equals $n^2 +n-2$ and its kernel is not included in $sl(V)\zdi V$. But, always in ${\mathcal Aff}(V)$, $sl(V)\zdi V=Ker(\zsu_{j=1}^{n}(v_j \zte v_{j}^{*})^* )$. Since $d\za_1 +td\zb_1$ equals $d\tilde\za+td\tilde\zb$ regarded on ${\mathcal A}(V,a)$ plus $(at/n)\zsu_{j=1}^{n}(v_j \zte v_{j}^{*})^* \zex e^*$, it follows that the rank of $d\za_1 +td\zb_1$ equals that of $d\tilde\za+td\tilde\zb$ plus two, that is $n^2 +n=dim{\mathcal A}(V,a)-1$.
Finally observe that ${\mathcal A}(V,a)$ is not unimodular if $a\znoi n$. Thus one may choose $a$ in such a way that $\zb_1$ is a contact form, $(d\za_1 ,d\zb_1 )$ is generic and ${\mathcal A}(V,a)$ is not unimodular; in this case proposition \[pro-3\] applied to ${\mathcal A}(V,a)$ and $(\za_1 ,\zb_1 )$ shows the existence on ${\mathcal A}(V,a)^*$ of linear Poisson pairs which are generic and non-flat.
Lie Poisson pairs and Nijenhuis torsion {#sec-8}
=======================================
In this section a method for constructing Lie Poisson pairs from an endomorphism with vanishing Nijenhuis is given. Recall that the Nijenhuis torsion of a $(1,1)$-tensor field $J$ on a differentiable manifold is the $(1,2)$-tensor field $N_J$ defined by $N_J (X,Y)=[JX,JY]+J^2 [X,Y]-J[X,JY]-J[JX,Y]$.
Let ${\mathcal A}$ be a Lie algebra and $\zf$ an endomorphism of ${\mathcal A}$ as vector space; since $\zf$ can be seen like a left invariant $(1,1)$-tensor field on some Lie group, we may define its Nijenhuis torsion, which in linear terms is given by the formula $N_\zf (a,b)=[\zf a,\zf b]+\zf ^2 [a,b]-\zf [a,\zf b]-\zf [\zf a,b]$.
Set $[a,b]_1 = [a,\zf b]+[\zf a,b]-\zf[a,b]$, $a,b\zpe{\mathcal A}$; then $[a,b]+t[a,b]_1 = [a,(I+t\zf )b]+[(I+t\zf )a,b]-(I+t\zf )[a,b]$. Now assume $N_\zf =0$; if $I+t\zf$ is invertible then $(I+t\zf )^{-1}[(I+t\zf )a,(I+t\zf )b]
= [a,(I+t\zf )b]+[(I+t\zf )a,b]-(I+t\zf )[a,b]$ since $N_{(I+t\zf )}=0$. Therefore $[\quad,\quad]+t[\quad,\quad]_1$ defines a structure of Lie algebra and $(I+t\zf)\zdp({\mathcal A},[\quad,\quad]+t[\quad,\quad]_1 )\zfl
({\mathcal A},[\quad,\quad])$ is an isomorphism of Lie algebras. As $(I+t\zf)$ is invertible for almost every $t\zpe\mathbb K$, it follows that $N_\zf =0$ implies that $[\quad,\quad]_1$ is a Lie bracket which is compatible with $[\quad,\quad]$. In this case the couple of Lie-Poisson structures $(\zL,\zL_1 )$, associated to $[\quad,\quad]$ and $[\quad,\quad]_1$ respectively, is a Lie Poisson pair. Moreover $(I+t\zf)^* \zdp({\mathcal A}^* ,\zL)\zfl({\mathcal A}^* ,\zL+t\zL_1 )$ is a Poisson diffeomorphism whenever $(I+t\zf)$ is invertible.
\[eje-8\]
Consider the truncated Lie algebra $ {\mathcal A}$ of dimension $m=2n-1\zmai 5$ and the basis $\{e_1 ,\zps,e_m \}$ given in example \[eje-1\]. Let $\zf=e_n \zte e_{m}^*$; then $N_\zf =0$. Besides the associated Lie algebra $({\mathcal A},[\quad,\quad]_1 )$ is unimodular and $e_n$ is a basis of its center.
In coordinates $(x_1 ,\zps,x_m )$ with respect to the dual basis $\{e_{1}^* ,\zps,e_{m}^* \}$ one has: $$\zL_1 = x_n \zsu_{i=2}^{n-1}2(i-n){\frac {\zpar} {\zpar x_{i}}}
\zex{\frac {\zpar} {\zpar x_{2n-i}}}\hskip 4 truecm$$ $$\hskip 1truecm+\zpizq(1-n)x_n {\frac {\zpar} {\zpar x_{1}}}+ \zsu_{i=2}^{n-1}(n-i)
x_{n+i-1}{\frac {\zpar} {\zpar x_{i}}} \zpder\zex {\frac {\zpar} {\zpar x_{2n-1}}}$$ while $$\zL_{1}^{n-1}=Cx_{n}^{n-1}{\frac {\zpar} {\zpar x_{1}}}\zex\zpu\zpu\zpu\zex
{\frac {\zpar} {\zpar x_{n-1}}}\zex{\frac {\zpar} {\zpar x_{n+1}}}
\zex\zpu\zpu\zpu\zex {\frac {\zpar} {\zpar x_{2n-1}}}$$ where $C$ is a non-vanishing constant. Thus $rank\zL_1 =2n-2$ if $x_n \znoi 0$. Observe that $dx_n$ is a Casimir of $\zL_1$.
On the other hand $$\zL= \zsu_{1\zmei i<j\zmei 2n-1;\, i+j\zmei 2n}(j-i)x_{i+j-1}{\frac {\zpar} {\zpar x_{i}}}
\zex{\frac {\zpar} {\zpar x_{j}}}\hskip 2truecm$$ $$= {\frac {\zpar} {\zpar x_{1}}}\zex\zpizq x_{2}{\frac {\zpar} {\zpar x_{2}}}+\zps
+(2n-2)x_{2n-1}{\frac {\zpar} {\zpar x_{2n-1}}}\zpder\hskip 2truecm$$ $$\hskip 1truecm+ {\frac {\zpar} {\zpar x_{2}}}\zex\zpizq x_{4}{\frac {\zpar}
{\zpar x_{3}}}+\zps+(2n-4)x_{2n-1}{\frac {\zpar} {\zpar x_{2n-2}}}\zpder$$ $$\hskip 2truecm+\zps+ {\frac {\zpar} {\zpar x_{n-1}}}\zex\zpizq x_{2n-2}
{\frac {\zpar} {\zpar x_{n}}}+2x_{2n-1}{\frac {\zpar} {\zpar x_{n+1}}}\zpder$$
Therefore $\zL^{n-1}(x)\znoi 0$ if and only if the vector fields given by the parentheses are linearly independent modulo $(\zpar/\zpar x_1 ),\zps,(\zpar/\zpar x_{n-1})$, which just happens when, at least, $x_{2n-2}\znoi 0$ or $x_{2n-1}\znoi 0$.
Let $A=\{x\zpe{\mathcal A}^* \zbv x_n \znoi 0,x_{2n-2}\znoi 0\}$. Since $\zf$ is nilpotent $(I+t\zf)$ is always invertible; moreover $(I+t\zf)^* (x)=(x_1 ,\zps,x_{2n-2},x_{2n-1}+tx_n )$, so $(I+t\zf)^* (A)=A$, which implies that $(\zL+t\zL_1 )^{n-1}(x)\znoi 0$, $t\zpe\mathbb K$, $x\zpe A$, since $(I+t\zf)^*$ transforms $\zL$ in $\zL+t\zL_1$. Therefore $(\zL,\zL_1 )$ is generic on $A$. Finally by lemmas \[lem-3\] and \[lem-5\] and remark \[rem-5\] (first paragraph) the Lie Poisson pair $(\zL,\zL_1 )$ is not flat at any point of $A$, because $dx_n$ is a Casimir of $\zL_1$ and $rank([e_n ,\quad])\zmai 2$.
\[pro-7\] Let ${\mathcal A}$ be a non-unimodular Lie algebra of dimension $m=2n-1\zmai 3$. If there exist $\za,\zb\zpe{\mathcal A}^*$ such that $\zb$ is a contact form and $(d\za,d\zb)$ is generic, then on the dual space of the product Lie algebra ${\mathcal A}\zpor {\mathcal Aff}({\mathbb K})$ there exists some generic and non-flat Lie Poisson pair.
Consider a basis $\{\tilde e_1 ,\zps,\tilde e_m \}$ of ${\mathcal A}$ and an another one $\{\tilde f_1 ,\tilde f_2 \}$ of ${\mathcal Aff}({\mathbb K})$ such that $\zb={\tilde e}_{m}^*$, $d\zb=\zsu_{i=1}^{n-1}{\tilde e}_{2i-1}^*
\zex {\tilde e}_{2i}^*$ and $[\tilde f_1 ,\tilde f_2 ]=\tilde f_1$. Set $e_i =(\tilde e_i ,0)$, $i=1,\zps,m$, $f_j =(0,\tilde f_j )$, $=1,2$, $\za_1 =\za+f_{1}^*$ and $\zb_1 =\zb$ where $\za,\zb$ are regarded this time like $1$-forms on ${\mathcal A}\zpor {\mathcal Aff}({\mathbb K})$ in the obvious way (that is $\za(\{0\}\zpor {\mathcal Aff}({\mathbb K}))
=\zb(\{0\}\zpor {\mathcal Aff}({\mathbb K}))=0$.) Then $\{e_1 ,\zps,e_m ,f_1 ,f_2 \}$ is a basis of ${\mathcal A}\zpor {\mathcal Aff}({\mathbb K})$; let $\{e_{1}^* ,\zps,e_{m}^* ,f_{1}^* ,f_{2}^* \}$ be its dual basis and $(x_1 ,\zps,x_m ,y_1 ,y_2 )$ the associated coordinates on $({\mathcal A}\zpor {\mathcal Aff}({\mathbb K}))^*$
It is easily checked that the Nijenhuis torsion of $f_1\zte\zb_1$ vanishes, so we have a second Lie bracket $[\quad,\quad]_1$, which is compatible with the product bracket $[\quad,\quad]$. Let $d,d_1$ the respective exterior derivatives and $\zL,\zL_1$ the Lie-Poisson structures on $({\mathcal A}\zpor {\mathcal Aff}({\mathbb K}))^*$. We shall show that the Lie Poisson pair $(\zL,\zL_1 )$ is generic and non-flat.
First one checks the genericity of $(\zL,\zL_1 )(\za_1 )$, which is equivalent to that of $(d\za_1 ,d_1 \za_1)$. As $d\za_1 =d\za -f_{1}^* \zex f_{2}^*$ and $d_1 \za_1 =-d\zb-e_{m}^* \zex f_{2}^*$, where $d\za$ and $d\zb$ are computed on ${\mathcal A}$ and then extended to ${\mathcal A}\zpor {\mathcal Aff}({\mathbb K})$ in the natural way, the rank of both $2$-forms equals $2n$. Therefore it will suffices to show that $rank(d\za_1 +td_1 \za_1)=2n$, $t\zpe{\mathbb C} -\{0\}$, which is obvious since $d\za_1 +td_1 \za_1
=(d\za -td\zb)-(f_{1}^* +te_{m}^* )\zex f_{2}^*$ and $(d\za,d\zb)$ is generic on ${\mathcal A}$.
Let $(\zw,\zw_1 ,\zW)$ be a representative of $(\zL,\zL_1 )$. As $({\mathcal A}\zpor {\mathcal Aff}({\mathbb K}),[\quad,\quad]_1 )$ is unimodular $d\zw_1 =0$; on the other hand $dy_1$ is a Casimir of $\zL_1$ since $f_1$ belongs to the center of this algebra. By lemma \[lem-5\] the vector field $X=\zsu_{j=1}^{m}(tr[e_j ,\quad])(\zpar/\zpar x_j )
+\zsu_{k=1}^{2}(tr[f_k ,\quad])(\zpar/\zpar y_k )$ is a basis of $Kerd\zw$; moreover since ${\mathcal A}$ is non-unimodular some $(tr[e_j ,\quad])(\zpar/\zpar x_j )$ does not vanish, so $X\zex(\zpar/\zpar y_1 )\znoi 0$. Assume that $(\zL,\zL_1 )$ is flat at $\za_1$; then, by lemma \[lem-3\], $X$ and $\zL(dy_1 ,\quad)=y_1 (\zpar/\zpar y_1 )$ have to be proportional, so $X\zex(\zpar/\zpar y_1 )=0$ [*contradiction*]{}.
\[rem-6\]
The hypotheses of propositions \[pro-4\] and \[pro-7\] are rather close and almost the same examples illustrate both results. Thus proposition \[pro-7\] may be applied to example \[eje-3\] when $[v,\quad]\zdp {\mathcal A}_0
\zfl {\mathcal A}_0$ has non-vanishing trace, to example \[eje-4\] if $a\znoi -1,0,(n-1)^{-1},(n-2)^{-1}$, always to example \[eje-5\] and, finally, to example \[eje-7\] when $a$ is chosen in such a way that ${\mathcal A}(V,a)$ is non-unimodular, $\zb_1$ is a contact form and $(d\za_1 ,d\zb_1 )$ generic.
Of course example \[eje-6\] has to be excluded since its Lie algebra is unimodular.
Generic non-flat linear Poisson pairs in dimension 3 {#sec-9}
====================================================
The purpose of this section and the next one is to illustrate proposition \[pro-2\]. Of course even if the our approach is new, as dimension three has been well studied, most of the results of both sections are known, perhaps stated in a different way. Let ${\mathcal A}$ be a $3$-dimensional non-unimodular Lie algebra and $I_0$ its unimodular ideal, whose dimension equals two (recall that unimodular implies flatness.) As Lie algebra $I_0$ itself is unimodular so abelian; therefore if $u,v\znope I_0$ then $[v,\quad]=s[u,\quad]$ for some $s\zpe{\mathbb K}-\{0\}$. Thus, up to non-vanishing multiplicative constant, one obtain a well-determined endomorphism $[u,\quad]_{\zbv I_0}\znoi 0$.
\[pro-8\] Consider a $3$-dimensional non-unimodular Lie algebra ${\mathcal A}$. Then on ${\mathcal A}^*$ there exists some generic linear Poisson pair that is non-flat if and only if the endomorphism $[u,\quad]_{\zbv I_0}$, $u\znope I_0$, is not a multiple of identity.
Let us see that. Consider a basis $\{e_1 ,e_2 ,e_3 \}$ of ${\mathcal A}$ such that $\{e_2 ,e_3 \}$ is a basis of $I_0$. Set $[e_1 ,e_2 ]=a_{22}e_2 +a_{23}e_3$ and $[e_1 ,e_3 ]=a_{32}e_2 +a_{33}e_3$; note that $a_{22}+a_{33}\znoi 0$ since ${\mathcal A}$ is non-unimodular. Moreover $\{e_{1}^* \zex e_{2}^* ,e_{1}^* \zex e_{3}^* \}$ is a basis of the vector space of $2$-cocycles. In coordinates $(x_1 ,x_2 ,x_3 )$ associates to the dual basis of $\{e_1 ,e_2 ,e_3 \}$ one has
$\zL=(\zpar/\zpar x_1 )\zex[(a_{22}x_2 +a_{23}x_3)(\zpar/\zpar x_2 )
+[(a_{32}x_2 +a_{33}x_3)(\zpar/\zpar x_3 )]$,
so $\zw= -(a_{32}x_2 +a_{33}x_3)dx_2 +(a_{22}x_2 +a_{23}x_3)dx_3$ and $\zW=dx_1 \zex dx_2 \zex dx_3$ represent $\zL$.
In turn, any constant and $\zL$-compatible Poisson structure $\zL_1$ writes $\zL_1 =(\zpar/\zpar x_1 )\zex[b_2 (\zpar/\zpar x_2 )
+b_3 (\zpar/\zpar x_3 )]$, $b_2 ,b_3 \zpe\mathbb K$, and $\zw_1 =-b_3 dx_2 +b_2 dx_3$, $\zW$ represent it (we do not specify the dependence on $(b_2 ,b_3 )$ of $\zL_1$.)
On the other hand $(\zL,\zL_1 )(x)$ is generic if and only if $(a_{22}b_{3}-a_{32}b_{2})x_2 +(a_{23}b_{3}-a_{33}b_{2})x_3 \znoi 0$; therefore the set of $(b_2 ,b_3 )\zpe{\mathbb K}^2$ such that $(\zL,\zL_1 )$ has no generic point is always included in a vector line of ${\mathbb K}^2$ (given for example by $a_{22}b_{3}-a_{32}b_{2}=0$ or by $a_{23}b_{3}-a_{33}b_{2}=0$; at least one of these equations is not trivial since some $a_{ij}$ does not vanish.)
As $(\zL,\zL_1 )$ is compatible and $d\zw_1 =0$, the $1$-form $\zl$ given by proposition \[pro-2\] is a functional multiple of $\zw_1$. Now a computation shows that
$\zl=(a_{22}+a_{33})[(a_{32}b_{2}-a_{22}b_{3})x_{2}
+(a_{33}b_{2}-a_{23}b_{3})x_{3}]^{-1}(-b_3 dx_2 +b_2 dx_3)$.
Note that $\zl$ is just defined at any generic point of $(\zL,\zL_1 )$. It is easily seen that $d\zl=0$ if and only if
3truecm $a_{32}b_{2}^{2}+(a_{33}-a_{22})b_{2}b_{3}-a_{23}b_{3}^{2}=0$.
When $[u,\quad]_{\zbv I_0}$, $u\znope I_0$ is a multiple of identity, automatically $d\zl=0$ and $(\zL,\zL_1 )$ is flat. Otherwise $a_{32}b_{2}^{2}+(a_{33}-a_{22})b_{2}b_{3}-a_{23}b_{3}^{2}$ can be regarded as a non-trivial quadratic form in $(b_2 ,b_3 )$, which allows to choose $(b_2 ,b_3 )\zpe{\mathbb K}^2$ in such a way that the set of generic points of $(\zL,\zL_1 )$ is not empty and $a_{32}b_{2}^{2}+(a_{33}-a_{22})b_{2}b_{3}-a_{23}b_{3}^{2}\znoi0$, so $d\zl\znoi 0$ and $(\zL,\zL_1 )$ is non-flat.
Generic non-flat Lie Poisson pairs in dimension 3 {#sec-10}
=================================================
This time consider a $3$-dimensional real or complex vector space ${\mathcal A}$ endowed with two compatible Lie brackets $[\quad,\quad]$, $[\quad,\quad]_1$, and their respective Lie-Poisson structures $\zL,\zL_1$ on ${\mathcal A}^*$. One wants to describe when $(\zL,\zL_1 )$ is generic and non-flat, therefore all flat cases will be put aside. Observe that at least one of the bracket, for example $[\quad,\quad]$, has to be non-unimodular, otherwise $(\zL,\zL_1 )$ is flat. Now replacing $[\quad,\quad]_1$ by $[\quad,\quad]_1 +s[\quad,\quad]$ for a suitable scalar $s$ if necessary, allows to suppose non-unimodular $[\quad,\quad]_1$ as well. Let $I$ be the unimodular ideal of $[\quad,\quad]$ and $I_1$ that of $[\quad,\quad]_1$; then $(I ,[\quad,\quad])$ and $(I_1 ,[\quad,\quad]_1 )$ are abelian and $2$-dimensional (see the foregoing section.) Moreover $I =I_1$.
Indeed, if $I \znoi I_1$ then ${\mathcal A}=I +I_1$ and there exists a basis $\{e_1 ,e_2 ,e_3 \}$ of ${\mathcal A}$ such that $\{e_1 ,e_2 \}$ is a basis of $I$ and $\{e_2 ,e_3 \}$ of $I_1$. Let $(x_1 ,x_2 ,x_3 )$ be the coordinates of ${\mathcal A}^*$ relative to the dual basis. With respect to $\zW=dx_1 \zex dx_2 \zex dx_3$ the Lie-Poisson structures $\zL$ and $\zL_1$ are represented by $\zw=(a_{11}x_{1}+a_{12}x_{2})dx_{1}
+(a_{21}x_{1}+a_{22}x_{2})dx_{2}$ and $\zw_1 =(b_{22}x_{2}+b_{23}x_{3})dx_{2}+(b_{32}x_{2}+b_{33}x_{3})dx_{3}$ respectively, where $a_{ij},b_{kr}$ are scalars (see section \[sec-9\] again.)
As $d\zw=adx_1 \zex dx_2 \znoi 0$ and $d\zw_1 =bdx_2 \zex dx_3 \znoi 0$ because $[\quad,\quad]$ and $[\quad,\quad]_1$ are non-unimodular, the $1$-form $\zl$ given by proposition \[pro-2\] necessarily equals $fdx_2$ for some function $f$. But $fdx_2 \zex\zw=d\zw=adx_1 \zex dx_2$ is closed, so $f=f(x_1 ,x_2 )$. On the other hand, reasoning with $\zw_1$ as before shows that $f=f(x_2 ,x_3 )$. Therefore $f=f(x_2 )$; but in this case $\zl=f(x_2 )dx_2$ is closed and $(\zL,\zL_1 )$ flat.
In other words, there exists a $2$-dimensional vector subspace $I$ of ${\mathcal A}$ which the unimodular ideal of both brackets. Therefore given any $u\znope I$ the structure is determined by the restriction of $[u,\quad]$ and $[u,\quad]_1$ to $I$. These two endomorphisms of $I$ have non-vanishing trace (obviously the trace before and after restriction to $I$ is the same), which shows the existence of an unimodular bracket $s[\quad,\quad]+s_1 [\quad,\quad]_1$, for some $s,s_1 \zpe{\mathbb K}-\{0\}$. Note that up to non-zero multiplicative constant this bracket is unique. Now replacing $[\quad,\quad]_1$ by $s[\quad,\quad]+s_1 [\quad,\quad]_1$ and calling it $[\quad,\quad]_1$ again, allows to suppose unimodular $[\quad,\quad]_1$ (of course one may consider $t[\quad,\quad]+t_1 [\quad,\quad]_1$, $t\zpe{\mathbb K}-\{0\}$ and $t_1 \zpe{\mathbb K}$, instead $[\quad,\quad]$ if desired.)
[*In short, up to linear combinations of brackets, our problem is reduced to consider two Lie brackets $[\quad,\quad]$ and $[\quad,\quad]_1$, non-unimodular the first one and unimodular but non-zero the second one, such that the unimodular ideal $I$ of $[\quad,\quad]$ is, at the same time, an abelian ideal of $[\quad,\quad]_1$. Note that $[\quad,\quad]$, $[\quad,\quad]_1$ automatically are compatible.*]{}
Observe that the endomorphism $[u,\quad]_{1\zbv I}$, $u\znope I$, is unique up to non-zero multiplicative constant, while $[u,\quad]_{\zbv I}$ is determined up to non-vanishing multiplicative constant plus any multiple of $[u,\quad]_{1\zbv I}$. Thus the existence of some eigenvector of $[u,\quad]_{1\zbv I}$ (perhaps with complex eigenvalue in the real case) which is not eigenvector of $[u,\quad]_{\zbv I}$ is independent of the choice of $u$, $[\quad,\quad]$ and $[\quad,\quad]_1$; in other words it is an intrinsic property of the Lie Poisson pair (more exactly of the Lie Poisson pencil.)
\[pro-9\] The foregoing Lie Poisson pair $(\zL,\zL_1 )$, on ${\mathcal A}^*$, is generic and non-flat if and only if there exists some eigenvector of $[u,\quad]_{1\zbv I}$ that is not an eigenvector of $[u,\quad]_{\zbv I}$.
Consider a basis $\{e_1 ,e_2 ,e_3 \}$ of ${\mathcal A}$ such that $\{e_2 ,e_3 \}$ is a basis of $I$. As $tr([e_1 ,\quad]_{1\zbv I})=0$ but $[e_1 ,\quad]_{1\zbv I}\znoi 0$, the vector space $I$ is cyclic and one can choose $\{e_2 ,e_3 \}$ in such a way that $[e_1 ,e_2 ]_1 =e_3$ and $[e_1 ,e_3 ]_1 =be_2$. Set $[e_1 ,e_2 ]=a_{22}e_{2}+a_{23}e_{3}$, $[e_1 ,e_3 ]=a_{32}e_{2}+a_{33}e_{3}$. In coordinates $(x_1 ,x_2 ,x_ 3)$ on ${\mathcal A}^*$ associated to the dual basis, and with respect to $\zW=dx_1 \zex dx_2 \zex dx_3$, the Poisson structures $\zL$ and $\zL_1$ are respectively represented by $\zw=-(a_{32}x_{2}+a_{33}x_{3})dx_2 +(a_{22}x_{2}+a_{23}x_{3})dx_3$ and $\zw_1 =-bx_2dx_2 +x_3 dx_3$.
Note that $(\zL,\zL_1 )(x)$ is generic just when $(\zw\zex\zw_1 )(x)\znoi 0$, that is to say just when $P(x)\znoi 0$ where $P=a_{22}bx_{2}^2 +(a_{23}b-a_{32})x_2 x_3
-a_{33}x_{3}^2$. Therefore $(\zL,\zL_1 )$ is generic if and only if $P$ is not identically zero.
As $d\zw_1 =0$ the $1$-form $\zl$ of proposition \[pro-2\] equals $f\zw_1$ for some function $f$. On the other hand $d\zw=(a_{22}+a_{33})dx_{2}\zex dx_3
=\zl\zex\zw=f\zw_1 \zex\zw$ and a computation shows that $f=-(a_{22}+a_{33})P^{-1}$. Thus $(\zL,\zL_1 )$ will be flat if and only if $\zl=-(a_{22}+a_{33})P^{-1}\zw_1$ is closed. Since $\zw_1 =-(1/2)dQ$ where $Q=bx_{2}^2 -x_{3}^2$, this is equivalent to say that $dP\zex dQ=0$; that is to say, if and only if $P$ and $Q$ are proportional as polynomials. Since $Q$ does not identically vanish, the foregoing condition is equivalent to the existence of $c\zpe\mathbb K$ such that $(a_{22}b,a_{23}b-a_{32},-a_{33})=c(b,0,-1)$.
First suppose $b=0$. Then $c$ does not exist just when $a_{32}\znoi 0$, that is just when $e_3$, which determines the single eigendirection of $[e_1 ,\quad]_{1\zbv I}$, is not an eigenvector of $[e_1 ,\quad]_{\zbv I}$. Observe that $a_{32}\znoi 0$ implies $P\znoi 0$.
Now assume $b\znoi 0$. Then $P\znoi 0$ since $a_{22}+a_{33}\znoi 0$. By replacing $[\quad,\quad]$ by $[\quad,\quad]-a_{23}[\quad,\quad]_1$ if necessary, one may suppose $a_{23}=0$ without lost of generality; in this case $e_2$ is an eigenvector of $[e_1 ,\quad]_{\zbv I}$. On the other hand $(a_{22}b,-a_{32},-a_{33})=c(b,0,-1)$ if and only if $a_{32}=0$ and $a_{22}=a_{33}$; that is if and only if $[e_1 ,\quad]_{\zbv I}$ is multiple of identity. Consequently any eigenvector of $[e_1 ,\quad]_{1\zbv I}$ is eigenvector of $[e_1 ,\quad]_{\zbv I}$ when $c$ exists.
Conversely, if any eigenvector of $[e_1 ,\quad]_{1\zbv I}$ is eigenvector of $[e_1 ,\quad]_{\zbv I}$ then, as $b\znoi 0$, there exist two eigendirections of $[e_1 ,\quad]_{\zbv I}$, coming from $[e_1 ,\quad]_{1\zbv I}$, which are different from the direction associated to $e_2$; therefore $[e_1 ,\quad]_{\zbv I}$ has three distinct eigendirections and necessarily is multiple of identity.
[99]{}
I.M. Gelfand, I.Ya. Dorfman, *Hamiltonian operators and algebraic structures related to them*, Funct. Anal. Appl., **13** (1979) 248-262.
I.M. Gelfand, I. Zakharevich, *Webs, Veronese curves and bi-Hamiltonian systems*, J. Funct. An., **99** (1991) 150-178.
I.M. Gelfand, I. Zakharevich, *On the local geometry of a bi-Hamiltonian structure*, Corvin, L. (ed) et al., The Gelfand Seminars, 1990-92, Birk$\rm\ddot{a}u$ser, Basel (1993) 51-112.
A. Izosimov, *Curvature of Poisson pencils in dimension three*, Differ. Geom. Appl. **31** (2013) 557-567.
A. Konyaev, *Algebraic and geometric properties of systems obtained by argument translation method*, PhD thesis, Moscow State University, 2011.
F. Magri, *A simple model of integrable Hamiltonian equations*, J. Math. Phys. **19** (2013) 1156-1162.
A. Panasyuk, *Veronese Webs for Bi-Hamiltonian Structures of Higher Corank*, Banach Center Publ. **51** (2000), 251-261.
M.-H. Rigal, *Systèmes bihamiltoniens en dimension impaire*, Ann. Scient. Éc. Norm. Sup., IV. Sér. **31** (1998) 345-359.
M.-H. Rigal, *Veronese webs and transversally Veronese foliations*, In: *Web theory and related topics* (Toulouse, 1996), 205-221, World Sci. Publishing, River Edge, NJ, 2001.
F.-J. Turiel *Classification locale simultanée de deux formes symplectiques compatibles*, Manuscripta Math. **82** (1994) 349-362.
F.-J. Turiel, *$C^{\zinf}$-équivalence entre tissus de Veronese et structures bihamiltoniennes*, C.R. Acad. Sci. Paris Série I **328** (1999) 891-894.
F.-J. Turiel, *On the local theory of Veronese webs*, arXiv: 1001.3098v1\[math.DG\], 2009.
F.-J. Turiel, *Décomposition locale d’une structure bihamiltonienne en produit Kronecker-symplectique*, C.R. Acad. Sci. Paris Série I **349** (2011) 85-87.
I. Zakharevich, *Kronecker webs, bi-Hamiltonian structures, and the method of argument translation*, Transform. Groups **6** (2001) 267-300.
|
---
abstract: 'We recently studied a doped two-dimensional bosonic Hubbard model with two hard-core species, with different masses, using quantum Monte Carlo simulations \[Phys. Rev. B 88, 161101(R) (2013)\]. Upon doping away from half-filling, we find several distinct phases, including a phase-separated ferromagnet with Mott behavior for the heavy species and both Mott insulating and superfluid behaviors for the light species. Introducing polarization, an imbalance in the population between species, we find a fully phase-separated ferromagnet. This phase exists for a broad range of temperatures and polarizations. By using finite size scaling of the susceptibility, we find a critical exponent which is consistent with the two-dimensional Ising universality class. Significantly, since the global entropy of this phase is higher than that of the ferromagnetic phase with single species, its experimental observation in cold atoms may be feasible.'
author:
- Kalani Hettiarachchilage
- 'Valéry G. Rousseau'
- 'Ka-Ming Tam'
- Mark Jarrell
- Juana Moreno
title: 'Ferromagnetic phase in the polarized two-species bosonic Hubbard Model'
---
Introduction
============
One of the frontiers of condensed matter physics is the study of competing quantum phases such as coexistent and inhomogeneous phases, quantum criticality, and secondary ordered phases close to quantum critical points. [@Yunoki; @Yunoki2; @Uehara; @Elbio1; @Coleman; @Si] These exotic phenomena in strongly correlated systems occur due to the competition and cooperation between the spin, charge, lattice, and orbital degrees of freedom. [@Elbio] Unfortunately, it is often difficult to differentiate the effect of these degrees of freedom in real materials. However, the advance of optical lattice experiments provides a tantalizing opportunity to study competing phases via controlled external parameters. [@Jaksch; @Hofstetter; @Esslinger]
The experimental tunability of Hamiltonian parameters using laser and magnetic fields [@Timmermans99; @kohler:1311] allows the realization of strongly correlated model Hamiltonians. The realization of the Bose-Hubbard model using ultra-cold atoms on optical lattices [@Greiner] has led to the observation [@Fisher; @Batrouni] of the Mott-insulator to superfluid phase transition. The Mott insulator phase is characterized by commensurate occupations, gapped excitations and incompressibility in the strong coupling regime. The superfluid phase is characterized by Bose-Einstein condensation, gapless excitations and finite compressibility in the weak coupling regime.
This success has spurred interest in mixtures of atoms which can give rise to even more interesting and complex phases. These include mixtures of bosonic and fermionic atoms [@Modugno; @Albus; @Modugno1; @Ospelkaus] (a Bose-Fermi mixture) and mixtures of two different bosonic species (a Bose-Bose mixture). [@Roati; @Thalhammer; @Papp] Moreover, experimental studies of $^{85}Rb$-$^{87}Rb$, $^{87}Rb$-$^{41}K$, $^{6}Li$-$^{40}K$ and different alkaline earth mixtures in optical lattices [@Catani; @Taie; @Taglieber] have motivated theoretical studies of the two species Bose-Hubbard model. [@Altman; @Soyler; @Soyler2; @Stephen; @Andrii; @Lv; @Trousselet; @Kuno; @Kalani] The zero-temperature phase diagram of the two-dimensional, two-species, hard-core bosonic Hubbard model has been studied at half-filling using a combination of mean field and variational methods, [@Altman] and by means of quantum Monte Carlo simulations. [@Soyler] The rich phase diagram found at half-filling in these studies shows ordered Mott insulating phases including anti-ferromagnetic and super-counter-fluid phases in the strong interaction limit. On the other hand, superfluid and antiferromagnetic/superfluid phases are found in the weak interaction limit. [@Altman; @Soyler; @Soyler2; @Stephen; @Andrii] Recently, we have included doping dependence as a control parameter to study this model using quantum Monte Carlo simulations. We found several distinct phases including a normal liquid at higher temperatures, an antiferromagnetically ordered Mott insulator, and a region of coexistent antiferromagnetic and superfluid order near half-filling. [@Kalani] We also reported a small dome containing a phase-separated ferromagnetic phase away from half-filling at zero polarization.
Though the realization of quantum magnetic phases has gained significant attention, the prominent experimental challenge is to reach the low temperatures and entropies needed to observe these phases. Several different experimental techniques have been proposed to reach such low entropies. [@Monroe; @Popp; @Li] Interestingly, a Bose-Fermi mixture may be used to squeeze the entropy of a Fermi gas into the surrounding Bose gas. [@Ho-Zhou] This can leave a low entropy heavy Fermi gas by evaporating the entropy absorbed by the light Bose gas. The relatively high global entropy of the phase-separated ferromagnetic phase we found away from half-filling in the two species Hubbard model [@Kalani] suggests that this ferromagnet should be easier to access experimentally.
In the experimental setup of boson-boson mixtures, the two species are not always perfectly balanced. [@Thalhammer; @Plich] The evaporative cooling leads to net losses of one of the species, due to the difference in the effective depth of the traps. This can be adjusted by loading different number of atoms for different species into the trap. [@Plich] This procedure can also be used to set an imbalance amount of atoms for the two species. Most of the previous theoretical or numerical studies on the two-species Bose-Hubbard model do not directly address this imbalance in the experimental conditions.
In this paper, we explore the extent of the phase-separated ferromagnetic phase as a function of [*finite*]{} polarization, i.e., with a different population for each species. When the polarization is positive (more of the light than heavy particles) we find a larger region of the ferromagnetic phase-separated order, with higher transition temperatures and greater extent in doping. Since this ferromagnetic phase exists for a broad range of sufficiently high temperatures and polarizations together with high global entropies, experimental observation in cold atoms may be achievable.
This manuscript is organized as follows. In section \[Sec:Model\] we describe our model and method. The density versus polarization phase diagram at low temperature is studied in section \[Sec:DPphasediagram\]. In section \[Sec:PTphasediagram\] we discuss the temperature versus polarization phase diagram along an optimal superfluid or maximum ferromagnetic phase line. The momentum distribution of the ferromagnetic and superfluid phases are presented in section \[Sec:Momentum\]. In section \[Sec:Entropy\] we calculate the entropy of the ferromagnetic, antiferromagnetic and superfluid phases. Finally we conclude in section \[Sec:Conclusion\].
Model and method {#Sec:Model}
================
The Hamiltonian for the two-species Hubbard model with hard-core heavy, [*a*]{}, and light, [*b*]{}, bosons confined on a two-dimensional square lattice takes the form: $$\begin{aligned}
\label{Eq:Hamiltonian} \nonumber \hat\mathcal H &=& -t_{a}\sum_{\langle i,
j\rangle}\Big(a_i^\dagger a_j^{\phantom\dagger}+H.c.\Big)\\
& & -t_{b}\sum_{\langle i,j\rangle}\Big(b_i^\dagger b_j^{\phantom\dagger}
+H.c.\Big)+{U_{ab}\sum_i \hat n_i^{a} \hat n_i^{b}},\end{aligned}$$ where $a_i^\dagger$ ($b_i^\dagger$) and $a_i$ ($b_i$) are the creation and annihilation operators of hard-core bosons [*a*]{} and ([*b*]{}), respectively, with number operators $\hat n_i^{a}=a_i^\dagger a_i^{\phantom\dagger}$, $\hat n_i^{b}=b_i^\dagger b_i^{\phantom\dagger}$. The sum $\sum_{\langle i,j\rangle}$ runs over all distinct pairs of first neighboring sites $i$ and $j$, $t_a (t_b)$ is the hopping integral between sites $i$ and $j$ for species [*a*]{} ([*b*]{}), and $U_{ab}$ is the strength of the on-site interspecies repulsion.
We perform a quantum Monte Carlo study of the model (\[Eq:Hamiltonian\]) by using the Stochastic Green Function algorithm [@SGF; @DirectedSGF] with global space-time updates [@SpaceTime] to solve the canonical ensemble on $L \times L$ lattices. We focus on the polarized phase diagram, with polarization $P=\frac {N_b -N_a}{L^2}$ and total density $\rho=\frac {N_b +N_a}{L^2}$, with $N_a$ and $N_b$ the number of heavy [*a*]{} and light [*b*]{} particles, respectively. For the other parameters we use the same values as in Ref. , namely $t_a=0.08\, t$, $t_b=t$, and $U_{ab}=6t$, where $t=1$.
Phase diagram at low temperature {#Sec:DPphasediagram}
================================
![(Color online) The total density, $\rho=\frac{N_b+N_a}{L^2}$, versus polarization, $P=\frac{N_b-N_a}{L^2}$, phase diagram at very low temperature. The transition temperatures associated with the data points are obtained from finite-size scaling calculations. The boundaries between the phases are estimated for $\beta t=60$ and $L=10$ with $t_a=0.08t, t_b=1.00t$, and $U_{ab}=6t$. The boundaries may slightly change for the ground state. The red area shows the phase-separated ferromagnet (FM). The green area shows the region of superfluidity of both [*a*]{} and [*b*]{} species (SF$_{ab}$). The white region represents the superfluidity of light [*b*]{} particles (SF$_b$)except that the system is an antiferromagnet at half-filling ($\rho=1$ and $P=0$), and there is an antiferromagnetic to superfluid phase-separated region near half-filling ($1.0<\rho<1.1$ and $P=0$) as discussed in Ref. . Both particles are in a Mott insulating phase whenever their individual densities are integers ($0$ or $1$). The blue squares indicate the transition temperatures from the light species superfluid to the ferromagnetic phase. The red circles correspond to the transition temperatures from the light species superfluid to the phase where both species are superfluid. The black dotted line, $N_b+\frac{N_a}{2}=L^2$, follows the highest ferromagnetic critical temperature. The phase diagram as a function of temperature along this black dotted line is shown in Fig. \[Fig:DensityTemp\]. The momentum distributions shown in Fig. \[Fig:Momentum\] are calculated along the purple dotted line ($N_a=0.625L^2$), which intersects the black dotted line. The momentum distributions for heavy and light species for three points along the purple dotted line are shown in the bottom panels of Fig. \[Fig:Momentum\]. The black line is the zero polarization axis. []{data-label="Fig:DensityDensity"}](DensityDensity.eps){width="50.00000%"}
Figure \[Fig:DensityDensity\] displays the total density $\rho$ versus polarization $P$ phase diagram at low temperature. In the thermodynamic limit, a ferromagnetic phase exists in a broad region of densities (red area), heavy [*a*]{} and light [*b*]{} particle superfluidity exists in a smaller region of densities (green area), and superfluidity of light [*b*]{} particles (with heavy [*a*]{} particles in the normal state) appears in most of the rest of the phase diagram (white area). Along the zero polarization axis there is an antiferromagnetic phase at half-filling ($\rho=1$ and $P=0$), and an antiferromagnetic to superfluid phase-separated region for $1.0<\rho<1.1$ and $P=0$ as discussed in Ref. . The black dotted line, $N_b+\frac{N_a}{2}=L^2$, follows the highest ferromagnetic critical temperatures (optimal superfluid line). Along this line the system shows fully phase-separated regions of average local densities $n_a\sim 0$ together with $n_b\sim 1$, and $n_a\sim1$ with $n_b\sim 0.5$. Therefore, the number of light particles, $N_b$, is given as $N_b= (L^2 -N_a) + \frac{N_a}{2}$ (or $\frac{N_a}{2}+N_b=L^2$). In our previous study we did not distinguish between the phase where only the light particles are superfluid from the one where both species are superfluid. The ferromagnetic phase boundaries at zero polarization have also changed slightly.
The blue squares in Fig. \[Fig:DensityDensity\] indicate the transition temperature from the light species superfluid to the ferromagnetic phase for the given densities and polarizations. To find these transition temperatures we calculate the ferromagnetic susceptibility for different system sizes and perform a finite-size scaling. The susceptibility is given as $\mathcal \chi({\mathbf{k}}) = \big\langle \vert \mathcal A({\mathbf{k}})\vert^2\big\rangle -\vert\big\langle \mathcal A({\mathbf{k}})\big\rangle\vert^2$ with $$\begin{aligned}
\label{Eq:Susceptibility} \mathcal A({\mathbf{k}}) &=& \frac{1}{\beta}\int_0^{\beta} \sum_j e^{i{\mathbf{k}}\cdot {\mathbf{r}}_j}(n_j^a(\tau)-n_j^b(\tau))\,d\tau.\end{aligned}$$ Following Ref. we calculate the ferromagnetic susceptibility ratio, $\mathcal R$, defined as $$\begin{aligned}
\label{Eq:SusceptibilityRato} \mathcal R&=& \frac{\chi(0,\varepsilon)+\chi(0,-\varepsilon)+\chi(\varepsilon,0)+\chi(-\varepsilon,0)}{\chi(\varepsilon,\varepsilon)+\chi(-\varepsilon,\varepsilon)+\chi(\varepsilon,-\varepsilon)+\chi(-\varepsilon,-\varepsilon)}, \end{aligned}$$ where $\varepsilon=\frac{2\pi}{L}$. We impose all the point group symmetries in ${\mathbf{k}}$-space near ${\mathbf{k}}\sim0$ to both the numerator and denominator to reduce the statistical noise associated with quantum Monte Carlo sampling.
![(Color online) Scaling behavior of the ferromagnetic susceptibility for the continuous transition from light species superfluid to ferromagnet at $\rho=1.4$ and $P=0$. The susceptibility ratios, $\mathcal R$ (Eq. \[Eq:SusceptibilityRato\]), versus temperature, $T/t$, for different system sizes cross at the critical temperature, $T_c=0.145t$. The inset shows the scaling near the critical temperature. The curves collapse onto a single curve with the critical exponent of correlation length $\nu=1$. The data points are based on simulation results, the lines are guides to the eye. []{data-label="Fig:Ferro"}](Ferro.eps){width="50.00000%"}
The scaling behavior of the ferromagnetic transition temperature is shown in Fig. \[Fig:Ferro\] where the susceptibility ratio $\mathcal R$ is plotted for different system sizes as a function of temperature $T/t$ at $\rho=1.4$ and $P=0$. In Fisher scaling [@Fisher1; @Fisher2; @Fisher3], the susceptibility at small wavenumber should scale as $\chi \sim L^{\frac{\gamma}{\nu}}g(L^{\frac{1}{\nu}}(T-T_c))$, where $\gamma$ and $\nu$ are the critical exponents for the ferromagnetic susceptibility and correlation length, respectively. By looking at the ratio of the susceptibilities $\mathcal R$, the $L^{\frac{\gamma}{\nu}}$ factor is canceled. At the transition, the scaling function $g(0)$ is independent of $L$. Thus, the susceptibility ratio $\mathcal R$ versus temperature $T$ for different system sizes should cross at the critical temperature $(T=T_c=0.145t)$ as it is shown in Fig. \[Fig:Ferro\]. By choosing the critical exponent of the correlation length as $\nu=1$, the value for a two-dimensional Ising transition, we find that the curves collapse onto one curve near the critical temperature (c.f., the inset of Fig. \[Fig:Ferro\]). If the transition is second order by considering symmetry arguments, it should belong to the Ising universality class. However, if we understand the polarized model as a Ising system within an external magnetic field, it is possible that the ferromagnetic transition is first order. Since it is very difficult to distinguish between first and second order phase transitions with our finite size calculations, we can not clarify this issue.
The red circles in Fig. \[Fig:DensityDensity\] indicate the superfluid transition temperature for the heavy species. The scaling behavior of the superfluid to normal liquid transition should follow that of the Kosterlitz-Thouless continuous transition. We note that the Hamiltonian (\[Eq:Hamiltonian\]) satisfies the condition (28) of Ref. , which allows one to relate the superfluid density to the fluctuations of the winding number. [@Pollock] In Fig. \[Fig:SFA\], we show the winding number of the [*a*]{} particles, $\langle W^2 \rangle$, as a function of temperature, $T/t$, for different system sizes. The order parameter, the superfluid density, has a universal jump of $\langle W^2 \rangle=\frac{4}{\pi}$ at the critical point. [@Nelson] The black dotted line shows $\frac{4}{\pi}T$ as a function of temperature. We read the crossing temperature, $T_L$, for different system sizes. Then we use the relation between the crossing temperature, $T_L$, and the cluster size, $L$, $T_L-T_c(\infty)\propto \displaystyle \frac {1}{ln^{2}(L)}$, [@Boninsegni05] to find the critical temperature, $T_c$, in the thermodynamic limit. The inset of Fig. \[Fig:SFA\] displays this scaling. We find $T_c=0.03t$ at $\rho=1.14$ and $P=0$.
![(Color online) Winding number of heavy [*a*]{} particles as a function of temperature for different system sizes at $\rho=1.14$ and $P=0$. The black dotted line corresponds to $\frac{4T}{\pi}$ and is used to find the crossing temperature for different system sizes. The inset shows the finite size scaling of the crossing temperatures to find the superfluid critical temperature, $T_c=0.03t$, in the thermodynamic limit. The data points are based on simulation results, the lines are guides to the eye. []{data-label="Fig:SFA"}](SFA.eps){width="50.00000%"}
Phase diagram on the optimal superfluid line {#Sec:PTphasediagram}
============================================
To better understand the phases of the polarized model we investigate snapshots of the average local densities. From the snapshots we propose that superfluid and ferromagnetic states are optimal along the black dotted line shown in Fig. \[Fig:DensityDensity\], where $\frac{N_a}{2}+N_b=L^2$, with $L^2$ the lattice size, and $N_a$ and $N_b$ the number of heavy and light atoms, respectively. Along this line the system shows fully phase-separated regions with average local densities $n_a\sim 0$ and $n_b\sim1$ in the Mott region, and $n_a\sim1$ and $n_b\sim 0.5$ in the Mott/superfluid region. The inset of Fig. \[Fig:DensityTemp\] displays snapshots of these average local densities for heavy (left panel) and light (right panel) particles. Physically, this optimal line is driven by the fact that a superfluid with $n_b\sim 0.5$ gains the most energy per particle. As an example, for $N_a=50$, $N_b=75$, $L=10$ ($\rho=1.25$ and $P=0.25$), half of the lattice is filled with [*a*]{} particles and the other half with [*b*]{} particles. The 25 remaining [*b*]{} particles will occupy the region filled by [*a*]{} particles and $n_b=\frac{25}{50}=0.5$ in that region. This reasoning is valid along this optimal superfluid line. However, when the system deviates far from $N_a=50\%$ of the number of lattice sites, it is difficult to stabilize small and large phase-separated regions. In this case, the pattern may break. This also explains why the ferromagnetic phase-separated phase is more stable for positive polarizations around $\rho =1.25$ and $P=0.25$. At half-filling of the heavy particles, this pattern is more stable since there are two large phase-separated regions reducing surface effects.
![(Color online) The temperature, $T/t$, versus polarization, $P=\frac{N_b-N_a}{L^2}$, phase diagram when $N_b+\frac{N_a}{2}=L^2$. The abscissa extends from $\rho=1, P=1$ ($N_a=0$, $N_b=L^2$) to $\rho=1.5, P=-0.5$ ($N_a=L^2$, $N_b=L^2/2$). The orange area corresponds to the phase-separated ferromagnet (FM). The green area is the region where the light [*b*]{} species displays superfluidity (SF). The white area is the normal liquid (NL). The blue squares and red circles indicating the boundaries between the phases are calculated by finite size scaling, see section \[Sec:DPphasediagram\]. The curves are guides to the eye. Since it is difficult to perform finite size scaling at very low temperatures, the edges of the phase diagram are estimations of the transition temperatures based on results of small clusters. The inset shows a snapshot of the average local densities versus lattice coordinates for $L=10$, $\rho=1.25$, $P=0.25$ and $\beta t=80$. Left panel: For [*a*]{} particles, the red regions have $\langle n_i^a\rangle \sim 0$ while the occupation of the blue region is $\langle n_i^a\rangle \sim 1$. Right panel: For [*b*]{} particles, the blue regions have $\langle n_i^b\rangle \sim 1$ while the occupation of the green region is $\langle n_i^b\rangle \sim 0.5$. The ferromagnetic phase separation occurs when the heavy species is in a Mott insulating state while the light one displays regions with either Mott insulating or superfluid behaviors. []{data-label="Fig:DensityTemp"}](DensityTemp.eps){width="50.00000%"}
Fig. \[Fig:DensityTemp\] shows the temperature, $T/t$, versus polarization, $P=\frac{N_b-N_a}{L^2}$, phase diagram on the optimal superfluid line, $\frac{N_a}{2}+N_b=L^2$. The blue squares are the ferromagnetic transition temperatures found by scaling as discussed in section \[Sec:DPphasediagram\]. The red circles indicate the transition temperatures for light species superfluid. Again, the scaling behavior of this light particle superfluid to normal liquid transition follows that of the Kosterlitz-Thouless transition as discuss earlier for the heavy particles. In Fig. \[Fig:SFB\], we show the winding of the [*b*]{} particles, $\langle W^2 \rangle$, as a function of temperature, $T/t$, for different system sizes. We find $T_c=0.245t$ at $\rho=1.25$ and $P=0.25$ as shown in Fig \[Fig:SFB\].
![(Color online) Winding number of light [*b*]{} particles as a function of temperature for different system sizes at $\rho=1.25$ and $P=0.25$. The black dotted line shows $\frac{4T}{\pi}$ and it is used to find the crossing temperatures for different system sizes. The lowering of the superfluid density at low temperatures occurs when the system enters the ferromagnetic phase where the light species displays both superfluid and Mott behaviors. The inset shows the finite size scaling of the crossing temperatures to find the superfluid critical temperature, $T_c=0.245t$, in the thermodynamic limit for the continuous transition. The data points are based on simulation results, the lines are guides to the eye. []{data-label="Fig:SFB"}](SFB.eps){width="50.00000%"}
![(Color online) The ${\mathbf{k}}$-space momentum distribution for $\beta t=50$ and $L=8$. Top panel: The momentum distribution at zero momentum ${\mathbf{k}} = (0,0)$ as a function of polarization for heavy (red circles) and light (blue squares) species along the purple dotted line shown in Fig. \[Fig:DensityDensity\]. The data points are based on simulation results, the lines are guides to the eye. Bottom panels: The momentum distributions for heavy and light species for three points along the purple dotted line shown in Fig. \[Fig:DensityDensity\]. Left panel: The momentum distribution at $\rho=1.19$ and $P=-0.06$ (open black square in Fig. \[Fig:DensityDensity\]). For both [*b*]{} (bottom) and [*a*]{} (top) particles, the distributions have a peak at ${\mathbf{k}} = (0,0)$ which corresponds to superfluid behavior. Middle panel: The momentum distribution for the ferromagnetic phase at $\rho=1.33$ and $P=0.08$ (open black circle in Fig. \[Fig:DensityDensity\]). For the [*b*]{} particles (bottom), the distribution has a peak at ${\mathbf{k}} = (0,0)$ which corresponds to superfluid behavior. For [*a*]{} particles (top), the distribution is uniform corresponding to Mott behavior. Right panel: The momentum distribution at $\rho=1.48$ and $P=0.23$ (open black triangle in Fig. \[Fig:DensityDensity\]). For both [*b*]{} (bottom) and [*a*]{} (top) particles, the distributions have a peak at ${\mathbf{k}} = (0,0)$ corresponding to superfluid behavior. []{data-label="Fig:Momentum"}](Momentum.eps){width="50.00000%"}
Momentum distribution {#Sec:Momentum}
=====================
A related experimentally accessible quantity that can distinguish different phases of bosons is the momentum distribution. It is defined as $$\begin{aligned}
\label{Eq:Momentum} \mathcal \displaystyle N({\mathbf{k}}) &=&\frac{1}{L^2}\sum\limits_{k,l}e^{i{\mathbf{k}} \cdot ({\mathbf{r}}_k-{\mathbf{r}}_l)}\langle a_{k}^\dagger a_{l}^{\phantom\dagger}\rangle, \end{aligned}$$ with the momentum $k_{x,y}=\frac{2\pi}{L}m$, $m=0, 1, ..., L-1$. The superfluid ground state is characterized by a peak at zero momentum, ${\mathbf{k}}= (0,0)$, while the Mott insulator phase has an uniform momentum distribution. [@Greiner; @Catani] Fig. \[Fig:Momentum\] displays the momentum distribution of heavy and light particles for the ferromagnetic and superfluid phases along the purple dotted line in Fig. \[Fig:DensityDensity\]. The momentum distribution at zero wavevector, ${\mathbf{k}} = (0,0)$, and $\beta t=50$ as a function of polarization for heavy and light species is shown in the top panel. The momentum distribution of the heavy particles at zero momentum is small in the ferromagnetic region but large in the superfluid phase. The light particles show significantly less variation with polarization and have a value which is consistently large compared to the ferromagnetic state of the heavy particles, indicative of a superfluid state. The momentum distributions for heavy and light species for three points along the purple dotted line are shown in the bottom panels. The left panels show that for $\rho=1.19$ and $P=-0.06$, heavy and light distributions display peaks at ${\mathbf{k}} = (0,0)$ indicating superfluidity of both species. The same happens at the right panels for $\rho=1.48$ and $P=0.23$. The middle panel at $\rho=1.33$ and $P=0.08$ displays a ${\mathbf{k}}=(0,0)$ peak in the momentum distribution of the light species while the momentum distribution of heavy particles is uniform. This behavior is consistent with the phase-separated ferromagnetic phase where the heavy species [*a*]{} becoming Mott while the light one [*b*]{} displays regions with Mott insulating and superfluid behaviors.
![(Color online) Entropy, $S(T)$, for $L=10$, $t_a=0.08$, $t_b=t=1$, and $U_{ab}=6$, as a function of temperature, $T/t$, for three different combinations of total densities, $\rho$, and polarizations, $P$. The red circles show the entropy of the phase-separated ferromagnetic phase at $\rho=1.25$ and $P=0.25$. The blue squares display the entropy of the antiferromagnetic phase at $\rho=1$ and $P=0$. The purple triangles show the entropy of the superfluid phase at $\rho=1.16$ for the non-polarized system. The data points are based on simulation results, the lines are guides to the eye. []{data-label="Fig:Entropy"}](Entropy.eps){width="50.00000%"}
Entropy {#Sec:Entropy}
=======
Reaching the low entropies and temperatures required to observe magnetically ordered or Mott insulating phases is still experimentally challenging. In the ferromagnetic phase separated region, the superfluid ordering of light [*b*]{} particles can carry most of the entropy, leaving the entropy of the heavy species in this phase essentially zero. Thus the ferromagnetic phase-separated phase can have large entropy. Fig. \[Fig:Entropy\] shows the entropy for an $L=10$ system calculated following Ref. for three different densities and polarizations, $\rho=1$ and $P=0$ (antiferromagnet), $\rho=1.16$ and $P=0$ (superfluid), and $\rho=1.25$ and $P=0.25$ (ferromagnet). The entropy of the ferromagnetic phase is greater than that of the antiferromagnetic phase and similar to that of the superfluid state, especially for low temperatures, indicating that it may be more accessible experimentally. The entropy, $S(T)$ is calculated by integrating the internal energy per site, $E(T)$, as: $$\begin{aligned}
\label{Eq:Entropy} \mathcal S(\beta, n) &=& S(0, n)+\beta E(\beta, n)-\int_0^{\beta} E(\beta^\prime,n)d\beta^\prime,\end{aligned}$$ where $S(0, n)$ depends on the possible per site occupation of [*a*]{} and [*b*]{} particles.
Conclusion {#Sec:Conclusion}
==========
By introducing a population imbalance between the two species, we find an extended region of phase-separated ferromagnetism in the two-dimensional two-species hard-core bosonic Hubbard model. The average local densities show that the heavy species has Mott-insulating behavior while the light species is phase separated into both Mott insulating and superfluid regions. This phase exists for a broad range of temperatures and polarizations. In this polarized model we find the optimal superfluid line, $\frac{N_a}{2}+N_b=L^2$, where the system shows high transition temperatures and fully phase-separated regions at low temperatures with average local densities $n_a\sim 0$ and $n_b\sim1$ on one of the regions, and $n_a\sim1$, $n_b\sim 0.5$ on the other. This line exists because the superfluidity of light species with $n_b\sim 0.5$ gains the most energy per particle. Further the ferromagnetic phase-separated phase is more stable for positive polarizations around $\rho =1.25$ and $P=0.25$. When the system deviates far from half-filling of the heavy [*a*]{} particles, the ferromagnetic phase vanishes since it is difficult to stabilize small and large phase-separated regions. By using finite-size scaling of ferromagnetic susceptibility ratios, we find the correlation length exponent $\nu \approx 1$ which is consistent with a two-dimensional Ising ferromagnet. Despite its ferromagnetic order, this phase has relatively high global entropy, which suggests that its experimental observation in cold atoms should be more accessible.
Acknowledgment
==============
This work is supported by NSF OISE-0952300 (KH, VGR and JM) and DMR-1237565 (KH and JM). Additional support was provided by the NSF EPSCoR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents (KMT and MJ). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation grant number ACI-1053575, and the high performance computational resources provided by the Louisiana Optical Network Initiative (http://www.loni.org).
[5]{}
S. Yunoki, J. Hu, A. L. Malvezzi, A. Moreo, N. Furukawa and E. Dagotto,Phys. Rev. Lett. [**80**]{}, 845(1998). A. Moreo, S. Yunoki and E. Dagotto, Science 2034, [**283**]{} (1999). M. Uehara, S. Mori, C. H. Chen, S. -W. Cheong, Nature [**399**]{}, 560 (1999). E. Dagotto, T. Hotta and A. Moreo, Phys. Rep. [**344**]{}, 1-153 (2001). P. Coleman and A. J. Schofield, Nature, [**433**]{}, 227 (2005). Q. Si and F. Steglich, Science [**329**]{}, 1161 (2010). E. Dagotto, Science [**309**]{}, 257 (2005). D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phy. Rev. Lett. [**81**]{}, 3108 (1998). W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin, Phys. Rev. Lett. [**89**]{}, 220407 (2002). T. Esslinger, Annual Review of Condensed Matter Physics [**1**]{}, 129 (2010). E. Timmermans, P. Tommasini, M. Hussein, A. Kerman, Phys. Rep. [**315**]{}, 199, (1999). T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys. [**78**]{}, 1311, (2006). M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature (London), [**415**]{}, 39 (2002). M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B [**40**]{}, 546 (1989). G. G. Batrouni and R. T. Scalettar, Phys. Rev. Lett. [**84**]{}, 1599 (2000). F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. [**87**]{}, 080403 (2001). A. Albus, F. Illuminati, and J. Eisert, Phys. Rev. A [**68**]{}, 023606 (2003). G. Modugno, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Science, [**297**]{} (2002). C. Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs, Phys. Rev. Lett. [**96**]{}, 020401 (2006). G. Roati, M. Zaccanti, C. D’Errico, J. Catani, M. Modugno, A. Simoni, M. Inguscio, and G. Modugno, Phys. Rev. Lett. [**99**]{}, 010403 (2007). G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani, F. Minardi, and M. Inguscio, Phys. Rev. Lett. [**100**]{}, 210402 (2008). S. B. Papp, J. M. Pino, and C. E. Wieman, Phys. Rev. Lett. [**101**]{}, 040402 (2008). J. Catani, L. De Sarlo, G. Barontini, F. Minardi, and M. Inguscio, Phys. Rev. A [**77**]{}, 011603 (2008). M. Taglieber, A.-C. Voigt, T. Aoki, T. W. Hänsch, and K. Dieckmann, Phys. Rev. Lett. [**100**]{}, 010401 (2008). S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsujimoto, R. Murakami, and Y. Takahashi, Phys. Rev. Lett. [**105**]{}, 190401 (2010). E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, New J. Phys. [**5**]{}, 113 (2003). S. G. Söyler, B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. Svistunov, New J. Phys. [**11**]{}, 073036 (2009). B. Capogrosso-Sansone, S. G. Söyler, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. A [**81**]{}, 053622 (2010). S. Powell, Phys. Rev. A [**79**]{}, 053614 (2009). A. Sotnikov, D. Cocks, and W. Hofstetter, Phys. Rev. Lett. [**109**]{}, 065301 (2012). Y. Kuno, K. Suzuki, and I. Ichinose, J. Phys. Soc. Jpn [**82**]{}, 12450 (2013). F. Trousselet, P. Rueda-Fonseca, and A. Ralko, Phys. Rev. B [**89**]{} 085104 (2014). J. P. Lv, Q. H. Chen, and Y. Deng, Phys. Rev. A [**89**]{}, 013628 (2014). K. Hettiarachchilage, V. G. Rousseau, K.-M. Tam, M. Jarrell, and J. Moreno, Phys. Rev. B [**88**]{}, 161101(R) (2013). C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev. Lett. [**75**]{}, 4011 (1995). M. Popp, J.-J. Garcia-Ripoll, K. G. Vollbrecht, and J. I. Cirac , Phys. Rev. A [**74**]{}, 013622 (2006). X. Li, T. A. Corcovilos, Y. Wang, and D. S. Weiss, Phys. Rev. Lett. [**108**]{}, 103001 (2012). T.-L. Ho and Q. Zhou, Proc. Natl. Acad. Sci. USA [**106**]{}, 6916 (2009). K. Pilch, A. D. Lange, A. Prantner, G. Kerner, F. Ferlaino, H.-C. Nägerl, and R. Grimm, Phys. Rev. A [**79**]{}, 042718 (2009). V. G. Rousseau, Phys. Rev. E [**77**]{}, 056705 (2008). V. G. Rousseau, Phys. Rev. E [**78**]{}, 056707 (2008). V. G. Rousseau and D. Galanakis, arXiv:1209.0946 (2012). R. A. Baños, A. Cruz, L.A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, M. Guidetti, D. Iñiguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. Muñoz Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, J. J. Ruiz-Lorenzo, S.F. Schifano, B. Seoane, A. Tarancon, P. Tellez, R. Tripiccione, D. Yllanes, Proc. Natl. Acad. Sci. U.S.A. [**109**]{}, 6452 (2012). M. E. Fisher, Rep. Prog. Phys. [**30**]{}, 615 (1967). M. E. Fisher, Rev. Mod. Phys. [**46**]{}, 597 (1974). M. E. Fisher, Rev. Mod. Phys. [**70**]{}, 653 (1998). V. G. Rousseau, Phys. Rev. B [**90**]{}, 134503 (2014). E. L. Pollock and D. M. Ceperley, Phys. Rev. B [**36**]{}, 8343 (1987). D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. [**39**]{}, 1201 (1977). M. Boninsegni and N. Prokof’ev, Phys. Rev. Lett. [**95**]{}, 237204 (2005). F. Werner, O. Parcollet, A. Georges, and S. R. Hassan, Phys. Rev. Lett. [**95**]{}, 056401 (2005).
|
---
author:
- |
: M. Dobbs$^1$, S. Frixione$^2$, E. Laenen$^3$, A. De Roeck$^4$, K. Tollefson$^5$\
: J. Andersen$^{6,7}$, C. Balázs$^8$, A. Banfi$^3$, S. Berge$^9$, W. Bernreuther$^{10}$, T. Binoth$^{11}$, A. Brandenburg$^{12}$, C. Buttar$^{13}$, Q-H. Cao$^5$, G. Corcella$^{4,14}$, A. Cruz$^{15}$, I. Dawson$^{13}$, V. Del Duca$^{16}$, A. De Roeck$^4$, V. Drollinger$^{17,18}$, L. Dudko$^{19}$, T. Eynck$^3$, R. Field$^{15}$, S. Frixione$^2$, M. Grazzini$^4$, J.P. Guillet$^{20}$, G. Heinrich$^{21}$, J. Huston$^5$, N. Kauer$^{10}$, N. Kidonakis$^6$, A. Kulesza$^{22}$, E. Laenen$^3$, K. Lassila-Perini$^{23}$, L. Magnea$^{16,24}$, F. Mahmoudi$^{20}$, E. Maina$^{16,24}$, F. Maltoni$^{25}$, M. Nolten$^{26}$, A. Moraes$^{13}$, S. Moretti$^{26}$, S. Mrenna$^{27}$, P. Nadolsky$^9$, Z. Nagy$^{28}$, F. Olness$^9$, I. Puljak$^{29}$, D.A. Ross$^{26}$, A. Sabio-Vera$^6$, G.P. Salam$^{30}$, A. Sherstnev$^{19}$, Z.G. Si$^{31}$, T. Sjöstrand$^{32}$, P. Skands$^{32}$, E. Thomé$^{32}$, Z. Trócsányi$^{33}$, P. Uwer$^4$, S. Weinzierl$^{14}$, C.P. Yuan$^5$, G. Zanderighi$^{27}$\
---
*Report of the Working Group on Quantum Chromodynamics and the Standard Model for the Workshop “Physics at TeV Colliders”, Les Houches, France, 26 May - 6 June, 2003.*
FOREWORD [^1]
=============
\[introduction\]
LES HOUCHES GUIDEBOOK TO MONTE CARLO GENERATORS FOR HADRON COLLIDER PHYSICS
===========================================================================
MULTIPLE INTERACTIONS AND BEAM REMNANTS [^2]
============================================
\[torbjorn\_multint3\]
DESCRIBING MINIMUM BIAS AND THE UNDERLYING EVENT AT THE LHC IN PYTHIA AND PHOJET [^3] {#amoraes-LH2003}
=====================================================================================
USING CORRELATIONS IN THE TRANSVERSE REGION TO STUDY THE UNDERLYING EVENT IN RUN 2 AT THE TEVATRON [^4] {#LH2003_field}
=======================================================================================================
SIMULATION OF THE QCD BACKGROUND FOR $t\bar{t}$ ANALYSES AT THE TEVATRON WITH A $l^\pm +$ JETS FINAL STATE [^5] {#volker_droll}
===============================================================================================================
MONTE-CARLO DATABASE [^6] {#lev_dudko_mcdb}
=========================
RESUMMATION AND SHOWER STUDIES [^7] {#torbjorn_resumandshower7}
===================================
NEW SHOWERS WITH TRANSVERSE-MOMENTUM-ORDERING [^8] {#torbjorn_shower3}
==================================================
MATCHING MATRIX ELEMENTS AND PARTON SHOWERS WITH HERWIG AND PYTHIA [^9] {#mrenna}
=======================================================================
$W$ BOSON, DIRECT PHOTON AND TOP QUARK PRODUCTION: SOFT-GLUON CORRECTIONS [^10] {#kid_softgluon}
===============================================================================
EXTENDING THRESHOLD EXPONENTIATION BEYOND LOGARITHMS FOR DIS AND DRELL-YAN [^11] {#sec:thresh-exp}
================================================================================
JOINT RESUMMATION FOR TOP QUARK PRODUCTION [^12] {#sec:joint-resumm-top}
================================================
A COMPARISON OF PREDICTIONS FOR SM HIGGS BOSON PRODUCTION AT THE LHC [^13] {#huston}
==========================================================================
MATRIX-ELEMENT CORRECTIONS TO $gg/q\bar q\to$ HIGGS IN HERWIG [^14]
===================================================================
CAESAR: AUTOMATING FINAL-STATE RESUMMATIONS [^15] {#sec:caes-autom-final}
=================================================
COMBINED EFFECT OF QCD RESUMMATION AND QED RADIATIVE CORRECTION TO $W$ BOSON MASS MEASUREMENT AT THE LHC [^16] {#paper_lhc}
==============================================================================================================
RESUMMATION FOR THE TEVATRON AND LHC ELECTROWEAK BOSON PRODUCTION AT SMALL $x$ [^17] {#sec:resumm-tevatr-lhc}
====================================================================================
THE HIGH ENERGY LIMIT OF QCD AND THE BFKL EQUATION [^18] {#LH_Smallx}
========================================================
PION PAIR PRODUCTION AT THE LHC: COMPARING QCD@NLO WITH PYTHIA [^19] {#2pi}
====================================================================
QCD-INDUCED SPIN PHENOMENA IN TOP QUARK PAIR PRODUCTION AT THE LHC [^20] {#sec:qcd-induced-spin}
========================================================================
QCD RADIATIVE CORRECTIONS TO PROMPT DIPHOTON PRODUCTION IN ASSOCIATION WITH A JET AT THE LHC [^21] {#sec:qcd-radi-corr}
==================================================================================================
ELECTROWEAK RADIATIVE CORRECTIONS TO HADRONIC PRECISION OBSERVABLES AT TEV ENERGIES [^22] {#SingleBosonLH}
=========================================================================================
TOWARDS AUTOMATED ONE-LOOP CALCULATIONS FOR MULTI-PARTICLE PROCESSES [^23] {#sec:towards-autom-one}
==========================================================================
INFRARED DIVERGENCES AT NNLO [^24] {#sec:infr-diverg-at}
==================================
[100]{}
T. Sj[ö]{}strand and M. van Zijl, [*Phys. Rev.*]{} [**D36**]{} (1987) 2019.
B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, [*Phys. Rept.*]{} [**97**]{} (1983) 31.
R.D. Field. See talks available from http://www.phys.ufl.edu/[$\sim$]{}rfield/cdf/.
T. Sj[ö]{}strand and P. Skands, [*Nucl. Phys.*]{} [**B659**]{} (2003) 243, \[[[hep-ph/0212264]{}](http://xxx.lanl.gov/abs/hep-ph/0212264)\].
T. Sj[ö]{}strand and P. Skands, [[hep-ph/0402078]{}](http://xxx.lanl.gov/abs/hep-ph/0402078).
J. Dischler and T. Sjostrand, [*Eur. Phys. J. direct*]{} [**C3**]{} (2001) 2, \[[[hep-ph/0011282]{}](http://xxx.lanl.gov/abs/hep-ph/0011282)\].
T. Sj[ö]{}strand, L. L[ö]{}nnblad, S. Mrenna, and P. Skands, [[hep-ph/0308153]{}](http://xxx.lanl.gov/abs/hep-ph/0308153).
P. D. B. Collins and A. D. Martin, [*Rep. Prog. Phys.*]{} [**45**]{} (1982) 335.
A. Capella [*et. al.*]{}, [*Phys. Rep.*]{} [**236**]{} (1994) 225.
T. Sjostrand [*et. al.*]{}, [*Comput. Phys. Commun.*]{} [**135**]{} (2001) 238–259, \[[[ hep-ph/0010017]{}](http://xxx.lanl.gov/abs/hep-ph/0010017)\].
T. Sjostrand, L. Lonnblad, and S. Mrenna, [[hep-ph/0108264]{}](http://xxx.lanl.gov/abs/hep-ph/0108264). PYTHIA 6.2 - Physics and Manual.
R. Engel, [*Z. Phys.*]{} [**C66**]{} (1995) 203.
R. Engel, [](http://xxx.lanl.gov/abs/{\rm PHOJET manual (program version
1.05)}).
Engel, R. [*H*adronic Interactions of Photons at High Energies]{}. Ph.D. thesis, University of Siegen, 1997.
A. Breakstone [*et. al.*]{}, [*Phys. Rev.*]{} [**D30**]{} (1984) 528.
G. J. Alner [*et. al.*]{}, [*Phys. Rept.*]{} [**154**]{} (1987) 247–383.
F. Abe [*et. al.*]{}, [*Phys. Rev.*]{} [**D41**]{} (1990) 2330.
T. Alexopoulos [*et. al.*]{}, [*Phys. Lett.*]{} [**B435**]{} (1998) 453–457.
S. G. Matinyan and W. D. Walker, [*Phys. Rev.*]{} [**D59**]{} (1999) 034022, \[[[hep-ph/9801219]{}](http://xxx.lanl.gov/abs/hep-ph/9801219)\].
Z. Koba, H. B. Nielsen, and P. Olesen, [*Nucl. Phys.*]{} [**B40**]{} (1972) 317–334.
R. E. Ansorge [*et. al.*]{}, [*Z. Phys.*]{} [**C43**]{} (1989) 357.
T. Affolder [*et. al.*]{}, [*Phys. Rev.*]{} [**D65**]{} (2002) 092002.
T. Sj[ö]{}strand, [*Phys. Lett.*]{} [**B157**]{} (1985) 321.
M. Bengtsson, T. Sj[ö]{}strand, and M. van Zijl, [*Z. Phys.*]{} [**C32**]{} (1986) 67.
J. Huston, *A Comparison of the Underlying Event in Jet and Min-Bias Events*. Talk presented at DPF2000, Columbus, OH, August 11, 2000.
V. Tano, *The Underlying Event in Jet and Minimum Bias Events at the Tevatron*. Talk presented at ISMD2001, Datong, China, September 1–7, 2001.
R.D. Field, *Min-Bias and the Underlying Event at the Tevatron and the LHC*. Talk presented at the Fermilab ME/MC Tuning Workshop, Fermilab, October 4, 2002.
R.D. Field, *Toward an Understanding of Hadron Collisions: From Feynman-Field until Now*. Talk presented at the Fermilab Joint Theoretical Experimental Seminar, Fermilab, October 4, 2002.
R.D. Field, *Min-Bias and the Underlying Event at CDF*. Talk presented at the Monte-Carlo Tools for the LHC Workshop, CERN, July 31, 2003.
G. Marchesini and B. R. Webber, [*Nucl. Phys.*]{} [**B310**]{} (1988) 461.
I. G. Knowles, [*Nucl. Phys.*]{} [**B310**]{} (1988) 571.
S. Catani, B. R. Webber, and G. Marchesini, [*Nucl. Phys.*]{} [**B349**]{} (1991) 635–654.
D. Acosta [*et. al.*]{}, [*Phys. Rev.*]{} [**D65**]{} (2002) 072005.
Collaboration [*FERMILAB*]{} [**390-E**]{} (1996).
M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau, and A. D. Polosa, [ *JHEP*]{} [**0307:001**]{} (2003) 35, \[[[hep-ph/0206293]{}](http://xxx.lanl.gov/abs/hep-ph/0206293)\].
V. Ilyin, [*[PEVLIB]{}, [`/afs/cern.ch/cms/physics/PEVLIB/`]{}*]{}.
A. Pukhov [*et. al.*]{}, [[ hep-ph/9908288]{}](http://xxx.lanl.gov/abs/hep-ph/9908288).
.
L. Dudko and S. Mrenna, [*[FNAL MCDB]{}, [`http://www-d0.fnal.gov/~dudko/mcdb`]{}*]{}.
L. Dudko and A. Sherstnev, [*[CMS MCDB]{}, [`http://cmsdoc.cern.ch/cms/generators/mcdb`]{}*]{}.
L. Dudko and A. Sherstnev, [*[CMS MCDB: direct AFS link]{}, [`/afs/cern.ch/cms/generators/{mcdb,mcdb2,mcdb3}`]{}*]{}.
P. Bartalini, L. Dudko, A. Kryukov, I. Seluzhenkov, A. Sherstnev, and A. Vologdin, [*[LCG Monte-Carlo Data Base, LCG note, in preparation]{}*]{}.
C. Balazs, J. Huston, and I. Puljak, [*Phys. Rev.*]{} [**D63**]{} (2001) 014021, \[[[hep-ph/0002032]{}](http://xxx.lanl.gov/abs/hep-ph/0002032)\].
J. Collins and D. Soper, [*Nucl. Phys.*]{} [**B193**]{} (1981) 381.
G. Miu and T. Sj[ö]{}strand, [*Phys. Lett.*]{} [**B449**]{} (1999) 313–320, \[[[hep-ph/9812455]{}](http://xxx.lanl.gov/abs/hep-ph/9812455)\].
E. Thom[é]{}, [[ hep-ph/0401121]{}](http://xxx.lanl.gov/abs/hep-ph/0401121).
H. L. Lai [*et. al.*]{},, [**CTEQ**]{} Collaboration [*Eur. Phys. J.*]{} [ **C12**]{} (2000) 375–392, \[[[ hep-ph/9903282]{}](http://xxx.lanl.gov/abs/hep-ph/9903282)\].
T. Sj[ö]{}strand, [](http://xxx.lanl.gov/abs/{\rm these proceedings}).
T. Affolder [*et. al.*]{},, [**CDF**]{} Collaboration [*Phys. Rev. Lett.*]{} [ **84**]{} (2000) 845–850, \[[[ hep-ex/0001021]{}](http://xxx.lanl.gov/abs/hep-ex/0001021)\].
C. Balazs [*et. al.*]{}, [](http://xxx.lanl.gov/abs/{\rm these
proceedings}).
G. Bozzi, S. Catani, D. de Florian, and M. Grazzini, [*Phys. Lett.*]{} [ **B564**]{} (2003) 65–72, \[[[ hep-ph/0302104]{}](http://xxx.lanl.gov/abs/hep-ph/0302104)\].
C. Balazs and C. P. Yuan, [*Phys. Lett.*]{} [**B478**]{} (2000) 192–198, \[[[hep-ph/0001103]{}](http://xxx.lanl.gov/abs/hep-ph/0001103)\].
A. Kulesza, G. Sterman, and W. Vogelsang, [*Phys. Rev.*]{} [**D69**]{} (2004) 014012, \[[[ hep-ph/0309264]{}](http://xxx.lanl.gov/abs/hep-ph/0309264)\].
E. Berger and J. Qiu, [*Phys. Rev.*]{} [**D67**]{} (2003) 034026, \[[[hep-ph/0210135]{}](http://xxx.lanl.gov/abs/hep-ph/0210135)\].
S. Frixione and B. Webber, [*JHEP*]{} [**06**]{} (2002) 029, \[[[hep-ph/0204244]{}](http://xxx.lanl.gov/abs/hep-ph/0204244)\].
S. Frixione, P. Nason, and B. Webber, [*JHEP*]{} [**08**]{} (2003) 007, \[[[hep-ph/0305252]{}](http://xxx.lanl.gov/abs/hep-ph/0305252)\].
G. Corcella [*et. al.*]{}, [*JHEP*]{} [**01**]{} (2001) 010, \[[[hep-ph/0011363]{}](http://xxx.lanl.gov/abs/hep-ph/0011363)\].
M. Bengtsson and T. Sj[ö]{}strand, [*Nucl. Phys.*]{} [**B289**]{} (1987) 810.
E. Norrbin and T. Sj[ö]{}strand, [*Nucl. Phys.*]{} [**B603**]{} (2001) 297–342, \[[[hep-ph/0010012]{}](http://xxx.lanl.gov/abs/hep-ph/0010012)\].
G. Marchesini and B. Webber, [*Nucl. Phys.*]{} [**B238**]{} (1984) 1.
L. L[ö]{}nnblad, [*Comput. Phys. Commun.*]{} [**71**]{} (1992) 15–31.
H. Kharraziha and L. L[ö]{}nnblad, [*JHEP*]{} [**03**]{} (1998) 006, \[[[hep-ph/9709424]{}](http://xxx.lanl.gov/abs/hep-ph/9709424)\].
S. Catani, F. Krauss, R. Kuhn, and B. R. Webber, [*JHEP*]{} [**11**]{} (2001) 063, \[[[hep-ph/0109231]{}](http://xxx.lanl.gov/abs/hep-ph/0109231)\].
L. L[ö]{}nnblad, [*JHEP*]{} [**05**]{} (2002) 046, \[[[hep-ph/0112284]{}](http://xxx.lanl.gov/abs/hep-ph/0112284)\].
T. Sj[ö]{}strand and P. Skands, [](http://xxx.lanl.gov/abs/{\rm these
proceedings}).
S. Moretti, L. L[ö]{}nnblad, and T. Sj[ö]{}strand, [*JHEP*]{} [**08**]{} (1998) 001, \[[[hep-ph/9804296]{}](http://xxx.lanl.gov/abs/hep-ph/9804296)\].
J. Huston, I. Puljak, T. Sj[ö]{}strand, and E. Thom[é]{}, [](http://xxx.lanl.gov/abs/{\rm these proceedings}).
T. Sjostrand and M. Bengtsson, [*Comput. Phys. Commun.*]{} [**43**]{} (1987) 367.
M. Bengtsson and T. Sjostrand, [*Z. Phys.*]{} [**C37**]{} (1988) 465.
G. Corcella [*et. al.*]{}, [[ hep-ph/0210213]{}](http://xxx.lanl.gov/abs/hep-ph/0210213).
M. H. Seymour, [*Z. Phys.*]{} [**C56**]{} (1992) 161–170.
M. H. Seymour, [*[contributed to 27th International Conference on High Energy Physics (ICHEP), Glasgow, Scotland, 20-27 Jul 1994]{}*]{}.
G. Corcella and M. H. Seymour, [*Phys. Lett.*]{} [**B442**]{} (1998) 417–426, \[[[hep-ph/9809451]{}](http://xxx.lanl.gov/abs/hep-ph/9809451)\].
G. Corcella and M. H. Seymour, [*Nucl. Phys.*]{} [**B565**]{} (2000) 227–244, \[[[hep-ph/9908388]{}](http://xxx.lanl.gov/abs/hep-ph/9908388)\].
M. H. Seymour, [*Comp. Phys. Commun.*]{} [**90**]{} (1995) 95–101, \[[[hep-ph/9410414]{}](http://xxx.lanl.gov/abs/hep-ph/9410414)\].
F. Krauss, [*JHEP*]{} [**08**]{} (2002) 015, \[[[hep-ph/0205283]{}](http://xxx.lanl.gov/abs/hep-ph/0205283)\].
F. Maltoni and T. Stelzer, [[ hep-ph/0208156]{}](http://xxx.lanl.gov/abs/hep-ph/0208156).
E. Boos [*et. al.*]{}, [[ hep-ph/0109068]{}](http://xxx.lanl.gov/abs/hep-ph/0109068).
N. Kidonakis and G. Sterman, [*Phys. Lett.*]{} [**B387**]{} (1996) 867.
N. Kidonakis and G. Sterman, [*Nucl. Phys.*]{} [**B505**]{} (1997) 321, \[[[hep-ph/9705234]{}](http://xxx.lanl.gov/abs/hep-ph/9705234)\].
N. Kidonakis, G. Oderda, and G. Sterman, [*Nucl. Phys.*]{} [**B531**]{} (1998) 365, \[[[hep-ph/9803241]{}](http://xxx.lanl.gov/abs/hep-ph/9803241)\].
N. Kidonakis, [*Int. J. Mod. Phys.*]{} [**A15**]{} (2000) 1245, \[[[hep-ph/9902484]{}](http://xxx.lanl.gov/abs/hep-ph/9902484)\].
N. Kidonakis, [[ hep-ph/0303186]{}](http://xxx.lanl.gov/abs/hep-ph/0303186).
P. B. Arnold and M. H. Reno, [*Nucl. Phys.*]{} [**B319**]{} (1989) 37.
R. J. Gonsalves, J. Pawlowski, and C.-F. Wai, [*Phys. Rev.*]{} [**D40**]{} (1989) 2245.
R. J. Gonsalves, J. Pawlowski, and C.-F. Wai, [*Phys. Lett.*]{} [**B252**]{} (1990) 663.
N. Kidonakis and V. Del Duca, [*Phys. Lett.*]{} [**B480**]{} (2000) 87, \[[[hep-ph/9911460]{}](http://xxx.lanl.gov/abs/hep-ph/9911460)\].
N. Kidonakis and A. Sabio Vera, [*JHEP*]{} [**02**]{} (2004) 027, \[[[hep-ph/0311266]{}](http://xxx.lanl.gov/abs/hep-ph/0311266)\].
A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, [*Eur. Phys. J.*]{} [**C28**]{} (2003) 455, \[[[ hep-ph/0211080]{}](http://xxx.lanl.gov/abs/hep-ph/0211080)\].
P. Aurenche, A. Douiri, R. Baier, M. Fontannaz, and D. Schiff, [*Phys. Lett.*]{} [**B140**]{} (1984) 87.
P. Aurenche, R. Baier, M. Fontannaz, and D. Schiff, [*Nucl. Phys.*]{} [ **B297**]{} (1988) 661.
L. E. Gordon and W. Vogelsang, [*Phys. Rev.*]{} [**D48**]{} (1993) 3136.
N. Kidonakis and J. F. Owens, [*Phys. Rev.*]{} [**D61**]{} (2000) 094004, \[[[hep-ph/9912388]{}](http://xxx.lanl.gov/abs/hep-ph/9912388)\].
N. Kidonakis and J. F. Owens, [*Int. J. Mod. Phys.*]{} [**A19**]{} (2004) 149, \[[[hep-ph/0307352]{}](http://xxx.lanl.gov/abs/hep-ph/0307352)\].
H. Baer, J. Ohnemus, and J. F. Owens, [*Phys. Rev.*]{} [**D42**]{} (1990) 61.
B. W. Harris and J. F. Owens, [*Phys. Rev.*]{} [**D65**]{} (2002) 094032, \[[[hep-ph/0102128]{}](http://xxx.lanl.gov/abs/hep-ph/0102128)\].
L. Bourhis, M. Fontannaz, J. P. Guillet, and M. Werlen, [*Eur. Phys. J.*]{} [**C19**]{} (2001) 89, \[[[ hep-ph/0009101]{}](http://xxx.lanl.gov/abs/hep-ph/0009101)\].
J. Pumplin [*et. al.*]{}, [*JHEP*]{} [**07**]{} (2002) 012, \[[[hep-ph/0201195]{}](http://xxx.lanl.gov/abs/hep-ph/0201195)\].
L. Apanasevich [*et. al.*]{}, [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 2642, \[[[hep-ex/9711017]{}](http://xxx.lanl.gov/abs/hep-ex/9711017)\].
G. Ballocchi [*et. al.*]{},, [**UA6**]{} Collaboration [*Phys. Lett.*]{} [ **B436**]{} (1998) 222.
N. Kidonakis, [*Phys. Rev.*]{} [**D64**]{} (2001) 014009, \[[[hep-ph/0010002]{}](http://xxx.lanl.gov/abs/hep-ph/0010002)\].
N. Kidonakis, E. Laenen, S. Moch, and R. Vogt, [*Phys. Rev.*]{} [**D64**]{} (2001) 114001, \[[[ hep-ph/0105041]{}](http://xxx.lanl.gov/abs/hep-ph/0105041)\].
N. Kidonakis and R. Vogt, [*Phys. Rev.*]{} [**D68**]{} (2003) 114014, \[[[hep-ph/0308222]{}](http://xxx.lanl.gov/abs/hep-ph/0308222)\].
N. Kidonakis and A. Belyaev, [*JHEP*]{} [**12**]{} (2003) 004, \[[[hep-ph/0310299]{}](http://xxx.lanl.gov/abs/hep-ph/0310299)\].
G. Sterman, [*Nucl. Phys.*]{} [**B281**]{} (1987) 310.
S. Catani and L. Trentadue, [*Nucl. Phys.*]{} [**B327**]{} (1989) 323.
M. Kramer, E. Laenen, and M. Spira, [*Nucl. Phys.*]{} [**B511**]{} (1998) 523–549, \[[[ hep-ph/9611272]{}](http://xxx.lanl.gov/abs/hep-ph/9611272)\].
G. Parisi, [*Phys. Lett.*]{} [**B90**]{} (1980) 295.
L. Magnea and G. Sterman, [*Phys. Rev.*]{} [**D42**]{} (1990) 4222–4227.
H. Contopanagos, E. Laenen, and G. Sterman, [*Nucl. Phys.*]{} [**B484**]{} (1997) 303–330, \[[[ hep-ph/9604313]{}](http://xxx.lanl.gov/abs/hep-ph/9604313)\].
R. Akhoury, M. Sotiropoulos, and G. Sterman, [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 3819–3822, \[[[ hep-ph/9807330]{}](http://xxx.lanl.gov/abs/hep-ph/9807330)\].
R. Akhoury and M. Sotiropoulos, [[hep-ph/0304131]{}](http://xxx.lanl.gov/abs/hep-ph/0304131).
Y. Dokshitzer, [*[private communication]{}*]{}.
E. Laenen, G. Sterman, and W. Vogelsang, [*Phys. Rev.*]{} [**D63**]{} (2001) 114018, \[[[ hep-ph/0010080]{}](http://xxx.lanl.gov/abs/hep-ph/0010080)\].
T. Eynck, E. Laenen, and L. Magnea, [*JHEP*]{} [**06**]{} (2003) 057, \[[[hep-ph/0305179]{}](http://xxx.lanl.gov/abs/hep-ph/0305179)\].
E. Gardi and R. Roberts, [*Nucl. Phys.*]{} [**B653**]{} (2003) 227–255, \[[[hep-ph/0210429]{}](http://xxx.lanl.gov/abs/hep-ph/0210429)\].
W. van Neerven and E. Zijlstra, [*Nucl. Phys.*]{} [**B382**]{} (1992) 11–62.
H.-n. Li, [*Phys. Lett.*]{} [**B454**]{} (1999) 328–334, \[[[hep-ph/9812363]{}](http://xxx.lanl.gov/abs/hep-ph/9812363)\].
A. Kulesza, G. Sterman, and W. Vogelsang, [*Phys. Rev.*]{} [**D66**]{} (2002) 014011, \[[[ hep-ph/0202251]{}](http://xxx.lanl.gov/abs/hep-ph/0202251)\].
E. Laenen, G. Sterman, and W. Vogelsang, [*Phys. Rev. Lett.*]{} [**84**]{} (2000) 4296–4299, \[[[ hep-ph/0002078]{}](http://xxx.lanl.gov/abs/hep-ph/0002078)\].
A. Banfi and E. Laenen, [*[in preparation]{}*]{}.
D. Cavalli [*et. al.*]{}, [[ hep-ph/0203056]{}](http://xxx.lanl.gov/abs/hep-ph/0203056).
S. Catani, D. de Florian, and M. Grazzini, [*JHEP*]{} [**05**]{} (2001) 025, \[[[hep-ph/0102227]{}](http://xxx.lanl.gov/abs/hep-ph/0102227)\].
R. Harlander and W. Kilgore, [*Phys. Rev.*]{} [**D64**]{} (2001) 013015, \[[[hep-ph/0102241]{}](http://xxx.lanl.gov/abs/hep-ph/0102241)\].
R. Harlander and W. Kilgore, [*Phys. Rev. Lett.*]{} [**88**]{} (2002) 201801, \[[[hep-ph/0201206]{}](http://xxx.lanl.gov/abs/hep-ph/0201206)\].
C. Anastasiou and K. Melnikov, [*Nucl. Phys.*]{} [**B646**]{} (2002) 220–256, \[[[hep-ph/0207004]{}](http://xxx.lanl.gov/abs/hep-ph/0207004)\].
V. Ravindran, J. Smith, and W. L. van Neerven, [*Nucl. Phys.*]{} [**B665**]{} (2003) 325–366, \[[[ hep-ph/0302135]{}](http://xxx.lanl.gov/abs/hep-ph/0302135)\].
S. Catani, D. de Florian, M. Grazzini, and P. Nason, [*JHEP*]{} [**07**]{} (2003) 028, \[[[hep-ph/0306211]{}](http://xxx.lanl.gov/abs/hep-ph/0306211)\].
G. Parisi and R. Petronzio, [*Nucl. Phys.*]{} [**B154**]{} (1979) 427.
Y. Dokshitzer, D. Diakonov, and S. I. Troian, [*Phys. Rept.*]{} [**58**]{} (1980) 269–395.
J. C. Collins, D. E. Soper, and G. Sterman, [*Nucl. Phys.*]{} [**B250**]{} (1985) 199.
S. Catani, D. de Florian, and M. Grazzini, [*Nucl. Phys.*]{} [**B596**]{} (2001) 299, \[[[hep-ph/0008184]{}](http://xxx.lanl.gov/abs/hep-ph/0008184)\].
S. Catani, E. D’Emilio, and L. Trentadue, [*Phys. Lett.*]{} [**B211**]{} (1988) 335.
W. Giele [*et. al.*]{}, [[ hep-ph/0204316]{}](http://xxx.lanl.gov/abs/hep-ph/0204316).
R. P. Kauffman, [*Phys. Rev.*]{} [**D45**]{} (1992) 1512–1517.
D. de Florian and M. Grazzini, [*Phys. Rev. Lett.*]{} [**85**]{} (2000) 4678, \[[[hep-ph/0008152]{}](http://xxx.lanl.gov/abs/hep-ph/0008152)\].
D. de Florian and M. Grazzini, [*Nucl. Phys.*]{} [**B616**]{} (2001) 247–285, \[[[hep-ph/0108273]{}](http://xxx.lanl.gov/abs/hep-ph/0108273)\].
A. Vogt, [*Phys. Lett.*]{} [**B497**]{} (2001) 228–234, \[[[hep-ph/0010146]{}](http://xxx.lanl.gov/abs/hep-ph/0010146)\].
S. Catani [*et. al.*]{}, [[ hep-ph/0005114]{}](http://xxx.lanl.gov/abs/hep-ph/0005114).
C. Balazs and C. P. Yuan, [*Phys. Rev.*]{} [**D56**]{} (1997) 5558–5583, \[[[hep-ph/9704258]{}](http://xxx.lanl.gov/abs/hep-ph/9704258)\].
G. Corcella and S. Moretti, [[ hep-ph/0402146]{}](http://xxx.lanl.gov/abs/hep-ph/0402146).
D. de Florian, M. Grazzini, and Z. Kunszt, [*Phys. Rev. Lett.*]{} [**82**]{} (1999) 5209–5212, \[[[ hep-ph/9902483]{}](http://xxx.lanl.gov/abs/hep-ph/9902483)\].
Z. Kunszt, S. Moretti, and W. J. Stirling, [*Z. Phys.*]{} [**C74**]{} (1997) 479–491, \[[[ hep-ph/9611397]{}](http://xxx.lanl.gov/abs/hep-ph/9611397)\].
A. Djouadi [*et. al.*]{}, [[ hep-ph/0002258]{}](http://xxx.lanl.gov/abs/hep-ph/0002258).
M. Dobbs, [*Phys. Rev.*]{} [**D64**]{} (2001) 034016, \[[[hep-ph/0103174]{}](http://xxx.lanl.gov/abs/hep-ph/0103174)\].
U. Baur and E. W. N. Glover, [*Nucl. Phys.*]{} [**B339**]{} (1990) 38–66.
S. Moretti, K. Odagiri, P. Richardson, M. H. Seymour, and B. R. Webber, [ *JHEP*]{} [**04**]{} (2002) 028, \[[[hep-ph/0204123]{}](http://xxx.lanl.gov/abs/hep-ph/0204123)\].
S. Frixione and B. Webber, [[ hep-ph/0309186]{}](http://xxx.lanl.gov/abs/hep-ph/0309186).
S. Bethke, [*J. Phys.*]{} [**G26**]{} (2000) R27, \[[[hep-ex/0004021]{}](http://xxx.lanl.gov/abs/hep-ex/0004021)\].
R. W. L. Jones, M. Ford, G. P. Salam, H. Stenzel, and D. Wicke, [*JHEP*]{} [ **12**]{} (2003) 007, \[[[ hep-ph/0312016]{}](http://xxx.lanl.gov/abs/hep-ph/0312016)\].
S. Kluth, P. A. Movilla Fernandez, S. Bethke, C. Pahl, and P. Pfeifenschneider, [*Eur. Phys. J.*]{} [**C21**]{} (2001) 199–210, \[[[hep-ex/0012044]{}](http://xxx.lanl.gov/abs/hep-ex/0012044)\].
M. Dasgupta and G. P. Salam, [[ hep-ph/0312283]{}](http://xxx.lanl.gov/abs/hep-ph/0312283).
S. Catani, L. Trentadue, G. Turnock, and B. R. Webber, [*Nucl. Phys.*]{} [ **B407**]{} (1993) 3–42.
A. Banfi, G. P. Salam, and G. Zanderighi, [[hep-ph/0304148]{}](http://xxx.lanl.gov/abs/hep-ph/0304148).
N. Brown and W. J. Stirling, [*Phys. Lett.*]{} [**B252**]{} (1990) 657–662.
M. Dasgupta and G. P. Salam, [*Phys. Lett.*]{} [**B512**]{} (2001) 323–330, \[[[hep-ph/0104277]{}](http://xxx.lanl.gov/abs/hep-ph/0104277)\].
A. Banfi, G. P. Salam, and G. Zanderighi, [*JHEP*]{} [**01**]{} (2002) 018, \[[[hep-ph/0112156]{}](http://xxx.lanl.gov/abs/hep-ph/0112156)\].
S. Haywood [*et. al.*]{}, [[ hep-ph/0003275]{}](http://xxx.lanl.gov/abs/hep-ph/0003275).
.
U. Baur, R.K. Ellis and D. Zeppenfeld, eds. Proceedings of Physics at Run II: QCD and Weak Boson Physics Workshop, Batavia, Illinois, 1999.
C. Balazs, J.-w. Qiu, and C. P. Yuan, [*Phys. Lett.*]{} [**B355**]{} (1995) 548–554, \[[[ hep-ph/9505203]{}](http://xxx.lanl.gov/abs/hep-ph/9505203)\].
T. Affolder [*et. al.*]{},, [**CDF**]{} Collaboration [*Phys. Rev.*]{} [**D64**]{} (2001) 052001, \[[[ hep-ex/0007044]{}](http://xxx.lanl.gov/abs/hep-ex/0007044)\].
S. Dittmaier and M. Kramer, [*Phys. Rev.*]{} [**D65**]{} (2002) 073007, \[[[hep-ph/0109062]{}](http://xxx.lanl.gov/abs/hep-ph/0109062)\].
U. Baur, S. Keller, and D. Wackeroth, [*Phys. Rev.*]{} [**D59**]{} (1999) 013002, \[[[hep-ph/9807417]{}](http://xxx.lanl.gov/abs/hep-ph/9807417)\].
Q.-H. Cao and C.-P. Yuan, [[ hep-ph/0401026]{}](http://xxx.lanl.gov/abs/hep-ph/0401026).
P. Nadolsky, D. R. Stump, and C. P. Yuan, [*Phys. Rev.*]{} [**D61**]{} (2000) 014003, \[[[ hep-ph/9906280]{}](http://xxx.lanl.gov/abs/hep-ph/9906280)\].
P. M. Nadolsky, D. R. Stump, and C. P. Yuan, [*Phys. Rev.*]{} [**D64**]{} (2001) 114011, \[[[ hep-ph/0012261]{}](http://xxx.lanl.gov/abs/hep-ph/0012261)\].
E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, [*Sov. Phys. JETP*]{} [**44**]{} (1976) 443–450.
I. I. Balitsky and L. N. Lipatov, [*Sov. J. Nucl. Phys.*]{} [**28**]{} (1978) 822–829.
F. Landry, R. Brock, P. M. Nadolsky, and C. P. Yuan, [*Phys. Rev.*]{} [**D67**]{} (2003) 073016, \[[[ hep-ph/0212159]{}](http://xxx.lanl.gov/abs/hep-ph/0212159)\].
D. Stump [*et. al.*]{}, [*JHEP*]{} [**10**]{} (2003) 046, \[[[hep-ph/0303013]{}](http://xxx.lanl.gov/abs/hep-ph/0303013)\].
J. Kodaira and L. Trentadue, [*Phys. Lett.*]{} [**B112**]{} (1982) 66.
C. T. H. Davies and W. J. Stirling, [*Nucl. Phys.*]{} [**B244**]{} (1984) 337.
R. P. Kauffman, [*Phys. Rev.*]{} [**D44**]{} (1991) 1415.
C. P. Yuan, [*Phys. Lett.*]{} [**B283**]{} (1992) 395.
Z. Bern, L. J. Dixon, and C. Schmidt, [*Phys. Rev.*]{} [**D66**]{} (2002) 074018, \[[[hep-ph/0206194]{}](http://xxx.lanl.gov/abs/hep-ph/0206194)\].
L. N. Lipatov, [*Sov. J. Nucl. Phys.*]{} [**23**]{} (1976) 338–345.
V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, [*Phys. Lett.*]{} [**B60**]{} (1975) 50–52.
E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, [*Sov. Phys. JETP*]{} [**45**]{} (1977) 199–204.
I. I. Balitsky and L. N. Lipatov, [*JETP Lett.*]{} [**30**]{} (1979) 355.
V. S. Fadin and L. N. Lipatov, [*Phys. Lett.*]{} [**B429**]{} (1998) 127–134, \[[[hep-ph/9802290]{}](http://xxx.lanl.gov/abs/hep-ph/9802290)\].
M. Ciafaloni and G. Camici, [*Phys. Lett.*]{} [**B430**]{} (1998) 349–354, \[[[hep-ph/9803389]{}](http://xxx.lanl.gov/abs/hep-ph/9803389)\].
D. A. Ross, [*Phys. Lett.*]{} [**B431**]{} (1998) 161–165, \[[[hep-ph/9804332]{}](http://xxx.lanl.gov/abs/hep-ph/9804332)\].
E. Levin, [[hep-ph/9806228]{}](http://xxx.lanl.gov/abs/hep-ph/9806228).
G. P. Salam, [*JHEP*]{} [**07**]{} (1998) 019, \[[[hep-ph/9806482]{}](http://xxx.lanl.gov/abs/hep-ph/9806482)\].
M. Ciafaloni and D. Colferai, [*Phys. Lett.*]{} [**B452**]{} (1999) 372–378, \[[[hep-ph/9812366]{}](http://xxx.lanl.gov/abs/hep-ph/9812366)\].
S. J. Brodsky, V. S. Fadin, V. T. Kim, L. N. Lipatov, and G. B. Pivovarov, [ *JETP Lett.*]{} [**70**]{} (1999) 155–160, \[[[hep-ph/9901229]{}](http://xxx.lanl.gov/abs/hep-ph/9901229)\].
C. R. Schmidt, [*Phys. Rev.*]{} [**D60**]{} (1999) 074003, \[[[hep-ph/9901397]{}](http://xxx.lanl.gov/abs/hep-ph/9901397)\].
J. R. Forshaw, D. A. Ross, and A. Sabio Vera, [*Phys. Lett.*]{} [**B455**]{} (1999) 273–282, \[[[ hep-ph/9903390]{}](http://xxx.lanl.gov/abs/hep-ph/9903390)\].
M. Ciafaloni, D. Colferai, and G. P. Salam, [*Phys. Rev.*]{} [**D60**]{} (1999) 114036, \[[[ hep-ph/9905566]{}](http://xxx.lanl.gov/abs/hep-ph/9905566)\].
R. D. Ball and S. Forte, [*Phys. Lett.*]{} [**B465**]{} (1999) 271–281, \[[[hep-ph/9906222]{}](http://xxx.lanl.gov/abs/hep-ph/9906222)\].
G. Altarelli, R. D. Ball, and S. Forte, [*Nucl. Phys.*]{} [**B575**]{} (2000) 313–329, \[[[ hep-ph/9911273]{}](http://xxx.lanl.gov/abs/hep-ph/9911273)\].
G. Altarelli, R. D. Ball, and S. Forte, [*Nucl. Phys.*]{} [**B599**]{} (2001) 383–423, \[[[ hep-ph/0011270]{}](http://xxx.lanl.gov/abs/hep-ph/0011270)\].
Y. V. Kovchegov and A. H. Mueller, [*Phys. Lett.*]{} [**B439**]{} (1998) 428–436, \[[[ hep-ph/9805208]{}](http://xxx.lanl.gov/abs/hep-ph/9805208)\].
N. Armesto, J. Bartels, and M. A. Braun, [*Phys. Lett.*]{} [**B442**]{} (1998) 459–469, \[[[ hep-ph/9808340]{}](http://xxx.lanl.gov/abs/hep-ph/9808340)\].
R. S. Thorne, [*Phys. Rev.*]{} [**D60**]{} (1999) 054031, \[[[hep-ph/9901331]{}](http://xxx.lanl.gov/abs/hep-ph/9901331)\].
R. S. Thorne, [*Phys. Lett.*]{} [**B474**]{} (2000) 372–384, \[[[hep-ph/9912284]{}](http://xxx.lanl.gov/abs/hep-ph/9912284)\].
J. R. Forshaw, D. A. Ross, and A. Sabio Vera, [*Phys. Lett.*]{} [**B498**]{} (2001) 149–155, \[[[ hep-ph/0011047]{}](http://xxx.lanl.gov/abs/hep-ph/0011047)\].
R. S. Thorne, [*Phys. Rev.*]{} [**D64**]{} (2001) 074005, \[[[hep-ph/0103210]{}](http://xxx.lanl.gov/abs/hep-ph/0103210)\].
M. Ciafaloni, M. Taiuti, and A. H. Mueller, [*Nucl. Phys.*]{} [**B616**]{} (2001) 349–366, \[[[ hep-ph/0107009]{}](http://xxx.lanl.gov/abs/hep-ph/0107009)\].
G. Altarelli, R. D. Ball, and S. Forte, [*Nucl. Phys.*]{} [**B621**]{} (2002) 359–387, \[[[ http://arXiv.org/abs/hep-ph/0109178]{}](http://xxx.lanl.gov/abs/http://arXiv.org/abs/hep-ph/0109178)\].
M. Ciafaloni, D. Colferai, G. P. Salam, and A. M. Stasto, [*Phys. Rev.*]{} [ **D66**]{} (2002) 054014, \[[[ hep-ph/0204282]{}](http://xxx.lanl.gov/abs/hep-ph/0204282)\].
M. Ciafaloni, D. Colferai, G. P. Salam, and A. M. Stasto, [*Phys. Lett.*]{} [**B541**]{} (2002) 314–326, \[[[ http://arXiv.org/abs/hep-ph/0204287]{}](http://xxx.lanl.gov/abs/http://arXiv.org/abs/hep-ph/0204287)\].
M. Ciafaloni, D. Colferai, D. Colferai, G. P. Salam, and A. M. Stasto, [ *Phys. Lett.*]{} [**B576**]{} (2003) 143–151, \[[[hep-ph/0305254]{}](http://xxx.lanl.gov/abs/hep-ph/0305254)\].
G. Altarelli, R. D. Ball, and S. Forte, [*Nucl. Phys.*]{} [**B674**]{} (2003) 459–483, \[[[ hep-ph/0306156]{}](http://xxx.lanl.gov/abs/hep-ph/0306156)\].
M. Ciafaloni, D. Colferai, G. P. Salam, and A. M. Stasto, [*Phys. Rev.*]{} [ **D68**]{} (2003) 114003, \[[[ hep-ph/0307188]{}](http://xxx.lanl.gov/abs/hep-ph/0307188)\].
M. Ciafaloni, D. Colferai, G. P. Salam, and A. M. Stasto,. hep-ph/0311325.
J. R. Andersen and A. Sabio Vera, [*Phys. Lett.*]{} [**B567**]{} (2003) 116–124, \[[[hep-ph/0305236]{}](http://xxx.lanl.gov/abs/hep-ph/0305236)\].
J. R. Andersen and A. Sabio Vera, [*Nucl. Phys.*]{} [**B679**]{} (2004) 345–362, \[[[hep-ph/0309331]{}](http://xxx.lanl.gov/abs/hep-ph/0309331)\].
J. Kwiecinski, C. A. M. Lewis, and A. D. Martin, [*Phys. Rev.*]{} [**D54**]{} (1996) 6664–6673, \[[[ hep-ph/9606375]{}](http://xxx.lanl.gov/abs/hep-ph/9606375)\].
C. R. Schmidt, [*Phys. Rev. Lett.*]{} [**78**]{} (1997) 4531–4535, \[[[hep-ph/9612454]{}](http://xxx.lanl.gov/abs/hep-ph/9612454)\].
L. H. Orr and W. J. Stirling, [*Phys. Rev.*]{} [**D56**]{} (1997) 5875–5884, \[[[hep-ph/9706529]{}](http://xxx.lanl.gov/abs/hep-ph/9706529)\].
J. R. Andersen and W. J. Stirling, [*JHEP*]{} [**02**]{} (2003) 018, \[[[hep-ph/0301081]{}](http://xxx.lanl.gov/abs/hep-ph/0301081)\].
L. H. Orr and W. J. Stirling, [*Phys. Lett.*]{} [**B429**]{} (1998) 135–144, \[[[hep-ph/9801304]{}](http://xxx.lanl.gov/abs/hep-ph/9801304)\].
J. R. Andersen, V. Del Duca, S. Frixione, C. R. Schmidt, and W. J. Stirling, [*JHEP*]{} [**02**]{} (2001) 007, \[[[hep-ph/0101180]{}](http://xxx.lanl.gov/abs/hep-ph/0101180)\].
J. R. Andersen, V. Del Duca, F. Maltoni, and W. J. Stirling, [*JHEP*]{} [ **05**]{} (2001) 048, \[[[ hep-ph/0105146]{}](http://xxx.lanl.gov/abs/hep-ph/0105146)\].
V. Del Duca and C. R. Schmidt, [*Nucl. Phys. Proc. Suppl.*]{} [**39BC**]{} (1995) 137–140, \[[[ hep-ph/9408239]{}](http://xxx.lanl.gov/abs/hep-ph/9408239)\].
W. J. Stirling, [*Nucl. Phys.*]{} [**B423**]{} (1994) 56–79, \[[[hep-ph/9401266]{}](http://xxx.lanl.gov/abs/hep-ph/9401266)\].
Collaboration. Technical Design Report, 1997.
Collaboration. Technical Design Report, 1997.
Collaboration. Technical Design Report, 1997, [CERN/LHCC 99-7]{}.
M. Rijssenbeek, talk contributed to the Xth Workshop on Elastic and Diffractive Scattering, Helsinki (2003).
A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, [*Eur. Phys. J.*]{} [**C14**]{} (2000) 133–145, \[[[hep-ph/9907231]{}](http://xxx.lanl.gov/abs/hep-ph/9907231)\].
M. A. Kimber, J. Kwiecinski, and A. D. Martin, [*Phys. Lett.*]{} [**B508**]{} (2001) 58–64, \[[[ hep-ph/0101099]{}](http://xxx.lanl.gov/abs/hep-ph/0101099)\].
A. De Roeck, [*Acta Phys. Polon.*]{} [**B33**]{} (2002) 3591–3597.
ATLAS: Detector and physics performance technical design report. Volume 1&2, CERN-LHCC-99-14,CERN-LHCC-99-15.
CMS DAQ TDR (CMS, The Trigger and Data Acquisition Project, Vol II: Data acquisition and High-Level Trigger, CERN-LHCC-2002-26, and S. Shevchenko, private communication.
T. Binoth, J. P. Guillet, E. Pilon, and M. Werlen, [*Eur. Phys. J.*]{} [ **C16**]{} (2000) 311–330, \[[[ hep-ph/9911340]{}](http://xxx.lanl.gov/abs/hep-ph/9911340)\].
T. Binoth, J. P. Guillet, E. Pilon, and M. Werlen, [*Eur. Phys. J. Direct*]{} [**C4**]{} (2002) 7, \[[[ hep-ph/0203064]{}](http://xxx.lanl.gov/abs/hep-ph/0203064)\].
T. Binoth, J. P. Guillet, E. Pilon, and M. Werlen, [*Eur. Phys. J.*]{} [ **C24**]{} (2002) 245–260.
T. Binoth, J. P. Guillet, E. Pilon, and M. Werlen, [*Phys. Rev.*]{} [**D63**]{} (2001) 114016, \[[[ hep-ph/0012191]{}](http://xxx.lanl.gov/abs/hep-ph/0012191)\].
T. Binoth, [[hep-ph/0005194]{}](http://xxx.lanl.gov/abs/hep-ph/0005194).
T. Binoth, [[hep-ph/0105149]{}](http://xxx.lanl.gov/abs/hep-ph/0105149).
J. F. Owens, [*Phys. Rev.*]{} [**D65**]{} (2002) 034011, \[[[hep-ph/0110036]{}](http://xxx.lanl.gov/abs/hep-ph/0110036)\].
A. Moraes, presented at IPPP Monte Carlo at Hadron Colliders Workshop, University of Durham $14^{th}$ - $17^{th}$ January 2003.
A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, [*Eur. Phys. J.*]{} [**C23**]{} (2002) 73–87, \[[[hep-ph/0110215]{}](http://xxx.lanl.gov/abs/hep-ph/0110215)\].
B. A. Kniehl, G. Kramer, and B. Potter, [*Nucl. Phys.*]{} [**B582**]{} (2000) 514–536, \[[[ hep-ph/0010289]{}](http://xxx.lanl.gov/abs/hep-ph/0010289)\].
J. Binnewies, B. A. Kniehl, and G. Kramer, [*Z. Phys.*]{} [**C65**]{} (1995) 471–480, \[[[ hep-ph/9407347]{}](http://xxx.lanl.gov/abs/hep-ph/9407347)\].
P. Nason, S. Dawson, and R. K. Ellis, [*Nucl. Phys.*]{} [**B303**]{} (1988) 607.
P. Nason, S. Dawson, and R. K. Ellis, [*Nucl. Phys.*]{} [**B327**]{} (1989) 49.
W. Beenakker, H. Kuijf, W. L. van Neerven, and J. Smith, [*Phys. Rev.*]{} [ **D40**]{} (1989) 54.
W. Beenakker, W. L. van Neerven, R. Meng, G. A. Schuler, and J. Smith, [ *Nucl. Phys.*]{} [**B351**]{} (1991) 507.
E. Laenen, J. Smith, and W. L. van Neerven, [*Phys. Lett.*]{} [**B321**]{} (1994) 254, \[[[hep-ph/9310233]{}](http://xxx.lanl.gov/abs/hep-ph/9310233)\].
R. Bonciani, S. Catani, M. L. Mangano, and P. Nason, [*Nucl. Phys.*]{} [ **B529**]{} (1998) 424, \[[[ hep-ph/9801375]{}](http://xxx.lanl.gov/abs/hep-ph/9801375)\].
W. Bernreuther and A. Brandenburg, [*Phys. Lett.*]{} [**B314**]{} (1993) 104.
W. Bernreuther and A. Brandenburg, [*Phys. Rev.*]{} [**D49**]{} (1994) 4481, \[[[hep-ph/9312210]{}](http://xxx.lanl.gov/abs/hep-ph/9312210)\].
W. Bernreuther, A. Brandenburg, and M. Flesch, [[hep-ph/9812387]{}](http://xxx.lanl.gov/abs/hep-ph/9812387).
W. Bernreuther, M. Flesch, and P. Haberl, [*Phys. Rev.*]{} [**D58**]{} (1998) 114031, \[[[ hep-ph/9709284]{}](http://xxx.lanl.gov/abs/hep-ph/9709284)\].
W. Bernreuther, A. Brandenburg, and Z. G. Si, [*Phys. Lett.*]{} [**B483**]{} (2000) 99, \[[[ hep-ph/0004184]{}](http://xxx.lanl.gov/abs/hep-ph/0004184)\].
W. Bernreuther, A. Brandenburg, Z. G. Si, and P. Uwer, [*Phys. Lett.*]{} [ **B509**]{} (2001) 53, \[[[ hep-ph/0104096]{}](http://xxx.lanl.gov/abs/hep-ph/0104096)\].
W. Bernreuther, A. Brandenburg, Z. G. Si, and P. Uwer, [*Phys. Rev. Lett.*]{} [**87**]{} (2001) 242002, \[[[ hep-ph/0107086]{}](http://xxx.lanl.gov/abs/hep-ph/0107086)\].
W. Bernreuther, A. Brandenburg, Z. G. Si, and P. Uwer, [](http://xxx.lanl.gov/abs/{\rm to be published}).
W. Beenakker, F. A. Berends, and A. P. Chapovsky, [*Phys. Lett.*]{} [**B454**]{} (1999) 129, \[[[ hep-ph/9902304]{}](http://xxx.lanl.gov/abs/hep-ph/9902304)\].
V. S. Fadin, V. A. Khoze, and A. D. Martin, [*Phys. Rev.*]{} [**D49**]{} (1994) 2247.
K. Melnikov and O. I. Yakovlev, [[hep-ph/9302311]{}](http://xxx.lanl.gov/abs/hep-ph/9302311).
A. Czarnecki, M. Jezabek, and J. H. Kuhn, [*Nucl. Phys.*]{} [**B351**]{} (1991) 70.
A. Brandenburg, Z. G. Si, and P. Uwer, [*Phys. Lett.*]{} [**B539**]{} (2002) 235, \[[[hep-ph/0205023]{}](http://xxx.lanl.gov/abs/hep-ph/0205023)\].
W. Bernreuther, A. Brandenburg, and P. Uwer, [*Phys. Lett.*]{} [**B368**]{} (1996) 153, \[[[ hep-ph/9510300]{}](http://xxx.lanl.gov/abs/hep-ph/9510300)\].
W. G. D. Dharmaratna and G. R. Goldstein, [*Phys. Rev.*]{} [**D53**]{} (1996) 1073.
G. Mahlon and S. J. Parke, [*Phys. Lett.*]{} [**B411**]{} (1997) 173, \[[[hep-ph/9706304]{}](http://xxx.lanl.gov/abs/hep-ph/9706304)\].
A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, [[hep-ph/0307262]{}](http://xxx.lanl.gov/abs/hep-ph/0307262).
M. Gluck, E. Reya, and A. Vogt, [*Eur. Phys. J.*]{} [**C5**]{} (1998) 461, \[[[hep-ph/9806404]{}](http://xxx.lanl.gov/abs/hep-ph/9806404)\].
M. N. Dubinin, V. A. Ilyin, and V. I. Savrin, [[hep-ph/9712335]{}](http://xxx.lanl.gov/abs/hep-ph/9712335).
S. Abdullin [*et. al.*]{}, [*Phys. Lett.*]{} [**B431**]{} (1998) 410–419, \[[[hep-ph/9805341]{}](http://xxx.lanl.gov/abs/hep-ph/9805341)\].
V. Del Duca, F. Maltoni, Z. Nagy, and Z. Trocsanyi, [*JHEP*]{} [**04**]{} (2003) 059, \[[[hep-ph/0303012]{}](http://xxx.lanl.gov/abs/hep-ph/0303012)\].
S. Catani, M. Fontannaz, J. P. Guillet, and E. Pilon, [*JHEP*]{} [**05**]{} (2002) 028, \[[[ hep-ph/0204023]{}](http://xxx.lanl.gov/abs/hep-ph/0204023)\].
S. Frixione, [*Phys. Lett.*]{} [**B429**]{} (1998) 369–374, \[[[hep-ph/9801442]{}](http://xxx.lanl.gov/abs/hep-ph/9801442)\].
Wielers, M., preprint ATL-PHYS-2002-004.
S. Catani and M. H. Seymour, [*Nucl. Phys.*]{} [**B485**]{} (1997, Erratum ibid. B510:503-504,1997) 291–419, \[[[hep-ph/9605323]{}](http://xxx.lanl.gov/abs/hep-ph/9605323)\].
Z. Nagy and Z. Trocsanyi, [*Phys. Rev.*]{} [**D59**]{} (1999) 014020, \[[[hep-ph/9806317]{}](http://xxx.lanl.gov/abs/hep-ph/9806317)\].
Z. Nagy, [*Phys. Rev. Lett.*]{} [**88**]{} (2002) 122003, \[[[hep-ph/0110315]{}](http://xxx.lanl.gov/abs/hep-ph/0110315)\].
G. C. Blazey [*et. al.*]{}, [[ hep-ex/0005012]{}](http://xxx.lanl.gov/abs/hep-ex/0005012).
D. de Florian and Z. Kunszt, [*Phys. Lett.*]{} [**B460**]{} (1999) 184–188, \[[[hep-ph/9905283]{}](http://xxx.lanl.gov/abs/hep-ph/9905283)\].
M. Ciafaloni, P. Ciafaloni, and D. Comelli, [*Phys. Rev. Lett.*]{} [**84**]{} (2000) 4810–4813, \[[[ hep-ph/0001142]{}](http://xxx.lanl.gov/abs/hep-ph/0001142)\].
E. Maina, S. Moretti, M. R. Nolten, and D. A. Ross, [*Phys. Lett.*]{} [ **B570**]{} (2003) 205–214, \[[[ hep-ph/0307021]{}](http://xxx.lanl.gov/abs/hep-ph/0307021)\].
J. H. Kuhn and G. Rodrigo, [*Phys. Rev.*]{} [**D59**]{} (1999) 054017, \[[[hep-ph/9807420]{}](http://xxx.lanl.gov/abs/hep-ph/9807420)\].
J. H. Kuhn and G. Rodrigo, [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 49–52, \[[[hep-ph/9802268]{}](http://xxx.lanl.gov/abs/hep-ph/9802268)\].
S. Frixione and M. L. Mangano, [*Nucl. Phys.*]{} [**B483**]{} (1997) 321–338, \[[[hep-ph/9605270]{}](http://xxx.lanl.gov/abs/hep-ph/9605270)\].
E. Maina, S. Moretti, and D. A. Ross, [*JHEP*]{} [**04**]{} (2003) 056, \[[[hep-ph/0210015]{}](http://xxx.lanl.gov/abs/hep-ph/0210015)\].
P. Arnold, R. K. Ellis, and M. H. Reno, [*Phys. Rev.*]{} [**D40**]{} (1989) 912.
W. T. Giele, E. W. N. Glover, and D. A. Kosower, [*Nucl. Phys.*]{} [**B403**]{} (1993) 633–670, \[[[ hep-ph/9302225]{}](http://xxx.lanl.gov/abs/hep-ph/9302225)\].
J. Campbell, R. K. Ellis, and D. Rainwater, [*Phys. Rev.*]{} [**D68**]{} (2003) 094021, \[[[ hep-ph/0308195]{}](http://xxx.lanl.gov/abs/hep-ph/0308195)\].
J. R. Ellis, S. Moretti, and D. A. Ross, [*JHEP*]{} [**06**]{} (2001) 043, \[[[hep-ph/0102340]{}](http://xxx.lanl.gov/abs/hep-ph/0102340)\].
G. Bunce, N. Saito, J. Soffer, and W. Vogelsang, [*Ann. Rev. Nucl. Part. Sci.*]{} [**50**]{} (2000) 525–575, \[[[hep-ph/0007218]{}](http://xxx.lanl.gov/abs/hep-ph/0007218)\].
T. Binoth, [*Nucl. Phys. Proc. Suppl.*]{} [**116**]{} (2003) 387–391, \[[[hep-ph/0211125]{}](http://xxx.lanl.gov/abs/hep-ph/0211125)\].
T. Binoth, J. P. Guillet, G. Heinrich, and C. Schubert, [*Nucl. Phys.*]{} [ **B615**]{} (2001) 385–401, \[[[ hep-ph/0106243]{}](http://xxx.lanl.gov/abs/hep-ph/0106243)\].
Z. Bern, L. J. Dixon, and D. A. Kosower, [*Nucl. Phys.*]{} [**B412**]{} (1994) 751–816, \[[[ hep-ph/9306240]{}](http://xxx.lanl.gov/abs/hep-ph/9306240)\].
T. Binoth, J. P. Guillet, and G. Heinrich, [*Nucl. Phys.*]{} [**B572**]{} (2000) 361–386, \[[[ hep-ph/9911342]{}](http://xxx.lanl.gov/abs/hep-ph/9911342)\].
Z. Bern, L. J. Dixon, and D. A. Kosower, [*Phys. Rev. Lett.*]{} [**70**]{} (1993) 2677–2680, \[[[ hep-ph/9302280]{}](http://xxx.lanl.gov/abs/hep-ph/9302280)\].
T. Binoth, J. P. Guillet, and F. Mahmoudi, [[hep-ph/0312334]{}](http://xxx.lanl.gov/abs/hep-ph/0312334).
D. E. Soper, [*Phys. Rev. Lett.*]{} [**81**]{} (1998) 2638–2641, \[[[hep-ph/9804454]{}](http://xxx.lanl.gov/abs/hep-ph/9804454)\].
D. E. Soper, [*Phys. Rev.*]{} [**D62**]{} (2000) 014009, \[[[hep-ph/9910292]{}](http://xxx.lanl.gov/abs/hep-ph/9910292)\].
Z. Nagy and D. E. Soper, [*JHEP*]{} [**09**]{} (2003) 055, \[[[hep-ph/0308127]{}](http://xxx.lanl.gov/abs/hep-ph/0308127)\].
D. A. Forde and A. Signer, [[ hep-ph/0311059]{}](http://xxx.lanl.gov/abs/hep-ph/0311059).
W. T. Giele and E. W. N. Glover, [*Phys. Rev.*]{} [**D46**]{} (1992) 1980–2010.
S. Keller and E. Laenen, [*Phys. Rev.*]{} [**D59**]{} (1999) 114004, \[[[hep-ph/9812415]{}](http://xxx.lanl.gov/abs/hep-ph/9812415)\].
S. Frixione, [*Nucl. Phys.*]{} [**B507**]{} (1997) 295–314, \[[[hep-ph/9706545]{}](http://xxx.lanl.gov/abs/hep-ph/9706545)\].
L. Phaf and S. Weinzierl, [*JHEP*]{} [**04**]{} (2001) 006, \[[[hep-ph/0102207]{}](http://xxx.lanl.gov/abs/hep-ph/0102207)\].
S. Catani, S. Dittmaier, M. H. Seymour, and Z. Trocsanyi, [*Nucl. Phys.*]{} [**B627**]{} (2002) 189–265, \[[[hep-ph/0201036]{}](http://xxx.lanl.gov/abs/hep-ph/0201036)\].
G. Passarino, [*Nucl. Phys.*]{} [**B619**]{} (2001) 257–312, \[[[hep-ph/0108252]{}](http://xxx.lanl.gov/abs/hep-ph/0108252)\].
A. Ferroglia, M. Passera, G. Passarino, and S. Uccirati, [*Nucl. Phys.*]{} [ **B650**]{} (2003) 162–228, \[[[ hep-ph/0209219]{}](http://xxx.lanl.gov/abs/hep-ph/0209219)\].
F. V. Tkachov, [*Nucl. Instrum. Meth.*]{} [**A389**]{} (1997) 309–313, \[[[hep-ph/9609429]{}](http://xxx.lanl.gov/abs/hep-ph/9609429)\].
T. Binoth, G. Heinrich, and N. Kauer, [*Nucl. Phys.*]{} [**B654**]{} (2003) 277–300, \[[[ hep-ph/0210023]{}](http://xxx.lanl.gov/abs/hep-ph/0210023)\].
N. Kauer, [*Comput. Phys. Commun.*]{} [**153**]{} (2003) 233–243, \[[[physics/0210127]{}](http://xxx.lanl.gov/abs/physics/0210127)\].
Z. Bern, L. J. Dixon, and A. Ghinculov, [*Phys. Rev.*]{} [**D63**]{} (2001) 053007, \[[[ hep-ph/0010075]{}](http://xxx.lanl.gov/abs/hep-ph/0010075)\].
Z. Bern, L. Dixon, and D. A. Kosower, [*JHEP*]{} [**01**]{} (2000) 027, \[[[hep-ph/0001001]{}](http://xxx.lanl.gov/abs/hep-ph/0001001)\].
C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, [*Nucl. Phys.*]{} [**B601**]{} (2001) 318–340, \[[[hep-ph/0010212]{}](http://xxx.lanl.gov/abs/hep-ph/0010212)\].
C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, [*Nucl. Phys.*]{} [**B601**]{} (2001) 341–360, \[[[hep-ph/0011094]{}](http://xxx.lanl.gov/abs/hep-ph/0011094)\].
C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, [*Phys. Lett.*]{} [**B506**]{} (2001) 59–67, \[[[hep-ph/0012007]{}](http://xxx.lanl.gov/abs/hep-ph/0012007)\].
C. Anastasiou, E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, [*Nucl. Phys.*]{} [**B605**]{} (2001) 486–516, \[[[hep-ph/0101304]{}](http://xxx.lanl.gov/abs/hep-ph/0101304)\].
E. W. N. Glover, C. Oleari, and M. E. Tejeda-Yeomans, [*Nucl. Phys.*]{} [ **B605**]{} (2001) 467–485, \[[[ hep-ph/0102201]{}](http://xxx.lanl.gov/abs/hep-ph/0102201)\].
Z. Bern, A. De Freitas, L. J. Dixon, A. Ghinculov, and H. L. Wong, [*JHEP*]{} [**11**]{} (2001) 031, \[[[ hep-ph/0109079]{}](http://xxx.lanl.gov/abs/hep-ph/0109079)\].
Z. Bern, A. De Freitas, and L. J. Dixon, [*JHEP*]{} [**09**]{} (2001) 037, \[[[hep-ph/0109078]{}](http://xxx.lanl.gov/abs/hep-ph/0109078)\].
Z. Bern, A. De Freitas, and L. Dixon, [*JHEP*]{} [**03**]{} (2002) 018, \[[[hep-ph/0201161]{}](http://xxx.lanl.gov/abs/hep-ph/0201161)\].
L. W. Garland, T. Gehrmann, E. W. N. Glover, A. Koukoutsakis, and E. Remiddi, [*Nucl. Phys.*]{} [**B627**]{} (2002) 107–188, \[[[hep-ph/0112081]{}](http://xxx.lanl.gov/abs/hep-ph/0112081)\].
L. W. Garland, T. Gehrmann, E. W. N. Glover, A. Koukoutsakis, and E. Remiddi, [*Nucl. Phys.*]{} [**B642**]{} (2002) 227–262, \[[[hep-ph/0206067]{}](http://xxx.lanl.gov/abs/hep-ph/0206067)\].
S. Moch, P. Uwer, and S. Weinzierl, [*Phys. Rev.*]{} [**D66**]{} (2002) 114001, \[[[hep-ph/0207043]{}](http://xxx.lanl.gov/abs/hep-ph/0207043)\].
T. Kinoshita, [*J. Math. Phys.*]{} [**3**]{} (1962) 650–677.
T. D. Lee and M. Nauenberg, [*Phys. Rev.*]{} [**133**]{} (1964) B1549–B1562.
S. Frixione, Z. Kunszt, and A. Signer, [*Nucl. Phys.*]{} [**B467**]{} (1996) 399–442, \[[[ hep-ph/9512328]{}](http://xxx.lanl.gov/abs/hep-ph/9512328)\].
S. Dittmaier, [*Nucl. Phys.*]{} [**B565**]{} (2000) 69–122, \[[[hep-ph/9904440]{}](http://xxx.lanl.gov/abs/hep-ph/9904440)\].
D. A. Kosower, [*Phys. Rev.*]{} [**D57**]{} (1998) 5410–5416, \[[[hep-ph/9710213]{}](http://xxx.lanl.gov/abs/hep-ph/9710213)\].
D. A. Kosower, [*Phys. Rev.*]{} [**D67**]{} (2003) 116003, \[[[hep-ph/0212097]{}](http://xxx.lanl.gov/abs/hep-ph/0212097)\].
D. A. Kosower, [*Phys. Rev. Lett.*]{} [**91**]{} (2003) 061602, \[[[hep-ph/0301069]{}](http://xxx.lanl.gov/abs/hep-ph/0301069)\].
D. A. Kosower, [*hep-ph/0311272*]{} \[[[hep-ph/0311272]{}](http://xxx.lanl.gov/abs/hep-ph/0311272)\].
K. Hepp, [*Commun. Math. Phys.*]{} [**2**]{} (1966) 301–326.
M. Roth and A. Denner, [*Nucl. Phys.*]{} [**B479**]{} (1996) 495–514, \[[[hep-ph/9605420]{}](http://xxx.lanl.gov/abs/hep-ph/9605420)\].
T. Binoth and G. Heinrich, [*Nucl. Phys.*]{} [**B585**]{} (2000) 741–759, \[[[hep-ph/0004013]{}](http://xxx.lanl.gov/abs/hep-ph/0004013)\].
S. Weinzierl, [*JHEP*]{} [**03**]{} (2003) 062, \[[[hep-ph/0302180]{}](http://xxx.lanl.gov/abs/hep-ph/0302180)\].
S. Weinzierl, [*JHEP*]{} [**07**]{} (2003) 052, \[[[hep-ph/0306248]{}](http://xxx.lanl.gov/abs/hep-ph/0306248)\].
Z. Bern, L. Dixon, D. C. Dunbar, and D. A. Kosower, [*Nucl. Phys.*]{} [ **B425**]{} (1994) 217–260, \[[[ hep-ph/9403226]{}](http://xxx.lanl.gov/abs/hep-ph/9403226)\].
Z. Bern, L. J. Dixon, and D. A. Kosower, [*Nucl. Phys.*]{} [**B513**]{} (1998) 3–86, \[[[hep-ph/9708239]{}](http://xxx.lanl.gov/abs/hep-ph/9708239)\].
D. A. Kosower, [*Nucl. Phys.*]{} [**B552**]{} (1999) 319–336, \[[[hep-ph/9901201]{}](http://xxx.lanl.gov/abs/hep-ph/9901201)\].
D. A. Kosower and P. Uwer, [*Nucl. Phys.*]{} [**B563**]{} (1999) 477–505, \[[[hep-ph/9903515]{}](http://xxx.lanl.gov/abs/hep-ph/9903515)\].
Z. Bern, V. Del Duca, W. B. Kilgore, and C. R. Schmidt, [*Phys. Rev.*]{} [ **D60**]{} (1999) 116001, \[[[ hep-ph/9903516]{}](http://xxx.lanl.gov/abs/hep-ph/9903516)\].
S. Catani and M. Grazzini, [*Nucl. Phys.*]{} [**B591**]{} (2000) 435–454, \[[[hep-ph/0007142]{}](http://xxx.lanl.gov/abs/hep-ph/0007142)\].
F. A. Berends and W. T. Giele, [*Nucl. Phys.*]{} [**B313**]{} (1989) 595.
A. Gehrmann-De Ridder and E. W. N. Glover, [*Nucl. Phys.*]{} [**B517**]{} (1998) 269–323, \[[[ hep-ph/9707224]{}](http://xxx.lanl.gov/abs/hep-ph/9707224)\].
J. M. Campbell and E. W. N. Glover, [*Nucl. Phys.*]{} [**B527**]{} (1998) 264–288, \[[[ hep-ph/9710255]{}](http://xxx.lanl.gov/abs/hep-ph/9710255)\].
S. Catani and M. Grazzini, [*Phys. Lett.*]{} [**B446**]{} (1999) 143–152, \[[[hep-ph/9810389]{}](http://xxx.lanl.gov/abs/hep-ph/9810389)\].
S. Catani and M. Grazzini, [*Nucl. Phys.*]{} [**B570**]{} (2000) 287–325, \[[[hep-ph/9908523]{}](http://xxx.lanl.gov/abs/hep-ph/9908523)\].
V. Del Duca, A. Frizzo, and F. Maltoni, [*Nucl. Phys.*]{} [**B568**]{} (2000) 211–262, \[[[ hep-ph/9909464]{}](http://xxx.lanl.gov/abs/hep-ph/9909464)\].
S. Moch, P. Uwer, and S. Weinzierl, [*J. Math. Phys.*]{} [**43**]{} (2002) 3363–3386, \[[[ hep-ph/0110083]{}](http://xxx.lanl.gov/abs/hep-ph/0110083)\].
S. Weinzierl, [*Comput. Phys. Commun.*]{} [**145**]{} (2002) 357–370, \[[[math-ph/0201011]{}](http://xxx.lanl.gov/abs/math-ph/0201011)\].
N. N. Bogoliubov and O. S. Parasiuk, [*Acta Math.*]{} [**97**]{} (1957) 227–266.
W. Zimmermann, [*Ann. Phys.*]{} [**77**]{} (1973) 536–569.
Smirnov, V. A. [*Renormalization and asymptotic expansions.*]{} Basel, Switzerland: Birkhaeuser (1991) 380p.
C. Itzykson and J. B. Zuber,. New York, USA: McGraw-Hill (1980) 705 p.
V. A. Smirnov, [*Phys. Lett.*]{} [**B524**]{} (2002) 129–136, \[[[hep-ph/0111160]{}](http://xxx.lanl.gov/abs/hep-ph/0111160)\].
G. Heinrich, [*Nucl. Phys. Proc. Suppl.*]{} [**116**]{} (2003) 368–372, \[[[hep-ph/0211144]{}](http://xxx.lanl.gov/abs/hep-ph/0211144)\].
A. Gehrmann-De Ridder, T. Gehrmann, and G. Heinrich, [[hep-ph/0311276]{}](http://xxx.lanl.gov/abs/hep-ph/0311276).
C. Anastasiou, K. Melnikov, and F. Petriello, [[hep-ph/0311311]{}](http://xxx.lanl.gov/abs/hep-ph/0311311).
T. Binoth and G. Heinrich, [*Nucl. Phys.*]{} [**B680**]{} (2004) 375, \[[[hep-ph/0305234]{}](http://xxx.lanl.gov/abs/hep-ph/0305234)\].
[^1]: M. Dobbs’ work was supported in part by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.
[^2]: Contributed by: [T. Sjöstrand and P. Skands ]{}
[^3]: Contributed by: [A. Moraes, C. Buttar, and I. Dawson]{}
[^4]: Contributed by: [A. Cruz and R. Field]{}
[^5]: Contributed by: [V. Drollinger]{}
[^6]: Contributed by: [L. Dudko and A. Sherstnev]{}
[^7]: Contributed by: [J. Huston, I. Puljak, T. Sjöstrand, E. Thomé]{}
[^8]: Contributed by: [T. Sjöstrand]{}
[^9]: Contributed by: [S. Mrenna]{}
[^10]: Contributed by: [N. Kidonakis]{}
[^11]: Contributed by: [T.O. Eynck, E. Laenen, L. Magnea]{}
[^12]: Contributed by: [A. Banfi, E. Laenen]{}
[^13]: Contributed by: [C. Balazs, M. Grazzini, J. Huston, A. Kulesza, I. Puljak]{}
[^14]: Contributed by: [G. Corcella, S. Moretti]{}
[^15]: Contributed by: [A. Banfi, G.P. Salam, G. Zanderighi]{}
[^16]: Contributed by: [Q.-H. Cao, C.-P. Yuan]{}
[^17]: Contributed by: [S. Berge, P. Nadolsky, F. Olness, C.-P. Yuan]{}
[^18]: Contributed by: [J. R. Andersen, V. Del Duca,A. De Roeck, A. Sabio Vera ]{}
[^19]: Contributed by: [T. Binoth, K. Lassila-Perini]{}
[^20]: Contributed by: [W. Bernreuther, A. Brandenburg, Z.G. Si, P. Uwer]{}
[^21]: Contributed by: [V. Del Duca, F. Maltoni, Z. Nagy, Z. Trócsányi]{}
[^22]: Contributed by: [E. Maina, S. Moretti, M.R. Nolten, D.A. Ross]{}
[^23]: Contributed by: [T. Binoth, J.Ph. Guillet, G. Heinrich, N. Kauer, F. Mahmoudi]{}
[^24]: Contributed by: [G. Heinrich, S. Weinzierl]{}
|
---
abstract: 'We report exact results for the Fermi Edge Singularity in the absorption spectrum of an out-of-equilibrium tunnel junction. We consider two metals with chemical potential difference $V$ separated by a tunneling barrier containing a defect, which exists in one of two states. When it is in its excited state, tunneling through the otherwise impermeable barrier is possible. We find that the lineshape not only depends on the total scattering phase shift as in the equilibrium case but also on the difference in the phase of the reflection amplitudes on the two sides of the barrier. The out-of-equilibrium spectrum extends below the original threshold as energy can be provided by the power source driving current across the barrier. Our results have a surprisingly simple interpretation in terms of known results for the equilibrium case but with (in general complex-valued) combinations of elements of the scattering matrix replacing the equilibrium phase shifts.'
author:
- 'B. Muzykantskii$^1$, N. d’Ambrumenil$^{1,2}$ and B. Braunecker$^{3,1}$'
bibliography:
- 'out\_of\_eqm.bib'
title: 'Fermi edge singularity in a non-equilibrium system'
---
Developments in the fabrication and manipulation of mesoscopic systems have allowed detailed and well-characterized transport measurements for a large range of devices including quantum pumps, tunnel junctions and carbon nanotubes. It is often the case that such measurements explore non-equilibrium effects particularly when the potential difference is dropped across a narrow potential barrier or over a short distance inside the metallic region [@RB94; @KGG01; @NCL00]. While there is often a very good theoretical description of much that has been observed for systems close to equilibrium, the theoretical picture for systems out of equilibrium is less clear with fewer established theoretical results.
A natural point to start, when looking for a description of non-equilibrium effects in many-electron systems is the Fermi Edge Singularity (FES), which is one of the simplest non-trivial many-body effects. The FES is characteristic of the response of a Fermi gas to a rapid switching process. Initially it was associated with the shape of the absorption edge and spectral line found when a core hole is created [@Mahan67]. However, it turns out to be a generic feature of a Fermi system’s response to any fast switching process and reflects the large number of low-energy (particle-hole) excitations which exist in Fermi liquids. It has also been shown to be related to Anderson’s orthogonality catastrophe [@ND69; @CN71] and can be used to reformulate the Kondo problem in terms of a succession of spin flips which are treated as the switching of a one-body potential between two different values [@YA70].
We consider a system at zero temperature with two Fermi surfaces separated by a barrier with a potential difference (bias) $V$ applied across the barrier (see Fig. \[fig:fig1\]). The barrier contains a defect, which exists in one of two states with energy separation $E_0$. Tunneling through the barrier is assumed to be possible only when the defect is in its excited state. We compute the absorption spectrum close to the threshold at $\omega_0=E_0-\mbox{Re}(\Delta(V))$, for frequencies $(\omega-\omega_0) \ll \xi_0$, where $\xi_0$ is of order the bandwidth and Re$(\Delta(V))$ is real part of the combined energy shift of the two Fermi seas when the defect is in its excited state. ($\Delta(V)$ is complex for non-zero $V$ on account of the dissipation in the system.)
Using an approach based on that of Nozières and de Dominicis (ND) [@ND69], we solve exactly for the asymptotic behavior of the absorption spectrum in two limiting cases: $(\omega-\omega_0)\gg V$ and $(\omega-\omega_0) \ll V$. Our results have a simple interpretation in terms of generalized (complex) phase-shifts at the Fermi energy. Typical lineshapes for the case $(\omega-\omega_0)\gg V$ illustrating the dependence on the reflection amplitudes and phases are shown in Figure \[fig:fig2\].
Our treatment of the problem is based on that of Muzykantskii and Adamov (MA) for the statistics of charge transfer in quantum pumps, which uses the relation between the many-particle response to the changing one-body potential and the solution of an associated matrix Riemann-Hilbert (RH) problem [@MA03]. This problem was also addressed perturbatively and using the ND approach in [@CR00; @Ng96], although the results in [@CR00] led the authors to question the validity of the ND approach of [@Ng96] (see also [@BB02]). Our solution shows clearly that the ND approach is valid, with the earlier difficulties probably associated with an incomplete analysis of the matrix RH problem associated with their singular integral equation.
We characterize the scattering at the interface between the two subsystems via the unitary $2\times2$ matrix, $S(\epsilon,t)$, connecting scattering states in the two wires for particles with energy $\epsilon$. This takes one of two values $S^g$, and $S^e$ depending on whether the the defect is in its ground ($g$) or excited state ($e$). In the following, we will take a row/column index equal to one (two) for the left (right) electrode so that the diagonal (off-diagonal) elements correspond to reflection (transmission) amplitudes (see Figure \[fig:fig1\]). We choose the scattering states to be the eigen states of the system when the defect is in its ground state and the barrier is totally reflecting hence: $S^g_{ij}=\delta_{ij}$. $S^e$ is an arbitrary unitary matrix with reflection probability $R=|S^e_{11}|^2 < 1$. We will assume that a negative potential $-V$ ($V>0$) has been applied to the left electrode with respect to the right electrode.
The spectral function, $\rho(\omega,V)$, for absorption by the local level is given by [@ND69]: $$\begin{aligned}
\rho(\omega,V) & \sim & \text{Re}
\int_{-\infty}^\infty \chi (t_f,V) e^{i\omega t_f} dt_f
\label{eq:rho_definition} \\
\chi (t_f,V) & = & \langle0|U(t_f,0)|0\rangle.
\label{eq:chi_definition}\end{aligned}$$ Here $|0\rangle$ is the ground state wavefunction of the complete system (the filled Fermi seas in the two electrodes and the defect in its ground state), while $U(t_f,0)$ is the time-evolution operator for the system between $t=0$ and $t=t_f$ with the defect in its excited state. $\chi(t_f,0)$ is the same as the core hole Green’s function computed in [@Mahan67; @ND69; @CN71].
Before discussing the full non-equilibrium case, we briefly review the known equilibrium results. When $V = 0$ the response of the system is that of the core hole problem in a non-separable potential considered in [@YY82; @Matveev-Larkin] $$\log{\chi(t_f,0)} = -i(E_0-\Delta(0))t_f - \beta \log{it_f\xi_0}
\label{eq:V=0}$$ where $\beta=
\sum_{j=1,2} \left(\frac{\delta_j}{\pi}\right)^2$. Here $e^{-i2\delta_j}$ are the eigen values of $S^e$. The threshold is shifted from $E_0$, the energy separation in the two-level system, by $\Delta(0)$, which is the shift of the ground state energy of the two Fermi seas when the scattering defect is in its excited state. This standard equilibrium result (\[eq:V=0\]) is well understood in terms of the low-lying particle-hole excitations created by the rapid switching of the potential, with the principal contributions to the logarithm in (\[eq:V=0\]) from excitations with frequencies between $t_f^{-1}$ and $\xi_0$.
When a voltage is applied across the barrier with the defect in its excited state and $R\neq1$, a current will flow and the system will become dissipative. For $t_f \ll V^{-1}$, the spectral response is dominated by excitations with frequencies $\omega \gg V$, involving states which do not sense the potential drop across the barrier. As a result $\chi(t_f,V)$ is unchanged from its value in equilibrium.
When $t_f \gg V^{-1}$, the response is controlled by electrons within the band of width $V$ about the mean Fermi energy. We find that $$\log{\chi(t_f,V)} = -i(E_0-\Delta(V)) t_f - \beta' \log{(Vt_f)} + D
\label{eq:Vneq0}$$ Here the function $\Delta(V)$ is given by: $$\Delta(V) = \int_{-\infty}^0 \frac{\mbox{tr}\log{(S(E))}}{2\pi i} dE +
\int_0^V \frac{\log{(S_{11}(E)) }}{2\pi i} dE
\label{eq:Delta(V)}$$ This expression (\[eq:Delta(V)\]) for the (in general complex) energy shift of the two Fermi seas, when the defect is in its excited state, can be thought of as the generalization of Fumi’s theorem [@Friedel52; @Fumi55] to the out-of-equilibrium case. The exponent $\beta'$ in (\[eq:Vneq0\]) is given by $$\beta' = \sum_{j=1,2}\left(-\log{(S^e_{jj})}/2\pi i\right)^2 .
\label{eq:beta'}$$ The constant term $D$ gives the contribution from excitations with frequencies between $V$ and $\xi_0$, which do not sense the potential drop across the barrier. To logarithmic accuracy [@Note_on_cutoff]: $$D = \beta \log{\xi_0/V}.
\label{eq:D}$$
Writing $S^e_{jj}=\sqrt{R}e^{i\alpha_j}$ and comparing the forms for $\beta$ and $\beta'$ in (\[eq:V=0\]) and (\[eq:beta’\]), we see that the quantity $-\log{(S^e_{jj})/2i}=-\alpha_j/2 + i(\log{R})/4\pi$ is acting as a complex phase shift. Its real part, $-\alpha_j/2$, characterizes the scattering in the $j$’th electrode and in (\[eq:Vneq0\]) describes the effect of particle-hole excitations in the band of width $V$ from the Fermi energy. Its imaginary part $(\log{R})/4\pi$ relates to the lifetime of the excitation.
The absorption spectrum is found from the Fourier transform of $\chi(t_f,V)$ in (\[eq:rho\_definition\]). Measuring $\omega$ from $\omega_0=E_0-\mbox{Re}(\Delta(V))$, it is given by [@note_on_Fourier_transform]: $$\rho(\omega) \sim \frac{1}{\Omega^{1-\beta'_1}}
e^{-\beta'_2\phi_\Omega} \sin{\left(\beta'_1\pi- (\beta'_1-1) \phi_\Omega
-\beta'_2 \log\Omega \right)}.
\label{eq:rho(omega)}$$ Here we have defined $\Omega\exp{i\phi_\Omega} \equiv \omega/V - i(\log{R})/4\pi$ and written $\beta'=\beta'_1 + i \beta'_2$. While the dependence on $\beta'_1$ reflects the total overall scattering on the two sides of the barrier as in equilibrium, $\beta'_2$ is proportional to the difference in the phases of the two reflection amplitudes $S^e_{11}$ and $S^e_{22}$ and its appearance in (\[eq:rho(omega)\]) is entirely an out-of-equilibrium effect.
When $R=1$, the term multiplying $\Omega^{1-\beta'_1}$ in (\[eq:rho(omega)\]) is proportional to the theta function $\theta(\omega)$ and describes the usual sharp threshold in $\rho(\omega)$. With $R<1$ it leads to a smearing of the threshold (see Figure \[fig:fig2\]). As pointed out in [@CR00], this broadening of the threshold reflects the existence of ‘negative energy excitations’ in the system involving a hole in the left electrode and a particle in the right electrode. From an experimental point of view, the below threshold broadening with its functional dependence on the phases of the reflection amplitudes and its overall energy scale fixed by the bias are probably the key signatures of the non-equilibrium effects we are describing. The sensitivity to the difference in scattering phase shifts (this difference is proportional to $\beta'_2$) would show up in changes in the line shape on reversing the bias and should also be observable.
The derivation of the overlap $\chi(t_f)$ follows quite closely that of MA [@MA03]. We introduce the operators $a_i(\epsilon)$ which annihilate particles on the $i$’th side of the barrier with energy $\epsilon$ in eigen states of the system with the defect in its ground state ($S=1$). The effect of the time-evolution operator $U$ acting between $t=0$ and $t_f$ on states $a^\dagger_i|\rangle$, where $|\rangle$ is the true vacuum with no particles, is given by $$Ua^\dagger_i(\epsilon) |\rangle = \sum_{i} \int d\epsilon'
\sigma_{ij}(\epsilon,\epsilon') a^\dagger_j(\epsilon') |\rangle.
\label{eq:sigma}$$ One can show that for states near the Fermi energy (see [@AM01] for example) $\sigma$ is given by: $$\sigma_{ij}(\epsilon,\epsilon')= e^{-iE_0t_f} \frac{1}{2\pi}
\int_{-\infty}^\infty S_{ij}(t) e^{i(\epsilon-\epsilon')t} dt
\label{eq:sigma=S}$$ provided that the adiabaticity condition $$\hbar \frac{\partial S}{\partial t} \frac{\partial S}{\partial E} \ll 1
\label{eq:adiabaticity}$$ is satisfied. In (\[eq:sigma\]) $S(t)=S^g$ for $t<0$ and $t>t_f$ and $S(t)=S^e$ for $0<t<t_f$ and we have suppressed the explicit dependence of $S$ on energy. When computing the low frequency asymptotics, this becomes a slow dependence on $(\epsilon + \epsilon')/2$, and can be neglected.
The overlap $\chi(t_f)$ can be written $$\chi(t_f) = \langle0|U|0\rangle = \mbox{det}' \sigma
\label{eq:determinant}$$ where the prime indicates that the operator determinant is to be taken only over the occupied states in the two filled Fermi seas. This reduces in the equilibrium case to the determinant in [@CN71]. With zero chemical potential in the right electrode and treating the (non-equilibrium) Fermi distribution as the diagonal operator $f_{ij}(\epsilon,\epsilon')= \delta_{ij}
\delta(\epsilon-\epsilon') \theta(-(\epsilon+V(2-i))$ allows us to write $$\begin{aligned}
\chi(t_f) & = & \mbox{det}(1-f+f\sigma) \label{eq:full_determinant} \\
\log{\chi(t_f)} & = & \mbox{Tr}
\left( \log{(1-f+f\sigma)} - f\log{\sigma} \right) + \mbox{Tr}f\log{\sigma}
\nonumber \\
& \equiv & C(V,t_f) + \mbox{Tr}f\log{\sigma}
\label{eq:log_chi}\end{aligned}$$ where the operator determinant is now the full determinant taken over all states and the trace, Tr, is the trace over energy and channels. The last term in the expression (\[eq:log\_chi\]) can be found by explicitly carrying out the integral in (\[eq:sigma=S\]). This gives that $\sigma_{ij}(\epsilon,\epsilon') = \delta_{ij}\delta{(\epsilon-\epsilon')}
- X_{ij}(\epsilon-\epsilon')$. The logarithm can then be expanded as a power series in the matrix $X$ [@Note_on_expanding_log_sigma]. After evaluating $X^n$ term by term and then resumming we obtain: $\mbox{Tr}f\log{\sigma}=-i(E_0 - \Delta(0))t_f + (V t_f/2\pi i)
(\log{S})_{11}$. The difference between this and $-i(E_0 - \Delta(V))t_f$ in (\[eq:Delta(V)\]) is contained in the function $C(t_f,V)$.
To evaluate $C(V,t_f)$ we introduce $\widetilde{S}(t,\lambda)$ where $$\widetilde{S}(t,\lambda) = \exp{(\lambda \log{S(t)}}),
\label{eq:S_matrix}$$ so that $\widetilde{S}(t,1)=S(t)$. We now apply the following gauge transformation: $$\begin{aligned}
\mathbf{a}(\epsilon) & \rightarrow & \mathbf{a}(\epsilon,t) = e^{iLVt}
\mathbf{a}(\epsilon)
\label{eq:gauge_transform} \\
\widetilde{S}(t,\lambda) & \rightarrow & \widetilde{S}(t,\lambda)=e^{iLVt}
\widetilde{S}(t,\lambda)
e^{-iLVt}
\label{eq:S(t,lambda)}\end{aligned}$$ Here $L$ is the diagonal matrix with $L_{11}=1$ and $L_{22}=0$. This has the advantage of eliminating the chemical potential difference between the two electrodes at the expense of an added time-dependence for $\widetilde{S}$ when $t\in [0,t_f]$. After switching to the time-representation (in which the trace, Tr, becomes a trace over channels and an integral over time) and substituting for $\sigma$ from (\[eq:sigma=S\]), $C(t_f,V)$ can be written $$C(t_f,V) = \mbox{Tr}
\int_0^\lambda d\lambda
\left[ \left((1-f+f\widetilde{S})^{-1}f -f\widetilde{S}^{-1}\right)
\frac{d\widetilde{S}}{d\lambda} \right].
\label{eq:integral_over_lambda}$$ Using a parallel argument to that of [@MA03], we find that $$(1-f+f\widetilde{S})^{-1} = Y_+\left((1-f)Y_+^{-1} + fY_-^{-1}\right).
\label{eq:RH_inverse_matrix}$$ where $Y_{\pm}=Y(t\pm i0,\lambda)$. Here $Y(z,\lambda)$ is an analytic (matrix) function of complex $z$ in the complement of the cut along the real axis between $z=0$ and $z=t_f$, and satisfies: $$Y_-Y_+^{-1} = \widetilde{S}(t,\lambda) \,\,\, \mbox{and} \,\,\,
Y(z,\lambda) \rightarrow \mbox{const} \,\,\, \mbox{for} \,\,\, |z| \rightarrow
\infty.
\label{eq:RH}$$ If there is no tunneling between electrodes ($S^e$ diagonal), this matrix RH problem can be shown to be the same as the homogeneous part of that solved in [@ND69]. After substituting (\[eq:RH\_inverse\_matrix\]) into (\[eq:integral\_over\_lambda\]), using the fact that in the time-representation (after the gauge transformation \[eq:gauge\_transform\]) $f(t,t')=i(2\pi(t-t'+i0))^{-1}$ and letting $t'\rightarrow t$ to compute the trace, Tr, we finally obtain $$C(t_f,V) = \frac{i}{2\pi} \int_0^1 d\lambda \int_0^{t_f}
\mbox{tr}\left\{\frac{dY_+}{dt} Y_+^{-1} S^{-1}\frac{dS}{d\lambda}\right\} dt.
\label{eq:logY_logS}$$ Here tr denotes a trace over channel indices.
Solving for $\chi(t_f,V)$ is equivalent to solving for the quantity $Y(z,\lambda)$. For small $V$, we can expand the exponential factors in $\widetilde{S}(z,\lambda)$ (see \[eq:S(t,lambda)\]) as $e^{\pm iVz} = 1 \pm iVz$. In this case $$Y(z,\lambda) = \exp{\left[ \frac{1}{2\pi i}\log{\left(\frac{z}{z-t_f}\right)
\log{\widetilde{S}(z,\lambda)}} \right]}
\label{eq:Y_small_t}$$ solves the RH problem. For $|z| \rightarrow \infty$, the exponent (and hence $Y$) tends to a constant as required. If $Vt_f \ll 1$ we can insert this result into (\[eq:logY\_logS\]) and compute the integrals over $t$ and $\lambda$. This yields the equilibrium result (\[eq:V=0\]). Although there are corrections to the equilibrium ($V=0$) solution for $Y_+$ which are linear in $Vt$, these cancel out after taking the trace in (\[eq:logY\_logS\]). Corrections to $C(t_f,V)$ can therefore only be of order $(Vt_f)^2$ or higher.
For times $t_f>V^{-1}$, a general solution to this type of matrix RH problem is not known. The form (\[eq:Y\_small\_t\]) for $Y_+$ is still valid for $0<t<V^{-1}$ and $t_f>t>t_f-V^{-1}$. The integral over times close to the branch points of $Y$ then gives the contribution varying as $D=\log{(\xi_0/V)}$ in (\[eq:D\]). However, although the form for $Y$ in (\[eq:Y\_small\_t\]) still satisfies the discontinuity condition along the cut, the exponent is unbounded for large $|z|$ and hence (\[eq:Y\_small\_t\]) is useless as a starting point for solving for $Y_+$ for $t\gg V^{-1}$. Following the derivation of [@MA03], we find that: $$Y_+(t,\lambda) = \left[
\begin{array}[c]{rcl}
\psi_+(t,\lambda) &\hbox{ when }& t< 0 \\
\begin{pmatrix}
1 & -\gamma(t,\lambda) \\
0 & 1
\end{pmatrix} \psi_+(t,\lambda) & \hbox{ when }& 0 < t < t_f \\
\psi_+(t,\lambda) &\hbox{ when } & t_f < t
\end{array} \right.
\label{eq:Y_large_t}$$ is asymptotically correct for $t\gg V^{-1}$. Here $\gamma(t,\lambda)=\widetilde{S}_{12}(t,\lambda)/
\widetilde{S}_{11}(t,\lambda)$ and $\psi_+(t,\lambda)=\psi(t+i0,\lambda)$ where $$\psi(z,\lambda) = \exp \left( \log \frac{z}{z-t_f}
\left[ \frac{ \log{ \widetilde{S}_{11}/\widetilde{S}^*_{22} } }{4\pi i}\tau_0
+ \frac{\log{(\widetilde{S}_{11}\widetilde{S}^*_{22})}}{4\pi i}
\tau_3 \right]\right).
\label{eq:psi}$$ The corresponding function $Y(z,\lambda)$ is not analytic across vertical cuts in the complex $z$-plane through the points $z=0$ and $z=t_f$, with discontinuities which decay as $e^{-V|z|}$ or $e^{-V|z-t_f|}$. (These factors show that we cannot describe the reverse bias case by taking $V<0$ in (\[eq:Y\_large\_t\]). Instead $Y_+$ takes a different form for negative $V$.) After inserting the solution (\[eq:Y\_large\_t\]) in (\[eq:logY\_logS\]) and computing the integrals over $\lambda$ and $t$, we obtain the first two terms in (\[eq:Vneq0\]). The term obtained after differentiating $\gamma$ in (\[eq:Y\_large\_t\]) and adding to the term from $\mbox{Tr}f\log \sigma$ in (\[eq:log\_chi\]), leads after some algebra to the term $-i(E_0-\Delta(V))t_f$. Differentiating $\psi_+(t_f,\lambda)$ in (\[eq:Y\_large\_t\]) leads to the term proportional to $\log{Vt_f}$. The constant term is derived using the form (\[eq:Y\_small\_t\]) for $Y_+$ valid for small $t$ and $t-t_f$ as discussed above.
|
---
title: '**Fractional D3-branes in diverse backgrounds**'
---
Introduction
============
During the last couple of years, the study of gauge/gravity duality has partly shifted from the original AdS/CFT correspondence [@mal] and its less supersymmetric cousins to more realistic, non-conformal cases. Different approaches has been proposed and explored, with configurations preserving ${\cal N}=2$, ${\cal N}=1$ or no supersymmetries : mass deformations of a conformal field theory [@gppz; @ps; @pw], NS5-branes or M5-branes wrapped on a Riemann surface [@mn; @gkmw; @bcz], fractional D3-branes on orbifold spaces or conifolds [@kn; @kt; @ks; @bgz; @bdflmp; @pol; @aha; @prz; @bgl; @grp; @bcffst; @bdflm] to cite several of those constructions. Typically, the “naive” supergravity solution displays naked singularities. However, the singularities are usually the manifestation of interesting and non-trivial phenomena in the infrared of the dual gauge theory and their resolutions in supergravity should take into account such effects. A typical example is the ${\cal N}=1$ solution of [@ks], where the chiral symmetry breaking of the field theory in the IR removes the singularity by deforming the conifold.
This example uses fractional D3-branes which first appeared the context of string theory defined on orbifold spaces [@dm]. On a conifold, one does not have anymore a microscopic description of the fractional D-branes and must rely on the pure supergravity solutions. One the other hand, for fractional D3-branes in type IIB theory on the orbifold space $\mathbb{C}^2/\mathbb{Z}_N$, one can use the conformal field theory technology. Indeed, these D-branes have an exact description in string theory as boundary states [@dg]. Despite being less interesting for real world applications than their ${\cal N}=1$ counterparts, these D3-branes configurations, whose world-volume gauge theory possesses an ${\cal N}=2$ supersymmetry, could be seen as tools to study and to test the gauge/gravity correspondence away from its conformal limit, like the original, maximally (super-)symmetric D3-branes were useful for the conformal case. Indeed, the larger amount of preserved supersymmetries gives a better control of the theories on both sides; in the gauge theory, we still have non-renormalization theorems, telling us, for instance, that the gauge couplings only receive perturbative corrections at one-loop. Moreover, its non-perturbative corrections could be, in principle, obtained using the Seiberg-Witten curve of the gauge theory [@sw]. On the gravity side, one could also expect that the $\alpha^\prime$ corrections are better controlled, as it is well known for solutions of heterotic string theories. Moreover, these ${\cal N}=2$ configurations display on their own interesting phenomena like the enhançon mechanism [@jpp] which was argued to solve the singular behaviour of their metrics; this characteristic is a common feature of several kinds of solutions preserving ${\cal N}=2$ supersymmetries presented in the literature, namely, mass deformation of an ${\cal
N}=4$ gauge theory [@bpp; @ejp], NS5-branes wrapped on a Riemann surface [@bcz] or fractional D3-branes [@bdflmp; @prz; @bgl]. Actually, the enhançon mechanism is the gravity manifestation of the existence of Landau poles in the gauge theory, when the one-loop correction cancels the tree-level contribution. However, we know that the complete metric on the moduli space of a ${\cal N}=2$ gauge theory is not singular when Yang-Mills instantons corrections are included. It is still a challenge to describe precisely these effects from the dual point-of-view.
Among ${\cal N}=2$ non-conformal configurations, the fractional D3-branes one seems to be the simplest. For instance, it is obvious to describe an arbitrary point on the Coulomb branch of the gauge theory since the [*vev*]{} of the adjoint scalars which parameterize the moduli space are proportional to the positions of the fractional D3-branes and are chosen arbitrarily to construct the solution. One the other hand, in mass deformed or wrapped NS5-branes configurations, the precise identification of the point on the Coulomb branch requires some work and, so far, it is not clear that any point can be obtained by deforming the known solutions[^1]. Moreover, we can also see a fractional D-brane configuration as a “mass deformed” conformal theory : one can always start with a stack of bulk D-branes and give masses to bifundamental hypermultiplets by pulling away some fractional D-branes. At large distance, the theory will be conformal but, in the IR of the gauge theory, the D-branes sources break the conformal symmetry.
The aim of this article, which is made of two independent parts, is two-folds. In the first part, we will generalize fractional D3-branes solutions to models with orientifolds. The presence of O-planes leads to symplectic or orthogonal gauge groups on the world-volume of the D-branes. After a short review of some basic facts on orientifold projections and existence of fractional D-branes, we describe the supergravity solutions for two possible cases. Such solutions are similar to their parents type IIB and display common features, namely singularities and enhançon; as expected, we show that the one-loop corrected gauge couplings for orthogonal and symplectic gauge groups are reproduced on the world-volume of fractional D3-branes probing these backgrounds. This result should be interpreted as a manifestation of the closed/open string duality. We also speculate on the relation between periodic monopoles and a refined solution close to the enhançon radius.
The second part of this paper is a first step toward a comparison of the alternative approaches to describe pure $SU(N)$ ${\cal N}=2$ gauge theory using supergravity. One way to study the possible connection between very different branes configurations is to uplift them to eleven dimensions where they should be described as M5-branes wrapped on manifolds in such a way that, in the IR, the world-volume theory flows to the four-dimensional gauge theory. Moreover, the M-theory point-of-view could be useful to understand the approximations made and the domains of validity of the different solutions. In particular, one can think about non-perturbative effects that are inaccessible in the type II description. Indeed, we know that such an eleven dimensional approach has been fruitful in the past to recover non-perturbative - D-instantons - corrections to string effective actions whose form was conjectured using symmetry - self-duality - arguments [@gg; @ggv; @bbg]. In this article, we study the case of the fractional D3-branes, first by constructing a solution on a two-centers Taub-NUT space which interpolates between the type IIB fractional D3-branes solution in the ALE limit and a partially smeared type IIA configuration in another limit. Then, we lift the solution to M-theory, where it describes an M5-brane wrapped on a Riemann surface. Finally, we will comment on the relation with M5-branes solutions proposed by [@bfms] to be dual to the pure $SU(N)$ ${\cal N}=2$ gauge theory but which has received much less attention so far than fractional D3-branes and wrapped NS5-branes.
[*Note added in proof :*]{} while this article was reaching its final stage, appeared the preprint [@dveil], which develops ideas similar to the ones considered in the second part by comparing fractional D-branes to wrapped branes solutions in the context of three dimensional gauge theories.
Fractional D-brane solutions with orthogonal and symplectic gauge groups
========================================================================
Orientifold projections and fractional D3-branes
------------------------------------------------
We start with type IIB string theory on the non-compact orbifold space $\mathbb{C}^2/\mathbb{Z}_2$; the directions parallel to the orbifold will be called $x^6, x^7, x^8, x^9$. The twisted fields of this theory, namely the NS-NS scalar $b$, the three blow-up modes, $\xi^\pm, \xi^3$, the R-R scalar $c$ and the R-R 2-form $A^{(2)}$ fill a six dimensional ${\cal N}_6=(2,0)$ tensor multiplet. In order to discuss the orientifold projection, it is convenient to use the ${\cal N}_6=(1,0)$ supersymmetry language where this tensor multiplet is split into a tensor multiplet and a hypermultiplet. The fields $b$ and $A^{(2)}$ make up the first while the four other scalars fill the hypermultiplet.
“Standard” orientifold projection [@bs; @gp] keeps the hypermultiplet and projected out the tensor multiplet. Therefore, the fractional D-branes which appear is this model always come in pairs. One can see this easily using a T-dual type IIA description. T-dualizing the orbifold space $\mathbb{C}^2/\mathbb{Z}_2$ along its $U(1)$ direction maps the $A_1$ singularity to a pair of NS5-branes separated along the T-dualized direction, $x^6$ [@ov; @ghm]. The orientifold 9-plane splits into two orientifolds 8-planes located at antipodal points on the circle parameterized by $x^6$. For the standard choice, each of the NS5-branes coincides with one $O8$-plane. Now, we remind the reader that, under T-duality, the twisted fields are supposed to be mapped to differences between the fields which live on each NS5-branes and which also fill ${\cal N}_6=(2,0)$ tensor multiplets [@kls]. For instance, the $b$ field is related to the distance between the two NS5-branes along $x^6$ while the blow-up modes are mapped to the relative positions of these branes along $x^7, x^8$ and $x^9$. Therefore, for symmetry reasons, we see that, for such orientifold choice, the field $b$ and $A^{(2)}$ are projected out.
One the other hand, the above T-dual picture suggests another choice for the orientifold projection [@dp; @bz; @bi] which will give fractional D-branes similar to their type IIB counterparts [@ep; @qs]. Indeed, it is possible to move the NS5-branes off the $O8$-planes in the direction $x^6$ and the symmetry of the problem imposes that the NS5-branes can not be moved anymore independently in the $x^7, x^8$ or $x^9$ directions. T-dualizing back to type IIB string theory, this means that the orientifold operation now projects out the hypermultiplet of the twisted sector. The orbifold space lacks its blow-up modes but has a twisted NS-NS $b$ field and its 2-form companion. In the following, we will always consider this second possibility and construct the fractional D-branes solutions on such backgrounds.
To get fractional D3-branes, we perform two T-dualities along two directions transverse to the orbifold; the R-R 2-form becomes a twisted 4-form under which the fractional D3-branes will be charged and which can also be Hodge-dualized to a scalar, called $c$ below. The theory will contain $O7$-planes, and depending on the orientifold projection, D7-branes to cancel the tadpole of the R-R 8-form. More precisely, we will consider in this paper two different orientifold projections which give rise to symplectic or/and orthogonal groups on the fractional D3-branes.
The first one has an orientifold projection defined by $P_1 \equiv (1+\alpha)(1+\Omega^\prime)/4$ where $\alpha$ is the generator of the $\mathbb{Z}_2$ orbifold group acting on the coordinates as $x^6, x^7, x^8, x^9 \rightarrow - (x^6, x^7, x^8, x^9)$ and $\Omega^\prime = \Omega R_{45}(-1)^{F_L}$ where $\Omega$ is the standard orientifold operator, $R_{45}$ is the reflection in $x^4, x^5$ directions and $(-1)^{F_L}$ acts as -1 on the left-moving Ramond sector. The presence of these last two operators is due to the T-duality transformations. This model contains four $O7^-$-planes whose total negative charge under $C^{(8)}$ is neutralized by adding 32 D7-branes. In this background, we can have fractional D3-branes and the gauge theory which lives on the world-volume of $N$ of those has a gauge group $Sp(2N)$, with a hypermultiplet in the fundamental representation, coming from strings stretched between the D7-branes and these D3-branes. More precisely, as in type IIB string theory, we have two different kinds of fractional D3-branes, whose world-volume theories are obtained by a projection of the gauge theories living on their type IIB parents. For the orientifold projection $P_1$ both types of fractional D-branes have the same massless field content. Therefore, for $2N_1$ fractional D3-branes of the first kind and $2N_2$ of the second kind (we are counting also their images under $\Omega^\prime$), the gauge theory will have a group $Sp(2N_1) \times Sp(2N_2)$ with one hypermultiplet in the bifundamental, two in the $\square_1$ and two in the $\square_2$. The minimally covariantized action of the linear couplings of these D3-branes to the background fields can be extracted from their boundary state description : $$\begin{aligned}
{\cal S}_{1/2} &=& - \frac{T_3}{\sqrt{2}\kappa_{\rm orb}}
\Bigl(\int d^4\sigma\, e^{-\phi}\,
\sqrt{-{\rm det}(g+F)}
\left(1\pm\frac{\tilde{b}}{2\pi^2\alpha^\prime}\right) \nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- \int C^{(4)}
\left(1\pm\frac{\tilde{b}}{2\pi^2\alpha^\prime}\right)
\pm \frac{A^{(4)}}{2\pi^2\alpha^\prime}\Bigr)
\label{d3action}\end{aligned}$$ where $g$ is the induced metric on the world-volume of the D-brane : $g_{\alpha\beta} \equiv G_{\mu\nu} \partial_\alpha X^\mu \partial_\beta
X^\nu$.
The second orientifold projection we will consider does not contain any D7-branes. The operator is defined by $P_2 \equiv
(1+\alpha)(1+\beta\Omega^\prime)$ where $\beta^2 = \alpha$ and does not induce any R-R tadpole. In the T-dual picture, the corresponding orientifold plane splits into two $O6$-planes, one negatively charged and the other oppositely charged under the R-R 7-form. Again, in this background, we have two kinds of fractional D3-branes; however, the orientifold projection now gives rise to a $Sp(2N)$ gauge theory on the first type and to a $SO(2N)$ gauge group on the other. The T-dual description gives again a simple geometrical interpretation of this result : the first case corresponds to D4-branes intersecting the $O6^-$-plane as for $P_1$, while the second comes from a D4-brane intersecting the positively charged $O$-plane. The most general configuration will have $N_1$ fractional D3-branes of the first kind and $N_2$ of the second type (plus their images under $\Omega^\prime$), leading to an $Sp(2N_1)\times SO(2N_2)$ gauge theory with one hypermultiplet in the bifundamental. The superconformal theories constructed using D3-branes in these models have been studied in [@pu; @gk]. In the context of the AdS/CFT correspondence, supergravity solutions for these conformal configurations have been given in [@elns].
Supergravity solutions and one-loop metric on the moduli space
--------------------------------------------------------------
### The field equations
Since we know their type IIB counterparts, the supergravity descriptions of these fractional D3-branes are easy to find. Indeed, we have just seen that they couple to the same supergravity fields. The fields projected out by the orientifold operators, namely, the untwisted NS-NS 2-form, the untwisted R-R 2-form, the twisted R-R 2-form and the blow-up modes were already inactive in the type IIB case. Therefore, the homogeneous field equations will be identical. These equations for the scalar fields can be read in [@grp; @bcffst] $$\begin{aligned}
&&d^\star d\tau = 0, \label{taueq} \\
&&d^{\star_6}d \gamma + i F_{(5)} \wedge (dc + \tau db) - b\, d^{\star_6}d
\tau =0 \label{gammaeq}\end{aligned}$$ where we have introduced the complex combinations $$\begin{aligned}
\tau = C^{(0)} + i e^{-\phi} \qquad {\rm and } \qquad
\gamma = c + \tau b \nonumber\end{aligned}$$ and defined the self-dual 5-form $$F^{(5)} = dC^{(4)} + \ ^\star dC^{(4)}. \nonumber$$ Requiring that a fraction of the supersymmetries is preserved by the solution implies that the scalar fields $\tau$ and $\gamma$ must be holomorphic functions of $z$. This constraint has been used to derive the equations (\[taueq\]) and (\[gammaeq\]). The two-dimensional homogeneous Laplace equation (\[taueq\]) is automatically satisfied and the precise form of $\tau$ must come from other arguments, one being that it has to reflect the presence of sources for the untwisted R-R scalar. An other requirement is that its imaginary part, $e^{-\phi}$, must stay positive. Finally, one can also use its modular properties under the group $SL(2, \mathbb{Z})$.
Using the standard metric and self-dual 5-form ansatz for D3-branes on top of D7-branes/O7-plane [@afm], $$\begin{aligned}
ds^2 &=& H^{-1/2}\eta_{\alpha\beta} dx^\alpha dx^\beta + H^{1/2}(e^{\psi} dz
d\bar{z} + \delta_{ij} dx^i dx^j), \nonumber \\
F^{(5)} &=& d(H^{-1} dx^0\wedge \ldots \wedge dx^3) + \ ^\star
d(H^{-1} dx^0\wedge \ldots \wedge dx^3),\end{aligned}$$ in the Einstein and 5-form equations leads[^2] $$dx^4\wedge dx^5\left(e^{\psi}\delta^{ij} \partial_i \partial_j +
\partial_z\partial_{\bar{z}} \right) H - \delta(x^6)
\ldots \delta(x^9) db \wedge dc
= 0
\label{heq}$$ for the harmonic function $H(z,\bar{z},x^i)$. Finally, $\psi$ is a function of $z$ and $\bar{z}$ which obeys the equation $$\partial_z\partial_{\bar{z}} \left(\psi - \log \tau_2\right)=0.$$ Compared to the fractional D3-branes on top of D7-branes studied in [@grp; @bcffst], the difference lies in the boundary conditions (or, equivalently, source terms in the right hand side of the above field equations) and in the geometrical symmetries imposed by the orientifold projections.
### $Sp(2N_1)\times Sp(2N_2)$ gauge theory
Besides the usual orbifold projection $(1+\alpha)$, the orientifold projection $P_1$ has a geometrical action on the coordinates $z=x^4+i x^5 \rightarrow
-z$, due to the presence of the reflection operator $R_{45}$ in $\Omega^\prime$. One of the four orientifold 7-plane is located at the fixed point $z=0$ of this operator, the three other being rejected at infinity in the decompactified limit of the theory. In this study, we will forget them and consider only the O7-plane at the origin and its eight D7-branes companions. If one only imposes the global cancellation of the untwisted R-R tadpole, these D7-branes can be located away from the origin of the $z$-plane. However, the $R_{45}$ symmetry imposes a constraint on their positions. Moreover, these D7-branes carry a charge under the twisted R-R field which, in absolute value, is one quarter of the charge of a fractional D3-brane. Before doing the T-dualities along $x^4$ and $x^5$, twisted R-R tadpole cancellation was a requirement for the consistency of the theory and imposed that one half of the D9-branes carries a positive charge and the other half a negative charge.
In the decompactified theory obtained after T-duality, this choice is no longer required. However, in this article, we will still assume that the twisted charge induced by the D7-branes vanishes locally : in other words, four pairs, each made of D7-branes with opposite twisted charges, will be located at points $z=\pm
m_i,\,i=1,2$. Therefore, all the twisted charge will be due to the presence of fractional D3-branes.
Separating the D7-branes from the O7-plane generates non trivial dilaton and untwisted R-R scalar. The presence of these sources for the R-R field leads to the following asymptotic behaviour : $$\tau(z) \sim i \left(1 + \frac{g^B_s}{2\pi} \left(8 \log (z) -
2\sum_{i=1}^2 \log (z^2-m_i^2) \right) \right)
\label{tauasymp}$$ which displays the expected monodromy. We have also taken into account a constant background value for $\tau_2$. However, the function (\[tauasymp\]) does not always have a positive imaginary part, and, therefore can not be the complete answer.
This problem has been solved in [@gsvy], where, using its modular properties, a formula has been proposed for $\tau$ : $$j(\tau(z)) = {z^{-8}}{\prod_{i=1}^{2}(z^2-m_i^2)^2}.$$ $j$ is the modular invariant function and $\tau$ is restricted to the fundamental domain ${\cal F}$ of the modular group. Indeed, $j$ induces a one-to-one mapping between ${\cal F}$ and the $z$-plane. At large $\tau_2$, one obtains the result (\[tauasymp\]) but, even if this approximation will be adequate for the purpose of this article, one should keep in mind that it is not the whole solution and that corrections could be needed to tackle other problems.
The only sources for the twisted field equation (\[gammaeq\]) are the actions (\[d3action\]) of the two kinds of fractional D3-branes. The positions of the D-branes of the first kind will be denoted $\pm z_{1p}$ for $p=1,\ldots,N_1$ while, for the other set, we will use $\pm z_{2q}$ for $q=1,\ldots,N_2$. As in the previous case, this information leads to the formula : $$\gamma(z) \sim 4 i \pi g^B_s \alpha^\prime \,\sum_{\epsilon=\pm
1}\left(\sum_{p=1}^{N_1} \log\frac{(z+\epsilon z_{1p})}{\Lambda} -
\sum_{q=1}^{N_2} \log\frac{(z+\epsilon z_{2q})}{\Lambda}\right)
\label{gammaasymp}$$ from which we can extract the twisted scalar $b$ and the twisted 4-form $A^{(4)}$ after an Hodge duality : $$\begin{aligned}
&&\tilde{b} \sim 4 \pi g^B_s \alpha^\prime \,{\rm Re}\sum_{\epsilon=\pm
1}\left(\sum_{p=1}^{N_1} \log\frac{(z+\epsilon z_{1p})}{\Lambda} -
\sum_{q=1}^{N_2} \log\frac{(z+\epsilon z_{2q})}{\Lambda}\right),
\nonumber\\
&&A^{(4)} \sim 4 \pi g^B_s \alpha^\prime \,{\rm Re}\sum_{\epsilon=\pm
1}\left(\sum_{p=1}^{N_1} \log\frac{(z+\epsilon z_{1p})}{\Lambda} -
\sum_{q=1}^{N_2} \log\frac{(z+\epsilon z_{2q})}{\Lambda}\right)\, dx^0
\wedge \ldots \wedge
dx^3. \nonumber \end{aligned}$$ As explained in the appendix of [@bdflmr], this profile only encodes one-loop open string effects. However, usually [@bdflmp; @pol], one takes this formula as the full solution to the field equation. This choice can be justified if one insists on preserving the rotational symmetry in the $z$-plane when all the D-branes are located at $z=0$. Indeed, other holomorphic solutions involving subleading corrections to (\[gammaasymp\]) breaks this symmetry, which, from the gauge theory point-of-view, corresponds to the $U(1)$ of the $R$-symmetry group. As we will see below, this solution contains enough information about the gauge coupling renormalization as predicted by open string/closed string duality and, from the gauge theory point-of-view, we do not expect any higher order perturbative corrections to it. However, this argument does not rule out the existence of exponentially vanishing corrections of order $(\Lambda/z)^n$, $n>0$, which could be attributed to D-instanton effects and, under the gauge/gravity correspondence, are expected to be mapped Yang-Mills instantons effects.
Actually, in the context of fractional D2-branes [@wz][^3], it has been argued recently that the leading order solution receives corrections which are exponentially small at infinity but become important at some finite distance from the core. The idea was to “improve” the effective description by going to a special point of the moduli space of the type IIA on $T^4/{\mathbf
Z}_2$ string theory where fractional D0-branes become massless. Indeed, the breakdown of the supergravity description in the case of fractional D2-branes corresponds to the appearance of new massless states at a special radius $r_e$, due to fractional D0-branes, which had been integrated out in the standard supergravity action. At this radius, a $U(1)$ gauge theory (corresponding to the R-R 1-form $A^{(1)}$ under which the fractional D2-branes are magnetically charged) gets enhanced to a $SU(2)$ group, the new massless states playing the role of W-bosons. A correct way to deal with this problem is to go directly to the $SU(2)$ point of the moduli space of the string theory and search for a solution of the new field equations. In this case, the twisted fields, $b$ and $A^{(1)}$, obey a $SU(2)$ Bogomolnyi equation. Its analytic solution is known for a magnetic charge equals to one. Asymptotically, the solution looks like the Dirac monopole but, close to the enhançon radius, non-perturbative effects due to the fractional D0-branes completely change the picture and lead to a smooth, non-singular metric (except obviously at the origin, where we always have a singularity, as for any D-brane). The twisted NS-NS field $b$ plays the role of a Higgs field and, away from the special radius $r_e$, the $SU(2)$ group is broken to a $U(1)$ by the Higgs mechanism and the fractional D0-branes get a mass proportional to $b$.
In the case of fractional D3-branes, we expect that similar effects will also solve the singularities of the metric, by taking into account the fact that fractional D-instantons become massless at $r=r_e$. More precisely, one expects the modified solution for the twisted fields to be given by the periodic monopole solutions considered by [@ck], in the limit of a vanishing circle : in this paper, the authors discuss solutions to the non-abelian Bogomolnyi equation on $\mathbb{R}^2 \times S^1$. In the language of the last paragraph, this would describe fractional D2-branes with a transverse direction (also transverse to the orbifold) compactified. In the zero radius limit, we obtain the T-dual configuration, namely fractional D3-branes. Obviously the asymptotics, given by the Dirac solution, coincide with (\[gammaasymp\]) but, unfortunately, the full solution is not known, even in the case of a charge one periodic monopole. However, one expects that the solution will be modified at short distances, like for the t’Hooft-Polyakov monopole, in such a way that the metric singularities are smooth out. More speculatively, one could conjecture that the exact solution is such that the cascading-like behaviour of the D3-brane charge discussed in [@pol] and refuted in [@bgl], completely disappears, since we do not expect such a Seiberg duality in a ${\cal N}=2$ gauge theory. It would be interesting to investigate these questions further but, for the purpose of this article, we stop these speculations now and consider only the formula (\[tauasymp\]) and (\[gammaasymp\]).
Using a D-brane probe, we will show that these asymptotics carry enough information to reproduce the one-loop corrected effective action on the moduli space of the $Sp(2N_1)$ gauge theory with $2N_2 + 2$ hypermultiplets in the fundamental representation. The Laplace equation (\[heq\]) for the harmonic function $H$ can not be solved explicitly in the case of D7-branes away from the origin of the $z$-plane but will not be needed. When the untwisted R-R charge is zero, one can choose $\psi$=0 and the harmonic function will be similar to the case of type IIB on $\mathbb{C}^2/\mathbb{Z}_2$ [@bdflmp].
To probe the background, we consider a fractional D3-brane of the first kind located at a position $z$. Inserting into its action ${\cal S}_1$ the supergravity fields, we can extract the gauge coupling and the kinetic term for the adjoint scalar fields $\Phi^a$ which belong to the vector multiplet. The expansion of the action to quadratic order in the gauge and scalar fields leads to : $$\begin{aligned}
{\cal S} &=& -\frac{1}{8\pi g^B_s} \int\, d^4x e^{-\phi}
\left(1+\frac{\tilde{b}}{2\pi^2\alpha^\prime} \right) \left(
\frac{1}{4} F_{\alpha\beta}F^{\alpha\beta}+
\frac{1}{2}\partial_\alpha{\Phi}^a \partial^\alpha{\Phi}^a \right)
\nonumber \\
&=& -\frac{1}{g^2_{\rm YM}(z)} \int\, d^4x \left(
\frac{1}{4} F_{\alpha\beta}F^{\alpha\beta}+
\frac{1}{2}\partial_\alpha{\Phi}^a \partial^\alpha{\Phi}^a \right) \end{aligned}$$ with $$\begin{aligned}
\frac{1}{g^2_{\rm YM}(z)} &=& \frac{1}{8\pi g^B_s} + \frac{1}{8\pi^2} {\rm
Re}
\sum_{\epsilon=\pm 1}
\left(
\sum_{p=1}^{N_1} 2\log\frac{(z+\epsilon z_{1p})}{\Lambda} -
\sum_{q=1}^{N_2} 2\log\frac{(z+\epsilon z_{2q})}{\Lambda}\right) \nonumber\\
&&~~~~~~~~~~~~~~~~+
4 \log(\frac{z}{\Lambda}) -
\sum_{i=1,2} \log\frac{(z^2-m_i^2)}{\Lambda^2}\end{aligned}$$ This formula is the one-loop corrected gauge coupling of the symplectic gauge theory on its Coulomb branch [@dkp], where the [ *vev*]{} of its adjoint scalars are proportional to $z_{1p}$, with $2N_2+2$ fundamental hypermultiplets. This one-loop renormalized gauge coupling displays Landau poles where the enhançon mechanism has been argued to take place. However, we know from the gauge theory point-of-view that the complete moduli space metric, which includes non-perturbative - Yang-Mills instantons - corrections is a smooth manifold, something that our solution is not able to capture.
### $Sp(2N_1)\times SO(2N_2)$ gauge theory
The set of fixed points of the other orientifold projection, $P_2$, reduces to the plane left $x^6=x^7=x^8=x^9=0$ left invariant by the orbifold operator $\alpha$. There is no dilaton tadpole and the complex scalar $\tau$ is a constant. However, the orientifold projection induces a background twisted charge [@dp; @bz; @pu], which is precisely twice the charge of a fractional D3-brane of the first kind. One of the consequences of this background charge is that the conformal field theory case requires fractional D-branes and not just bulk D-branes : it is obtained for $N_2 = N_1 + 1$. A solution to the twisted field equation which correctly encodes the presence of these sources reads $$\gamma(z) = 4 i \pi g^B_s \alpha^\prime \,\left(\sum_{\epsilon=\pm
1}\left(\sum_{p=1}^{N_1} \log\frac{(z+\epsilon z_{1p})}{\Lambda} -
\sum_{q=1}^{N_2} \log\frac{(z+\epsilon z_{2q})}{\Lambda}\right) + 2
\log\frac{z}{\Lambda}\right).$$ Again, we can probe this supergravity background using the two different kinds of fractional D3-branes. Extracting the kinetic terms for scalar fields on the first kind of probe located at $z$ leads to $$\begin{aligned}
{\cal S}_1 &=& -\left(\frac{1}{8\pi g^B_s}+\frac{1}{8\pi^2} {\rm Re}
\sum_{\epsilon=\pm
1}\left(\sum_{p=1}^{N_1} 2\log\frac{(z+\epsilon z_{1p})}{\Lambda} -
\sum_{q=1}^{N_2} 2\log\frac{(z+\epsilon z_{2q})}{\Lambda}\right) + 4
\log\frac{z}{\Lambda}\right) \nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\int\, d^4\sigma \left(
\frac{1}{4} F_{\alpha\beta}F^{\alpha\beta}+
\frac{1}{2}\partial_\alpha{\Phi}^a \partial^\alpha{\Phi}^a \right) \end{aligned}$$ while, for the other probe, $$\begin{aligned}
{\cal S}_2 &=& -\left(\frac{1}{8\pi g^B_s}+\frac{1}{8\pi^2} {\rm Re}
\sum_{\epsilon=\pm
1}\left(\sum_{q=1}^{N_2} 2\log\frac{(z+\epsilon
z_{2q})}{\Lambda}) -
\sum_{p=1}^{N_1} 2\log\frac{(z+\epsilon z_{1p})}{\Lambda}\right)
- 4
\log\frac{z}{\Lambda}\right) \nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\int\, d^4\sigma \left(
\frac{1}{4} F_{\alpha\beta}F^{\alpha\beta}+
\frac{1}{2}\partial_\alpha{\Phi}^a \partial^\alpha{\Phi}^a \right) \end{aligned}$$ In these expressions, we recognize respectively the one-loop corrected metric on the moduli space of the ($Sp(2N_1) + 2 N_2\, \square_1$) and ($SO(2N_2) + 2 N_1\, \square_2$) gauge theories.
To conclude this first part, we must mention that there is (at least) another way to obtain fractional D3-branes with orthogonal or symplectic gauge groups. The construction, which has been described in [@uranga] for the case of type IIB string theory on $\mathbb{C}^2/\mathbb{Z}_{2N}$, uses an $O3$-plane. For $N=1$, the gauge theory which lives on the two possible kinds of fractional D-branes is $Sp(2N_1)\times SO(2N_2)$ with a hypermultiplet in the bifundamental. In this paper, one can also see that the orientifold projection, called $\Omega \Pi$, does not introduce any untwisted tadpoles but gives the same background twisted charge in the sector $k=N$ as the $P_2$ projection we considered in this section. Therefore, the supergravity solution and the gauge couplings renormalizations will be identical.
Fractional D-branes versus wrapped M5-branes
============================================
D3-branes on a two-centers Taub-NUT space
-----------------------------------------
Alternative approaches to describe non-conformal pure ${\cal
N}=2$ $SU(N)$ super-Yang-Mills theories using supergravity solutions have been discussed recently [@pw; @gkmw; @bcz; @bfms]. It would be interesting to make connections between these various models and to completely understand which points on the moduli space of the gauge theory they describe. The aim of this section is to provide a first step in this direction by comparing the fractional D3-brane supergravity solution of type IIB on the non-compact orbifold $\mathbb{C}^2/\mathbb{Z}_2$ to another proposal, namely the M5-brane wrapped on a Riemann surface of [@bfms].
The strategy we will adopt is to lift the type IIB solution to eleven dimensions. To do this, we will first replace the orbifold singularity by the two-centers Taub-NUT space. Indeed, we know that, on one hand, a $k$-centers Taub-NUT manifold degenerates into a $\mathbb{C}^2/\mathbb{Z}_k$ orbifold in a specific limit and, on the other hand, it is T-dual to $k$ NS5-branes delocalized along the T-dualized direction [@ov; @ghm]. Then, we construct fractional D3-brane solutions on this ALF space, using the method described in [@bcffst] to obtain the supergravity solution for D3-branes with fluxes on the Eguchi-Hanson manifold. Performing a T-duality along the fibered $U(1)$ of the Taub-NUT manifold, we will obtain a type IIA solution describing D4-branes stretched between NS5-branes. Finally, using the type IIA/M-theory duality dictionary, we will lift the solution to eleven dimensions.
Let us first begin by reviewing some properties of the two-centers Taub-NUT space [@egh], which will be useful to find the D3-brane solution. The metric of the Taub-NUT manifold parameterized by the coordinates $(x^6, \ldots, x^9)$ and with centers located at $x^i=x^i_1,\, x^i_2$ for $i=7, 8 ,9$, can be written as : $$ds_{\rm Taub-NUT}^2 = V\delta_{ij} dx^i dx^j + V^{-1} (dx^6 + A_i
dx^i)^2
\label{taubnut}$$ with $$\begin{aligned}
V=1+\sum_{a=1,2}\frac{R}{\vert x^i - x^i_a \vert} \qquad {\rm and }
\qquad \partial_i V = \epsilon_{ijk} \partial_j A_k.
\label{vtaubnut}\end{aligned}$$ In order to avoid conical singularities at $r=0$, the coordinate $x^6$ must have periodicity $4\pi R$. The metric is asymptotically flat and approaches the cylinder $\mathbb{R}^3 \times S^1_{4\pi R}$ at large $|x|$. When the centers become coincident, this metric has a $\mathbb{Z}_2$ singularity at $r=0$. In this case, the orbifold limit is obtained by sending the radius $R$ to infinity. More precisely, if one defines the new coordinates $(r, \theta, \phi, \psi)$ $$\begin{aligned}
&&x^6 = 2 R \psi,~~~~x^7 = r \cos\theta,~~~~x^8 = r \sin\theta,
\cos\phi~~~~{\rm and}~~~~x^9 = r \sin\theta \sin\phi \nonumber \end{aligned}$$ in the limit $R \rightarrow \infty$, the metric (\[taubnut\]) with $x_1=x_2=0$ degenerates to $$\begin{aligned}
ds^2 \simeq 2R \left(\frac{1}{r}
\left(dr^2 + r^2(d\theta^2 + \sin^2 \theta d\phi^2) \right)
+ r (d\psi + \cos \theta d\phi)^2 \right)\end{aligned}$$ where $\psi \sim \psi + 2 \pi$. We recognize the metric of an ALE space with an $A_1$-type singularity. If we define the radial coordinate $\tilde{r} = 2\sqrt{2R r}$, we obtain a flat space metric with a $\mathbb{Z}_2$ identification.
The two-centers Taub-NUT manifold (\[taubnut\],\[vtaubnut\]) possesses a homology 2-cycle given by the fibration of the $S^1$ parameterized by $x^6$ over the segment joining the two centers. For coincident centers, in the limit $R \rightarrow \infty$, it degenerates to the vanishing 2-cycle of the $\mathbb{Z}_2$ orbifold located at the origin of the space. The dual of this cycle is an anti-self-dual harmonic 2-form, $G^{(2)}$, defined as $$G^{(2)} = d(V^{-1}(dx^6+2R\cos \theta d\phi))$$ and which satisfies the relation $$G^{(2)}\wedge G^{(2)} = - \Delta(r)\, \Omega_{\rm Taub-NUT}$$ where $\Delta(r) \equiv (\partial V^{-1})^2$ and $\Omega_{\rm Taub-NUT}$ is the volume form of Taub-NUT.
To recover the fractional D3-brane solution in the orbifold limit, we will search for a D3-brane metric of the form $$ds_{\rm IIB}^2 = H^{-1/2} dx^2_{1+3} + H^{1/2} \left(dz d\bar{z} +
ds^2_{\rm Taub-NUT} \right).$$ induced by the presence of fluxes of NS-NS and R-R 2-forms through the 2-cycle of the Taub-NUT space[^4]. In the orbifold limit, they become twisted NS-NS and R-R scalars. They are expressed in term of the harmonic 2-form : $$C^{(2)} + i B^{(2)} = \gamma\, G^{(2)}$$ and the complex scalar $\gamma$ is a solution of the two dimensional Laplace equation $$\square_{\mathbb{R}^2} \gamma = 0$$ in the space transverse to the D3-branes and to the Taub-NUT manifold. As in the previous section, requiring the preservation of supersymmetries imposes that $\gamma$ is a holomorphic function. Moreover, we would like to have $N$ fractional D3-branes located at positions $z_p$. A solution satisfying these constraints is given by $$\begin{aligned}
\gamma(z) = 4 i \pi g^B_s \alpha^\prime \,
\sum_{p=1}^{N} \log\frac{(z+ z_{p})}{\Lambda}\end{aligned}$$ but, as discussed in the previous section, it could receive corrections which improve its behaviour close to the D-branes.
Finally, the Einstein equation implies that the harmonic function $H(r,z,\bar{z})$ must verify : $$\left( \square_{\mathbb{R}^2} + \square_{\rm Taub-NUT} \right) H = -
\vert \partial_z \gamma \vert^2 \Delta(r).
\label{einstein}$$ A standard way to obtain solutions to this equation is to perform a Fourier transformation over $z$ and $\bar{z}$. For simplicity, we will consider only the case of $z_p=0$ for which the rotational symmetry in the $z$-plane is preserved. We call $\rho =
\sqrt{z\bar{z}}$. Therefore, the Fourier transform of the harmonic function $H$ is defined by $$\hat{H}(r, \mu) = \frac{1}{2\pi} \int_0^\infty d\rho \, \rho \,
J_0(\mu \rho) \, \left(H(r, \rho)-1 \right).$$ $\mu$ runs from $0$ to infinity. For convenience, the constant part of the harmonic function has been subtracted to define its Fourier transform. Then, the field equation (\[einstein\]) becomes $$\left(\partial_r^2 + \frac{2}{r} \partial_r - \mu^2
\left(1+\frac{2R}{r}\right) \right) \hat{H} + \hat{J} = 0,
\label{eqfourier}$$ where $\hat{J}$ is the Fourier transform of $V(\partial_r V^{-1})^2
|\partial_z \gamma|^2$.
The general solution of this equation is the sum of the homogeneous equation solutions and of a particular solution. One can see that the homogeneous equation is related to the Whittaker equation; its generic solution is given by $$\hat{H}_h(r, \mu) = \frac{1}{r} \left(\alpha(R, \mu)
W_{-\mu R,-\frac{1}{2}}(2 \mu r) +
\beta(R, \mu) W_{\mu R,-\frac{1}{2}}(-2 \mu r) \right).$$ To fix the integration constants $\alpha$ and $\beta$, we will impose (hopefully) reasonable D3-brane boundary conditions in the asymptotic regions $r\rightarrow \infty$ ($R$ small) and $r \rightarrow
0$ ($R$ large).
First, one would like to have an asymptotically flat metric when $r\rightarrow \infty$. This implies that $\lim_{r\rightarrow \infty}
\hat{H}(r, \mu) = 0$. On the other hand, we know the behaviour of the Whittaker function when the norm of its argument is large [@gr] : $$W_{a, b}(x) \sim x^a \, e^{-x/2}~~~~{\rm for}~~~~|x| \rightarrow \infty.$$ This means that $\beta$ must vanish.
Then, we can consider the ALE space limit, namely $R$ large, with $\tilde{r}=2\sqrt{2R r}$ fixed, arbitrary, being the radial coordinate of the four-dimensional flat space. At leading order, we would like to recover the first term of the harmonic function of fractional D3-branes [@bdflmp] at the orbifold conformal field theory point where they carry half of the mass and of the charge of bulk D3-branes [^5], $$H_{h}\left(\frac{\tilde{r}^2}{{8R}},\rho\right) \simeq 1+\frac{4\pi
N g^B_s
\alpha^{\prime 2}}{(\tilde{r}^2 +
\rho^2)^2},$$ whose Fourier transform is expressed in term of the Bessel function $K_1$ as : $$\hat{H}_{h}\left(\frac{\tilde{r}^2}{8R},\mu\right)
\simeq \frac{4\pi N g^B_s \alpha^{\prime
2}}{\tilde{r}}\,\frac{\mu K_{1}(\mu\tilde{r})}{2}.$$ Using the following property of the Whittaker function [@gr], $$\lim_{R \rightarrow \infty} \left(\Gamma(1+\mu R)\, W_{-\mu R,
-\frac{1}{2}}\left(\frac{\mu \tilde{r}^2}{4R}\right)\right) =
\mu \tilde{r} K_1(\mu\tilde{r}),$$ one finds that $\alpha(R, \mu) \sim
4\pi N g^B_s \alpha^{\prime 2}\,{\Gamma(1+\mu R)}/{16 R}$ when $R$ is large. A priori, this formula is valid only in the ALE space limit and does not say anything when $R$ is small. However, one can try to see if we obtain a sensible answer when we extrapolate this expression to small $R$, where the Taub-NUT space is asymptotically of the form $\mathbb{R}^3 \times S^1_{4\pi R}$.
Therefore, let us consider as the full solution of the homogeneous equation the following expression : $$\hat{H}_h(r, \mu) = \frac{4\pi N g^B_s \alpha^{\prime 2}\,{\Gamma(1+\mu
R)}}{16Rr}
W_{-\mu R,-\frac{1}{2}}(2 \mu r)
\label{solution}$$ and study its small $R$ behaviour. If we expand this formula in the asymptotic region $R$ small, $r\rightarrow \infty$, we obtain : $$\hat{H}_h(r, \mu) \sim \frac{4\pi N g^B_s \alpha^{\prime
2}\,{\Gamma(1+\mu R)}}{16Rr} \, e^{-\mu r}\,(1+\mu R \log(2\mu r)+\ldots)$$ Its leading order in an expansion in series of $R$ can be Fourier transform back to : $$H_h(r, \rho) \sim 1 + \frac{4\pi N g^B_s \alpha^{\prime 2}}{16 R
(r^2+\rho^2)^{3/2}}
\label{leadharm}$$ In this expression, we recognize the smeared harmonic function of $N/2$ D3-branes delocalized along the circle $S^1_{4\pi
R}$. We see that the division by two of the D3-brane charge imposed in the ALE space limit by choosing a fractional D3-brane boundary condition has propagated to the other limit where the Taub-NUT space degenerates to $\mathbb{R}^3 \times S^1_{4\pi R}$ with $R$ small. Actually, when $R$ is sent to zero, the type IIB description is not the correct one anymore and one must go to its type IIA dual. In the next section, we will perform explicitly this T-duality and we will show that the division by two confirms the interpretation of the T-dual configuration as D4-branes stretched between NS5-branes, carrying half of the charge of standard D4-branes.
However, before doing this, we must mention that the full solution could be obtained by adding to (\[solution\]) a particular solution of the equation (\[eqfourier\]). To find such a solution one can use the method of [@bcffst] : one constructs first the Green function $G(\mu| r, r^\prime)$ with the appropriate boundary conditions using the solutions of the homogeneous equation. Then, the particular solution will be given by $$\hat{H}_{n-h}(r, \mu) = - \int_0^\infty G(\mu| r, r^\prime)
\hat{J}(r^\prime, \mu) (r^\prime)^2 dr^\prime.$$
Uplift to eleven dimensions and comments
----------------------------------------
Using transformation rules for NS-NS and R-R fields given in [@bho], we can perform a T-duality along the isometric direction $x^6$ to map the above solution to a type IIA one. Discussions in the literature have argued that the dual of the two-centers Taub-NUT space is given by two NS5-branes whose separation in the $x^6$ direction is related to the twisted scalar $b$ and that the fractional D3-branes become D4-branes stretched between the two NS5-branes. The T-dual coordinate $\tilde{x}^6$ has periodicity $\pi\alpha^\prime/R$ and the type IIA gauge coupling constant is $g^A_s = g^B_s \sqrt{\alpha^\prime} / 2 R$. The D4-branes/NS5-branes metric, dilaton, NS-NS and R-R field strength read : $$\begin{aligned}
&&ds_{\rm IIA}^2 = H^{-1/2} dx^2_{1+3} + H^{1/2} \left(dz d\bar{z} +
V\delta_{ij} dx^i dx^j \right) + H^{-1/2} V (d\tilde{x}^6 - b\, dV^{-1})^2,
\nonumber \\
&&e^{\phi_A} = V H^{-1/2}, \qquad H^{(3)} = \frac{\alpha^\prime}{\tilde{R}}
\sin \theta\, d(\tilde{x}^6 -
b\, V^{-1})\wedge d\theta \wedge d\phi, \nonumber \\
&& F^{(2)} = - dc \wedge dV^{-1}, \nonumber \\
&& F^{(4)} = \frac{\alpha^\prime}{\tilde{R}} \sin \theta \, V^{-1}
\left(
(d\tilde{x}^6 - b \, dV^{-1}) \wedge dc + 2
c\, db \wedge dV^{-1}
\right)\wedge d\theta \wedge d\phi, \nonumber \\
&& F^{(6)} = dH^{-1} \wedge dx^0 \wedge \ldots \wedge dx^3 \wedge
(d\tilde{x}^6 - b \, dV^{-1}), \nonumber \\
&&\gamma(z) = (c + i b)(z) = 4\pi i N g^A_s
\frac{(\alpha^\prime)^{3/2}}{\tilde{R}} \log
\frac{z}{\Lambda},
\label{tdsol}\end{aligned}$$ where we have defined the dual radius $\tilde{R} =
\alpha^\prime/2R$. As always after T-duality, the $p$-forms appear at the same time as the dual $10-p$-forms. However, one can transform the 4-form $F^{(4)}$ to its Hodge dual and combine the result with the $F^{(6)}$ of (\[tdsol\]). More surprising is the existence of a non trivial R-R 2-form $F^{(2)}$. It comes from the dualization of the 2-form potential $C^{(2)}_{x^6r}$ which was required to obtain a solution which preserves supersymmetry. Dualizing this 2-form field strength to an 8-form gives : $$F^{(8)} = (HV)^{-1} dx^0 \wedge \ldots \wedge dx^3 \wedge db \wedge
H^{(3)}.$$ Therefore, we see that the R-R 7-form potential vanishes and the 8-form is induced by the non-trivial NS-NS 3-form field strength $H^{(3)}$ and R-R 5-form potential $C^{(5)}_{x^0x^1x^2x^3\rho}$.
Since the type IIA solution has been obtained using T-duality in the $x^6$ direction, the two NS5-branes are delocalized along this coordinate and their harmonic function is smeared : $$V=1+\frac{\alpha^\prime}{\tilde{R}
r}.$$ In particular, momentum modes along $\tilde{x}^6$ have been neglected [@ghm]. Moreover, this NS5-brane harmonic function does not include the back-reaction due to the stretched D4-branes.
Now, we can express the leading order (\[leadharm\]) of the harmonic function in term of type IIA variables : $$H_h(r, \rho) \sim 1+\frac{\pi N g_s^A (\alpha^\prime)^{3/2}}{2
(r^2+\rho^2)^{3/2}}.
\label{leadiia}$$ This harmonic function corresponds to $N$ D4-branes with half of the mass and charge of standard D4-branes as one can expect for D4-branes stretched between NS5-branes located at antipodal points of the circle[^6]. This result is an argument in favor of the conjecture that the expression (\[solution\]) is correct not only for large $R$ but also for any radius.
Then, we can uplift this solution to eleven dimensions. The type IIA and eleven dimensional metrics are related by $$ds^2_{11d} = e^{-2\phi_A/3} ds^2_{\rm IIA} + e^{4\phi_A/3} (dx^{10} +
C^{(1)}_\mu dx^{\mu})^2$$ which leads to the result $$\begin{aligned}
ds^2_{11d} &=& (HV)^{-1/3} dx^2_{1+3} + (H V)^{2/3} \left(dr^2 + r^2
d\Omega^2_2 \right)
+ H^{2/3} V^{-1/3} dz d\bar{z} \nonumber \\
&&~~~~~~~~~~~~~~+ H^{-1/3} V^{2/3}\left((d\tilde{x}^6-b\,
dV^{-1})^2+(dx^{10}+c\,
dV^{-1})^2\right).
\label{11dmetric}\end{aligned}$$ To relate this solution to the wrapped M5-branes configurations considered in [@bfms], we rewrite the metric as a function of the new complex coordinate $z^2=s$ defined as : $$s \equiv \tilde{x}^6 + i x^{10} + i \gamma(z)\,V^{-1}(r).$$ Renaming $z$ into $z^1$, the metric takes the form of the Fayyazuddin/Smith ansatz [@fs1; @fs2] : $$\begin{aligned}
ds^2_{11d} &=& g^{-1/3} dx^2_{1+3} + g^{2/3} \left(dr^2 + r^2
d\Omega^2_2 \right) + g^{-1/3} g_{m\bar{n}}dz^m dz^{\bar{n}}\end{aligned}$$ where $g = H \, V$ is the determinant of the Kähler metric $$\begin{aligned}
g_{z\bar{z}} = H + V^{-1} \vert\partial \gamma\vert^2, \qquad
g_{s\bar{s}} = V,\qquad
g_{z\bar{s}} = -i \partial \gamma, \qquad g_{s\bar{z}} = i
\bar{\partial} \bar{\gamma}.\end{aligned}$$ It is easy to check that this metric verifies the equations : $$\partial_m \partial_{\bar{n}} g + \square_{3} g_{m\bar{n}} = 0.
\label{kahler}$$ In particular, the equation (\[kahler\]) for $(m, \bar{n})=(1,
\bar{1})$ is just a rewriting of (\[einstein\]).
To summarize, starting with the type IIB supergravity solution which describes fractional D3-branes on a two-centers Taub-NUT space, we have obtained a partially smeared supergravity solution representing M5-branes wrapped on a two dimensional surface parameterized by the coordinates $s$ and $z$. Since the gauge theory living on the world-volume of the D3-branes is a pure ${\cal N}=2$ $SU(N)$ SYM theory, the eleven dimensional solution should also correspond to this theory and one expects that the the Riemann surface is given by the Seiberg-Witten curve of this gauge theory.
This problem has been studied directly in eleven dimensions in [@bfms], where a completely localized solution has been proposed. Let us remind in few words their result. The authors of [@bfms] considered a configuration of M5-branes wrapped on the Riemann surface defined by $$f(t,z) = t + 2 B(z) + \frac{1}{t} = 0
\label{seibergwitten}$$ which is the Seiberg-Witten curve of the pure ${\cal N}=2$ $SU(N)$ SYM theory when $B(z)=\prod_{p=1}^N (z-z_p)$ [@wit]. In this case, only the eleventh coordinate is compactified $x^{10} \sim x^{10} + 2\pi R^\prime$ and $t=e^{
s/R^\prime}$. They consider a “decoupling” limit, where field theory quantities are held fixed while the eleven dimensional Planck scale is sent to zero. The new variables $Z$, $S$ and $u$ defined by $$\begin{aligned}
Z = \frac{z}{\alpha^\prime} = \frac{z R^\prime}{l^3_P}, \qquad S =
\frac{s}{R^\prime}, \qquad u^2 = \frac{r}{l^3_P}\end{aligned}$$ are fixed under the scaling. In term of these new variables, the Kähler potential which defines the metric appears as the sum of two terms $K=K^{(1)}(u,F(Z,S),\bar{F}(\bar{Z},\bar{S})) + |G(Z,S)|^2$ : $$\begin{aligned}
&&K^{(1)}(u,F(Z,S),\bar{F}(\bar{Z},\bar{S})) = \frac{\pi N}{u^2} \ln
\frac{\sqrt{u^4+|F|^4}+u^2}{\sqrt{u^4+|F|^4}-u^2}, \qquad
F=f^{1/N},\nonumber \\
&&G(Z,S) = - \frac{Z N f^{1-\frac{1}{N}}}{4} \int_0^1
\frac{dt}{\sqrt{\left(\frac{f}{2} - B(t Z) \right)^2-1}}.\end{aligned}$$ Using this Kähler potential, one can calculate the determinant of the metric : $$\begin{aligned}
g=\frac{\pi N}{2\left(u^4+|F|^4\right)^{3/2}}.
\label{bfmssol}\end{aligned}$$ This metric is singular at the location of the M5-brane, namely on the Seiberg-Witten curve at $r=0$. However, it does not have the repulsive singularities displayed by other ${\cal N}=2$ solutions. It would be interesting to probe this background to see if one can reproduce the running of the coupling constant on the world-volume of the probe. To do this, one should consider the action of M5-brane wrapped on the Riemann surface defined by (\[seibergwitten\]).
Unfortunately, we see that the two solutions are obtained in two very different approximations. Indeed, in the D3-brane case, $s$ parameterizes a 2-torus while the Seiberg-Witten curve (\[seibergwitten\]) is valid only when $\tilde{x}^6$ is a non-compact coordinate. Moreover, the M5-brane solution considered here is smeared over this 2-torus, whereas the solution (\[bfmssol\]) of [@bfms] was completely localized. Since the functions $V$ and $\gamma$ are smeared and depend explicitly on the radius of compactification, the decompactification limit is meaningless. The last difference is that our metric is asymptotically flat while the M5-brane of [@bfms] is only a “near-horizon” solution. One can take also a near horizon limit in (\[11dmetric\]) but the other approximations and hypothesis make the comparison impossible. Obviously, it would be very interesting to find a solution which goes beyond the limits of these two solutions, namely which is fully localized on the 2-torus and is not just valid in a decoupling region but the task seems to be difficult. A better understanding of the properties of the solution of [@bfms], in particular in the optic of the gauge/gravity correspondence, is also desirable.
[**Acknowledgements**]{}
I would like to thank M. Bianchi and A. Sagnotti for having given me the opportunity to present these results to the String Theory group in [*Tor Vergata*]{} and for discussions. I also wish to thank C. Bachas for the invitation to visit the Laboratoire de Physique Théorique de l’École Normale Supérieure and for his remarks on this work. This research is supported by a PPARC fellowship.
[99]{}
J. M. Maldacena, [*The Large N Limit of Superconformal Field Theories and Supergravity*]{}, Adv. Theor. Math. Phys. [**2**]{} (1998) 231, .
L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, [*Confinement and Condensates Without Fine Tuning in Supergravity Duals of Gauge Theories*]{}, JHEP [**9905**]{} (1999) 026, .
J. Polchinski and M. Strassler, [*The String Dual of a Confining Four-Dimensional Gauge Theory*]{}, .
K. Pilch and N. P. Warner, [*$N=1$ Supersymmetric Renormalization Group Flows from IIB Supergravity*]{}, , [*$N=2$ Supersymmetric RG Flows and the IIB Dilaton*]{}, Nucl. Phys. [**B594**]{} (2001) 209, .
J. M. Maldacena and C. Nunez, [*Towards the large N limit of pure N=1 super Yang Mills*]{}, Phys. Rev. Lett. [**86**]{} (2001) 588, .
J. P. Gauntlett, N. Kim, D. Martelli and D. Waldram, [*Wrapped fivebranes and $N=2$ super Yang-Mills theory*]{}, Phys. Rev. [**D64**]{} (2001) 106008, .
F. Bigazzi, A. L. Cotrone and A. Zaffaroni, [*$N=2$ Gauge Theories from Wrapped Five-branes*]{}, Phys. Lett. [**B519**]{} (2001) 269, .
I. R. Klebanov and N. A. Nekrasov, [*Gravity Duals of Fractional Branes and Logarithmic RG Flow*]{}, Nucl. Phys. [**B574**]{} (2000) 263, .
I. R. Klebanov and A. A. Tseytlin, [*Gravity Duals of Supersymmetric $SU(N) \times SU(N+M)$ Gauge Theories*]{}, Nucl. Phys. [**B578**]{} (2000) 123, .
I. R. Klebanov and M. J. Strassler, [*Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities*]{}, JHEP [**0008**]{} (2000) 052, .
F. Bigazzi, L. Girardello and A. Zaffaroni, [*A note on regular type 0 solutions and confining gauge theories*]{}, Nucl. Phys. [**B598**]{} (2001) 530, .
M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta and I. Pesando, [*Fractional D-branes and their gauge duals*]{}, JHEP [**0102**]{} (2001) 014, .
J. Polchinski, [*$N=2$ Gauge/Gravity Duals*]{}, Int. J. Mod. Phys. [**A16**]{} (2001) 707, .
O. Aharony, [*A note on the holographic interpretation of string theory backgrounds with varying flux*]{}, JHEP [**0103**]{} (2001) 012, .
M. Petrini, R. Russo and A. Zaffaroni, [*$N=2$ Gauge Theories and Systems with Fractional Branes*]{}, Nucl. Phys. [**B608**]{} (2001) 145, .
M. Billó, L. Gallot and A. Liccardo, [*Classical geometry and gauge duals for fractional branes on ALE orbifolds*]{}, Nucl. Phys. [**B614**]{} (2001) 254, .
M. Graña and J. Polchinski, [*Gauge/Gravity Duals with Holomorphic Dilaton*]{}, .
M. Bertolini, V. L. Campos, G. Ferretti, P. Fré, P. Salomonson and M. Trigiante, [*Supersymmetric 3-branes on smooth ALE manifolds with flux*]{}, .
M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda and R. Marotta, [*N=2 Gauge theories on systems of fractional D3/D7 branes*]{}, .
M. R. Douglas and G. Moore, [*D-branes, Quivers, and ALE Instantons*]{}, .
D. Diaconescu and J. Gomis, [*Fractional Branes and Boundary States in Orbifold Theories*]{}, JHEP [**0010**]{} (2000) 001, .
N. Seiberg and E. Witten, [*Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD*]{}, Nucl. Phys. [**B431**]{} (1994) 484, .
C. V. Johnson, A. W. Peet and J. Polchinski, [*Gauge Theory and the Excision of Repulson Singularities*]{}, Phys. Rev. [**D61**]{} (2000) 086001, .
A. Buchel, A. W. Peet and J. Polchinski, [*Gauge Dual and Noncommutative Extension of an N=2 Supergravity Solution*]{}, Phys. Rev. [**D63**]{} (2001) 044009, .
N. Evans, C. V. Johnson and M. Petrini, [*The Enhancon and N=2 Gauge Theory/Gravity RG Flows*]{}, JHEP [**0010**]{} (2000) 022, .
M. B.Green and M. Gutperle, [*Effects of D-instantons*]{}. Nucl. Phys. [**B498**]{} (1997) 195, .
M. B. Green, M. Gutperle and P. Vanhove, [*One loop in eleven dimensions*]{}, Phys.Lett. [**B409**]{} (1997) 177, .
C.P. Bachas, P. Bain and M.B. Green, [*Curvature terms in D-brane actions and their M-theory origin*]{}, JHEP [**9905**]{} (1999) 011, .
B. Brinne, A. Fayyazuddin, S. Mukhopadhyay and D. J. Smith, [*Supergravity M5-branes wrapped on Riemann surfaces and their QFT duals*]{}, JHEP [**0012**]{} (2000) 013, .
M. Bianchi and A. Sagnotti, [*Twist symmetry and open string wilson lines*]{}, Nucl. Phys. [**B361**]{} (1991) 519.
E. G. Gimon and J. Polchinski, [*Consistency Conditions for Orientifolds and D-Manifolds*]{}, Phys. Rev. [**D54**]{} (1996) 1667, .
H. Ooguri and C. Vafa, [*Two-Dimensional Black Hole and Singularities of CY Manifolds*]{}, Nucl. Phys. [**B463**]{} (1996) 55, .
R. Gregory, J. A. Harvey and G. Moore, [*Unwinding strings and T-duality of Kaluza-Klein and H-Monopoles*]{}, Adv. Theor. Math. Phys. [**1**]{} (1997) 283, .
A. Karch, D. Lust and D.J. Smith, [*Equivalence of Geometric Engineering and Hanany-Witten via Fractional Branes*]{}, Nucl. Phys. [**B533**]{} (1998) 348, .
A. Dabholkar and J. Park, [*Strings on Orientifolds*]{}, Nucl. Phys. [**B477**]{} (1996) 701, .
J. D. Blum and A. Zaffaroni, [*An Orientifold from F Theory*]{}, Phys. Lett. [**B387**]{} (1996) 71, .
J. D. Blum and K. Intriligator, [*Consistency Conditions for Branes at Orbifold Singularities*]{}, Nucl. Phys. [**B506**]{} (1997) 223, .
E. Eyras and S. Panda, [*Non-BPS Branes in a Type I Orbifold*]{}, JHEP [**0105**]{} (2001) 056, .
N. Quiroz and B. Stefanski Jr, [*Dirichlet Branes on Orientifolds*]{}, .
J. Park and A. M. Uranga, [*A Note on Superconformal N=2 theories and Orientifolds*]{}, Nucl. Phys. [**B542**]{} (1999) 139, .
S. Gukov and A. Kapustin, [*New N=2 superconformal field theories from M/F theory orbifolds*]{}, Nucl. Phys. [**B545**]{} (1999) 283, .
I. Ennes, C. Lozano, S. Naculich and H. Schnitzer, [*Elliptic Models, Type IIB Orientifolds and the AdS/CFT Correspondence*]{}, Nucl. Phys. [**B591**]{} (2000) 195, .
O. Aharony, A. Fayyazuddin and J. Maldacena, [*The Large N Limit of ${\cal N} =2,1 $ Field Theories from Threebranes in F-theory*]{}, JHEP [**9807**]{} (1998) 013, .
B. R. Greene, A. D. Shapere, C. Vafa and S.-T. Yau, [*Stringy Cosmic String and non-compact Calabi-Yau Manifolds*]{}, Nucl. Phys. [**B337**]{} (1990) 1.
M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta and R. Russo, [*Is a classical description of stable non-BPS D-branes possible?*]{}, Nucl. Phys. [**B590**]{} (2000) 471, .
M. Wijnholt and S. Zhukov, [*Inside an Enhançon: Monopoles and Dual Yang-Mills Theory*]{}, .
J. Harvey and J. Liu, [*Magnetic Monopoles in N=4 Supersymmetric Low-Energy Superstring Theory*]{}, Phys. Lett. [**B268**]{} (1991) 40.
S. A. Cherkis and A. Kapustin, [*Nahm Transform For Periodic Monopoles And N=2 Super Yang-Mills Theory*]{}, Commun. Math. Phys. [**218**]{} (2001) 333, .
E. D’Hoker, I. M. Krichever and D. H. Phong, [*The Effective Prepotential of N=2 Supersymmetric $SO(N_c)$ and $Sp(N_c)$ Gauge Theories*]{}, Nucl. Phys. [**B489**]{} (1997) 211, .
A. M. Uranga, [*A New Orientifold of $C^2/Z_N$ and Six-dimensional RG Fixed Points*]{}, Nucl.Phys. [**B577**]{} (2000) 73, .
S.W. Hawking, [*Gravitational Instantons*]{}, Phys. Lett. [**A60**]{} (1977) 81.
G.W. Gibbons and S.W. Hawking, [*Gravitational multi-Instantons*]{}, Phys. Lett. [**B78**]{} (1978) 430.
T. Eguchi, P. B. Gilkey and A. J. Hanson, [*Gravitation, Gauge Theories and Differential Geometry*]{}, Phys. Rept. [**66**]{} (1980) 213.
Gradshteyn and Ryzhik, [*Table of integrals series and products*]{}, Academic Press.
E. Bergshoeff, C.M. Hull and T. Ortin, [*Duality in the Type–II Superstring Effective Action*]{}, Nucl. Phys. [**B451**]{} (1995) 547, .
A. Fayyazuddin and D. J. Smith, [*Localized intersections of M5-branes and four-dimensional superconformal field theories*]{}, JHEP [**9904**]{} (1999) 030, .
A. Fayyazuddin and D. J. Smith, [*Warped AdS near-horizon geometry of completely localized intersections of M5-branes*]{}, JHEP [**0010**]{} (2000) 023, .
E. Witten, [*Solutions Of Four-Dimensional Field Theories Via M Theory*]{}, Nucl. Phys. [**B500**]{} (1997) 3, .
P. Di Vecchia, H. Enger, E. Imeroni and E. Lozano-Tellechea, [*Gauge theories from wrapped and fractional branes*]{}, .
[^1]: An example of such deformation have been given in [@bcz], where the [*vev*]{} are linearly distributed.
[^2]: In this article, we will only write the homogeneous equations. The presence of the D-brane sources for the fields will be imposed at the level of the boundary conditions.
[^3]: In the paper [@wz], it was believed that the Harvey-Liu solution [@harveyliu] was mapped under heterotic on $T^4$/type IIA on $K3$ duality to the D6-brane wrapped on $K3$ configuration of [@jpp] but, actually, it corresponds to the fractional D2-brane case. The wrapped D6-brane configuration is mapped to a superposition of H and KK-monopoles in the heterotic string and its “non-abelian” generalization is not known so far.
[^4]: An alternative approach would be to search for a “true” D5-brane solution wrapped on the 2-cycle. This has been explored in [@gkmw; @bcz]. There, the NS-NS 2-form is zero while the dilaton has a non-trivial profile. Moreover, the solution does not carry any D3-brane charge.
[^5]: This expression is the same as for $N$ standard D3-branes in a flat space. However, one must remember that the $\mathbb{Z}_2$ identification $\psi \sim \psi +2\pi$ divides by two the volume of the surrounding 5-sphere used to compute the (untwisted) D3-brane charge and, therefore, gives only an $N/2$ D3-brane charge.
[^6]: The approximation (\[leadiia\]) corresponds to this symmetric configuration since the bending of the NS5-branes due to the stretched D4-branes appears only when one takes into account the non-homogeneous solution.
|
---
abstract: 'We present a new numerical scheme to solve the transfer of diffuse radiation on three-dimensional mesh grids which is efficient on processors with highly parallel architecture such as recently popular GPUs and CPUs with multi- and many-core architectures. The scheme is based on the ray-tracing method and the computational cost is proportional to $N_{\rm m}^{5/3}$ where $N_{\rm m}$ is the number of mesh grids, and is devised to compute the radiation transfer along each light-ray completely in parallel with appropriate grouping of the light-rays. We find that the performance of our scheme scales well with the number of adopted CPU cores and GPUs, and also that our scheme is nicely parallelized on a multi-node system by adopting the multiple wave front scheme, and the performance scales well with the amount of the computational resources. As numerical tests to validate our scheme and to give a physical criterion for the angular resolution of our ray-tracing scheme, we perform several numerical simulations of the photo-ionization of neutral hydrogen gas by ionizing radiation sources without the “on-the-spot” approximation, in which the transfer of diffuse radiation by radiative recombination is incorporated in a self-consistent manner.'
author:
- 'Satoshi <span style="font-variant:small-caps;">Tanaka</span> Kohji <span style="font-variant:small-caps;">Yoshikawa</span> Takashi <span style="font-variant:small-caps;">Okamoto</span> and Kenji <span style="font-variant:small-caps;">Hasegawa</span>'
title: 'A new ray-tracing scheme for 3D diffuse radiation transfer on highly parallel architectures'
---
Introduction
============
Radiation transfer (RT) has been long recognized as a indispensable ingredient in numerically simulating many astrophysical phenomena including the reionization of intergalactic medium (IGM) in the early universe, radiative feedback during the galaxy formation, and others. So far, varieties of numerical schemes for solving the RT in three dimensions are proposed during the last two decades [@iliev2006], and some of them can be coupled with the hydrodynamic simulations [@iliev2009] thanks to not only the increase of the available computational resources, but also the improvement of numerical algorithms to solve the RT in many astrophysical conditions.
Most of the numerical schemes for RT can be divided into two groups: one is the moment-based schemes which solve the moment equation of the RT equation instead of solving the RT equation directly, and the other is the ray-tracing schemes. As for the moment-based schemes, the important advantage is that the computational costs scale with the number of mesh grids, $N_{\rm m}$ and hence can be easily coupled with hydrodynamic simulations. The flux-limited diffusion (FLD) scheme, which adopts the closure relation valid in the diffusion limit, is the most common among the moment-based schemes, while there are a number of more sophisticated schemes which close the moment equations with the optically thin variable Eddington tensor approximation [@gnedin2001] and the locally evaluated Eddington tensor (the M$_1$ model) [@gonzalez2007; @skinner2013; @kanno2013]. The accuracy and validity of the moment-based schemes are, however, problem-dependent. For example, the FLD scheme has a problem in handling shadows formed behind opaque objects [@gonzalez2007]. While schemes with M$_1$ model are capable of simulating shadows sucessfully, they cannot solve the crossing of multiple beamed lights, where the beamed lights unphysically merge into one beam [@rosdahl2013]. Therefore, the ray-tracing schemes are naturally chosen for solving the RT in situations that we are considering in the studies of galaxy formation and cosmic reionization, in which there exist a number of radiation sources.
In ray-tracing schemes, emission and absorption of radiation are followed along the light-rays that extend through the computational domain. As for the long-characteristics schemes [@abel1999; @sokasian2001] in which light-rays between all radiation sources and all other relevant meshes are considered, the computational cost scales with $N_{\rm m}^{2}$ in general cases and $N_{\rm
m}^{4/3}N_{\rm s}$ when we consider only the RT from point radiating sources, where $N_{\rm s}$ is the number of point sources. On the other hand, for the short-characteristics schemes [@kunasz1988; @stone1992] which are similar to the long-characteristics schemes but integrate the RT equation only along paths connecting nearby mesh grids, the computational cost scales with $N_{\rm m}^{5/3}$ in general and $N_{\rm
m}N_{\rm s}$ for the RT from point sources. Ray-tracing schemes are in principle versatile for any physical settings but computationally much more expensive than the moment-based schemes. Due to such huge computational costs, RT simulations with the ray-tracing schemes have been applied only to static conditions or snapshots of hydrodynamical simulations in a post-process manner in many previous studies.
Some of the ray-tracing schemes are now coupled with hydrodynamical simulations adopting smoothed particle hydrodynamics (SPH) codes [@susa2006; @hasegawa2010; @pawlik2011] and mesh-based codes [@rijkhorst2006; @wise2011], and they can handle the RT and its hydrodynamical feedback in a self-consistent manner. Majority of these radiation hydrodynamics codes, however, consider the transfer of radiation only from point sources and ignore the effect of radiation transfer from spatially extended diffuse sources, such as the recombination radiation emitted from ionized regions and infrared radiation emitted by dust grains, since the computational costs for computing the transfer of diffuse radiation is prohibitively large.
Specifically, in the numerical RT calculations of the hydrogen ionizing radiation, we usually adopt the on-the-spot approximation in which one assumes that the ionizing photons emitted by radiative recombinations in ionized regions are absorbed by neutral atoms in the immediate vicinity of the recombining atoms. However, adopting the on-the-spot approximation can fail to notice the important effects of diffuse recombination radiation in some situations. The roles of ionizing recombination photons in the epoch of cosmic reionization is discussed by a number of works [@ciardi2001; @miralda-escude2003; @dopita2011; @rahmati2013a]. @dopita2011 proposed the recombination photons produced in the fast accretion shocks in the structure formation in the universe as an possible source of ionizing photons responsible for the cosmic reionization, though @wyithe2011 showed that its impact on the cosmic reionization is not very significant. It is also reported that the recombination radiation plays an important role at transition regions between highly ionized and self-shielded regions [@rahmati2013a]. As for the effect of recombination photons on the galaxy-size scales, @inoue2010 showed that the recombination radiation produces the Lyman-‘bump’ feature in the spectral energy distributions of high-$z$ galaxies, and also that the escaping ionizing photons from high-$z$ galaxies are to some extent contributed by the recombination radiation. @rahmati2013b also pointed out that the recombination radiation makes the major contribution to the photo-ionization at regions where the gas is self-shielded from the UV background radiation.
The RT of infrared diffuse radiation emitted by dust grains plays an important roles in the evolution of star-forming galaxies, in which the radiation pressure exerted by multi-scattered infrared photons drives stellar winds. In most of numerical simulations of galaxy formation, however, such momentum transfer is treated only in a phenomenological manner (e.g. Okamoto et al. 2014).
In this paper, we present a new ray-tracing scheme to solve the RT of diffuse radiation from spatially extended radiating sources efficiently on processors with highly parallel architectures such as graphics processing units (GPUs) and multi-core CPUs which are recently popular or available in near future. The basic idea of the scheme is based on the scheme presented by @razoumov2005 and ‘Authentic Radiation Transfer’ (ART) scheme @nakamoto2001b. Generally speaking, development of such numerical schemes with high concurrency is of critical importance because the performance improvement of recent processors are achieved by the increase of the number of processing elements or CPU cores integrated on a single processor chip rather than the improvement of the performance of individual processing elements.
The rest of the paper is organized as follows. Section 2 is devoted to describe the numerical scheme to simulate the radiation transfer. In section 3, we present our implementation of the scheme suitable to highly parallel architectures such as GPUs and CPUs with multi-core architectures. We present the results of numerical test suits of RT of diffuse radiation in Section 4. The computational performance of our implementation is shown in Section 5. Finally, we summarize our results in Section 6.
Methodology
===========
In this section, we describe our ray-tracing scheme of diffuse radiation transfer. Generally, the radiation field can be decomposed into two components. One is the direct incident radiation from point radiation sources, and the other is the diffuse radiation emerged from spatially extended regions. In our implementation, the RT of photons emitted by point radiation sources is computed separately from that of diffuse radiation. Throughout in this paper, we consider the RT of hydrogen ionizing photons emitted by point radiation sources, and recombination photons emerged from the ionized regions as the diffuse radiation. We use the steady state RT equation for a given frequency $\nu$: $$\frac{dI_{\nu}}{d\tau_{\nu}} = -I_{\nu} + \mathcal{S}_{\nu} ,
\label{eq:rad_tr1}$$ where $I_{\nu},\tau_{\nu}$ and $\mathcal{S}_{\nu}$ are the specific intensity, the optical depth and the source function, respectively. The source function is given by $\mathcal{S}_{\nu} =
\varepsilon_{\nu}/\kappa_{\nu},$ where $\kappa_{\nu}$ and $\varepsilon_{\nu}$ are the absorption and emission coefficients, respectively. The formal solution of this equation is given by $$I_{\nu}(\tau_{\nu}) = I_{\nu}(0)\,e^{-\tau_{\nu}} + \int^{\tau_{\nu}}_{0} \mathcal{S}_{\nu}(\tau'_{\nu}) e^{-(\tau_{\nu}-\tau'_{\nu})}d\tau'_{\nu} ,
\label{eq:rad_tr2}$$ where $\tau'_{\nu}$ is the optical depth at a position along the ray. When we adopt the “on-the-spot” approximation in which recombination photons emitted in ionized regions are assumed to be absorbed where they are emitted, we neglect the source function, $\mathcal{S}_\nu$, and the formal solution is simply reduced to $$I_{\nu}(\tau_{\nu}) = I_{\nu}(0)\,e^{-\tau_{\nu}} .
\label{eq:on_the_spot}$$
RT from point radiation sources\[sub:rt\_point\]
------------------------------------------------
To solve the RT from point radiation sources, we compute the optical depth between each pair of a point radiation source and a target mesh grid, i.e. an end point of each light-ray (see Figure \[fig:long\]). Instead of solving equation (\[eq:on\_the\_spot\]), we compute the radiation flux density at the target mesh grid as $$f_\alpha(\nu) = \frac{L_\alpha(\nu)}{4\pi r_\alpha^2}\,\exp\left[-\tau_\alpha(\nu)\right],$$ where $L_\alpha(\nu)$ is the intrinsic luminosity of the $\alpha$-th point radiation source, and $r_\alpha$ and $\tau_\alpha(\nu)$ are the distance and the optical depth between the point radiation source and the target mesh grid, respectively. Then, the photo-ionization and photo-heating rates of the $i$-th species contributed by the $\alpha$-th point radiation source are computed by $$\Gamma^\alpha_{i,\gamma} = \int^\infty_{\nu_i}
\frac{f_\alpha(\nu)}{h\nu}\sigma_i(\nu)\,d\nu,$$ and $$\mathcal{H}^\alpha_{i,\gamma} = \int^\infty_{\nu_i} \frac{f_\alpha(\nu)}{h\nu}(h\nu-h\nu_i)\sigma_i(\nu)\,d\nu$$ respectively, where $\sigma_i(\nu)$ and $\nu_i$ are the ionization cross section and the threshold frequency of the $i$-th species, respectively. In the test simulations desribed in this paper, we compute these photo-ionization and photo-heating rates in a photon-conserving manner [@abel1999] as described in appendix \[app:photon\_conserving\].
![Schematic illustration of the ray-tracing method for the radiation emitted by a point radiation source in the two-dimensional mesh grids.\[fig:long\]](f1.eps){width="5cm"}
For a single point radiation source, the number of rays to be calculated is $N_{\rm m}$, and the number of mesh grids traveled by a single light-ray is in the order of $N^{1/3}_{\rm m}$. Thus, the computational cost for a single point radiation source is proportional to $N^{4/3}_{\rm m}$. Therefore, the total computational cost scales as $N_{\rm m}^{4/3} N_{\rm s}$, where $N_{\rm s}$ is the number of point radiation sources. For a large number of point radiation sources, we can mitigate the computational costs by adopting more sophisticated scheme such as the ARGOT scheme [@okamoto2012] in which a distant group of point radiation sources is treated as a bright point source located at the luminosity center with a luminosity summed up for all the sources in the group to effectively reduce the number of radiation sources and hence the computational cost is proportional to $N^{4/3}_{\rm m} \log
N_{\rm s}$.
RT of the diffuse radiation \[sub:rt\_rec\]
-------------------------------------------
We solve the equation (\[eq:rad\_tr2\]) to compute the RT of the diffuse radiation. The numerical scheme we adopt in this work is based on the method developed by @razoumov2005 and ART scheme [@nakamoto2001b; @iliev2006], which is reported to have little numerical diffusion in the searchlight beam test like the long characteristics method although its computational cost is proportional to $N_{\rm m}^{5/3}$ similarly to that of the short characteristic method [@nakamoto2001b]. In this scheme, we solve the equation (\[eq:rad\_tr2\]) along equally spaced parallel rays as schematically shown in Figure \[fig:parallel\_ray\].
For a given incoming radiation intensity $I^{\rm in}_\nu$ along a direction $\hat{\mathbf{n}}$, the outgoing radiation intensity $I^{\rm
out}_\nu$ after getting through a path length $\Delta L$ of a single mesh is computed by integrating equation (\[eq:rad\_tr2\]) as $$I^{\rm out}_\nu (\hat{\mathbf{n}}) = I^{\rm in}_\nu(\hat{\mathbf{n}})\,e^{-\Delta\tau_\nu} + \mathcal{S}_\nu
(1-e^{-\Delta\tau_\nu}),
\label{eq:rt}$$ where $\Delta\tau_\nu$ is the optical depth of the path length $\Delta
L$ (i.e. $\Delta\tau_\nu = \kappa_\nu \Delta L$), and $\mathcal{S}_\nu$ and $\kappa_\nu$ are the source function and the absorption coefficient of the mesh grid, respectively.
The intensity of the incoming radiation averaged over the path length $\Delta L$ across a single mesh grid can be calculated as $$\bar{I}^{\rm in}_\nu(\hat{\mathbf{n}}) = \frac{1}{\Delta L}\int_0^{\Delta L} I^{\rm
in}_\nu (\hat{\mathbf{n}})e^{-\kappa_\nu l}\,dl = I_\nu^{\rm in}(\hat{\mathbf{n}})\frac{1-e^{-\Delta
\tau_\nu}}{\Delta \tau_\nu}.$$ In addition to this, we have a contribution to the radiation intensity from the source function which we set constant in each mesh grid, and the total intensity averaged over the path length is given by $$\bar{I}_\nu(\hat{\mathbf{n}}) = \bar{I}^{\rm in}_\nu(\hat{\mathbf{n}}) + \mathcal{S}_\nu = I_\nu^{\rm in}(\hat{\mathbf{n}}_i)\frac{1-e^{-\Delta
\tau_\nu}}{\Delta \tau_\nu} + \mathcal{S}_\nu$$ For those mesh grids through which multiple parallel light-rays pass, the averaged intensity can be given by $$\bar{I}_\nu^{\rm ave}(\hat{\mathbf{n}}) =
\frac{\sum_j \Delta \tau_{\nu,j} \bar{I}_{\nu,j}(\hat{\mathbf{n}})}{\sum_j \Delta
\tau_{\nu,j}}
= \bar{I}_{\nu}^{\rm ave, in}(\hat{\mathbf{n}}) + \mathcal{S}_\nu,
\label{eq:path_averaged_intensity}$$ where $\bar{I}_{\nu,i}$ and $\Delta \tau_{\nu, i}$ are the intensity averaged over the $i$-th light-ray and the optical depth of $i$-th light-ray in the mesh grids, respectively, $\bar{I}^{\rm ave, in}_\nu$ is a contribution from the incoming radiation given by $$\bar{I}^{\rm ave, in}_\nu(\hat{\mathbf{n}}) = \frac{\sum_j \Delta \tau_{\nu,j}
\bar{I}^{\rm in}_{\nu,j}(\hat{\mathbf{n}})}{\sum_j \Delta
\tau_{\nu,j}},
\label{eq:averaged_intensity}$$ and the summation is over all the parallel light-rays in the same mesh grid. Then, the mean intensity can be computed by averaging $\bar{I}^{\rm ave}_\nu$ described above over all the directions as, $$J_\nu = \frac{1}{N_{\rm d}}\sum_{i=1}^{N_{\rm d}} \bar{I}_{\nu}^{\rm
ave}(\hat{\mathbf{n}}_i) = J^{\rm in}_\nu + \mathcal{S}_\nu,
\label{eq:mean_intensity}$$ where $\hat{\mathbf{n}}_i$ describes a vector toward the $i$-th direction and $N_{\rm d}$ is the number of directions of light-rays to be considered, $\bar{I}_{\nu}^{\rm ave}(\hat{\mathbf{n}}_i)$ is the averaged intensity along the $i$-th direction calculated with equation (\[eq:path\_averaged\_intensity\]), and $J^{\rm in}_\nu$ is given by $$J^{\rm in}_\nu=\frac{1}{N_{\rm d}}\sum_{i=1}^{N_{\rm d}} \bar{I}^{\rm
ave, in}_{\nu}(\hat{\mathbf{n}}_i).$$ Then, the photo-ionization and photo-heating rates of the $i$-th species contributed by the diffuse radiation in each mesh grid can be computed as $$\Gamma_{i,\gamma}^{\rm diff} = 4\pi\int_{\nu_i}^\infty \frac{J_\nu}{h\nu}\sigma_i(\nu)\,d\nu$$ and $$\mathcal{H}_i^{\rm diff} = 4\pi \int_{\nu_i}^\infty
\frac{J_\nu}{h\nu}(h\nu-h\nu_i) \sigma_i(\nu)\,d\nu$$
As for the recombination radiation of ionized hydrogen (HII) regions, the number of recombination photons to the ground state per unit time per unit volume, $\dot{N}^{\rm rec}$, can be expressed in terms of the emissivity coefficient $\varepsilon_\nu$ as $$% \dot{N}^{\rm rec}=4\pi
% \int_{\nu_i}^{\nu_i+\Delta\nu}\frac{\varepsilon_\nu}{h\nu}\,d\nu =
\dot{N}^{\rm rec}=4\pi \int_{\nu_0}^{\infty}\frac{\varepsilon_\nu}{h\nu}\,d\nu =
[\alpha_{\rm A}(T)-\alpha_{\rm B}(T)]n_{\rm e}n_{\rm HII},$$ where $\nu_0$ is the Lyman limit frequency, $\alpha_{\rm A}(T)$ and $\alpha_{\rm B}(T)$ are the recombination rates of HII as functions of temperature $T$ in the case-A and case-B approximations, respectively, and $n_{\rm e}$ and $n_{\rm HII}$ are the number densities of the electrons and HII, respectively. In this work, we adopt a rectangular functional form of $\varepsilon_\nu/(h\nu)$ as $$\frac{\varepsilon_\nu}{h\nu} = \left\{
\begin{array}{ll}
\displaystyle \frac{\Delta\alpha(T)n_{\rm
e}n_{\rm HII}}{4\pi \Delta\nu_{\rm th}} &
(\nu_0\le\nu\le\nu_0+\Delta\nu_{\rm th}) \\
0 & (\mbox{otherwise}),
\end{array}
\right.$$ where $\Delta\alpha(T)=\alpha_{\rm A}(T)-\alpha_{\rm B}(T)$ and $\Delta\nu_{\rm th}$ is the frequency width of the recombination radiation and given by $\Delta\nu_{\rm th} = k_{\rm B}T/h$. Thus, the source function is given by $$\mathcal{S}_\nu = \frac{\varepsilon_\nu}{\kappa_\nu} =
\left\{
\begin{array}{ll}
\displaystyle \frac{\Delta\alpha(T)n_{\rm e}n_{\rm HII}h\nu}{4\pi n_{\rm
HI}\sigma_{\rm HI}(\nu) \Delta\nu_{\rm th}} &
(\nu_0\le\nu\le\nu_0+\Delta\nu_{\rm th} )\\
0& ({\rm otherwise}).
\end{array}
\right.$$ This spectral shape is the same as adopted in @kitayama2004 and @hasegawa2010, the results of which are compared with our results to verify the validity of our scheme. Note that for the typical temperature of HII regions, $T=10^4\,\,{\rm K}$, we have $\Delta\nu_{\rm
th}\ll \nu_0$. Calculations of the mean intensity based on equations (\[eq:rt\]) to (\[eq:mean\_intensity\]) are done in a monochromatic manner at the Lyman limit frequency $\nu_0$. In computing photo-ionization and photo-heating rates, we assume that the mean radiation intensity $J^{\rm in}_\nu$ has a rectangular functional form as $$J^{\rm in}_\nu = \left\{
\begin{array}{ll}
\displaystyle \mathcal{J}^{\rm in} &
(\nu_0\le\nu\le\nu_0+\Delta\nu_{\rm th}) \\
0 & (\mbox{otherwise}).
\end{array}
\right.$$ Therefore, the photo-ionization and photo-heating rates of neutral hydrogen can be rewritten as $$\Gamma^{\rm diff}_{\rm HI,\gamma} = 4\pi\mathcal{J}^{\rm
in}\int_{\nu_0}^{\nu_0+\Delta\nu_{\rm th}}\frac{\sigma_{\rm HI}(\nu)}{h\nu} d\nu + \frac{\Delta\alpha(T)n_{\rm
e}n_{\rm HII}}{n_{\rm HI}},
\label{eq:gamma_diff}$$ and $$\mathcal{H}^{\rm diff}_{\rm HI,\gamma} = 4\pi\mathcal{J}^{\rm
in}\int_{\nu_0}^{\nu_0+\Delta\nu_{\rm th}}\left(1-\frac{\nu_0}{\nu}\right)\,\sigma_{\rm HI}(\nu)\,d\nu +
\frac{\Delta\alpha(T)n_{\rm e}n_{\rm HII}}{2n_{\rm HI}}h\Delta\nu_{\rm
th},
\label{eq:heat_diff}$$ respectively. In the test simulations described in section \[sec:test\], we fix the frequency width $\Delta\nu_{\rm th}$ by assuming temperature of HII regions to be $10^4$ K, and the integrals in equations (\[eq:gamma\_diff\]) and (\[eq:heat\_diff\]) can be estimated prior to the simulations. For the transfer of diffuse radiation with more general spectral form, we can easily extend our method described above by adopting a nonparametric functional form of radiation spectra as $$I_{\nu}=\sum_i \mathcal{I}^i \Pi(\nu-\nu_i,\Delta\nu),$$ where $\Pi(x,y)$ is the rectangular function given by $$\Pi(x,y) = \left\{\begin{array}{ll}
1 & -y/2 \le x \le y/2\\
0 & {\rm otherwise} \\
\end{array}\right.,$$ and $\nu_i$ is the central frequency of the $i$-th frequency bin.
The number of light-rays parallel to a specific direction is proportional to $N_{\rm m}^{2/3}$, and the number of mesh grids traversed by a single light-ray is in the order of $N_{\rm
m}^{1/3}$. Therefore, the total computational cost is proportional to $N_{\rm m}N_{\rm d}$.
![Schematic illustration of the ray-tracing scheme for the diffuse radiation in the two-dimensional mesh grid. For a given direction, equally-spaced parallel light-rays are cast from boundaries of the simulation volume and travel to the other boundaries. Note that gray mesh grids are traversed by multiple parallel light-rays, while the subsets of light-rays depicted by blue or red get through them only once. []{data-label="fig:parallel_ray"}](f2.eps){width="5cm"}
Angular resolution for RT of the diffuse radiation {#sub:direction}
--------------------------------------------------
The number of the directions of light-rays, $N_{\rm d}$, determines angular resolution of the RT of the diffuse radiation. In order to guarantee that light-rays from a mesh grid on a face of the simulation box reach all the mesh grids on the other faces, $N_{\rm d}$ should be in the order of $N_{\rm m}^{2/3}$. In the case that the mean free path of the diffuse photons is sufficiently shorter the simulation box size, however, such a large $N_{\rm d}$ is redundant because only a small fraction of diffuse photons reach the other faces, and we can reduce the total computational cost by decreasing the number of directions, $N_{\rm
d}$, while keeping the reasonable accuracy of the diffuse RT. Thus, the number of directions should be flexibly changed depending on the physical state.
To achieve this, we use the HEALPix (Hierarchical Equal Area isoLatitude Pixelization) software package [@gorski2005] to set up the directions of the light-rays. The HEALPix is suitable to our purposes in the sense that each direction corresponds to exactly the same solid angle and that the directions are nearly uniformly sampled. Furthermore, it can provide a set of directions with these properties in arbitrary resolutions, each of which contains $12N_{\rm side}^2$ directions, where $N_{\rm side}$ is an angular resolution parameter. Since it is larger than the number of mesh grids on six faces of a cube with a side length of $N_{\rm side}$ mesh spacings, $6N_{\rm side}^2$, it is expected that a set of light-rays originated from a single point with directions generated by the HEALPix with an angular resolution parameter of $N_{\rm
side}$ get through all the mesh grids within a cube centered by the point with a side length of $N_{\rm side}$ mesh spacings. Thus, the optimal number of directions should be chosen so that the mean free path of the recombination photons is sufficiently shorter than $N_{\rm
side}\Delta H$, where $\Delta H$ is the mesh spacing.
Chemical reactions and radiative heating and cooling
----------------------------------------------------
With photo-ionization and photo-heating rates computed with the prescription described above, time evolutions of chemical compositions and thermal states of gas are computed in the same manner as adopted in @okamoto2012. Details of the numerical schemes are briefly described in appendices \[app:reaction\],\[app:heating\_cooling\] and \[app:timestep\].
The chemical reaction rates and radiative cooling rates adopted in this paper are identical to those adopted in @okamoto2012, and the literatures from which we adopt these rates are summarized in Table \[tab:rate\].
physical process literature
------------------ ---------------
RR (case-A) (1), (1), (2)
RR (case-B) (3), (3), (3)
CIR (7), (7), (1)
RCR (case-A) (2), (2), (2)
RCR (case-B) (3), (5), (3)
CICR (2), (2), (2)
CECR (2), (2), (2)
BCR (4)
CCR (6)
CS (8), (8), (8)
: Rates of chemical reactions and radiative cooling processes adopted in this paper. Reference for radiative recombination rates (RR) of HII, HeII and HeIII in the case-A and case-B approximation; collisional ionization rates (CIR) of HI, HeI, and HeII; recombination cooling rates (RCR) of HII, HeII and HeIII in the case-A and case-B approximation; collisional ionization cooling rates (CICR) of HI, HeI and HeII; collisional excitation cooling rates (CECR) of HI, HeI and HeII; bremsstrahlung cooling rate; inverse Compton cooling rate (CCR); photoionization cross sections (CS) of HI, HeI and HeII. \[tab:rate\]
\(1) @abel1997; (2) @cen1992; (3) @hui1997; (4) @hummer1994; (5) @hummer1998; (6) @ikeuchi1986; (7) @janev1987; (8) @agnagn;
Implementation on Highly Parallel Architectures
===============================================
In this section, we describe the details of the implementation of the RT calculation of the diffuse radiation which performs effectively on recently popular processors with highly parallel architecture, such as GPUs, multi-core CPUs, and many-core processors. Throughout this paper, we present the results using the implementation with the `OpenMP` and `CUDA` technologies. The former is supported by most of the multi-core processors, and the many-core processors such as the Intel Xeon-Phi processor, while the latter is the parallel programming platform for GPUs by NVIDIA.
In the implementation on GPUs with the `CUDA` platform, the fluid dynamical and chemical data in all the mesh grids are transferred from the memory attached to CPUs to those of GPUs prior to the RT calculations. After the RT calculations, ionization rates and heating rates in all the mesh grids computed on GPUs are sent back to the CPU memory.
Ray Grouping {#ss:ray_grounping}
------------
In the calculations of the transfer of the diffuse radiation described in the previous section, many parallel light-rays travel from boundaries of the simulation volume until they reach the other boundaries. On processors with highly parallel architecture, a straightforward implementation is to assign a single thread to compute the RT along each light-ray and calculate the RT along multiple light-rays in parallel. Such a simple implementation, however, does not work because some mesh grids are traversed by multiple parallel light-rays (see gray mesh grids in Figure \[fig:parallel\_ray\]), and in computing equation (\[eq:path\_averaged\_intensity\]), multiple computational threads write data to the identical memory addresses. Thus, equation (\[eq:path\_averaged\_intensity\]) has to be computed not in parallel but in a exclusive manner using the “atomic operations”. The use of the atomic operations, however, significantly degrade the parallel efficiency and computational performance in many architectures.
To avoid such use of atomic operations and the deterioration of the parallel efficiency, we split the parallel light-rays into several groups so that parallel light-rays in each group do not traverse any mesh grid more than once. For example, in two-dimensional mesh grids in Figure \[fig:parallel\_ray\], parallel light-rays are split into two groups each of which are depicted by blue and red arrows. One can see that light-rays in each groups do not intersect any mesh grids more than once. We can extend this technique to the three-dimensional mesh grids by splitting the parallel light-rays into four groups, where the light-rays in each group are cast from the two-dimensionally interleaved mesh grids as depicted by the same color in Figure \[fig:grouping\].
![Schematic illustration of light-ray grouping for the three-dimensional mesh grids. Light-rays in each group start from boundary faces of mesh grids painted with the same color. Only the light-rays in one group are shown in this figure. \[fig:grouping\]](f3.eps){width="6.9cm"}
Efficient Use of Multiple External Accelerators
-----------------------------------------------
Many recent supercomputers are equipped with multiple external accelerators such as GPUs on a single computational node, each of which has an independent memory space. To attain the maximum benefit of the multiple accelerators, we decompose calculations of the diffuse radiation transfer according to the directions of the light-rays, and assign the decomposed RT calculation to the multiple accelerators. After carrying out the RT calculation for the assigned set of directions, and computing the mean intensity with equation (\[eq:mean\_intensity\]) averaged over the partial set of directions on each external accelerator, the results are transferred to the memory on the hosting nodes. Then, we obtain the mean intensity averaged over all directions.
Node Parallelization
--------------------
In addition to the thread parallelization within processors, we implement the inter-node parallelization using the Message Passing Interface (MPI). In the inter-node parallelization, the simulation box is evenly decomposed into smaller rectangular blocks with equal volumes along the Cartesian coordinate.
For the inter-node parallelization of the calculations of the diffuse radiation transfer, we adopt the multiple wave front (MWF) scheme developed by @nakamoto2001, in which light-ray directions are classified into eight groups according to signs of their three direction cosines, and for each group of light-ray directions, the RT calculations along each direction are carried out in parallel on a “wave front” in the node space, while the RT for different directions are computed on the other wave fronts simultaneously. By transferring the radiation intensities at the boundaries from upstream nodes to downstream ones, one can sequentially compute the RT of diffuse radiation along all the directions in each group of light-ray directions. See @nakamoto2001 for more detailed description of MWF scheme.
Test Simulation {#sec:test}
===============
In this section, we present a series of test simulations to validate our RT code. All the test simulations are carried out with $128^3$ mesh grids and the angular resolution parameter of $N_{\rm side}=8$ unless otherwise stated.
Test-1 : HII region expansion
-----------------------------
The first test is the simple problem of a HII region expansion in a static homogeneous gas which consists of only hydrogen around a single ionizing source. We adopt the same initial condition as that of Test-2 in Cosmological Radiative Transfer Codes Comparison Project I [@iliev2006], where the hydrogen number density is $n_{\rm
H}=10^{-3}\,{\rm cm}^{-3}$ and the initial gas temperature is $T=100\,{\rm K}$. The ionizing source emits the blackbody radiation with an effective temperature of $10^5\,{\rm K}$, and $5\times 10^{48}$ ionizing photons per second and located at a corner of simulation box with a side length of 6.6 kpc. In this initial condition, the recombination time is $t_{\rm rec}=122.4{\rm\, Myr}$ and the Strömgren radius is estimated to be 5.4 kpc. Figure \[fig:test-1\] shows the radial profiles of ionization fraction and gas temperature at $t=30{\rm Myr}$, $100{\rm Myr}$ and $500{\rm
Myr}$. The solid lines with and without circles indicate the results with and without the on-the-spot approximation (OTSA) , respectively. In the calculation with the effect of recombination radiation, the ionized regions are more extended than those computed with the on-the-spot approximation , especially at later stages ($t=100{\rm Myr}$ and $500{\rm Myr}$) because of the additional ionization of hydrogen by the recombination photons.
To verify the validity of our scheme for the transfer of diffuse recombination radiation, we compare our results with the ones obtained with the one-dimensional spherically symmetric RT code by @kitayama2004, which also incorporates the transfer of recombination photons emitted by the ionized hydrogen using the impact-parameter method. We find that the one-dimensional results with the effect of recombination radiation denoted by dashed lines show a good agreement with our three-dimensional results, indicating that our treatment of diffuse radiation transfer is consistent with that of well-established impact-parameter method.
![Test-1: Radial profiles of neutral and ionized fractions of hydrogen and gas temperature at $t=30{\rm Myr}$, $100{\rm Myr}$ and $500{\rm Myr}$. Solid lines with and without circles indicates the results with and without the on-the-spot approximation (OTSA), respectively. Dashed lines show the results obtained with one-dimensional spherically symmetric code without the OTSA presented in @kitayama2004. \[fig:test-1\]](f4.eps){width="15cm"}
Test-2 : Shadow by a dense clump {#sub:angular_resolution}
--------------------------------
In the second test, we compute the RT from point radiation source in the presence of a dense gas clump. A point radiation source is located at the center of the simulation box with the same side length as the Test-1 (6.6kpc), and surrounded by the ambient uniform gas with the same hydrogen number density and temperature as the Test-1 ($n_{\rm
H}=10^{-3}\,{\rm cm}^{-3}$ and $T=100\,{\rm K}$, respectively). In addition, we set up a spherical dense gas clump with a radius of 0.56kpc centered at 0.8 kpc apart from the point radiation source along the $x$-direction. We set the density of the dense clump to 200 times higher than that of the ambient gas, and the temperature is set to 100K. The spectrum and luminosity of the point radiation source is the same as that in Test-1.
In Figure \[fig:test-2-map\], maps of the neutral fraction of hydrogen in the mid-plane of the simulation volume at $t=30\,{\rm Myr}$, 100Myr and 500Myr are shown. One can see that the ionizing photons are strongly absorbed by the dense gas clump and conical shadows are created behind the gas clump in the both runs with and without the effect of recombination radiation. In the run without the on-the-spot approximation (upper panels of Figure \[fig:test-2-map\]), the recombination photons emitted by the ionized gas gradually ionize the neutral gas behind the dense gas clump. On the other hand, in the run with the on-the-spot approximation, the boundaries of neutral and ionized regions are kept distinct because of the lack of recombination photons.
This test is identical to Test-6 in @hasegawa2010 calculated with the START code. In the START code, the RT is solved with a ray-tracing scheme based on the SPH technique, and the transfer of diffuse recombination radiation can be handled by allowing each SPH paricle to radiate recombination photons. Figure \[fig:test-2-section\] shows profiles of gas temperature and hydrogen neutral fraction along the lines across the conical shadow shown in Figures \[fig:test-2-map\] and \[fig:test-2-t\] as well as the results obtained with the START code. One can see that both the results with and without the OTSA are in fairly good agreement with each other, which supports the validity of our scheme for diffuse radiation transfer.
![Test-2: Maps of neutral fraction of hydrogen in the mid-plane of the simulation box at $t=30{\,\rm Myr}$, $100{\,\rm Myr}$ and $500{\,\rm Myr}$. The lower and upper panels show the results with and without the on-the-spot approximation (OTSA), respectively. Dashed vertical lines indicate the location along which the profiles of temperature and neutral fractions are presented in Figure \[fig:test-2-section\]. \[fig:test-2-map\]](f5.eps){width="10cm"}
![Test-2: Same as Figure \[fig:test-2-map\] but shows the gas temperature maps in the mid-plane of the simulation box. Dashed vertical lines indicate the location along which the profiles of temperature and neutral fractions are presented in Figure \[fig:test-2-section\].\[fig:test-2-t\]](f6.eps){width="10cm"}
![Profiles of temperatures and hydrogen neutral fractions along the dashed lines shown in Figures \[fig:test-2-map\] and \[fig:test-2-t\] with and without the OTSA. Results in @hasegawa2010 are also shown for comparison.\[fig:test-2-section\]](f7.eps){width="10cm"}
In this test, we also perform runs with various angular resolution parameter $N_{\rm side}$ in the RT calculation of diffuse radiation to see the effect of angular resolution. Figure \[fig:test-2-resolution\] shows maps of neutral fraction of hydrogen in the mid-plane of the simulation box at $t=30\,{\rm Myr}$ with angular resolution parameter $N_{\rm side}$ of 16, 4 and 1. The results with $N_{\rm side}=16$ and $4$ are in good agreement with one with $N_{\rm side}=8$ in Figure \[fig:test-2-map\], indicating that the angular resolution with $N_{\rm side}=4$ is sufficient for the current RT calculations. The results with $N_{\rm side}=1$, however, have spurious features in the map of neutral fraction. These numerical artifacts can be ascribed to the low angular resolution of light-rays by the comparison of the mean free path of the diffuse recombination photons and $N_{\rm side}\Delta H$. As described in § \[sub:angular\_resolution\], the mean free path of the diffuse photons should be sufficiently smaller than $N_{\rm side}\Delta H$ to compute the RT of diffuse photons accurately. For the recombination photons emitted by ionized hydrogens in the current setup, the mean free path in the neutral ambient gas is estimated as $$\lambda_{\rm mfp} = \frac{1}{n_{\rm HI}\sigma_{\rm HI}(\nu_0)}=51.4
\left(\frac{n_{\rm HI}}{10^{-3} {\rm cm}^{-3}}\right)^{-1} {\rm pc},$$ and the mesh spacing is $\Delta H=6.6{\,\rm kpc}/128=51.5{\,\rm pc}$. Thus, it is quite natural to have strong numerical artifacts in the results with $N_{\rm side}=1$, because the mean free path is almost equal to $N_{\rm side}\Delta H$, and the condition for the accurate RT calculation ($\lambda_{\rm mfp} \ll N_{\rm side}\Delta H$) is not satisfied.
![Test-2: Maps of neutral fraction of hydrogen in the mid-plane of the simulation box at $t=30{\,\rm Myr}$ for different angular resolution parameter, $N_{\rm side}=16$, 4 and 1 from left to right. \[fig:test-2-resolution\]](f8.eps){width="10cm"}
Test-3 : Ionization front trapping and shadowing by a dense clump
-----------------------------------------------------------------
The third test computes the transfer of ionizing radiation incident to a face of the rectangular simulation box and the propagation of ionized region into a spherical dense clump. This test is indentical to the Test-3 in @iliev2006. The size of the simuation box is $6.6$ kpc, and hydrogen number density and initial temperature are set to $n_{\rm
H}=2\times 10^{-4}$ cm$^{-3}$ and $T=8000$ K, except that a spherical dense clump with a radius of 0.8 kpc located at 1.7 kpc apart from the center of the simulation volume has a uniform hydrogen number density of $n_{\rm H,c}=200n_{\rm H}=0.04$ cm$^{-3}$ and a temperature of $T_{\rm
c}=40$ K. The ionizing radiation has the blackbody spectrum with a tempearature of $T=10^5 {\rm K}$ and constant ionizing photon flux of $F=10^6$ s$^{-1}$ cm$^{-2}$ at a boundary of the simulation box.
![Test-3: Maps of neutral fraction of hydrogen in the mid-plane of the simulation box at $t=1$ Myr, 3 Myr and 15 Myr. The lower and upper panels show the results with and without the on-the-spot approximation, respectively.\[fig:test-3-fmap\]](f9.eps){width="10cm"}
![Test-3: Same as Figure \[fig:test-3-fmap\] but shows the gas temperature maps in the mid-plane of the simulation box.\[fig:test-3-tmap\]](f10.eps){width="10cm"}
![Test-3: Profiles of hydrogen neutral and ionized fractions and gas temperature along the axis of symmetry at $t=1$ Myr, 3 Myr and 15 Myr. Solid lines with and without circles indicate the results with and without the on-the-spot approximation, respectively. Dotted lines shows the results with RSPH code [@susa2006] presented in @iliev2006 with the on-the-spot approxiamtion. \[fig:test-3-prof\]](f11.eps){width="12cm"}
Figures \[fig:test-3-fmap\] and \[fig:test-3-tmap\] shows the maps of hydrogen neutral fraction and gas temperature in the mid-plane of the simulation volume at $t=1$ Myr, 3 Myr and 15 Myr from left to right, where the ionizing photons enter from the left boundary of the figures. We show the results with and without the on-the-spot approximation in the lower and upper panels, respectively.
At $t=1$ Myr, the ionization front enters the spherical clump and a cylindrical shadow is formed behind the clump. At $t=3$ Myr and 15 Myr, the spherical clump is slightly ionized and the boundary of the shadow is ionized and photo-heated by the hard photons which penetrate the edge of the clump. These overall ionization and temperature structures with the on-the-spot approximation are consistent with the ones presented in @iliev2006. The effect of the recombination radiation is clearly seen in the results at 15 Myr, in which the cylindrical shadow is significantly ionized and heated by the recombination photons emitted at the ambient ionized region.
Figure \[fig:test-3-prof\] shows the profiles of ionized fraction and gas temperature along the axis of symmetry at $t=1$ Myr, 3 Myr and 15 Myr, where we also plot the results of the `RSPH` code [@susa2006] which are computed with the on-the-spot approximation and presented in @iliev2006. Our results with the on-the-spot approximation are consistent with the ones computed with the `RSPH` code in @iliev2006. The effect of the recombination radiation is siginificant at $t=3$ Myr and 15 Myr in the ionized fraction profiles, in which the recombination photons accelerate the propagation of the ionization front in the run without the on-the-spot approximation.
Performance
===========
In this section, we show the performance of our RT calculations of diffuse radiation. The code for the transfer of diffuse radiation is designed so that it can be run both on multi-core CPUs and GPUs produced by NVIDIA. The performance is measured on the HA-PACS system installed in Center for Computational Sciences, University of Tsukuba. Each computational node of the HA-PACS system consists of two sockets of 2.6 GHz Intel Xeon processor E5-2670 with eight cores based on the Sandy-Bridge microarchitecture and four GPU boards of NVIDIA Tesla M2090, each of which is connected to the CPU sockets through PCI Express Gen2 $\times$ 16 link. Thus, a single computational node provides 2.99 Tflops (0.33 Tflops by CPUs and 2.66 Tflops by GPUs) of computing capability in double precision.
The upper panel of Figure \[fig:time\_single\] shows wallclock time for a iteration of the diffuse RT calculation on a single node with various numbers of CPU cores and GPU boards. The wallclock times are measured for $N_{\rm m}=64^3$, $128^3$ and $256^3$. The angular resolution parameter $N_{\rm side}$ is set to $N_{\rm side} = N_{\rm m}^{1/3}/16$ so that $N_{\rm side}\Delta H$ is kept constant. Note that the wallclock times are nearly proportional to $N_{\rm m}^{5/3}$ as theoretically expected. The lower panel of Figure \[fig:time\_single\] shows the performance gain of the diffuse RT calculation with multiple CPU cores and GPU boards relative to the performance with a single CPU core and a single GPU board, respectively. Use of the multiple CPU cores and multiple GPU boards provides the efficient performance gains nearly proportional to the adopted numbers of CPU cores and GPU boards for $N_{\rm m}=128^3$ and $256^3$ except for the fact that those with 16 CPU cores (2 CPU sockets) is not very impressive even for $N_{\rm m}=256^3$ because of the relatively slow memory access across the CPU sockets. On the other hand, the performance gain for $N_{\rm m}=64^3$ is somewhat degraded because of the overheads for invoking the multiple threads and communication overhead for data exchange between CPUs and GPUs. The performance with the aid of four GPU boards is nearly 7 times better than that with 16 CPU cores for $256^3$ mesh grids, while it is only 3.5 times better for $64^3$ mesh grids due to the communication overhead between CPUs and GPUs.
![Wallclock times of diffuse RT calculation with various numbers of CPU cores and GPU boards for $N_{\rm m}=64^3$, $128^3$ and $256^3$ are shown in the upper panel. A dotted line indicate the dependence of computational cost on a number of mesh grids, $\propto N_{\rm
m}^{5/3}$. In the lower panel, we present the performance gains of diffuse RT calculation with multiple CPU cores and GPU boards relative to the performance with a single CPU core and GPU board, respectively. Horizontal dotted lines indicates the performance gains of 2, 4, 8 and 16 from bottom to top. \[fig:time\_single\]](f12.eps){width="7cm"}
We compare the performance of our diffuse RT calculations with and without the ray grouping technique on GPUs. In the implementation without the ray grouping technique, we utilize the atomic operation provided by the `CUDA` programming platform in computing the averaged radiation intensity (equation (\[eq:averaged\_intensity\])). Figure \[fig:time\_atomic\] shows the performance gains obtained by the use of the ray grouping technique, where the individual performance is measured with a single computational node and four GPUs. One can see that the use of the ray grouping technique significantly improves the performance of diffuse RT more than by a factor of two irrespective of the number of mesh grids.
![Performance gains obtained by the use of ray grouping for $N_{\rm m}=64^3$, $128^3$ and $256^3$ on GPUs. The performances are measured on a single computational node and four GPUs. \[fig:time\_atomic\]](f13.eps){width="7cm"}
![Wallclock times for diffuse RT calculation with various number of mesh grids computed with 1, 8 and 64 nodes are shown in the upper panel. The results with and without the use of four GPU boards are depicted. The dashed line indicates an analytic scaling of computational cost for the RT calculation of diffuse radiation, $N_{\rm m}^{5/3}$. \[fig:time\_multi\]](f14.eps){width="7cm"}
![Upper panel: Wallclock times for MPI communication in diffuse radiation transfer calculations for various number of mesh grids with 8 and 64 nodes. A scaling relation of $N_{\rm m}^{4/3}$ is shown in a dashed line. Lower panel: fractions of wallclock times for MPI communication relative to total wallclock time elapsed in the calculations of diffuse radiation transfer in the runs with the use of GPUs.\[fig:time\_comm\]](f15.eps){width="7cm"}
The upper panel of Figure \[fig:time\_multi\] shows the wallclock time of diffuse RT calculation performed on a single and multiple computational nodes with and without GPU boards, where we invoke one MPI process on each computational node. In the runs without the use of GPUs, each MPI process invokes 16 `OpenMP` threads, while in the runs with the aid of GPUs, we utilize four GPU boards on each computational node. We measure the wallclock time consumed for a single iteration of diffuse RT calculation for $64^3$–$1024^3$ mesh grids on 1, 8, and 64 computational nodes. The lower panel depicts the performance gain of the runs with 8 and 64 computational nodes relative to those with 1 and 8 computational nodes, respectively, where the ideal performance gain of 8 is shown by a dotted line. As for the runs without the use of GPUs, the parallel efficiency is reasonable when $N_{\rm m}/N_{\rm node} \ge
64^3$, where $N_{\rm node}$ is the number of computational nodes in use. For a given number of computational nodes, the runs with the use of GPUs have poorer performance gains than those without it, mainly beacause the computational times in the runs with GPUs are significantly shorter than those withtout GPUs, and the MPI data communication, as well as the commnication between CPUs and GPUs, gets more salient. Such communication overhead is proportional to the number of light-rays getting through the surface of the decomposed computational domains, $\propto N_{\rm m}^{2/3}N_{\rm d}\propto N_{\rm
m}^{4/3}$. Figure \[fig:time\_comm\] shows that the time consumed by the MPI communication is nearly proportional to $N_{\rm m}^{4/3}$, and that it occupies a significant fraction of the total wallclock time for a small $N_{\rm m}$. This scaling with respect to $N_{\rm m}$ has weaker dependence on $N_{\rm m}$ than the computational costs, $\propto N_{\rm
m}^{5/3}$. Therefore, the overhead can be concealed for a sufficient number of mesh grids, and we have better parallel efficiency for a larger $N_{\rm m}/N_{\rm node}$.
Summary & Discussion
====================
In this paper, we present a new implementation of the RT calculation of diffuse radiation field on three-dimensional mesh grids, which is suitable to be run on recent processors with highly-parallel architecture such as multi-core CPUs and GPUs. The code is designed to be run on both of ordinary multi-core CPUs and GPUs produced by NVIDIA by utilizing the `OpenMP` application programming interface and the `CUDA` programming platform, respectively.
Since our RT calculation is based on the ray-tracing scheme, the RT calculation itself can be carried out concurrently by assigning the RT calculation along each light-ray to individual software threads. To avoid the atomic operations in computing the averaged intensity (equation (\[eq:path\_averaged\_intensity\])) which can potentially degrade the efficiency of the thread parallelization, we devise a new scheme of the RT calculations in which a set of parallel light-rays are split into 4 groups so that parallel light-rays in each group do not get through any mesh grids more than once. As well as the thread parallelization inside processors or computational nodes, we also parallelize our code on a multi-node system using the MWF scheme developed by @nakamoto2001.
We perform several test simulations where the transfer of photo-ionizing radiation emitted by a point radiating source and recombination radiation from ionized regions as diffuse radiation are solved. We verify the validity of our RT calculation of the diffuse radiation by comparing our results with the effect of recombination radiation and the ones with other two independent codes, the one-dimensional spherical code by @kitayama2004 and START code by @hasegawa2010. We also clarify the condition of the required angular resolution in our diffuse radiation transfer scheme based on the mean free path of the diffuse photons and the mesh spacing.
We show good parallel efficiency of our implementation for intra- and inter-node parallelizations. As for the intra-node parallelization, the performance scales well with the number of CPU cores and GPU boards in use, except for the one in the case that multiple CPU sockets are used as a single shared-memory system. The scalability of the inter-node parallelization with the MWF scheme is also measured for $64^3$ to $1024^3$ mesh grids on up to 64 computational nodes and it is found that the inter-node parallelization is efficient when we have a sufficient number of mesh grids per node, $N_{\rm m}/N_{\rm node}\ge 128^3$ and $N_{\rm m}/N_{\rm node}\ge 64^3$ for the runs with and without GPUs, respectively. The ray-grouping technique described in \[ss:ray\_grounping\] is effective and significantly improves the performance of our RT calculations by a factor of more than two, at least on GPUs (NVIDIA Tesla M2090).
With our implementation presented in this paper, we are able to perform the diffuse RT calculations in a reasonable wallclock time comparable to that of other physical processes such as hydrodynamical calculations. This means that the calculations of the diffuse radiation transfer can be coupled with hydrodynamic simulations and we are able to conduct radiation hydrodynamical simulations with the effect of diffuse radiation transfer as well as the radiation transfer from point radiating sources in three-dimensional mesh grids. Currently, we are developing such a radiation hydrodynamic code and, based on this, we will address astrophysical problems in which diffuse radiation transfer plays important roles.
It should be noted that, though we present the implementations and the performance on the multi-core CPUs and GPUs produced by NVIDIA, our approaches presented in this paper can be readily applied to other processors with similar architecture, such as the Intel Xeon-Phi processor or GPUs by other vendors. In addition, our approach can be easily extended to adaptively refined mesh grids using the prescription described in @razoumov2005, although we present the implementation for uniform mesh grids in this paper.
We would like to thank Tetsu Kitayama for providing us with his radiation transfer code. We are also grateful to the anonymous referee for helpful comments. Numerical simulations in this work have been carried out on the HA-PACS supercomputer system under the “Interdisciplinary Computational Science Program” in the Center for Computational Sciences, University of Tsukuba. This work was partially supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (S) (20224002). TO acknowledges the financial support of JSPS Grant-in-Aid for Young Scientists (B: 24740112). KH acknowledges the support of MEXT SPIRE Field 5 and JICFuS and the financial support of JSPS Grain-in-Aid for Young Scientists (B: 24740114).
Photon-conserving estimation of photo-heating and radiative cooling rates\[app:photon\_conserving\]
===================================================================================================
We describe the photon-conserving evaluation of photo-ionization and photo-heating rates of a mesh grid contributed by a point radiating source, where the inner and outer intersections of a light-ray from the point raidating source and the mesh grid are located at $r_{\rm in}$ and $r_{\rm out}$ from the point radiating source. We consider a imaginary spherical shell centered at the point radiating source with inner and outer radii of $r_{\rm in}$ and $r_{\rm out}$, respectively. The incoming photon number per unit time $\dot{N}_{{\rm in},\nu}$ is given by $$\dot{N}_{{\rm in}, \nu} = \frac{L_{\nu} \exp(-\tau_{\nu})}{h\nu},$$ where $L_{\rm \nu}$ is the luminosity density at a frequency $\nu$, and $\tau_{\nu}$ is the optical depth between the point source and the inner side of the shell. The outgoing photon number per unit time from the outer side of the shell is written as $$\dot{N}_{{\rm out}, \nu} = \frac{L_{\nu} \exp(-(\tau_{\nu} +\Delta \tau_{\nu}))}
{h\nu} ,$$ where $\Delta \tau_\nu$ is the radial optical depth of the shell. Then, the number of absorped photons per unit time $\dot{N}_{\rm abs}$ is given by $$\dot{N}_{{\rm abs}, \nu} = \dot{N}_{{\rm out},\nu} - \dot{N}_{{\rm in},\nu} = \frac{L_{\nu} \exp(-\tau_{\nu})}{h\nu} \left[1-\exp(-\Delta \tau_{\nu}) \right] .$$ When we consider multiple chemical components, the absoption rate of the $i$-th species is rewritten as $$\dot{N}^i_{{\rm abs}, \nu} = \frac{\Delta \tau^i_\nu}{\Delta \tau_{\nu}}\frac{L_{\nu} \exp(-\tau_{\nu})}{h\nu} \left[1-\exp(-\Delta \tau_{\nu}) \right],$$ where $\Delta \tau^i_\nu$ is a optical depth contributed by the $i$-th component, and $\Delta \tau_\nu = \sum_i \Delta \tau^i_\nu$ is the total optical depth. Since $\dot{N}^i_{{\rm abs},\nu}$ is equal to the number of ionization of the $i$-th species, the photo-ionization rate of the $i$-th component can be written as $$\Gamma_{i,\gamma} = \frac{1}{N_{i}} \int^{\infty}_{\nu_{i}}
\dot{N}^{i}_{{\rm abs},\nu} d\nu ,
\label{eq:pc_gamma}$$ where $\nu_i$ is the threshold frequecy of the $i$-th species and $N_i$ is the number of $i$-th species in the shell. The photo-heating rate is similarly calculated in terms of $\dot{N}_{{\rm abs},\nu}$ as $${\cal H}_{i,\gamma} = \frac{1}{N_i}\int_{\nu_i}^\infty \dot{N}_{{\rm abs},\nu} (h\nu-h\nu_i) d\nu.$$
Ionization Balance\[app:reaction\]
==================================
The time evolution of the number density of the $i$-th chemical species can be schematically described by $$\label{eq:app_chem}
\frac{dn_i}{dt} = C_i(T,n_j)-D_i(T,n_j)n_i,$$ where $C_i(T,n_j)$ is the collective production rate of the $i$-th species and $D_i(T, n_j)n_i$ is the destruction rate of the $i$-the species. For example, in the case of atomic hydrogen, $C_{\rm HI}$ and $D_{\rm HI}$ is given by $$\begin{aligned}
C_{\rm HI} &=& \alpha_{\rm HII}\,n_{\rm e}n_{\rm HII}\\
D_{\rm HI} &=& \Gamma_{\rm HI} n_{\rm e} + \Gamma_{\rm HI,\gamma}\end{aligned}$$ where $\alpha_{\rm HII}(T)$ is the radiative recombination rate of HII, $\Gamma_{\rm HI}$ is the collisional ionization rate and $\Gamma_{\rm
HI,\gamma} = \sum_\alpha\Gamma^\alpha_{\rm HI,\gamma} + \Gamma_{\rm
HI,\gamma}^{\rm diff}$ is the photoionization rate of HI.
These equations are numerically solved using the backward difference formula (BDF) [@anninos1997; @yoshikawa2006], in which the number densities of the $i$-th chemical species at a time $t+\Delta t$, $n_i^{t+\Delta t}$, is computed as $$n_i^{t+\Delta t} = \frac{C_i\Delta t+n_i^t}{1+D_i\Delta t},$$ where, $C_i$ and $D_i$ are estimated with the number densities of each species at the advanced time, $n_j^{t+\Delta t}$. However, the number densities in the advanced time step are not available for all the chemical species in evaluating $C_i$ due to the intrinsic non-linearity of equation \[eq:app\_chem\]. Thus, we sequentially update the number densities of each chemical species in the increasing order of ionization levels rather than updating all the species simultaneously. It is confirmed that this scheme is stable and accurate [@anninos1997; @yoshikawa2006].
Photo-heating and radiative cooling\[app:heating\_cooling\]
===========================================================
The specific energy change for each mesh by the photo-heating and radiative cooling is followed by the energy equation $$\label{eq:app_energy}
\frac{du}{dt} = \frac{\mathcal{H}-\mathcal{C}}{\rho},$$ where $u$ is the specific internal energy and $\mathcal{H}$ and $\mathcal{C}$ are the photo-heating and cooling rate, respectively and $\mathcal{H}$ is given by $$\mathcal{H} = \sum_i n_i \left(\sum_\alpha \mathcal{H}_i^\alpha + \mathcal{H}_i^{\rm diff}\right).$$ The specific internal energy for each mesh is updated implicitly by solving the equation $$u^{t+\Delta t} = u^t + \frac{\mathcal{H}^{t+\Delta t}
-\mathcal{C}^{t+\Delta t}}{\rho^t}\Delta t$$ for $u^{t+\Delta t}$, where the photo-heating $\mathcal{H}^{t+\Delta t}
= \mathcal{H}(n^{t+\Delta t})$ and cooling rates $\mathcal{C}^{t+\Delta
t} = \mathcal{C}(n^{t+\Delta t}, u^{t+\Delta t})$ are evaluated at the advanced time $t+\Delta t$.
Timestep constraints\[app:timestep\]
====================================
Since we solve the static RT equation (\[eq:rad\_tr1\]), equations (\[eq:app\_chem\]) for chemical reactions and (\[eq:app\_energy\]) for photo-heating and radiative cooling have to be solved iteratively until the electron number density and specific internal energy in each mesh grid converges: $|n_{\rm e}^{(i)}-n_{\rm e}^{(i-1)}|<\epsilon n_{\rm
e}^{(i)}$ and $|u^{(i)}-u^{(i-1)}|<\epsilon u^{(i)}$, where $\epsilon$ is set to $10^{-3}$, and $n_{\rm e}^{(i)}$ and $u^{(i)}$ indicates the specific internal energy and the electron number density after the $i$-th iteration, respectively.
The timestep in solving chemical reactions and energy equation, $\Delta
t_{\rm chem}$, is set to $$\Delta t_{\rm chem} = \epsilon_{\rm e}\left|\frac{n_{\rm e}}{\dot{n}_{\rm
e}}\right|+\epsilon_{\rm HI} \left|\frac{n_{\rm
HI}}{\dot{n}_{\rm HI}}\right|,$$ where the second term on the right hand side prevents the timestep from getting prohibitively short in the case that the gas is almost neutral, and $\epsilon_{\rm e}$ and $\epsilon_{\rm HI}$ are set to 0.2 and 0.002, respectively.
The timestep, $\Delta t$, with which we update the radiation field can be larger than the chemical timestep, $\Delta t_{\rm chem}$ by subcycling the rate and energy equations (\[eq:app\_chem\]) and (\[eq:app\_energy\]). Throughout in this paper, the timestep for the RT calculation is set to $$\Delta t = F \min_i \Delta t_{\rm chem,\it i},$$ where $\Delta t_{\rm chem,\it i}$ is the chemical timestep for the $i$-th mesh grid, and we typically set $F=1\sim10$ so that the radiation field successfully converges with a reasonable number of iterations.
Abel, T., Anninos, P., Zhang, Y., Norman, M. L., 1997, New Astronomy, 2, 181
Abel, T., Norman,M. L., Madau, P., 1999, , 523, 66
Anninos, P., Zhang, Y., Abel, T., Norman, M. L., 1997, New Astronomy, 2, 209
Cen, R., 1992, , 78, 341
Ciardi, B., Ferrara, A., Marri, S., & Raimondo, G. 2001, , 324, 381
Dopita, M. A., Krauss, L. M., Sutherland, R. S., Kobayashi, C., Lineweaver, C. H., 2011, , 335, 345
Gnedin, N. Y., Abel, T., 2001, New Astronomy, 6, 437
Gonz[á]{}lez, M., Audit, E., Huynh, P., 2007, , 464, 429
G[ó]{}rski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., Hansen, F. K., Reinecke, M., Bartelmann, M., 2005, , 622, 759
Hasegawa, K., Umemura, M., 2010, , 407, 2632
Hui, L., Gnedin, N. Y., 1997, , 292, 27
Hummer, D. G., 1994, , 268, 109
Hummer, D. G., Storey, P. J., 1998, , 297, 1073
Ikeuchi, S., Ostriker, J. P., 1986, , 301, 522
Iliev, I. T. et al., 2006, , 371, 1057
Iliev, I. T. et al., 2009, , 400, 1283
Inoue, A. K. 2010, , 401, 1325
Janev R. K., Langer W. D., Post Jr D. E., Evans Jr K., 1987, in Janev R.K., Lnger W.D., Evans, K., eds, Elementary Processes in Hydrogen-Helium Plasmas – Cross-section and Reaction Rate Coefficients. Springer-Verlag , Berlin
Kanno, Y., Harada, T., & Hanawa, T. 2013, , 65, 72
Kitayama, T., Yoshida, N., Susa, H., Umemura, M., 2004, , 613, 631
Kunasz, P., Auer, L. H., 1988, J. Quant. Spectr. Radiative Transfer, 39, 67
Miralda-Escud[é]{}, J., 2003, , 597, 66
Nakamoto, T., Umemura, M., Susa, H., 2001, , 321, 593
Nakamoto, T., Umemura, M., & Susa, H. 2001, in ASP Conf. Ser. 222, The Physics of Galaxy Formation, ed. M. Umemura & H. Susa (San Francisco: ASP), 109
Okamoto, T., Yoshikawa, K., Umemura, M., 2012, , 419, 2855
Okamoto, T., Shimizu, I., & Yoshida, N. 2014, , 66, 70
Osterbrock, D. E., 2006, Astrophysics Of Gaseous Nebulae And Active Galactic Muclei. University Science Books
Pawlik, A. H., Schaye, J., 2011, , 412, 1943
Rahmati, A., Pawlik, A. H., Raičević, M., & Schaye, J. 2013, , 430, 2427
Rahmati, A., Schaye, J., Pawlik, A. H., & Raičević, M. 2013, , 431, 2261
Razoumov, A. O., Cardall, C. Y., 2005, , 362, 1413
Rijkhorst, E.-J., Plewa, T., Dubey, A., Mellema, G., 2006, A&A, 452, 907
Rosdahl, J., Blaizot, J., Aubert, D., Stranex, T., & Teyssier, R. 2013, , 436, 2188
Skinner, M. A., & Ostriker, E. C. 2013, , 206, 21
Sokasian, A., Abel, T., Hernquist, L. E., 2001, New Astronomy, 6, 359
Stone, J. M., Mihalas, D., Norman, M. L., 1992, , 80, 819
Susa, H., 2006, , 58, 445
Wise, J. H., Abel, T., 2011, , 414, 3458
Wyithe, J. S. B., Mould, J., & Loeb, A. 2011, , 743, 173
Yoshikawa, K., Sasaki, S., 2006, , 58, 641
|
---
abstract: 'In this work we consider general facility location and social choice problems, in which sets of agents $\mathcal{A}$ and facilities $\mathcal{F}$ are located in a metric space, and our goal is to assign agents to facilities (as well as choose which facilities to open) in order to optimize the social cost. We form new algorithms to do this in the presence of only [*ordinal information*]{}, i.e., when the true costs or distances from the agents to the facilities are [*unknown*]{}, and only the ordinal preferences of the agents for the facilities are available. The main difference between our work and previous work in this area is that while we assume that only ordinal information about agent preferences in known, we know the exact locations of the possible facilities $\mathcal{F}$. Due to this extra information about the facilities, we are able to form powerful algorithms which have small [*distortion*]{}, i.e., perform almost as well as omniscient algorithms but use only ordinal information about agent preferences. For example, we present natural social choice mechanisms for choosing a single facility to open with distortion of at most 3 for minimizing both the total and the median social cost; this factor is provably the best possible. We analyze many general problems including matching, $k$-center, and $k$-median, and present black-box reductions from omniscient approximation algorithms with approximation factor $\beta$ to ordinal algorithms with approximation factor $1+2\beta$; doing this gives new ordinal algorithms for many important problems, and establishes a toolkit for analyzing such problems in the future.'
author:
- Elliot Anshelevich and Wennan Zhu
bibliography:
- 'ref.bib'
title: |
Ordinal Approximation for Social Choice, Matching,\
and Facility Location Problems given Candidate Positions
---
Introduction
============
Many important problems involve assigning agents to facilities. For example, assigning patients to hospitals, students to universities, people to houses, etc. The target of assignment problems is usually to minimize social cost or maximize social welfare. When we consider the social cost of assignment problems, it is natural to assume the agents prefer facilities that are “closer" to them in some sense, thus the social cost of an agent is often represented by the distance between the agent and the facility it is assigned to. Besides the cost of distances, there are many other cost functions and constraints for different problems; for example, in the capacitated facility assignment problem, each facility has a maximum number of agents it can accommodate. In this work we consider general facility location problems, in which sets of agents $\mathcal{A}$ and facilities $\mathcal{F}$ are located in a metric space, and our goal is to assign agents to facilities (as well as choose which facilities to open) so that agents are assigned to facilities which are close to them. For example, $\mathcal{F}$ may be possible locations for opening new stores, and the goal may be that all agents have a store near them, or that the sum of agent distances to the stores they are assigned to is small, etc. This setting also captures many social choice problems, in which the facilities correspond to candidates, and the goal would be to choose a single candidate (and assign all agents to this candidate) so that the distances from the agents to the chosen candidate are small. Here the distances correspond to [*spatial preferences*]{}, i.e., the metric space represents the ideological space in which a more preferred candidate would be closer to me; see [@enelow1984spatial; @anshelevich2015approximating] for discussion of such spatial preferences in social choice. Our setting also captures matching and many related problems, in which we would open all facilities, but are only able to assign one agent to each facility, thus forming a matching between agents and facilities; facilities here could correspond to houses or items, for example.
If the distances between agents and facilities are known, then we can calculate the optimal solution for these assignment problems. Note that many of the facility location problems are NP-Complete, but at least it is possible to compute optimum assignments of agents to facilities (or the optimum candidates to select for social choice settings) given unlimited computational resources. For many of the settings we mentioned above, however, it is unlikely that we know the exact distances from the agents to the facilities. For social choice these distances would correspond to the cardinal preferences of voters for candidates, for example, “My cost for candidate X winning is exactly 2.35." It is far more common that only [*ordinal*]{} preferences of the agents for the candidates are known, i.e., “I prefer X to Y". Similarly, when trying to form a matching, or even in general facility location problems where we survey the agents to find out their preferences, it is much easier to elicit ordinal preferences (“I prefer to be matched with X over Y") over precise numerical preferences. These observations have recently led to a large body of work using the [*utilitarian approach*]{}, in which we assume that some latent numerical costs or utilities exist, but we only know the [*ordinal*]{} preferences of the agents, not their underlying numerical costs. See for example [@anshelevich2015approximating; @anshelevich2015randomized; @boutilier2015optimal; @goel2016metric; @feldman2016voting; @skowron2017social; @cheng2018distortion] for the social choice setting, [@anshelevich2015blind; @anshelevich2016truthful; @anshelevich2017tradeoffs; @abramowitz2017utilitarians] for matching and other graph problems, and [@caragiannis2016truthful] for facility location. These works focus on measuring the [*distortion*]{} of various algorithms: a measure of how well an algorithm behaves when using only ordinal information, as compared to the optimum algorithm which has access to the true underlying numerical information. More formally, the *distortion* [@procaccia2006distortion; @anshelevich2015approximating] of an assignment is defined as the worst-case ratio of its social cost to the social cost of the optimal solution.
As in the work mentioned above, we assume that only ordinal information about the distances between agents and facilities is known. However, although the locations and numerical preferences of the agents are usually difficult to obtain, the locations of facilities are mostly public information. The locations of political candidates in ideological space can be reasonably well estimated based on their voting records and public statements. When forming a survey about new stores to open, we may not know exactly how much the customers would prefer one store over the other since the customer locations may be private, but the locations of the possible stores themselves are public knowledge. The main difference between our work and previous work in this area is that we assume:\
[*While only ordinal information about agent preferences in known, we know the exact locations of the possible facilities $\mathcal{F}$.*]{}\
As we discuss below, this extra information about the locations of the facilities relative to each other allows us to produce much stronger algorithms, and show much nicer bounds on distortion. In fact, in many cases, we do not even need the full information about the locations of the facilities. The main message of this paper is that having a small amount of information about the candidates in social choice settings, or the facilities in facility location, allows us to obtain solutions which are provably [*close to optimal*]{} for a large class of problems even though the only information we have about the agent preferences is ordinal, and thus it is impossible (even given unlimited computational resources) to compute the [*true*]{} optimum solution.
Our Contributions
-----------------
We begin by looking at the social choice setting, in which we have agents $\mathcal{A}$ and candidates $\mathcal{F}$ in a metric space, and we are given an ordinal ranking of each agent for the candidates. This setting was considered in e.g., [@anshelevich2015approximating; @anshelevich2015randomized; @goel2016metric; @feldman2016voting; @skowron2017social; @cheng2018distortion; @gross2017vote]. In particular, for the objective of minimizing the total distance from the agents to the chosen candidate, [@anshelevich2015approximating] showed that Copeland and similar voting mechanisms always have distortion of at most 5, while no deterministic voting mechanism can achieve a worst-case distortion of less than 3. Finding a deterministic mechanism with distortion less than 5 has been an open problem for several years [@goel2016metric]. In this paper, we show that if we know the exact locations of the candidates in addition to the ordinal ranking of the agents, then there is a simple algorithm which achieves a distortion of 3, and no better bound is possible. In other words, while we do not know the true distances from agents to candidates, we can compute an outcome which is a 3-approximation [*no matter what*]{} the true distances are, as long as they are consistent with the ordinal preferences given to us. Moreover, this approximation is possible even if for each agent we are only given their favorite (i.e., top-choice) candidate: there is no need for the agents to submit a full preference ranking over all the alternatives.
We also study other objective functions in addition to minimizing the total distance from agents to the chosen alternative. We give a natural deterministic voting mechanism which has distortion at most 3 for objectives such as minimizing the median voter cost, the egalitarian objective of minimizing maximum voter cost, and many other objectives. This mechanism achieves all these approximation guarantees [*simultaneously*]{}, and moreover it does not need the exact locations of the candidates: it suffices to be given an ordinal ranking of the distances from each candidate to each other candidate. In other words, this mechanism is especially suitable for the case when candidates are a subset of voters, as our mechanism will obtain the ordinal ranking of each voter for all the candidates, and this is the only information which would be required. Note that [@anshelevich2015approximating] proved that [*no*]{} deterministic mechanism can achieve a distortion of better than 5 for the median objective; the reason why we are able to achieve a distortion of 3 here is precisely because we also know how each candidate ranks all the other candidates, in addition to how each voter ranks all the candidates.
We then proceed to our general facility assignment model. We are given a set of agents and a set of facilities in a metric space. The distances between facilities are given, but the distances between agents and facilities are unknown; instead we only know ordinal preferences of agents over facilities which are consistent with the true underlying distances. There could be arbitrary constraints on the assignment, such as facility capacities, or constraints enforcing that some agents cannot be (or must be) assigned to the same facility, etc. A valid assignment is to assign each agent to a facility without violating the constraints. We consider many different social cost functions to optimize. For a general class of cost functions (essentially ones which are monotone and subadditive), we give a black-box reduction which converts an algorithm for the omniscient version of this problem (i.e., the version where the true distances are known) to an ordinal algorithm with small distortion. Specifically, if we have an omniscient algorithm which always produces an assignment which is a $\beta$-approximation to the optimum, then using it we can create an ordinal algorithm which only knows the ordinal preferences of the agents instead of their true distances to the facilities, but has distortion of at most $1+2\beta$.
-------------------------------- ---------------------------- ------------------------ ---------------------- --
Omniscient: Agents’ ordinal prefs Only agents’ ordinal
full distances and facility locations prefs (lower bounds)
Total (Sum) Social Choice 1 3 5(3)
Median Social Choice 1 3 5(5)
Min Weight Bipartite Matching 1 3 $n$(3)
Egalitarian Bipartite Matching 1 3 -(2)
Facility Location 1.488 [@li20111] 3.976 $\infty$ ($\infty$)
$k$-center 2 [@hochbaum1985best] 5 - (-)
$k$-median 2.675 [@byrka2014improved] 6.35 - ($\Omega(n)$)
-------------------------------- ---------------------------- ------------------------ ---------------------- --
: Best known distortion of polynomial-time algorithms in different settings. “Omniscient” stands for the setting where all the distances between agents and facilities are known, and the numbers represent the best-known approximation ratios. The second column represent our setting, in which the ordinal preferences of the agents, and the numerical distances between facilities are known. The last column represents the pure ordinal setting in which only the agent ordinal preferences are known, but the distances between facilities are unknown; this setting has been previously studied, and we include the known lower bounds on the possible distortion in parentheses, including some which we prove in the Appendix.[]{data-label="table_results"}
Many well-known problems fall into our facility assignment model; Table \[table\_results\] summarizes some of our results. For example, classic facility location with facility costs, minimum weight bipartite matching, egalitarian bipartite matching, $k$-center, and $k$-median are all special cases. In particular our results show that if we are given unbounded computational resources, then it is always possible to form an assignment with distortion of at most 3 for these problems, and no better bound is possible simply due to the fact that we do not possess all the relevant information to compute the true optimum. This is a large improvement over previously known distortion bounds: for minimum cost ordinal matching the best-known distortion bound is $n$ using random serial dictatorship (RSD) [@caragiannis2016truthful]; by using the knowledge of facility locations we are able to reduce this approximation ratio to 3.
Discussion and Related Work
---------------------------
Ordinal approximation [@anshelevich2016ordinal] for the minimum social cost (or maximum social welfare) with underlying utilities/distances between agents and alternatives has been studied in many settings including social choice [@procaccia2006distortion; @boutilier2015optimal; @anshelevich2015approximating; @anshelevich2015randomized; @goel2016metric; @feldman2016voting; @caragiannis2017subset; @skowron2017social; @cheng2018distortion], matchings [@bhalgat2011social; @filos2014social; @anshelevich2015blind; @anshelevich2016truthful; @caragiannis2016truthful; @christodoulou2016social; @anshelevich2017tradeoffs], secretary problems [@hoefer2017combinatorial], participatory budgeting [@benade2017preference], general graph problems [@anshelevich2015blind; @abramowitz2017utilitarians] and many other models in recent years. The general assumption of the ordinal setting is that we only have the ordinal preferences of agents over alternatives, and the goal is to form a solution that has close to optimal social cost. There are different models: social choice, matching, facility location, etc.; different objectives: minimizing social cost, maximizing social welfare, total cost objective, median objective, egalitarian objective, etc.; different assumptions on utility or cost functions: unit-sum, unit-range, metric space, etc. In this paper, we study general facility assignment problems in a metric space, and assume that the ordinal preferences of agents over alternatives are given. Unlike previous work on this topic, we also assume the locations of the alternatives are known; we show that this extra information enables us to achieve much better approximation ratios than in the pure ordinal setting for many problems.
The distortion of social choice functions was first introduced in [@procaccia2006distortion], to describe the ratio between the total utility of the optimal candidate and the candidate selected by a mechanism using only ordinal preferences. [@anshelevich2015approximating; @skowron2017social; @goel2016metric] studied the distortion of social choice functions in a metric space; the assumption that the underlying numerical costs have this metric property allows for much better results than more general costs. In particular, for the objective of minimizing the total distance from the agents to the chosen candidate, the above papers were able to show good distortion bounds for many well-known mechanisms, in particular a bound of 5 for Copeland [@anshelevich2015approximating], a bound of $O(\ln m)$ for Single Transferable Vote (STV) [@skowron2017social], and many others. In addition, [@anshelevich2015approximating] proved that no deterministic mechanism can have worst-case distortion better than 3, and [@skowron2017social] showed that all scoring rules for $m$-candidates have a distortion of at least $1 + 2 \sqrt{\ln m - 1}$. Goel et al. [@goel2016metric] showed that Ranked Pairs, and the Schulze rule have a worst-case distortion of at least 5, and the expected worst-case distortion of any (weighted)-tournament rule is at least 3. They also introduced the notion of “fairness” of social choice rules, and discussed the fairness ratio of Copeland, Randomized Dictatorship, and a general class of cost functions. Finding a deterministic mechanism with distortion less than 5 has been an open problem for several years. In this paper, we show that if we know the exact locations of the candidates in addition to the ordinal ranking of the agents, then there is a simple algorithm which achieves a distortion of 3, and no better bound is possible.
While the above work, as well as our paper, only focuses on deterministic algorithms, the distortion of randomized algorithms in social choice has also been considered, see for example [@anshelevich2015randomized; @fain2017sequential; @gross2017vote; @feldman2016voting]. In a slightly different flavor of result, [@cheng2017people; @cheng2018distortion] consider the special case where candidates are randomly and independently drawn from the set of voters. While we leave the analysis of randomized algorithms which know the location of the facilities to future work, and consider the worst-case candidate locations, it is worth pointing out that our [*deterministic*]{} algorithm achieves a distortion of 3, which is also the best known distortion bound for any [*randomized*]{} mechanism which only knows the ordinal preferences of the agents. Similarly, another common goal is to form [*truthful*]{} mechanisms with small distortion for matching and social choice, as in [@feldman2016voting; @anshelevich2016truthful; @caragiannis2016truthful]; we focus on general mechanisms in this paper in order to understand the limitations of knowing only certain kinds of ordinal information, and leave the goal of forming truthful mechanisms for future work.
For the median objective of social choice problems, [@anshelevich2015approximating] showed that Copeland gives a distortion of at most 5, while [*no*]{} deterministic mechanism can achieve a distortion of better than 5 . [@anshelevich2015randomized] also gave a randomized algorithm that has a distortion of at most 4. In this paper, we are able to improve this bound to a tight worst-case distortion of 3 by a deterministic mechanism, because we also know how each candidate ranks all the other candidates, in addition to how each voter ranks all the candidates.
The distortion of matching in a metric space has received far less attention than social choice questions. [@anshelevich2015blind; @anshelevich2016truthful; @anshelevich2017tradeoffs] analyzed maximum-weight metric matching; the maximization objective makes this problem far easier, and even choosing a uniformly random matching yields a distortion of a small constant. This is very different from our goal of computing a minimum-cost matching, for which no ordinal approximations better than $O(n)$ are known. [@caragiannis2016truthful] studied facility assignment problems in a metric space; they considered the problem with or without resource augmentation, and the cases without augmentation are exactly the minimum weight bipartite matching problem. [@caragiannis2016truthful] showed that the approximation ratio of random serial dictatorship (RSD) is at most $n$, and gave a lower bound of $2^n - 1$ for the approximation ratio of serial dictatorship (SD), and a lower bound of $n^{0.26}$ for RSD. Their results are the best known ordinal approximations for this problem. In this paper, we are able to give a tight 3-approximation for the minimum weight matching problem, given the locations of facilities in addition to the agents’ ordinal preferences.
Model and Notation: Social Choice {#section-social-model}
=================================
For the social choice problems studied in this paper, we let $\mathcal{A} = \{ 1, 2, \dots, n\}$ be a set of agents, and let $\mathcal{F} = \{ F_1, F_2, \dots, F_m\}$ be a set of alternatives, which we will also refer to sometimes as [*candidates*]{} or [*facilities*]{}. We will typically use $i$ and $j$ to refer to agents and $W, X, Y, Z$ to refer to alternatives. Let $\mathcal{S}$ be the set of total orders on the set of alternatives $\mathcal{F}$. Every agent $i \in \mathcal{A}$ has a preference ranking $\sigma \in \mathcal{S}$; by $X \succ_i Y$ we will mean that $X$ is preferred over $Y$ in ranking $\sigma$. Although we assume that each agent has a total order of preference over the alternatives and that this order is known to us, for many of our results it is only necessary that the top choice of each agent is known. We say $X$ is $i$’s top choice if $i$ prefers $X$ to every other alternative in $\mathcal{F}$. We call the vector $\sigma = (\sigma_1, \dots, \sigma_n) \in \mathcal{S}^n$ a preference profile. We say that an alternative $X$ pairwise defeats $Y$ if $| \{i \in \mathcal{A}: X \succ_i Y \}| > \frac{n}{2}$. The goal is to choose a single winning alternative.
#### Cardinal Metric Costs.
In this work we take the utilitarian view, and assume that the ordinal preferences $\sigma$ are derived from underlying (latent) cardinal agent costs. Formally, we assume that there exists an arbitrary metric $d: (\mathcal{A} \cup \mathcal{F})^2 \rightarrow \mathbb{R}_{\ge 0}$ on the set of agents and alternatives. The cost incurred by agent $i$ of alternative $X$ being selected is represented by $d(i, X)$, which is the distance between $i$ and $X$. Such spatial preferences are relatively common and well-motivated, see for example [@enelow1984spatial; @anshelevich2015approximating] and the references therein. The underlying distances $d(i, X)$ are [*unknown*]{}, but unlike most previous work we [*do*]{} assume the distances between alternatives are given. For example, when alternatives represent facilities or stores to be opened, it makes sense that their specific locations would be known, while the distances from the customers to the stores may be private. Similarly, when the alternatives represent political candidates, it may be easy to estimate their locations in ideological space (for example based on their voting records and public statements), but the ideology of the voters is much harder to estimate, with mechanism designers only knowing which candidates the voters prefer but not how much they prefer them. The distance between two alternatives $X$ and $Y$ is denoted by $l(X, Y)$. We say that $d$ is [*consistent*]{} with $l$ if $\forall X,\ Y \in \mathcal{F}$, $d(X, Y) = l(X, Y)$. The metric costs $d$ naturally give rise to a preference profile. We say that $d$ is [*consistent*]{} with $\sigma$ if $\forall i \in \mathcal{A}$, $\forall X,\ Y \in \mathcal{F}$, if $d(i, X) < d(i, Y)$, then $X \succ_i Y$. It means that the cost of $X$ is less than the cost of $Y$ for agent $i$, so agent $i$ prefers $X$ over $Y$. As described above, we know exactly the distances $l$ and the preferences $\sigma$, but do not know the true costs $d$ which give rise to $\sigma$. Let $\mathcal{D}(\sigma, l)$ be the set of metrics that are consistent with $\sigma$ and $l$; we know that one of the metrics from this possibly infinite space captures the true costs, but do not know which one.
#### Social Cost Distortion
We study several objective functions for social cost in this paper. First, the most common notion of social cost is the sum objective function, defined as $SC_{\sum}(X, \mathcal{A}) = \sum_{i \in \mathcal{A}} d(i, X)$. We also study the median objective function, $SC_{\text{med}}(X, \mathcal{A}) = \text{med}_{i \in \mathcal{A}} (d(i, X))$, as well as the egalitarian objective and many others (see Section \[subsection-social-median\]). We use the notion of distortion to quantify the quality of an alternative in the worst case, similar to the notation in [@boutilier2015optimal; @procaccia2006distortion]. For any alternative $W$, we define the distortion of $W$ as the ratio between the social cost of $W$ and the optimal alternative:
$$\begin{aligned}
dist_{\sum}(W, \sigma, l) &= \sup\limits_{d \in \mathcal{D}(\sigma, l)} \frac{SC_{\sum}(W, \mathcal{A})}{\min_{X \in \mathcal{F}}SC_{\sum}(X, \mathcal{A})} \\
dist_{\text{med}}(W, \sigma, l) &= \sup\limits_{d \in \mathcal{D}(\sigma, l)} \frac{SC_{\text{med}}(W, \mathcal{A})}{\min_{X \in \mathcal{F}}SC_{\text{med}}(X, \mathcal{A})}\end{aligned}$$
In other words, saying that the distortion of $W$ is at most 3 means that, no matter what the true costs $d$ are (as long as they are consistent with the $\sigma$ and $l$ which we know), it must be that the social cost of $W$ is within a factor of 3 of the true optimum alternative, which is impossible to compute without knowing the true costs. Because of this, a small distortion value means that there is no need to obtain the true agent costs, and the ordinal information $\sigma$ (together with information $l$ about the alternatives) is enough to form a good solution.
A social choice function $f$ on $\mathcal{A}$ and $\mathcal{F}$ takes $\sigma$ and $l$ as input, and returns the winning alternative. We say the distortion of $f$ is the same as the distortion of the winning alternative chosen by $f$ on $\sigma$ and $l$. In other words, the distortion of a social choice mechanism $f$ on a profile $\sigma$ and facility distances $l$ is the worst-case ratio between the social cost of $W = f(\sigma, l)$, and the social cost of the true optimal alternative.\
Distortion of Social Choice Mechanisms {#section-social-choice}
======================================
Distortion of Total Social Cost {#subsection-social-sum}
-------------------------------
In this section, we study the sum objective and provide a deterministic algorithm that gives a distortion of at most 3. According to [@anshelevich2015approximating], the lower bound on the distortion for deterministic social choice functions with only ordinal preferences (without knowing $l$) is 3. This occurs in the simple example with 2 alternatives which are tied with approximately half preferring each one. No matter which one is chosen, the true optimum could be the other one, and its social cost can be as much as 3 times better. Because the example in Theorem 3 from [@anshelevich2015approximating] only has two alternatives, knowing $l$ does not provide any extra information, and thus that example also provides a lower bound of 3 in our setting, although we assume the distances $l$ between facilities are known in this paper. Therefore, our mechanism achieves the best possible distortion in this setting. Note that if we only have ordinal preferences of the agents without the distances between facilities, then the best known approach so far is Copeland, which gives a distortion at most 5. Thus our results establish that by knowing the distances $l$ between alternatives, it is possible to reduce the distortion from 5 to 3, and no better deterministic mechanism is possible.
\[lemma-social-basic\] Let $W,\ X$ be alternatives. If $W\succ_i X$, then $d(i, X) \ge \frac{d(X, W)}{2}$. \[Lemma 5 in [@anshelevich2015approximating]\]
In the following algorithm, we generate a set of projected agents as follows: Given agents $\mathcal{A}$, alternatives $\mathcal{F}$, and the preference profile $\sigma$, for each agent $i$ denote alternative $X_i$ as $i$’s top choice. Then we create a new agent $\tilde{i}$ at the location of $X_i$ in the metric space, as shown in Figure \[fig:projected\] (a); consequently, $\forall \ Y \in \mathcal{F}$, $d(\tilde{i}, Y) = d(X_i, Y)$. Denote the set of the new agents as $\tilde{\mathcal{A}} = \{ \tilde{1}, \tilde{2}, \dots, \tilde{n} \}$. For any metric $d$ consistent with $l$, $d(\tilde{i}, Y) = d(X_i, Y) = l(X_i, Y)$, so the distances between agents in $\tilde{\mathcal{A}}$ and alternatives in $\mathcal{F}$ are known to us, unlike the true distances between $\mathcal{A}$ and $\mathcal{F}$.
![(a) For each agent, generate a projected agent at the location of its top choice alternative. (b) A figure demonstrating agent $i$, $i$’s top choice alternative $Y$, $i$’s projected agent $\tilde{i}$ located at $Y$, the winner $W$, and the optimal alternative $X$ for the proof of Theorem \[thm-social-sum\].[]{data-label="fig:projected"}](figures/projected1.png)
Generate projected agent set $\tilde{\mathcal{A}}$. For each alternative $X \in \mathcal{F}$, calculate the total social cost on $\tilde{\mathcal{A}}$ by choosing $X$, i.e., $SC_{\sum}(X, \tilde{\mathcal{A}}) = \sum_{\tilde{i} \in \tilde{\mathcal{A}} } d(\tilde{i}, X) = \sum_{\tilde{i} \in \tilde{\mathcal{A}} } l(\tilde{i}, X)$ .
**Final Output:** Return the alternative $W$ that has the minimum social cost $SC_{\sum}(W, \tilde{\mathcal{A}})$ .
\[thm-social-sum\] The distortion of Algorithm \[alg-social-sum\] for minimum total social cost on $\mathcal{A}$ is at most 3.
Let $W$ denote the winning alternative. $W$ has the minimum social cost on the agent set $\tilde{\mathcal{A}}$, so for any alternative $Y$, it must be that
$$\frac{SC_{\sum}(W, \tilde{\mathcal{A}})}{SC_{\sum}(Y, \tilde{\mathcal{A}})} = \frac{\sum_{\tilde{i} \in \tilde{\mathcal{A}} } d(\tilde{i}, W)}{\sum_{\tilde{i} \in \tilde{\mathcal{A}} } d(\tilde{i}, Y)} = \frac{\sum_{i \in \mathcal{A} } d(\tilde{i}, W)}{\sum_{i \in \mathcal{A} } d(\tilde{i}, Y)} \le 1 \label{lemma-loc-eq1}$$
Let $X$ denote the true optimal alternative for $\mathcal{A}$. We want to get $dist_{\sum}(W, \sigma, l)$ by upper bounding the cost incurred by $W$ compared to $X$:
$$\begin{aligned}
\frac{SC_{\sum}(W, \mathcal{A})}{SC_{\sum}(X, \mathcal{A})} &= \frac{\sum_{i \in \mathcal{A}} d(i, W)}{\sum_{i \in \mathcal{A}} d(i, X)} \nonumber \\
&\le \frac{\sum_{i \in \mathcal{A}} (d(i, \tilde{i}) + d(\tilde{i}, W))}{\sum_{i \in \mathcal{A}} d(i, X)} \nonumber \\
&= \frac{\sum_{i \in \mathcal{A}} d(i, \tilde{i})}{\sum_{i \in \mathcal{A}} d(i, X)} + \frac{\sum_{i \in \mathcal{A}} d(\tilde{i}, W)}{\sum_{i \in \mathcal{A}} d(i, X)} \label{lemma-loc-eq2}
\end{aligned}$$
The inequality $d(i, W) \le d(i, \tilde{i}) + d(\tilde{i}, W)$ is due to the triangle inequality since $d$ is a metric, as shown in Figure \[fig:projected\] (b). $\forall i \in \mathcal{A}$, we know that $\tilde{i}$ is located at $i$’s top choice alternative, so the distance between $i$ and $\tilde{i}$ must be less than (or equal to) the distance between $i$ and any alternative; thus $d(i, \tilde{i}) \le d(i, X)$. Summing up for all $i \in \mathcal{A}$, we get that $\frac{\sum_{i \in \mathcal{A}} d(i, \tilde{i})}{\sum_{i \in \mathcal{A}} d(i, X)} \le 1$. For any agent $i$ such that $X$ is not $i$’s top choice, suppose alternative $Y$ is $i$’s top choice, then $\tilde{i}$ has the same location as $Y$ and $d(\tilde{i}, X) = d(X, Y)$. By Lemma \[lemma-social-basic\], $d(i, X) \ge \frac{d(X, Y)}{2}$, thus $d(i, X) \ge \frac{d(\tilde{i}, X)}{2}$. For all $i$ that $X$ is $i$’s top choice, $d(\tilde{i}, X) = 0$, so the inequality $d(i, X) \ge \frac{d(\tilde{i}, X)}{2}$ holds for all $i \in \mathcal{A}$. Together with inequality \[lemma-loc-eq1\] and \[lemma-loc-eq2\],
$$\frac{SC_{\sum}(W, \mathcal{A})}{SC_{\sum}(X, \mathcal{A})} \le 1 + \frac{\sum_{i \in \mathcal{A}} d(\tilde{i}, W)}{\sum_{i \in \mathcal{A}} \frac{d(\tilde{i}, X)}{2}}
= 1 + 2\frac{\sum_{i \in \mathcal{A}} d(\tilde{i}, W)}{\sum_{i \in \mathcal{A}} d(\tilde{i}, X)}
\le 3$$
Distortion of Median Social Cost {#subsection-social-median}
--------------------------------
In this section, we study the median objective function, and provide a deterministic mechanism that gives a distortion of at most 3. Recall that we define the median social cost of an alternative $X$ as $SC_{\text{med}}(X, \mathcal{A}) = \text{med}_{i \in \mathcal{A}} (d(i, X))$. We will refer to this as $\text{med}(X)$ when $d$ and $\mathcal{A}$ are fixed. If $n$ is even, we define median to be the $(\frac{n}{2} + 1)^{th}$ smallest value of the distances. Note that no deterministic mechanism which only knows ordinal preferences can have worst-case distortion better than 5 (Theorem 14 in [@anshelevich2015approximating]). With known distances between facilities, we are able to provide a natural social choice function with distortion of 3, which is also provably the best possible distortion in our setting (consider the example in Theorem 3 from [@anshelevich2015approximating] again). Moreover, our social choice function only uses ordinal information about the alternatives, and not the full distances $l$; in particular as long as we have ordinal preferences of each alternative for each other alternative (and thus a total order of the distances from each alternative to the others), then our mechanism will work properly. Such ordinal information may be easier to obtain than full distances $l$; for example candidates can rank all the other candidates. In particular, given agents with ordinal preferences such that the candidates are a subset of the agents, our mechanism will always form an outcome with small distortion, even if we do not know the distances $l$.
Note that using only agents’ top choices over alternatives and the distances between alternatives, as Algorithm \[alg-social-sum\] does for the total social cost objective, is not enough to give a worst-case distortion of 3 for the median objective. Consider the following example: there are 4 alternatives $W, X, Y, Z$, the distances between them are: $d(W, Y) = d(Y, X) = d(X, Z) = d(Z, W) = 2$ and $d(W, X) = d(Y,Z) = 4$. Suppose $W$ is agents 1, 2’s top choice, $X$ is agent 3, 4’s top choice, $Y$ is agent 5, 6’s top choice, and $Z$ is agent 7, 8’s top choice. This graph is symmetric, so we choose an arbitrary alternative as the winner. Suppose we choose $W$ as the winner, and the distances between agents and facilities are: the distances from agents 1, 2 to $W$ are both 100, the distances from agents 1, 2 to $X, Y, Z$ are all 102. The distances from agents 5, 6 to $Y, X$ are all 1, and the distances from agents 5, 6 to $W, Z$ are all 3. The distances from agents 7, 8 to $Z, X$ are all 1, and the distances from agents 7, 8 to $Y, W$ are all 3. The distances from agents 3, 4 to $X$ are both 1, the distances from 3, 4 to $Y, Z$ are all 3, and the distances from 3, 4 to $W$ are both 5. In this example, the median is the distance from $5^{th}$ closest agent to the winning alternative. $X$ is the optimal alternative with $\text{med}(W) = 1$, while $\text{med}(W) = 5$ has a distortion of 5.\
We will use the following Lemmas from [@anshelevich2015approximating] in the proof of our algorithm:
\[lemma-social-median1\] For any two alternatives $W$ and $Y$, we have $\text{med}(W) \le \text{med}(Y) + d(Y, W)$. \[Lemma 11 in [@anshelevich2015approximating]\]
\[lemma-social-median2\] For any two alternatives $Y$ and $P$, if $P$ pairwise defeats (or pairwise ties) $Y$, then $\text{med}(Y) \ge \frac{d(Y, P)}{2}$. \[Proved in Theorem 16 in [@anshelevich2015approximating]\]
\[lemma-social-median3\] Let $W, Y$ be an alternatives $\in \mathcal{F}$, if $W$ pairwise defeats (or pairwise ties) $Y$, then $\text{med}(W) \le 3\text{med}(Y)$. \[Proved in Theorem 8 in [@anshelevich2015approximating]\]
The main easy insight which we use in the formation of our algorithm comes from the following lemma.
\[lemma-social-median4\] For any three alternatives $W$, $Y$, and $P$, if $P$ pairwise defeats (or pairwise ties) $Y$, and $d(Y, W) \le d(Y, P)$, then $\text{med}(W) \le 3 \text{med}(Y)$.
By Lemma \[lemma-social-median1\], $\text{med}(W) \le \text{med}(Y) + d(Y, W)$. By Lemma \[lemma-social-median2\], $\text{med}(Y) \ge \frac{d(Y, P)}{2}$. And we know that $d(Y, P) \ge d(Y, W)$, thus
$$\begin{aligned}
\text{med}(W) &\le \text{med}(Y) + d(Y, W)\\
&\le \text{med}(Y) + d(Y, P)\\
&\le \text{med}(Y) + 2\text{med}(Y)\\
&\le 3\text{med}(Y)
\end{aligned}$$
We use a natural Condorcet-consistent algorithm to approximate the minimum median social cost with the agents’ preference rankings $\sigma$ and the ordinal preferences of every alternative over other alternatives. First, create the majority graph $G=(\mathcal{F}, E)$, i.e., a graph with alternatives as vertices and an edge $(Y, Z) \in E$ if $Y$ pairwise defeats or pairwise ties $Z$. If a Condorcet winner (i.e. an alternative which pairwise defeats all others) exists, then we return it immediately.
Otherwise, we consider each pair of alternatives. By Lemma \[lemma-social-median3\], if the edge $(W, Y) \in E$, then $\text{med}(W) \le 3\text{med}(Y)$. When considering an alternative pair $W, Y$, if $(W, Y) \not\in E$ and we know that there exists another alternative $P$ which meets the conditions of Lemma \[lemma-social-median4\], then we add an edge $(W, Y)$ to $G$. It is not difficult to see that whenever the edge $(W, Y)$ is in our graph, this means that $\text{med}(W) \le 3\text{med}(Y)$. As we prove below, at the end of this process there always exists at least one alternative which has edges to all the other alternatives, and thus the distortion obtained from selecting it is at most 3, no matter which alternative is the true optimal one.
Note that from the ordinal preferences of alternatives over each other, we can get a partial order of distances between the alternatives. Denote this partial order as $\preceq$, i.e., we say that $d(W,Y)\preceq d(W,Z)$ if we know that $W$ prefers $Y$ to $Z$ (we do not have information about strict preference). This is the information we have on hand: we only know the partial order of distances between pairs of alternatives which share an alternative in common. Note, however, that if there exists a cycle in this partial order, i.e., $d(Y_1,Y_2) \preceq d(Y_2,Y_3) \preceq d(Y_3,Y_4) \preceq \dots \preceq d(Y_k,Y_1) \preceq d(Y_1,Y_2)$, then this implies that all the distances in the cycle are actually equal, and thus we can also add the relations $d(Y_1,Y_2) \succeq d(Y_2,Y_3) \succeq d(Y_3,Y_4) \succeq \dots \succeq d(Y_k,Y_1) \succeq d(Y_1,Y_2)$. Such cycles are easy to detect (e.g., by forming a graph with a node for every alternative pair and then searching for cycles), and thus we can assume that whenever a cycle exists in our partial order, then for every pair of distances $d(W,Y)$ and $d(W,Z)$ in the cycle, we have both $d(W,Y)\preceq d(W,Z)$ and $d(W,Y)\succeq d(W,Z)$.
If there is a Condorcet winner $W$, **return $W$ as the winner**.
There must exists an alternative $W$ such that $\forall Y \in \mathcal{F} - \{ W \}$, $(W, Y) \in E$. Return $W$ as the winner.
\[lemma-social-median5\] Consider the modified majority graph $G=(\mathcal{F}, E)$ at any point during Algorithm \[alg-social-median\]. For any edge $(W, Y) \in E$, we have that $\text{med}(W) \le 3\text{med}(Y)$.
By Lemma \[lemma-social-median3\], for any edge $(W, Y)$ in the original majority graph, $\text{med}(W) \le 3\text{med}(Y)$.
Now consider an edge $(W, Y)$ added to $E$ when processing the alternative pair $W, Y$. It must be the case that there exists an alternative $P$, such that $d(Y, W) \le d(Y, P)$ and $P$ pairwise defeats (or ties) $Y$. By Lemma \[lemma-social-median4\], $\text{med}(W) \le 3\text{med}(Y)$.
\[lemma-social-median6\] At the end of Algorithm \[alg-social-median\], there must exist an alternative $W$ such that $\forall Y \in \mathcal{F} - \{ W \}$, $(W, Y) \in E$.
We prove this lemma by contradiction. Suppose no such alternative $W$ exists. Then for each alternative $Y$, there is at least one alternative $Z$, such that only $(Z, Y) \in E$ and $(Y, Z) \not\in E$. This is because we start with the majority graph, so at least one edge always exists between every pair. We create another directed graph $G' = (\mathcal{F}, E')$, with $E'$ being all the edges $(Z,Y)$ such that $Y,Z\not\in E$. Thus any pair of alternatives in $G'$ have at most one direction of edge between them. And by our assumption, each alternative $Y$ has at least one incoming edge in $G'$. Since the in-degree of each node is at least 1 in $G'$, there must be at least one cycle in $G'$. To see this, one can for example take the edge $(Y_2,Y_1)$ coming into $Y_1$, then the edge $(Y_3,Y_2)$ coming into $Y_2$, and proceed in this way until a cycle is formed. Note that every edge in $G'$ must be in the original majority graph, because if we add an edge when processing a pair of alternatives in our algorithm, that pair must have edges in both directions.
Consider a cycle formed by edges $(Y_1, Y_2)$, $(Y_2, Y_3)$, …, $(Y_{k-1}, Y_k)$, $(Y_k, Y_1)$. When processing the alternative pair $Y_2, Y_3$ in Algorithm \[alg-social-median\], we did not add edge $(Y_3, Y_2)$ to $E$, so it must be the case that no alternative $P$ exists such that $d(Y_2, Y_3) \preceq d(Y_2, P)$ and $P$ pairwise defeats (or ties) $Y_2$. But we know that $Y_1$ pairwise defeats (or ties) $Y_2$, because edge $(Y_1, Y_2)$ is in the original majority graph. Then the only possibility is we don’t know if $d(Y_2, Y_3) \le d(Y_1, Y_2)$, i.e., either $d(Y_2, Y_3)$ and $d(Y_1, Y_2)$ are incomparable in our partial order, or we only know that $d(Y_2, Y_3) \succeq d(Y_1, Y_2)$. They cannot be incomparable, since we have the ordinal preferences of $Y_2$ for $Y_1$ and $Y_3$, thus our partial order must state that $d(Y_2, Y_3) \succeq d(Y_1, Y_2)$, i.e., $Y_2$ prefers $Y_1$ to $Y_3$. By the same reasoning, we also get that $Y_3$ prefers $Y_2$ to $Y_4$, and more generally that $Y_i$ prefers $Y_{i-1}$ to $Y_{i+1}$ for all $i$, where $Y_0=Y_k$ and $Y_{k+1}=Y_1$ since it is a cycle. This means that in our partial order, we have that $d(Y_1, Y_2) \preceq d(Y_2, Y_3) \preceq \dots \preceq d(Y_{k-1}, Y_k) \preceq d(Y_k, Y_1) \preceq d(Y_1, Y_2)$. Recall, however, that this means we know $d(Y_1, Y_2) = d(Y_2, Y_3) = \dots = d(Y_{k-1}, Y_k) = d(Y_k, Y_1)$, and before running Algorithm \[alg-social-median\], we detect cycles in the partial order of alternative distances, and add the equality information to the partial order. This means that whenever $d(Y_1, Y_2) \preceq d(Y_2, Y_3) \preceq \dots \preceq d(Y_{k-1}, Y_k) \preceq d(Y_k, Y_1) \preceq d(Y_1, Y_2)$ exists in our partial order, we also have $d(Y_1, Y_2) \succeq d(Y_2, Y_3) \succeq \dots \succeq d(Y_{k-1}, Y_k) \succeq d(Y_k, Y_1) \succeq d(Y_1, Y_2)$ in the partial order as well. But this gives us a contradiction, since having $d(Y_2, Y_3) \preceq d(Y_1, Y_2)$ in the partial order, combined with the fact that $Y_1$ pairwise defeats $Y_2$, would cause us to add the edge $(Y_3,Y_2)$ in our algorithm, which contradicts the statement that only the edge $(Y_2,Y_3)$ is in the final graph produced by the algorithm, but not $(Y_3,Y_2)$. Thus there must exist at least one alternative with edges from it to all the others.
\[thm-social-median\] The distortion of Algorithm \[alg-social-median\] for minimum median social cost is at most 3.
If there is a Condorcet winner, by Lemma \[lemma-social-median3\], the distortion is at most 3.
Otherwise, by Lemma \[lemma-social-median6\], the algorithm always returns a winner. Suppose it returns alternative $W$ as the winner, by Lemma \[lemma-social-median5\], $W$ has a distortion at most 3 with any alternative $X$ as the optimal solution.
### Generalizing Median: Percentile Distortion
Instead of just considering the median objective, we also consider a more general objective: the $\alpha$-percentile social cost. Let $\alpha\text{-PC}(Y)$ denote the value from the set $\{ d(i, Y) : i \in \mathcal{A} \}$, that $\alpha$ fraction of the values lie below $\alpha\text{-PC}(Y)$. Thus median is a special case when $\alpha=\frac{1}{2}$, $\text{med}(Y) = \frac{1}{2}\text{-PC}(Y)$. It was shown in [@anshelevich2015approximating] Theorem 17 that the worst-case distortion when $\alpha \in [0, \frac{1}{2}]$ in that setting (only have agent’s ordinal preferences over alternatives) is unbounded, and the same example shows $\alpha \in [0, \frac{1}{2}]$ in our setting is also unbounded. However, we are able to give a distortion of 3 for $\alpha \in [\frac{1}{2}, 1]$ in this paper, while for the setting in [@anshelevich2015approximating], the lower bound for distortion when $\alpha \in [\frac{1}{2}, \frac{2}{3}]$ is 5. The reason is that the ordinal preferences between alternatives are also available in our setting. We will show that Algorithm \[alg-social-median\] gives a distortion of at most 3 not only for the median objective, but also for the general $\alpha$-percentile objective, because all the lemmas we used to prove the conclusion for the median objective could be generalized to $\alpha$-percentile.
We use the following lemma from [@anshelevich2015approximating] in the proof of our algorithm:
\[lemma-social-median-alpha1\] For any two alternatives $W$ and $Y$, we have $\alpha\text{-PC}(W) \le \alpha\text{-PC}(Y) + d(Y,W)$. \[Lemma 18 in [@anshelevich2015approximating]\]
We can generalize Lemma \[lemma-social-median4\] to the following lemma, and the proof is by using Lemma \[lemma-social-median-alpha1\] instead of Lemma \[lemma-social-median1\] in the proof of Lemma \[lemma-social-median4\],
\[lemma-social-median-alpha2\] For any three alternatives $W$, $Y$, and $P$, if $P$ pairwise defeats (or pairwise ties) $Y$, and $d(Y, W) \le d(Y, P)$, then $\alpha\text{-PC}(W) \le 3 \alpha\text{-PC}(Y)$.
\[thm-social-median-alpha\] The distortion of Algorithm \[alg-social-median\] for the $\alpha\text{-PC}$ objective social cost with $\frac{1}{2} \le \alpha \le 1$ is at most 3.
Note that Lemma \[lemma-social-median-alpha1\] is actually a generalization of Lemma \[lemma-social-median1\], and Lemma \[lemma-social-median-alpha2\] is a generalization of Lemma \[lemma-social-median4\]. Lemma \[lemma-social-median2\] and Lemma \[lemma-social-median3\] also generalize to the $\alpha\text{-PC}$ objective, because when $\frac{1}{2} \le \alpha \le 1$, for any alternative $Y$, we know $\alpha\text{-PC}(Y) \ge \text{med}(Y)$. Then Lemma \[lemma-social-median5\] also generalizes to the $\alpha\text{-PC}$ objective, because it only uses Lemma \[lemma-social-median3\] and Lemma \[lemma-social-median4\] in the proof. And Lemma \[lemma-social-median6\] still holds for the same algorithm. Thus all the lemmas and properties of the median objective used in the proof of Theorem \[thm-social-median\] could be generalized into the $\alpha\text{-PC}$ objective, so the conclusion still holds for the $\alpha\text{-PC}$ objective when $\frac{1}{2} \le \alpha \le 1$.
### Algorithm \[alg-social-median\] and the Total Social Cost
Although Algorithm \[alg-social-median\] is designed for the median objective, it also performs quite well for the sum objective. Interestingly, the distortion of this algorithm for the minimum total social cost is at most 5, which is the same as Copeland (the best known deterministic algorithm with no knowledge of candidate preferences). Thus this algorithm gives a distortion of 3 for median (and in fact for all $\alpha$-percentile objectives) and distortion of 5 for sum simultaneously. In settings where we are not sure which objectives to optimize, or ones where we care both about the total social good, and about fairness, this social choice mechanism provides the best of both worlds. The lemmas and proofs for this result are similar to Theorem \[thm-social-median\], as follows.
\[lemma-social-medians-sum1\] Let $W, Y$ be alternatives $\in \mathcal{F}$. If $W$ pairwise defeats (or pairwise ties) $Y$, then $SC_{\sum}(W, \mathcal{A}) \le 3 SC_{\sum}(Y, \mathcal{A})$. \[Proved in Theorem 7 in [@anshelevich2015approximating]\]
\[lemma-social-median-sum2\] For any three alternatives $W$, $Y$, and $P$, if $P$ pairwise defeats (or pairwise ties) $Y$, and $d(Y, W) \le d(Y, P)$, then $SC_{\sum}(W, \mathcal{A}) \le 5 SC_{\sum}(Y, \mathcal{A})$.
For all $i \in \mathcal{A}$, we know $d(i, W) \le d(i, Y) + d(Y, W)$ by the triangle inequality. Summing up for all $i \in \mathcal{A}$, we get $SC_{\sum}(W, \mathcal{A}) \le SC_{\sum}(Y, \mathcal{A}) + n \cdot d(Y, W)$.
$P$ pairwise defeats (or pairwise ties) $Y$, so at least half of the agents prefer $P$ to $Y$; thus the total social cost of $Y$ is at least the sum of the social cost of these half of agents. By Lemma \[lemma-social-basic\], we get $SC_{\sum}(Y, \mathcal{A}) \ge \frac{n}{2} \frac{d(Y, P)}{2} = \frac{n}{4} d(Y, P)$. Thus,
$$\begin{aligned}
SC_{\sum}(W, \mathcal{A}) &\le SC_{\sum}(Y, \mathcal{A}) + n \cdot d(Y, W)\\
&\le SC_{\sum}(Y, \mathcal{A}) + n \cdot d(Y, P)\\
&\le SC_{\sum}(Y, \mathcal{A}) + 4 SC_{\sum}(Y, \mathcal{A})\\
&\le 5 SC_{\sum}(Y, \mathcal{A})
\end{aligned}$$
\[lemma-social-median-sum3\] Consider the modified majority graph $G=(\mathcal{F}, E)$ at any point during Algorithm \[alg-social-median\]. For any edge $(W, Y) \in E$, we have that $SC_{\sum}(W, \mathcal{A}) \le 5 SC_{\sum}(Y, \mathcal{A})$.
By Lemma \[lemma-social-medians-sum1\], for any edge $(W, Y)$ in the original majority graph, $SC_{\sum}(W, \mathcal{A}) \le 3 SC_{\sum}(W, \mathcal{A})$.
Now consider an edge $(W, Y)$ added to $E$ when processing the alternative pair $W, Y$. It must be the case that there exists an alternative $P$, such that $d(Y, W) \le d(Y, P)$ and $P$ pairwise defeats (or ties) $Y$. By Lemma \[lemma-social-median-sum2\], $SC_{\sum}(W, \mathcal{A}) \le 5 SC_{\sum}(Y, \mathcal{A})$.
\[thm-social-median-sum\] The distortion of Algorithm \[alg-social-median\] for minimum total social cost is at most 5, and this bound is tight.
If there is a Condorcet winner, by Lemma \[lemma-social-medians-sum1\], the distortion is at most 3. Otherwise, suppose the algorithm returns alternative $W$ as the winner; by Lemma \[lemma-social-median-sum3\] $W$ has a distortion at most 5 with any alternative $X$ as the optimal solution.
To see that this bound is tight, consider the following example. There are three facilities $W$, $Y$, and $P$. There are $q$ agents who prefer $Y$ to $W$ to $P$, $q$ agents who prefer $P$ to $Y$ to $W$, and 1 agent who prefers $W$ to $P$ to $Y$. We denote these three sets of agents as $\mathcal{A}_Y$, $\mathcal{A}_P$ and $\mathcal{A}_W$ separately. By the preferences of agents, we know that $Y$ pairwise defeats $W$, $W$ pairwise defeats $P$, and $P$ pairwise defeats $Y$. The distances between facilities are: $d(Y, W) = 2 - 2\epsilon$, $d(W, P) = 2 - \epsilon$, $d(P, Y) = 2$, where $\epsilon$ is a very small positive number. $\mathcal{A}_Y$ is located at the same location as $Y$, so $d(\mathcal{A}_Y, Y) = 0$, $d(\mathcal{A}_Y, P)$ = 2, and $d(\mathcal{A}_Y, W) = 2 - 2\epsilon$. The distances between $\mathcal{A}_P$ and the alternatives are: $d(\mathcal{A}_P, Y) = d(\mathcal{A}_P, P) = 1$, $d(\mathcal{A}_P, W) = 3 - 2\epsilon$. $\mathcal{A}_W$ has a distance of 1 to all alternatives. Run Algorithm \[alg-social-median\] on this example, and consider the alternative pair $W$, $Y$. Because $P$ pairwise defeats $Y$ and $d(Y, W) \preceq d(Y, P)$, we add edge $(W, Y)$ to the graph and make $W$ the winner. The total social cost of $W$ is $q * (2-2\epsilon) + q * (3 - 2\epsilon) + 1 = q(5-4\epsilon) + 1$. While the optimal solution is to choose $Y$ as the winner, and get a total social cost of $q + 1$. When $q$ is very large and $\epsilon$ is very small, the distortion in this example approaches 5.
Model and Notation: Facility Assignment Problems {#section-general-model}
================================================
The mechanism we used for approximation of total social cost in Theorem \[thm-social-sum\] can be applied to far more general problems. In this section, we describe a set of facility assignment problems that fit in this framework. As before, let $\mathcal{A} = \{ 1, 2, \dots, n\}$ be a set of agents, and $\mathcal{F} = \{ F_1, F_2, \dots, F_m\}$ be a set of facilities, with each agent $i$ having a preference ranking $\sigma_i$ over the facilities, and $\sigma = (\sigma_1, \dots, \sigma_n)$.
As in the social choice model, we assume that there exists an arbitrary unknown metric $d: (\mathcal{A} \cup \mathcal{F})^2 \rightarrow \mathbb{R}_{\ge 0}$ on the set of agents and facilities. The distances $d(i, F_j)$ between agents and facilities are unknown, but the ordinal preferences $\sigma$ and the distances $l$ between facilities are given. Let $\mathcal{D}(\sigma, l)$ be the set of metrics consistent with $\sigma$ and $l$, as defined previously in Section \[section-social-model\].
Unlike for social choice, our goal is now to choose which facilities to open, and which agents should be assigned to which facilities. Formally, we must choose an assignment $x:\mathcal{A}\rightarrow \mathcal{F}$, where $x(i)$ is the facility that $i$ is assigned to. Every $i \in \mathcal{A}$ must be assigned to one (and only one) facility in $\mathcal{F}$; other than that, there could be arbitrary constraints on the assignment. Here are some examples of constraints which fall into our framework: each facility $F_i$ has a capacity $c_i$, which is the maximum number of agents that can be assigned to $F_i$; at least (or at most) $p$ facilities should have agents assigned to them; agents $i$ and $j$ must be (or must not be) assigned to the same facility, etc. The social choice model is a special case of this one with the constraint that exactly one facility must be opened, and all agents must be assigned to it. Note that the constraints are only on the assignment, and independent of the metric space $d$. An assignment $x$ is valid if it satisfies all constraints. Let $\mathcal{X}$ be the set of all valid assignments.
#### The cost function of assignments.
The cost of an assignment $x$ consists of two parts. The first part is the distance cost between agents and facilities. $\forall i \in \mathcal{A}$, let $s_i$ denote the distance between $i$ and the facility it is assigned to, i.e., $s_i=d(i,x(i))$. For a given metric $d$ and assignment $x$, let $s(x, d)$ denote the vector of distances between each $i \in \mathcal{A}$ and $x(i)$, i.e., $s(x, d) = (s_1, s_2, \dots, s_n)$. Let $c_d: \mathbb{R}_{\ge 0}^n \rightarrow \mathbb{R}_{\ge 0}$ be a cost function that takes a vector of distances as input. For example, this could simply sum up all the distances, take the maximum distance for an egalitarian objective, etc. To be as general as possible, instead of fixing a specific function $c_d$ we consider the set of distance cost functions that are monotone nondecreasing and subadditive. Formally, $c_d$ is monotonically nondecreasing means that for any vectors $s$ and $s'$ such that $s\leq s'$ componentwise, we have that $c_d(s) \le c_d(s')$. Any reasonable cost function should satisfy this property if agents desire to be assigned to closer facilities. $c_d$ being subadditive means that for any vectors $s$ and $s'$, we have that $c_d(s+s') \le c_d(s) + c_d(s')$. While not all functions are subadditive, many important ones are, as they represent the concept of “economies of scale", a common property of realistic costs. The second part of the assignment cost is the facility cost. Let $c_f(x)$ denote the facility cost for assignment $x$. $c_f$ can be an [*arbitrary*]{} function over the assignments, for example, the opening cost of facilities, the penalty (or reward) for assigning certain agents to the same facility, etc. Our framework includes all such functions, and thus is quite general, as we discuss below. The main components needed for our framework to work is that the function $c_f$ does not depend on the distances, only on $x$, and that the function $c_d$ is subadditive.
The total cost $c(x, d)$ of an assignment $x$ is the sum of the distance cost and the facility cost, i.e. $c(x, d) = c_d(s(x, d)) + c_f(x)$. We study algorithms to approximate the minimum cost assignment given only agents’ ordinal preferences over facilities, and the distances between facilities, as described above.
#### Social Cost Distortion
As for social choice, we use the notion of distortion to measure the quality of an assignment in the worst case, similar to the notation in [@boutilier2015optimal; @procaccia2006distortion]. For any assignment $x$, we define the distortion of $x$ as the ratio between the social cost of $x$ and the optimal assignment:
$$dist(x, \sigma, l) = \sup\limits_{d \in \mathcal{D}(\sigma, l)} \frac{c(x, d)}{\min_{x' \in \mathcal{X}}c(x', d)}$$
A social choice function $f$ on $\mathcal{A}$ and $\mathcal{F}$ takes $\sigma$ and $l$ as input, and returns a valid assignment on $\mathcal{A}$ and $\mathcal{F}$. We say the distortion of $f$ on $\sigma$ and $l$ is the same as the distortion of the assignment returned by $f$. In other words, the distortion of an assignment function $f$ on a profile $\sigma$ and facility distances $l$ is the worst-case ratio between the social cost of $x = f(\sigma, l)$, and the social cost of the true optimal assignment, to obtain which we would need the true distances $d$.
#### Approximation ratio of omniscient algorithms
Consider omniscient algorithms which know the true numerical distances between agents and facilities for the facility assignment problems, in other words, the metric $d$. In some sense, the goal of our work is to determine when algorithms with only limited information can compete with such omniscient algorithms. With the full distances information, we can of course obtain the optimal assignment using brute force, while for our algorithms with limited knowledge this is impossible even given unlimited computational resources. Nevertheless, we are also interested in what is possible to achieve if we restrict ourselves to polynomial time. To differentiate traditional approximation algorithms from algorithms with small distortion, suppose that an omniscient approximation algorithm $\tilde{f}$ returns assignment $x$. Then we denote the approximation ratio of a valid assignment $x$ as:
$$ratio(x) = \frac{c(x, d)}{\min_{x' \in \mathcal{X}}c(x', d)}$$
Thus we say the approximation ratio of an omniscient algorithm $\tilde{f}$ is $\beta$ if for any input of the problem, the assignment $x$ returned by $\tilde{f}$ has $ratio(x) \le \beta$.
Examples of Facility Assignment Problems {#subsection-examples}
----------------------------------------
In this section we illustrate that our framework is quite general by giving various important examples which fit into our framework. In the section which follows, we prove a general black-box reduction theorem for our framework, and thus immediately obtain mechanisms with small distortion for all these examples simultaneously.
The total social cost problem we discussed in Section \[subsection-social-sum\] is a special case of the facility assignment problem such that the constraint is only one facility (alternative) is chosen, and all agents are assigned to it. For any assignment $x$, the facility cost function $c_f(x) = 0$, and the distance cost function $c_d(s(x, d))$ is the sum of distances from the winning alternative to all agents in the metric $d$. $c_d$ is monotone and additive (thus subadditive). Here are some other examples that fit in our framework:\
**Minimum weight metric bipartite matching.** Given a set of agents $\mathcal{A}$ and a set of facilities $\mathcal{F}$ such that $|\mathcal{A}| = |\mathcal{F}| = n$. $G = (\mathcal{A}, \mathcal{F}, E)$ is an undirected complete bipartite graph. The facilities and agents lie in a metric space $d$. The weight of each edge $(i, F) \in E$ is the distance between $i$ and $F$, $w(i, F) = d(i, F)$. The goal is to find a minimum weight perfect matching of the bipartite graph given only ordinal information. This setting has been studied before, and the best distortion bound known is $n$ [@caragiannis2016truthful] given by RSD for the case when only the ordinal preferences $\sigma$ are known. Our results show that if we also know the distances $l$ between facilities, then even without knowing the distances $d$ between agents and facilities, it is possible to create simple mechanisms with distortion at most 3 (we can show that no better bound is possible for this setting). Thus having a bit more information about the facilities immediately improves the distortion bound by a very large amount. We show this result by using our facility assignment framework above: the constraint here is that each facility has a capacity of 1, thus a valid assignment is a perfect matching of the bipartite graph. For any assignment $x$, the facility cost function is $c_f(x) = 0$, and the distance cost function $c_d(s(x, d))$ is the total edge weight in the assignment. $c_d$ is monotone and additive (thus subadditive).\
**Egalitarian bipartite matching.** With the same bipartite graph as in minimum weight matching problems, the only difference is that the goal of egalitarian bipartite matching is to find a perfect matching such that maximum edge weight (instead of the total weight) in the matching is minimized [@bogomolnaia2004random].
The egalitarian bipartite matching problem is the same as minimum weight bipartite matching except the distance cost function $c_d(s(x, d))$ is the maximum edge weight in the assignment. This function is also monotone and subadditive.\
**Metric Facility Location.** In this problem, one is given a set of agents $\mathcal{A}$ and a set of facilities $\mathcal{F}$ such that $|\mathcal{A}| = n$, $|\mathcal{F}| = m$. The facilities and agents lie in a metric space $d$. Each facility $F_j \in \mathcal{F}$ has an opening cost $f_j$. Each agent is assigned to a facility; in different versions there may be capacities on the number of agents assigned to a facility, lower bounds on the number of agents assigned to a facility, or various other constraints [@farahani2009facility]. The goal is to find a subset of facilities $\mathcal{\hat{F}} \subseteq \mathcal{F}$ to open, so that the sum of opening costs for facilities in $\mathcal{\hat{F}}$ and total distance of the assignment is minimized.
Our framework allows arbitrary constraints on what constitutes a valid assignment, which captures facilities with capacities or lower bounds if needed. For any assignment $x$, the facility cost function $c_f(x)$ is the sum of the opening costs $f_j$ for those facilities $F_j$ that have at least one agent assigned to it. The distance cost function $c_d(s(x, d))$ is the total distances in the assignment, which is monotone increasing and additive (thus subadditive).\
**$k$-center problem.** The goal in this classic problem is to open a set of $k$ facilities, with each agent assigned to the closest one. The optimal solution is the subset of $\mathcal{\hat{F}}$ which minimizes $\max_{i \in \mathcal{A}} d(i, x(i))$. To express this in our framework, the constraint is that no more than $k$ facilities have agents assigned to them. For any assignment $x$, the facility cost function $c_f(x) = 0$, and the distance cost function $c_d(s(x, d))$ is the maximum distance between any agent and facility in the assignment.\
**$k$-median problem.** This classic problem is the same as $k$-center, except the goal is to minimize the sum of distances of agents to the facilities instead of the maximum distance.
Distortion of Facility Assignment Problems {#section-facility-assignment}
==========================================
In this section, we study general facility assignment problems, as described in Section \[section-general-model\], and form mechanisms with small distortion. First, we construct a projected problem such that the distances between agents and facilities are known, so it could be solved by an omniscient algorithm. Then we map the result of the projected problem to the original problem and bound the distortion of the original problem.
Given agents $\mathcal{A} = \{ 1, 2, \dots, n\}$ and facilities $\mathcal{F} = \{ F_1, F_2, \dots, F_m\}$, suppose facility $F'$ is $i$’s top choice in $\mathcal{F}$. We create a new agent $\tilde{i}$ at the location of $F'$ in the metric space. Consequently, $\forall F \in \mathcal{F}$, $d(\tilde{i}, F) = d(F', F)$. Denote the set of the new agents as $\tilde{\mathcal{A}} = \{\tilde{1}, \tilde{2}, \dots, \tilde{n} \}$.
The original assignment problem is on agents $\mathcal{A}$ and facilities $\mathcal{F}$, and only ordinal preferences of agents in $\mathcal{A}$ over facilities are given. The projected problem is on agents $\tilde{\mathcal{A}}$ and facilities $\mathcal{F}$, and we know the actual distances between agents in $\tilde{\mathcal{A}}$ and facilities $\mathcal{F}$, since we know the distances $l$ between facilities. The constraints and costs $c_d$ and $c_f$ remain the same for both the original and the projected problem; the only difference is in the distances $d$. Our main result is that if we have a $\beta$-approximation assignment to the minimum assignment cost on the projected problem, then we can get an assignment that has a distortion of $2\beta + 1$ for the original problem in polynomial time.\
\[thm-facility-projection\] Given a valid assignment $\tilde{x}$ for the projected problem on $\tilde{\mathcal{A}}$ and $\mathcal{F}$, with $ratio(\tilde{x}) \le \beta$, the assignment $x(i)=\tilde{x}(\tilde{i})$ has distortion of at most $(1 + 2\beta)$ for original assignment problem on $\mathcal{A}$ and $\mathcal{F}$.
First, $\tilde{x}$ is a valid assignment for the projected problem on $\tilde{\mathcal{A}}$ and $\mathcal{F}$, so $x$ must also be a valid assignment for the original problem on $\mathcal{A}$ and $\mathcal{F}$. This is because the constraints are only on the assignment, and are independent of the metric space $d$. For the same reason, the facility cost of $x$ equals the facility cost of $\tilde{x}$, $c_f(x) = c_f(\tilde{x})$.
Now consider the distance cost of $x$. Let $x^*$ denote the optimal assignment for the original problem. $\forall i \in \mathcal{A}$, let $s_i = d(i, x(i))$, $t_i = d(i, \tilde{i})$, $b_i = d(\tilde{i}, x(i))$. Similarly, let $s_i^* = d(i, x^*(i))$, $b_i^* = d(\tilde{i}, x^*(i))$.
For any agent $i$ and facility $x(i)$, by triangle inequality,
$$\begin{aligned}
s_i = d(i, x(i)) &\le d(i, \tilde{i}) + d(\tilde{i}, x(i)) = t_i + b_i
\end{aligned}$$
Because $c_d$ is monotonically nondecreasing and subadditive,
$$\begin{aligned}
c_d(s_1, s_2, \dots, s_n) &\le c_d(t_1 + b_1, t_2 + b_2, \dots, t_n + b_n) \\
&\le c_d(t_1, t_2, \dots, t_n) + c_d(b_1, b_2, \dots, b_n)
\end{aligned}$$
Therefore, the cost of our assignment $x$ is bounded as follows: $$\begin{aligned}
c_f(x) + c_d(s(x, d)) &= c_f(x) + c_d(s_1, s_2, \dots, s_n) \\
&= c_f(\tilde{x}) + c_d(s_1, s_2, \dots, s_n) \\
&\le c_f(\tilde{x}) + c_d(t_1, t_2, \dots, t_n) + c_d(b_1, b_2, \dots, b_n)
\end{aligned}$$
Because $\tilde{i}$ is located at $i$’s top choice facility, and $x^*(i)$ is a facility, we thus know that $t_i \le s_i^*$, and by monotonicity $c_d(t_1, t_2, \dots, t_n) \le c_d(s_1^*, s_2^*, \dots, s_n^*)$. Thus,
$$\begin{aligned}
c_f(x) + c_d(s(x, d)) &\le c_f(\tilde{x}) + c_d(t_1, t_2, \dots, t_n) + c_d(b_1, b_2, \dots, b_n)\\
&\le c_f(\tilde{x}) + c_d(s_1^*, s_2^*, \dots, s_n^*) + c_d(b_1, b_2, \dots, b_n)
\end{aligned}$$
We know that $\tilde{x}$ is a $\beta$-approximation to the optimum assignment for the projected problem. Its total cost is exactly $c_f(\tilde{x}) + c_d(b_1, b_2, \dots, b_n)$, since the distance from $\tilde{i}$ to $\tilde{x}(\tilde{i})=x(i)$ is exactly $b_i$. Now consider another assignment for the projected problem, in which $\tilde{i}$ is assigned to $x^*(i)$. The cost of this assignment is $c_f(x^*) + c_d(b_1^*, b_2^*, \dots, b_n^*)$, by definition of $b_i^*$. Since $\tilde{x}$ is a $\beta$-approximation, we therefore know that
$$\begin{aligned}
c_f(\tilde{x}) + c_d(b_1, b_2, \dots, b_n) \leq \beta c_f(x^*) + \beta c_d(b_1^*, b_2^*, \dots, b_n^*),
\end{aligned}$$
and thus
$$\begin{aligned}
c_f(x) + c_d(s(x, d)) &\le c_f(\tilde{x}) + c_d(s_1^*, s_2^*, \dots, s_n^*) + c_d(b_1, b_2, \dots, b_n)\\
&\le c_d(s_1^*, s_2^*, \dots, s_n^*) + \beta c_f(x^*) + \beta c_d(b_1^*, b_2^*, \dots, b_n^*)
\end{aligned}$$
For any agent $i$ and facility $x^*(i)$ in $x^*$, by triangle inequality,
$$\begin{aligned}
b_i^* = d(\tilde{i}, x^*(i)) \leq d(i, x^*(i)) + d(i, \tilde{i}) \leq 2 d(i, x^*(i)) = 2s_i^*
% d(i, F_i^*) + d(i, \tilde{i}) &\ge d(\tilde{i}, F_i^*) \\
% 2 d(i, F_i^*) &\ge d(\tilde{i}, F_i^*) \\
% 2 s_i^* &\ge b_i^*\\
% 2 c_d(s_1^*, s_2^*, \dots, s_n^*) & \ge c_d(b_1^*, b_2^*, \dots, b_n^*)
\end{aligned}$$
$d(i, \tilde{i})\leq d(i, x^*(i))$ above since $\tilde{i}$ is located at the closest facility to $i$. Because $c_d$ is monotone and subadditive, we also have that
$$\begin{aligned}
c_d(b_1^*, b_2^*, \dots, b_n^*) \leq c_d(2s_1^*, 2s_2^*, \dots, 2s_n^*) \leq 2 c_d(s_1^*, s_2^*, \dots, s_n^*)
\end{aligned}$$
Putting everything together,
$$\begin{aligned}
c_f(x) + c_d(s(x, d)) %&= c_f(x) + c_d(s_1, s_2, \dots, s_n) \\
% &= c_f(\tilde{x}) + c_d(s_1, s_2, \dots, s_n) \\
% &\le c_f(\tilde{x}) + c_d(t_1, t_2, \dots, t_n) + c_d(b_1, b_2, \dots, b_n) \\
% &\le c_f(\tilde{x}) + c_d(s_1^*, s_2^*, \dots, s_n^*) + c_d(b_1, b_2, \dots, b_n)\\
&\le c_d(s_1^*, s_2^*, \dots, s_n^*) + \beta c_f(x^*) + \beta c_d(b_1^*, b_2^*, \dots, b_n^*)\\
&\le \beta c_f(x^*) + c_d(s_1^*, s_2^*, \dots, s_n^*) + 2\beta c_d(s_1^*, s_2^*, \dots, s_n^*)\\
&= \beta c_f(x^*) + (1 + 2\beta) c_d(s_1^*, s_2^*, \dots, s_n^*)\\
&\le (1 + 2\beta) (c_f(x^*) + c_d(s(x^*, d)))
\end{aligned}$$
Note that the above theorem immediately implies that if we are only concerned with what is possible to achieve given limited ordinal information in addition to distances between facilities, and are not worried about our algorithms running in polynomial time, then we can always form an assignment with distortion of at most 3 from knowing only $\sigma$ and $l$. This is because we can solve the projected problem with brute force, and then we have $\beta=1$. This bound of 3 is tight for many facility assignment problems: consider for example an instance of min-cost metric matching with two agents and two facilities, with both preferring $F_1$ to $F_2$. One of the agents has distance to $F_1$ of 0, and one is located halfway between $F_1$ and $F_2$, but since we only have ordinal information we do not know which agent is which. If we assign the wrong agent to $F_1$, then we end up with distortion of 3, and it is impossible to do better for any deterministic mechanism.
If on the other hand we want to form poly-time algorithms with small distortion, the above theorem gives a black-box reduction: if we have a $\beta$-approximation algorithm for the omniscient case, then we can form a $1+2\beta$-distortion algorithm for the ordinal case. Actually, we get a $1+2\beta$-distortion for the distance cost, and a $\beta$-distortion for the facility cost, which is shown in the second-to-last line of the proof for Theorem \[thm-facility-projection\]. This leads to the following corollaries:
We can achieve the following distortion in polynomial time:
1. At most 3 for the minimum weight bipartite matching problem.
2. At most 3 for Egalitarian bipartite matching.
3. At most 3.976 for the facility location problem (1.488-approximation for the facility cost, and 3.976-approximation for the distance cost).
4. At most 5 for the k-center problem.
5. At most 6.35 for the k-median problem.
Min-cost matching and egalitarian matching are poly-time solvable, so $\beta=1$. For the latter, one can fix the threshold weight $t$ such that every edge chosen should be at most $t$, and then determine if such a matching exists. Performing a binary search on $t$ gives an efficient algorithm. For facility location, one can use the omniscient algorithm which is a 1.488-approximation in [@li20111]. For the $k$-center problem, a greedy algorithm [@hochbaum1985best] gives a 2-approximation for the setting that agents are a subset of facilities, which is the case in our projected problem. [@byrka2014improved] gives a 2.675-approximation omniscient algorithm for the $k$-median problem when agents are a subset of facilities, thus it also gives a 2.675-approximation for our projected problem.
Note that the median function, unlike sum and maximum, is not subadditive, and thus does not fit into our framework. In fact, while both min-cost and egalitarian matching problems have algorithms with small distortion in our setting, the same is not possible for forming a matching where the objective function is the cost of the [*median*]{} edge: see the Appendix for a lower bound.
Conclusion
==========
In this paper, we provided two mechanisms to solve different social cost problems. The first one makes use of the distances between facilities and an omniscient algorithm to get a low distortion for general facility assignment problems. The second mechanism is a new voting rule for social choice which simultaneously achieves a distortion of 3 for many objectives, including the cost of the median voter, and a distortion of 5 for the total social cost at the same time. The first mechanism requires the full distances $l$, but only needs the top choice from each agent. Thus, it puts only a small load on the agents which submit their preferences, but requires the mechanism designer to collect more information about the facilities and their distances to each other. The second mechanism, on the other hand, only requires ordinal preference information from the facilities, but needs the full preference ranking from the agents instead of just the top choice. It is especially appropriate for settings in which the candidates or alternatives are agents themselves.
Many open questions remain for our setting. How well can facility location problems be approximated given information about facilities? While we established upper bounds on distortion, we have no lower bounds besides the trivial bound of 3. What about randomized mechanisms, or what if the mechanisms must be truthful? And more generally, exactly what information is enough to guarantee mechanisms with small distortion? While our results show that knowing information about facility locations is enough to result in small distortion, it may be possible that obtaining even a bit of targeted information would result in powerful approximation algorithms. We look forward to future work on this topic.
Bad Examples and Lower Bounds
=============================
Note that our Algorithm \[alg-social-median\] is only for social choice problems, and does not fit in the definition of our general facility assignment problems. This is because the median cost function, unlike sum and maximum, is not subadditive. In fact, while both min-cost and egalitarian matching problems have algorithms with small distortion in our setting, the same is not possible for forming a matching where the objective function is the cost of the [*median*]{} edge.
The worst-case distortion of the median-cost bipartite matching problem in a metric space (given both agent preference profiles and distances between facilities) is unbounded.
Consider the following example: there are three agents $a$, $b$, $c$, and three facilities $X$, $Y$, $Z$. The preferences of agents are: $a, b \in XYZ$, while $c \in ZXY$. The distances between facilities are: $l(X, Y) = 2$, $l(X, Z) = l(Y, Z) = 1000$. The distances between the agents and facilities are, of course, unknown. Consider the instance $d(c, Z) = \epsilon$, $d(a, X) = 2\epsilon$, and $d(b, X) = d(b, Y) = 1$. $\epsilon$ is a very small positive real number, and other distances not given obey triangle inequality. In this instance, the optimal solution is $x^* = \{ (a, X), (b, Y), (c, Z) \}$, which gives a median value of $2\epsilon$. But because $a$ and $b$ have the same preference profile, the instance could also be $d(c, Z) = \epsilon$, $d(b, X) = 2\epsilon$, and $d(a, X) = d(a, Y) = 1$. If we still return the assignment $x^*$ for this instance, the median would be $1$. The distortion is arbitrarily bad when $\epsilon$ approaches 0.
The following Theorems show some of the lower bounds mentioned in Table \[table\_results\].
The worst-case distortion for the facility location problem in a metric space (given only agents’ preference profiles) is unbounded.
Consider the following example: there are two agents 1, 2, and two facilities $X$, $Y$. Agent 1 prefers $X$ to $Y$, while agent 2 prefers $Y$ to $X$. The opening costs are: $c_f(X) = 1$, $c_f(Y) = 100$. We can choose to open one facility or both of them.
**Case 1.** Suppose we only open $X$. Consider the following distances between the agents and facilities: $d(1, X) = d(2, Y) = 1$, $d(1, Y) = d(2, X) = L$, for some very large $L$. If we only open $X$, then the total cost is $>L$. While the optimal solution is to open both $X$ and $Y$, which has a total cost of 103. The distortion is unbounded.
**Case 2.** Suppose we only open $Y$. Consider the same distances as in **Case 1**, then the total cost is also $L$. And the optimal solution still has a total cost of 103. The distortion is unbounded.
**Case 3.** Suppose we open both facilities. Consider the following distances between the agents and facilities: $d(1, X) = d(1, Y) = d(2, X) = d(2, Y) = \epsilon$, where $\epsilon$ is a very small positive real number. If we open both facilities, the total cost is $101 + 2\epsilon$. While the optimal solution is to only open $X$ , which has a total cost of $1 + 2\epsilon$. If we increase $c_f(Y)$, the approximation ratio is unbounded.
The worst-case distortion for the k-median problem in a metric space (given only agents’ preference profiles) is at least $\Omega(n)$.
Consider the following example: There are three facilities $X$, $Y$, and $Z$. There are $q$ agents who prefer $X$ to $Y$ to $Z$, $q$ agents who prefer $Y$ to $X$ to $Z$, and 1 agent who prefers $Z$ to $X$ to $Y$. We denote these three sets of agents as $\mathcal{A}_X$, $\mathcal{A}_Y$ and $\mathcal{A}_Z$ separately. Suppose $k=2$, then we have three choices of the winners:
**Case 1.** Choose $X, Y$ as the winners. Consider the following distances between agents and facilities: $d(X, Y) = 1$, $d(Y, Z) = d(X, Z) = L$ for some very large $L$. $\mathcal{A}_X$ is located at the same location as $X$, $\mathcal{A}_Y$ is at the same location as $Y$, and $\mathcal{A}_Z$ is at the same location as $Z$. The cost of choosing $X, Y$ as the winners is $L$ because we need to assign the agent in $\mathcal{A}_Z$ to $X$ or $Y$. While the optimal solution is to choose $Y, Z$ as the winners, and get a total cost of $q$. So the distortion in this case is unbounded.
**Case 2.** Choose $X, Z$ as the winners. Consider the following distances between agents and facilities: $d(X, Y) = d(Y, Z) = d(X, Z) = 1$, and $\mathcal{A}_X$ locate on top of $X$, $\mathcal{A}_Y$ locate on top of $Y$, and $\mathcal{A}_Z$ locate on top of $Z$. The cost of choosing $X, Z$ as the winner is $q$, while the optimal solution is to choose $X, Y$ as the winners, and get a total cost of $1$. The distortion is $q$ in this case.
**Case 3.** Choose $Y, Z$ as the winners. Consider the same distances as in **Case 2**. If we choose $Y, Z$ as the winners, the total cost is still $q$, and the distortion of this case is also $q$.
The total number of agents is $n = 2q + 1$, so we can conclude that the distortions in all these three cases are at least $\Omega(n)$.
The worst-case distortion of the egalitarian bipartite matching problem in a metric space (given only agents’ preference profiles) is at least $2$.
Consider the following example: there are two agents 1, 2, and two facilities $X$, $Y$. Both agents prefer $X$ to $Y$. W.L.O.G., assume we match agent 1 to $X$, and agent 2 to $Y$. Suppose the distances between agents and facilities are: $d(1, X) = d(1, Y) = 1$, $d(2, X) = \epsilon$, $d(2, Y) = 2$, where $\epsilon$ is a very small positive real number. The egalitarian cost of our matching is $2$, while the optimal solution is to match agent 1 to $Y$, and agent 2 to $X$, which has a cost of $1$.
|
---
abstract: 'The Fano line profiles, originally discovered in the context of photoionization, have been found to occur in a large class of systems like resonators, metamaterials, plasmonics. We demonstrate the existence of such resonances in cavity optomechanics by identifying the interfering contributions to the fields generated at antiStokes and Stokes frequencies. Unlike the atomic systems, the optomechanical systems provide great flexibility as the width of the resonance is controlled by the coupling field. We further show how the double cavities coupled by a single optomechanical mirror can lead to the splitting of the Fano resonance and how the second cavity can be used to tune the Fano resonances. The Fano resonances are quite sensitive to the decay parameters associated with cavities and the mechanical mirror. Such resonances can be studied by both pump probe experiments as well as the spectrum of the quantum fluctuations in the output fields.'
author:
- 'Kenan Qu and G. S. Agarwal'
title: Fano resonances and their control in optomechanics
---
Introduction
============
The Fano line shapes [@Fano] have played an important role in our understanding of the photoelectron spectra [@Bransden] in atomic physics and more recently in the field of plasmonics [@plasmonics]. In atomic systems the interference minimum in the Fano line shape has been extensively used in extracting information on the relative strengths of the different decay channels which are responsible for interference effects. The Fano interference has been at the forefront in producing lasing without population inversion [@LWI]. The Fano line shapes have been considered as a probe of decoherence [@note1]. The Fano interference is leading to other remarkable possibilities like the improvement in the efficiency of the heat engines [@heatengine]. In plasmonics, interference effects play a very important role as photons can travel along different interfering paths and thus the Fano line shapes become quite common. The asymmetry of the Fano line shape can be continuously tuned by controlling the rate of saturation of two interference paths [@fanoRMP]. Such line shapes are also used in obtaining information on the interaction between nano particles with light [@plasmonics]. Interestingly enough the Fano line shapes can be understood rather easily by using a two coupled oscillator model in which the two oscillators have quite different lifetimes [@AJP]. Thus the lifetimes of the modes are important in the realization of the Fano line shapes. In fact, plasmonic systems and metamaterial structures are designed keeping the lifetime issue in mind [@plasmonics]. It turns out that the lifetime conditions are well satisfied for the optomechanical systems (OMS) since the damping of the mirror is much smaller than the cavity damping.
In the last few years the OMS have shown the possibility of a variety of interference effects [@OMIT; @Painter]. Clearly such a system, under certain condition, can exhibit Fano line shape [@Fano-OMS1; @Fano-OMS2] and in fact [@Painter] reports the observation of the Fano line shape. In this paper, we analyze the physics and mathematics behind Fano resonances in optomechanics and report the possibility of double Fano resonance in coupled OMS. The double Fano resonances which are interesting in their own right can provide information on a variety of optomechanical couplings. The Fano minima can also be exploited for the successful transfer of states and transduction of photons. The double cavity set up can lead to the tunability of the Fano resonances, which in fact has been an important issue in other contexts [@gu]. The double-cavity OMS are not the only ones for the observation of double Fano resonances, for example, we can use electromechanical systems with several superconducting capacitors [@qlimit].
The outline of this paper is as follows: In Sec. \[analyze\], we recall the Fano line shapes and physically explain how these can arise in OMS. In Sec. \[model\], we introduce the model for OMS and derive the line profiles under steady state conditions. In Sec. \[single\], we examine the Fano resonance in a single cavity OMS and derive its resonance frequency center as well as its width. We also quantify the Fano shape by using the Fano asymmetry parameter. In Sec. \[double\], we consider the double-cavity OMS and show the existence of the double Fano lineshape in different coupling regimes. We explain how we can tune Fano resonances by changing the coupling fields. In Sec. \[quantum\], we calculate the quantum fluctuations of the fields in OMS, and demonstrate the existence of Fano resonances in the spectrum of quantum fluctuations.
Fano resonances from interfering paths in cavity optomechanics {#analyze}
==============================================================
In the classic work [@Fano], Fano considered photoionization process when a weakly bound state $|a\rangle$ lies in the continuum $|E\rangle$. The weakly bound state has a finite life time due to its coupling with the continuum. Thus, in the simplest case, there are two transition amplitudes leading to photoionization—one involves a direct transition to the continuum and the other involves transition via the autoionizing state to the continuum. These two transition amplitudes interfere leading to the famous Fano formula for the probability $p$ for the photoionization $$\label{0}
p(E)=\frac{(\epsilon+q)^2}{\epsilon^2+1}, \qquad \epsilon=\frac{2(E-E_a)}{\Gamma},$$ where $q$ is called the Fano asymmetry parameter and $\Gamma$ is the width of the state $|a\rangle$. The Fano minimum occurs at $\epsilon=-q$. The parameter $q$ depends on the relative strengths of the independent transitions to the state $|a\rangle$ and $|E\rangle$. If $q$ is large, interference disappears.
Clearly, to produce Fano line shapes, we need two paths which interfere. Let us see, in terms of the physical processes, the possible physical mechanisms which lead to the building up of the cavity field in an OMS as shown in Fig. \[Fig1\]a.
![\[Fig1\][(Color online) (a) Schematic of OMS and (b) photon-phonon interaction.]{}](fig-1.eps){width="40.00000%"}
Here the field $\mathcal{E}_p$ with frequency $\omega_p$ is close to the cavity frequency $\omega$ and the field $\mathcal{E}_c$ is red detuned from the cavity resonance by an amount close to the oscillating frequency of the mirror. If the mirror position $x$ were fixed, the cavity field at $\omega_p$ will have the form $\mathcal{E}_p/[\kappa+i(\omega-\omega_p)]$, where $2\kappa$ is the rate at which the photons leak out of the left mirror. If the mirror is active, due to the radiation pressure interaction, we have the possibility of Stokes and antiStokes processes. The fields $\mathcal{E}_c$ and $\mathcal{E}_p$ can produce a coherent photon $x(\neq0)$. Once the coherent phonon is produced it can combine with $\mathcal{E}_c$ to produce the cavity field at $\omega_p$. In the case of a moving mirror, the two successive nonlinear frequency conversion processes between two light fields and a phonon as illustrated in Fig. \[Fig1\]b would produce the cavity field. Therefore, in OMS, we have two coherent processes leading to the building up of the cavity field. These are: (a) direct building up due to the application of $\mathcal{E}_p$; and (b) building up due to the nonlinear processes in Fig. \[Fig1\]b. These two processes interfere leading to the Fano resonances in OMS. The precise mathematical treatment is given in Sec. IV, where we introduce the analog of the Fano asymmetry parameter $q$.
In atomic physics, Fano profile have been extensively studied. Many generalizations to several resonances in continuum or interactions with different continua exist. The different continua can be either radiative or ionizing. Radiative continuum tends to make the value of the minimum nonzero [@note1].
Basic Model {#model}
===========
We study the system in Fig. \[Fig2\], in which a mechanical resonator, coated with perfect reflecting films on both sides, is coupled to two cavities.
![\[Fig2\][(Color online) Schematic double-cavity OMS.]{}](fig-2.eps){width="35.00000%"}
The mechanical resonator is modeled as a harmonic oscillator with mass $m$, frequency $\omega_m$ and momentum decay rate $\gamma_m$. For each cavity, we denote its field by $a_i$, frequency $\omega_i$ and decay rate $\kappa_i$, $i=1,2$. The field annihilation and creation operators satisfy the commutation relation $[a_i,a^\dag_j]=\delta_{ij}$. The probe laser with frequency $\omega_p$, is sent into cavity $1$. Other possible realizations would be using “membrane in the middle” setup [@membrane]. The two cavities are coupled only via the oscillations of the mechanical mirror which are produced by the applied strong laser fields $\mathcal{E}_{ci}$’s. Further the two cavities can be in different frequency regimes. We introduce normalized coordinates $Q=\sqrt{m\omega_m/\hbar}x$ and $P=\sqrt{1/(m\hbar\omega_m)}p$ for the mechanical oscillator with commutation relations $[Q,P]=[x,p]/\hbar=i$. The Hamiltonian for this system is given by $$\begin{aligned}
\label{1}
H &= H_1 + H_2 + H_m + H_\text{diss}, \nonumber \\
H_1 &= \hbar(\omega_1-\omega_{c1})a_1^\dag a_1 - \hbar g_1 a_1^\dag a_1 Q + i\hbar\mathcal{E}_{c1}(a_1^\dag-a_1) \nonumber \\
&\qquad + i\hbar(\mathcal{E}_p a_1^\dag \mathrm{e}^{-i\delta t} - \mathcal{E}_p^* a_1 \mathrm{e}^{i\delta t}) \nonumber \\
H_2 &= \hbar(\omega_2-\omega_{c2})a_2^\dag a_2 + \hbar g_2 a_2^\dag a_2 Q + i\hbar\mathcal{E}_{c2}(a_2^\dag-a_2) \nonumber \\
H_m &= \frac12\hbar\omega_m(P^2+Q^2),\end{aligned}$$ where $\delta=\omega_p-\omega_{c1}$ is the detuning between the probe field and the coupling field in cavity $1$; and coupling coefficients are defined by $g_i=(\omega_{i}/L_i)\sqrt{\hbar/(m\omega_m)}$ with $L_i$ being the length of the $i$th cavity. The $\mathcal{E}$ terms in (\[1\]) denote the coupling of the cavity field to the applied laser fields. The coupling and probe fields are related to the power of the applied laser fields via $\mathcal{E}_{ci} = \sqrt{2\kappa_iP_i/(\hbar\omega_i)}$, and $i=1,2$ and $\mathcal{E}_p = \sqrt{2\kappa_1P_{p}/(\hbar\omega_p)}$, respectively. In Eq. (\[1\]), all the dissipative interactions are denoted by $H_\text{diss}$. These include the leakage of photons from both cavities and the Brownian motion of the mirror. The Hamiltonian (\[1\]) has been written by working in a picture so the very fast frequencies $\omega_{ci}$’s are removed. This results in detuning terms like $(\omega_i-\omega_{ci})a_i^\dag a_i$.
Using Eq. (\[1\]) we can derive the quantum Langevin equations for the operators $Q$, $P$, $a_i$ and $a_i^\dag$. However, for the purpose of this paper, we will work with semiclassical equations so that all operator expectation values are replaced by their mean values. Thus in the rest of the paper, all the quantities $Q$, $P$, $a_i$ and $a_i^\dag$ will be numbers. The equations of motion for $Q$, $P$, $a_i$ and $a_i^\dag$ are found to be $$\label{2}
\begin{aligned}
\dot{Q} & =\omega_{m}P, \\
\dot{P} & =(g_{1}a_{1}^{\dagger}a_{1}-g_{2}a_{2}^{\dagger}a_{2})-\omega_{m}Q-\gamma_{m}P, \\
\dot{a}_{1} & = -i(\omega_1-\omega_{c1}-g_1Q)a_{1} -\kappa_{1}a_{1} +\mathcal{E}_{c1} +\mathcal{E}_{p}\mathrm{e^{-i\delta t}}, \\
\dot{a}_{2} & = -i(\omega_2-\omega_{c2}+g_2Q)a_{2}-\kappa_{2}a_{2}+\mathcal{E}_{c2}.
\end{aligned}$$ Eqs. (\[2\]) involve periodically oscillating terms hence in the long time limit, any of the fields and the mechanical coordinates will have a solution of the form $A=\sum_{n=-\infty}^{+\infty} \mathrm{e}^{-in\delta t}A_n$. The $A_n$’s can be obtained by the Floquet analysis. We assume that the probe is much weaker than the coupling field, then $A_n$’s can be obtained perturbatively. In particular, we find the steady state results to first order in $|\mathcal{E}_{p}/\mathcal{E}_{ci}|$ $$\begin{aligned}
a_{10} &= \frac{\mathcal{E}_{c1}}{\kappa_1+i\Delta_1}, \qquad a_{20} = \frac{\mathcal{E}_{c2}}{\kappa_2+i\Delta_2}, \label{7}\\
Q_0 &= \frac{1}{\omega_m}(g_1|a_{10}|^2-g_2|a_{20}|^2), \label{8}\\
Q_+ &= -\frac{1}{d(\delta)}\frac{g_1a_{10}^*\mathcal{E}_{p}}{(\kappa_1+i\Delta_1 -i\delta)}, \label{9}\\
d(\delta) &= \sum_{i=1,2} \frac{2\Delta_ig_i^2|a_{i0}|^2}{(\kappa_i-i\delta)^2 + \Delta_i^2} - \frac{\omega_m^2-\delta^2-i\delta\gamma_m}{\omega_m}, \label{10} \\
a_{1+} &= \frac{ig_1a_{10}}{(\kappa_1+i\Delta_1-i\delta)}Q_+ + \frac{\mathcal{E}_p}{(\kappa_1+i\Delta_1-i\delta)}, \label{11} \\
a_{1-} &= \frac{ig_1a_{10}}{(\kappa_1+i\Delta_1+i\delta)}Q_+^*, \label{12} \\
a_{2+} &=\frac{-ig_2a_{20}}{(\kappa_2+i\Delta_2-i\delta)}Q_+, \label{13}\\
a_{2-} &=\frac{-ig_2a_{20}}{(\kappa_2+i\Delta_2+i\delta)}Q_+^*, \label{14}\end{aligned}$$ where $\Delta_1=\omega_1-\omega_{c1}-g_1Q_0$ and $\Delta_2=\omega_2-\omega_{c2}+g_2Q_0$ are the detunings of the coupling lasers to the effective cavity frequencies. $Q_0$, which is typically small, denotes the displacement of the mechanical resonator under radiation pressure. The cavity field $a_{i0}$ is the field in the $i$th cavity at the frequency of the coupling laser. The coupling efficients are enhanced by the cavity photon number hence we define $G_i=g_i|a_{i0}|/\sqrt2$. The fields $a_{i\pm}$’s are the antiStokes and Stokes fields in the $i$’th cavity. The output fields from the two cavities are given by $$\begin{aligned}
\mathcal{E}_{1out} &= 2\kappa_1(a_{10}\mathrm{e}^{-i\omega_{c1} t} + a_{1+}\mathrm{e}^{-i(\omega_{c1}+\delta) t} + a_{1-}\mathrm{e}^{-i(\omega_{c1}-\delta) t}) \nonumber \\
&\quad - \mathcal{E}_p\mathrm{e}^{-i\omega_p t} \label{15}\\
\mathcal{E}_{2out} &= 2\kappa_2(a_{20}\mathrm{e}^{-i\omega_{c2} t} + a_{2+}\mathrm{e}^{-i(\omega_{c2}+\delta) t} + a_{2-}\mathrm{e}^{-i(\omega_{c2}-\delta) t}) \label{16}\end{aligned}$$
We now concentrate on the output fields from the cavity $1$. The output fields in the form of Eq. (\[15\]) contain components at three different frequencies: the coupling frequency $\omega_{c1}$; the antiStokes frequency, which is also the probe frequency, $\omega_{c1}+\delta=\omega_p$; and the Stokes frequency $\omega_{c1}-\delta$. Among these three components, we are most interested in the generated antiStokes and Stokes sidebands and we display them as the normalized quantities defined by $\mathcal{E}_{ias} = 2\kappa_ia_{i+}/\mathcal{E}_p$ and $\mathcal{E}_{is} = 2\kappa_ia_{i-}/\mathcal{E}_p$. The actual normalized output field at the antiStokes frequency from the cavity $1$ is given by $(\mathcal{E}_{1as}-1)$, cf. Eq. (\[15\]). The antiStokes field would be resonantly enhanced in the vicinity of the cavity frequency $\omega_1$, when both the coupling fields are tuned by an amount close to the mechanical frequency below their corresponding cavity frequency, i.e. $\Delta_1\sim\omega_m$. We work in the regime with cooperativity $C_i=G_i^2/\kappa_i\gamma_m>1$ in which the OMS is strongly coupled, then the antiStokes and Stokes fields in cavity $1$ are given by $$\begin{aligned}
\mathcal{E}_{1as} &= \frac{2\kappa_1}{[\kappa_1-i(\delta-\Delta_1)] + \frac{G_1^2}{[\frac{\gamma_m}{2}-i(\delta-\omega_m)] + \frac{G_2^2}{[\kappa_2-i(\delta-\Delta_2)]}}}, \label{17} \\
\mathcal{E}_{1s} &= \frac{-i\kappa_1/\omega_m}{1 + \frac{[\kappa_1+i(\delta-\Delta_1)]}{G_1^2}\left\{ [\frac{\gamma_m}{2}+i(\delta-\omega_m)] + \frac{G_2^2}{\kappa_2+i(\delta-\Delta_2)]}\right\} }. \label{18}\end{aligned}$$ Similarly, the antiStokes and Stokes fields in cavity $2$ are found to be $$\begin{aligned}
\mathcal{E}_{2as} &= \frac{-2\kappa_2 \frac{G_1}{[\kappa_1-i(\delta-\Delta_1)]} \frac{G_2}{[\kappa_2-i(\delta-\Delta_2)]}}{\frac{G_1^2}{[\kappa_1-i(\delta-\Delta_1)]} + \frac{G_2^2}{[\kappa_2-i(\delta-\Delta_2)]} + [\frac{\gamma_m}{2}-i(\delta-\omega_m)]}, \label{19} \\
\mathcal{E}_{2s} &= \frac{\kappa_2\frac{G_2}{\omega_m} \frac{G_1}{[\kappa_1-i(\delta-\Delta_1)]}}{\frac{G_1^2}{[\kappa_1+i(\delta-\Delta_1)]} + \frac{G_2^2}{[\kappa_2+i(\delta-\Delta_2)]} + [\frac{\gamma_m}{2}+i(\delta-\omega_m)]}. \label{20}\end{aligned}$$
The Fano Resonance in the output fields {#single}
=======================================
We examine now Fano resonances in the output fields. We have four different fields $\mathcal{E}_{ias}$, $\mathcal{E}_{is}$, and $i=1,2$. We first examine $\mathcal{E}_{1as}$ if $G_2=0$, i.e. the case of OMS in Fig. \[Fig1\]. For this system, the antiStokes field is $$\label{21}
\mathcal{E}_{1as} = \frac{2\kappa_1[\frac{\gamma_m}{2}-i(\omega_p-\omega_{c1}-\omega_m)]}{[\kappa_1-i(\omega_p-\omega_1)][\frac{\gamma_m}{2}-i(\omega_p-\omega_{c1}-\omega_m)] + G_1^2}.$$ Typically the mechanical damping is much smaller than the cavity damping, $\gamma_m\ll\kappa_1$, and we work in the resolved sideband limit, $\omega_m\gg\kappa_1$. We expect two resonances at $\omega_p\approx\omega_1$, and at $\omega_p\approx\omega_{c1}+\omega_m=\omega_1+(\omega_m-\omega_1+\omega_{c1})=\omega_1-\Omega_1$. In order to keep these two resonances distinct, we keep $\omega_1\neq\omega_{c1}+\omega_m$, i.e. $\Omega_1\neq0$. For clarity, we show the relations between different frequencies in Fig. \[Fig3\].
![\[Fig3\][(Color online) Schematic illustration of frequencies used in obtaining Fano lineshapes. Fano asymmetry parameter $q$ is defined in terms of detuning $q=-\Omega_1/\kappa_1$. The effective damping is defined by $\Gamma_1=G_1^2/[\kappa_1(1+q^2)]$.]{}](fig-3.eps){width="40.00000%"}
The resonance at $\omega_p=\omega_{c1}+\omega_m$ would be very narrow since $\gamma_m\ll\kappa_1$. The frequency offset factor $\Omega_1$ plays important role in the production of the Fano line shapes. Physically it means that the antiStokes process is not resonant with the cavity frequency. We examine the structure of $\mathcal{E}_{as1}$ near the resonance $\omega_p=\omega_{c1}+\omega_m=\omega_1-\Omega_1$ for a fixed value of $\Omega_1$, and we define $x=\omega_p-\omega_1+\Omega_1$. In the vicinity of this resonance, $x\sim0$ and Eq. (\[21\]) can be approximated to $$\begin{aligned}
\label{22}
\mathcal{E}_{1as} &\approx \frac{ 2\kappa_1(\gamma_m/2-ix)}{(\kappa_1+i\Omega_1)(\gamma_m/2-ix) + G_1^2} \nonumber \\
&\approx \frac{2\kappa_1}{\kappa_1+ i \Omega_1} \cdot \frac{x }{x + \frac{i G_1^2}{\kappa_1+i\Omega_1} } \nonumber \\
&= \frac{2}{1+ i\frac{\Omega_1}{\kappa_1}} \cdot \frac{\frac{\kappa_1^2+\Omega_1^2}{\kappa_1G_1^2}x }{ \frac{\kappa_1^2+\Omega_1^2}{\kappa_1G_1^2}x+\frac{\Omega_1}{\kappa_1}+i},\end{aligned}$$ and hence $$\label{23}
Re[\mathcal{E}_{1as}] = \frac{2}{1+q^2}\cdot\frac{(\bar{x}+q)^2}{\bar{x}^2+1},$$ where $\bar{x}=x/\Gamma-q$, $\Gamma\cong\frac{\kappa_1G_1^2}{\Omega_1^2+\kappa_1^2}$, and $q=-\Omega_1/\kappa_1$. The profile (\[23\]) has exactly the same form as the classic profile of Fano resonance with maximum at $\bar{x}=1/q$ and zero at $\bar{x}=-q$. The asymmetry parameter $q$ is related to the frequency offset $\Omega_1$. Keep in mind that this is derived in the vicinity of $x\sim0$, i.e. $\omega_p\simeq\omega_1-\Omega_1$. In order to see explicitly the nature of the output fields, we use the following set of experimentally realizable parameters $\omega_m=2\pi\times10$MHz, $\gamma_m=2\pi\times0.01$MHz, $\kappa_1=2\pi\times1$MHz, and $G_1=2\pi\times0.3$MHz. We display the full profile of the output fields in Fig. \[Fig4\]a as a function of $(\omega_p-\omega_1)/\kappa_1$ for a single cavity OMS.
![\[Fig4\][(Color online) The antiStokes field $\mathcal{E}_{as}$ (a) and the Stokes field $\mathcal{E}_{s}$ (b) as a function of frequency of the probe laser input $\omega_p$ for the OMS in Fig. \[Fig1\]a. The black dotted, blue dashed and red solid curves are corresponding to $\Omega_1=0$, $\Omega_1=0.5\kappa_1$ and $\Omega_1=\kappa_1$, respectively.]{}](fig-4.eps){width="45.00000%"}
It shows the narrow Fano profile as well as the relatively broad resonance near $\omega_p\sim\omega_1$. For detuning $\Delta_1=\omega_m$ (dotted curve), we obtain the standard EIT profiles [@OMIT; @Painter]. As we increase the detuning, the Fano resonance shifts away from the cavity resonance frequency and becomes asymmetric. Each of these Fano lineshapes has a zero point exactly at the frequency $\bar{x}=-q$ or equivalently $\omega_p-\omega_1=-\Omega_1$. Our approximation formula (\[23\]) and the numerical curves obtained directly from (\[17\]) agree well.
Safavi-Naeini et al [@Painter] have observed such profiles for a broad range of $q$ values. What we have demonstrated in this section is how Fano line shapes can arise in OMS under the condition $\gamma_m\ll\kappa_1$. When $\gamma_m$ starts increasing, the character of the line shape starts changing in a manner similar to changes in the Fano line profiles when the radiative effects are included.
It is also noteworthy to study the Stokes sideband generated by the coupling laser and the mechanical oscillator, although it is suppressed because it is an off-resonantly process. In Fig. \[Fig4\]b, we plot the Stokes sideband. The line shape is asymmetric though a good signature of interference is missing. This is because Fano resonance requires two coherent routes for building up the cavity field, which can interfere with each other, whereas the only route producing Stokes sideband is via the combination of coupling field and the mechanical phonons.
Double Fano Resonances in Cavity Optomechanics {#double}
==============================================
Recently double cavity configurations have attracted a lot of attention because of their wide applicability in photon switching [@PPT; @KQ], state transfer [@Tian; @Clerk] and transduction of photons [@Vitali]. We discuss yet another possibility, making use of double cavities to tune the Fano resonances. In this section, we will show how we can change and control the Fano resonance by adding a second cavity in OMS. When the coupling fields exist in both cavities, the denominator of Eq. (\[17\]) becomes cubic and hence the number of roots increases from two to three. This is because we have three coupled systems: two cavity modes and one mechanical mode. At the same time, the numerator in (\[17\]) becomes a quadratic function of $x$ suggesting the possibility of two different minima in the output fields. Therefore, a single Fano resonance goes over to a double Fano resonance.
Next we examine the quantitative features of the double Fano resonance in OMS. The parameter space is large and therefore we begin by fixing the detuning in cavity $1$ as $\Delta_1=\omega_m+\kappa_1$ so that its Fano asymmetry parameter $q=-1$, and we let the detuning of cavity $2$ arbitrary such that $\Delta_2=\omega_m+\Omega_2$. In the vicinity of $x=\omega_p-\omega_1+\kappa_1\sim0$, the roots of the numerator in Eq. (\[17\]) determine the existence of the Fano minima. We first discuss the case when $G_2^2\gg\Omega_2^2$. Then to first order in dampings, the roots are $$\label{32}
x_\pm \simeq \pm G_2+\frac{\Omega_2}{2} - i\frac{\kappa_2+\gamma_m/2}{2},$$ leading to the splitting of the Fano resonances into two. The power of the coupling field in cavity $2$ determines their frequency splitting.
![\[Fig5\][(Color online) The antiStokes fields $\mathcal{E}_{as}$ in cavity $1$ (a) and in cavity $2$ (b) as a function of frequency of the probe laser input in a double-cavity OMS. The thin black, blue dashed and red solid curves are corresponding to different coupling strength of cavity $2$ that $G_2/\kappa_1=0$, $0.15$ and $0.3$, respectively. We set $\Omega_2=0.1\kappa_1$ and $\kappa_2=0.05\kappa_1$.]{}](fig-5.eps){width="45.00000%"}
In Fig. \[Fig5\]a, we explicitly show the splitting of the Fano resonance in the double-cavity OMS using the same parameters for cavity $1$ and $\kappa_2=0.05\kappa_1$, $\Omega_2=0.1\kappa_1$ for cavity $2$ with different coupling strengths. In Fig. \[Fig5\]a, the thin curve shows a single Fano resonance when the coupling field in cavity $2$ is absent. As we increase the coupling field in cavity $2$, the Fano resonance splits and the splitting increases linearly as we increase the coupling power. Apart from the splitting, their resonance frequency center is shifted by an amount $\Omega_2/2$. The frequency splittings of the two Fano resonance are $0.3\kappa_1$ and $0.6\kappa_1$, which respectively equals to $2G_2$. The frequency splitting is independent of the detuning of cavity $2$, as long as it is close to $\omega_m$. Therefore, one can always obtain the coupling strength, as well as the coupling power, by measuring the double Fano resonances. In the figure, the minimum values of the double Fano resonances do not go to zero due to the finite values of $\kappa_2$ and $\gamma_m$. In an OMS with lower $\kappa_2$, we should be able to obtain a lower minimum and a higher maximum in the double Fano resonances. This is reminiscent of the result in the context of photoionization in which the value of the minimum depends on the radiative effects [@note1].
In Fig. \[Fig5\]b, we plot the antiStokes field in cavity $2$ in response to the probe laser input in cavity $1$. We see asymmetric peaks generated around the frequency of the Fano resonances in $\mathcal{E}_{1as}$ and their widths are similar to the corresponding Fano resonances. Both the peaks heights and peaks splitting increase with the increasing of the coupling power. Physically, this can be interpreted as the probe energy in cavity $1$ is transferred to cavity $2$ via the mechanical resonator. The antiStokes field in cavity $2$ shows anti-symmetric split Fano resonances.
The characteristics of the double Fano resonances are different in the weak coupling limit. When $G_2^2 \ll \Omega_2^2$, the roots of the numerator in Eq. (\[17\]) determining the Fano minima are $$\label{32}
\begin{aligned}
x_+ &\simeq -\frac{G_2^2}{\Omega_2} - i\kappa_2\frac{G_2^2}{\Omega_2^2} - i\frac{\gamma_m}{2}(1-\frac{G_2^2}{\Omega_2^2}) \\
x_- &\simeq \Omega_2 - i\kappa_2(1-\frac{G_2^2}{\Omega_2^2}) - i\frac{\gamma_m}{2}\frac{G_2^2}{\Omega_2^2} .
\end{aligned}$$ The root $x_+$ indicates a frequency shift of the Fano resonance with an amount $-G_2^2/\Omega_2$, and the root $x_-$ implies the emergence of a new Fano resonance around $x\sim\Omega_2$ besides the original Fano resonance around $x\sim0$. In Fig. \[Fig6\], we illustrate both the antiStokes and Stokes field in cavity $1$ using the following parameters $\kappa_2=0.5\gamma_m=0.005\kappa_1$, $\Omega_2=-5\gamma_m=-0.05\kappa_1$, and $G_2=0.02\kappa_1$, (compared with $G_2=0$ for the single cavity case as dashed curves) and parameters for cavity $1$ are identical to Fig. \[Fig4\].
![\[Fig6\][(Color online) The antiStokes field (a) and Stokes field (b) in cavity $1$ as a function of frequency of the probe laser input in a double-cavity OMS. The solid curves: double-cavity OMS with $G_2=0.02\kappa_1$ shows the emergence of a the second Fano resonance, compared to the dashed curve: single cavity OMS. The inset in (a) shows the full profile in a large scale. We set $\Omega_2=-0.05\kappa_1$ and $\kappa_2=0.5\gamma_m=0.005\kappa_1$. ]{}](fig-6.eps){width="45.00000%"}
Using these parameters, the zero point frequency shift of the original Fano resonance is calculated to be $\sim0.008\kappa_1$ and the width increase to be negligible. In Fig. \[Fig6\]a, the new Fano resonance emerges around $\omega_p-\omega_1+\Omega_1 \simeq -0.06\kappa_1$ which matches our calculation.
In Fig. \[Fig6\]b, we also plot the Stokes field in cavity $1$. It is interesting that a narrow dip is created inside the original single-peak lineshape when cavity $2$ is coupled to the OMS. The widths of the broad lineshape and the narrow dip are close to the widths of the original and newly-emerged Fano resonances of the antiStokes field in cavity $1$, respectively. The dip is caused by cavity $2$ adding an extra damping mechanism to the mechanical resonator and destructively interfering with the mechanical damping, so that it prevents the mechanical mode from aiding the generation of the Stokes field in cavity $1$.
Fano resonances in quantum fluctuations of fields {#quantum}
=================================================
In the previous sections, we studied the OMS when the optical cavity is fed by both a detuned coupling field and a weak probe field, and found its output exhibits Fano resonance. Now we will study the quantum fluctuation of the cavity field without any input probe field, as illustrated in Fig. \[Fig7\].
![\[Fig7\][(Color online) Schematic double-cavity OMS. Here $\mathcal{E}_{ci}$’s are coherent fields and $a_{i\text{in}}$’s are the quantum vacuum fields. $\xi$ is the Brownian noise.]{}](fig-7.eps){width="35.00000%"}
The quantum fluctuation of the cavity fields arises (i) direct from the fluctuation of the vacuum input and (ii) from the process of photon creation via oscillating mirror subjected to thermal noise. Those two mechanisms can interfere destructively creating a zero amplitude in the fluctuations of the cavity field.
In a double-cavity OMS, the quantum Langevin equations governing the operators $Q$, $P$, $a_i$ and $a_i^\dag$ are given by $$\label{40}
\begin{aligned}
\dot{Q} & =\omega_{m}P, \\
\dot{P} & =(g_{1}a_{1}^{\dagger}a_{1}-g_{2}a_{2}^{\dagger}a_{2})-\omega_{m}Q-\gamma_{m}P + \xi, \\
\dot{a}_{1} & = -i(\omega_1-\omega_{c1}-g_1Q)a_{1} -\kappa_{1}a_{1} +\mathcal{E}_{c1} + \sqrt{2\kappa_1} a_\text{1in}, \\
\dot{a}_{2} & = -i(\omega_2-\omega_{c2}+g_2Q)a_{2}-\kappa_{2}a_{2}+\mathcal{E}_{c2} + \sqrt{2\kappa_2} a_\text{2in}.
\end{aligned}$$ where $a_\text{1in}$ and $a^\dag_\text{1in}$ are the input noise from cavity $1$ with correlation function $\langle a_\text{1in}(t)a^\dag_\text{1in}(t')\rangle =\delta(t-t')$ and $\xi$ stems form the thermal noise of the mechanical resonator at finite temperature with correlation function in frequency domain $$\label{4ins}
\langle\xi(\omega)\xi(\Omega)\rangle = 2\pi \frac{\gamma_{m}}{\omega_{m}}\omega\left[1+\coth\left(\frac{\hbar\omega}{2k_B T}\right)\right]\delta(\omega+\Omega),$$ where $k_B$ is the Boltzmann constant and $T$ is the temperature of the environment of the mirror.
Eqs. (\[40\]) are difficult to solve because they are nonlinear. However, considering that the quantum fluctuation values around their steady states are relatively small, we can adopt the standard linearization method by separating the fluctuations from their mean values, $$\label{41}
Q=Q_0 + \delta Q, \quad P=P_0 + \delta P, \quad a_i = a_{i0} + \delta a_i,$$ for $i=1,2$. When expanding the products of two operators $A$ and $B$, we can make the approximation $\delta(AB)\approx A_0\delta B + B_0\delta A$ so that quantum Langevin equations are modified as $$\label{42}
\begin{aligned}
\delta\dot{Q} & =\omega_{m}\delta P, \\
\delta\dot{P} & =g_{1}(a_{10}^*\delta a_1+a_{10}\delta a_1^\dag)-g_{2}(a_{20}^*\delta a_2+a_{20}\delta a_2^\dag) \\
& \qquad -\omega_{m}\delta Q-\gamma_{m}\delta P + \xi, \\
\delta\dot{a}_{1} & = -(\kappa_1 + i\Delta_1)\delta a_1 + ig_1a_{10}\delta Q + \sqrt{2\kappa_1} a_\text{1in}, \\
\delta\dot{a}_{2} & = -(\kappa_2 + i\Delta_2)\delta a_1 - ig_2a_{20}\delta Q + \sqrt{2\kappa_2} a_\text{2in},
\end{aligned}$$ The coupling fields are absorbed in the steady state mean values, so they do not show explicitly in Eq. (\[42\]). The $a_{i0}$’s and $\Delta_i$’s are defined identically to Sec. \[model\]. In order to get the spectra of the fluctuations in the quantities $\delta Q$, $\delta P$, $\delta a_i$ and $\delta a_i^\dag$, we Fourier transform them into the frequency domain using $f(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}f(\omega)\mathrm{e}^{-i\omega t}\mathrm{d}\omega$. By solving them, we obtain the fields in cavity $1$ containing the signature of the quantum fluctuations $$\begin{aligned}
\label{43}
&\sqrt{2\kappa_1} \delta a_1(\omega) = E_1(\omega)a_\text{1in}(\omega) + F_1(\omega)a^\dag_\text{1in}(-\omega) \nonumber \\
& \quad + E_2(\omega)a_\text{2in}(\omega) + F_2(\omega)a^\dag_\text{2in}(-\omega) + V(\omega)\xi(\omega),\end{aligned}$$ and fluctuations $a_2(\omega)$ in cavity $2$ can be calculated similarly using the symmetry property of the double-cavity configuration. Note that the term $E_1(\omega)$ physically means that a noise photon at $\omega+\omega_{c1}$ produces a photon at frequency $\omega+\omega_{c1}$ where as the term $F_1(\omega)$ corresponds to the four wave mixing process where a photon of frequency $\omega_{c1}-\omega$ produces a photon of frequency $\omega_{c1}+\omega$. Similar interpretations apply to $E_2(\omega)$ and $F_2(\omega)$. Thus $E_1(\omega)$ and $F_1(-\omega)$ would have direct relation to the antiStokes and Stokes fields discussed in the earlier sections. The mechanical noise can be suppressed by cooling down the environment temperature, though it is the dominant contribution to the fluctuations at high temperatures, and hence we omit the $\xi(\omega)$ term. Here $E_i(\omega)$’s, and $F_i(\omega)$’s are the functions given by $$\label{44}
\begin{aligned}
E_1(\omega) &= -\frac{2\kappa_1}{D(\omega)} \frac{2iG_1^2}{(\kappa_1+i\Delta_1-i\omega)^2} + \frac{2\kappa_1}{\kappa_1+i\Delta_1-i\omega}, \\
F_1(\omega) &= -\frac{2\kappa_1}{D(\omega)} \frac{2iG_1^2}{(\kappa_1-i\omega)^2+\Delta_1^2}, \\
E_2(\omega) &= \frac{\sqrt{2\kappa_12\kappa_2}}{D(\omega)} \frac{2iG_1G_2}{(\kappa_1+i\Delta_1-i\omega)(\kappa_2+i\Delta_2-i\omega)}, \\
F_2(\omega) &= \frac{\sqrt{2\kappa_12\kappa_2}}{D(\omega)} \frac{2iG_1G_2}{(\kappa_1+i\Delta_1-i\omega)(\kappa_2-i\Delta_2-i\omega)}, \\
D(\omega) &= \sum_{i=1,2} \frac{4\Delta_iG_i^2}{(\kappa_i-i\omega)^2 + \Delta_i^2} - \frac{\omega_m^2-\omega^2-i\omega\gamma_m}{\omega_m}.
\end{aligned}$$
The quadratures of the field in cavity $1$, which can be measured using homodyne detection, have the spectra defined as $\langle X_1(\Omega)X_1(\omega)\rangle = 2\pi S_{1X}\delta(\omega+\Omega)$ and $\langle Y_1(\Omega)Y_1(\omega)\rangle = 2\pi S_{1Y}\delta(\omega+\Omega)$ with $X_1=(a_1^\dag+a_1)/\sqrt{2}$ and $Y_1=i(a_1^\dag-a_1)/\sqrt{2}$. Now we calculate the fluctuation spectrum in the $X$ quadrature as $$\begin{aligned}
\label{45}
&2\kappa_1 S_{1X}(\omega) \nonumber \\
&= \frac12|E_1^*(-\omega)+F_1(\omega)|^2 + \frac12|E_2^*(-\omega)+F_2(\omega)|^2 \nonumber\\
&=\frac12 \left|\frac{2\kappa_1}{\kappa_1+i\Delta_1+i\omega}\right|^2 \left|1-\frac{1}{D(\omega)}\frac{4\Delta_1G_1^2}{(\kappa_1+i\omega)^2+\Delta_1^2}\right|^2 \nonumber\\
&\quad +\frac12 \left|\frac{\sqrt{2\kappa_12\kappa_2}}{\kappa_2+i\Delta_2+i\omega}\right|^2 \left|\frac{1}{D(\omega)}\frac{4\Delta_1G_1G_2}{(\kappa_1+i\omega)^2+\Delta_1^2}\right|^2.\end{aligned}$$ We first study single-cavity OMS when $G_2=0$. The cavity field fluctuation in $X$ quadrature is given as $$\label{46}
2\kappa_1 S_{1X}(\omega) \approx \frac{\kappa_1^2}{2\omega_m^2} \frac{(\omega-\omega_m)^2}{ (\omega-\omega_m-\frac{\Omega_1G_1^2}{\Omega_1^2+\kappa_1^2})^2 + (\frac{\kappa_1G_1^2}{\Omega_1^2+\kappa_1^2})^2}.$$ We do not show the expressions of fluctuations $S_{1Y}(\omega)$ or $S_{1a}(\omega)$ since they do not exhibit Fano minimum. Equation (\[46\]) indicates a Fano lineshape, which has a minimum at $\omega=\omega_m$ and a maximum at $\omega=\omega_m + \frac{\Omega_1G_1^2}{\Omega_1^2+\kappa_1^2}$ with width $\Gamma_\text{qu}=\frac{\kappa_1G_1^2}{\Omega_1^2+\kappa_1^2}$ and asymmetry parameter $q=-\Omega_1/\kappa_1$. To see the Fano resonance, it is important to have $\kappa_1\gg\gamma_m$ and $G_i^2\gg\kappa_i\gamma_m$. The spectra $S_\text{1Xout}(\omega)$ and $S_\text{1Yout}(\omega)$ of the output field are different from the cavity fields by an amount of $a_\text{1in}$ using the input-output relation $a_\text{1out}=\sqrt{2\kappa_1}\delta a_1 - a_\text{1in}$.
We illustrate the spectra of the quadrature $S_\text{1X}(\omega)$, $S_\text{1Y}(\omega)$, $S_\text{1Xout}(\omega)$, and $S_\text{1Yout}(\omega)$ for both single-cavity OMS (solid curves) and double-cavity OMS (dashed curves) in Fig. \[Fig8\] using parameters as in Fig. \[Fig5\].
![\[Fig8\][(Color online) The spectra of the quadratures for the cavity fields and output fields for both single-cavity OMS (solid curves) and double-cavity OMS (dashed curves). The parameters used are the same to Fig. \[Fig3\]. ]{}](fig-8.eps){width="48.00000%"}
From the solid curves in the figure, we see that the $S_\text{1X}(\omega)$ quadrature exhibits a clear Fano resonance. The Fano resonance has a zero point at frequency $\omega-\omega_m=0$ and has width $\Gamma_\text{qu}=0.1\kappa_1$, both of which match our calculation. The spectrum of the quadratures $S_\text{1Xout}(\omega)$ and $S_\text{1Yout}(\omega)$ of the output field also have typically asymmetric line shapes which are signatures of interferences. These spectra have similarities to the spectra for the Stokes field (Fig. \[Fig4\]b). Note that the formula like Eq. (\[46\]) shows that the quadrature spectra are determined by the interference of the Stokes and antiStokes terms. The reason is that in the region of interest in the spectrum $S_\text{1X}(\omega)$, the term $E_1^*(-\omega)$ is approximately flat. When the second cavity is coupled to the system, we expect a splitting of the Fano lineshape in the spectrum of fluctuations following the the classical analysis of Sec. \[double\]. The splittings of the resonances separated by $0.6\kappa_1=2G_2$ appear in the dashed curves in Fig. \[Fig8\]. The splittings are due to the enhanced coupling strength by the increasing photon number in the cavities, which induced normal mode splitting of the cavity states.
Conclusion
==========
In conclusion, we have shown how the asymmetric Fano line shapes can arise in optomechanics. We identify interfering pathways leading to the Fano resonances. In contrast to atomic systems, the coupling field can be used to tune Fano resonances using both the frequency and the power of coupling field. In fact, as displayed in Fig. \[Fig1\]b the coupling field opens up another coherent path way. We give explicit expressions for the width and the asymmetry parameter. The Fano resonances can be studied both via pump probe experiment and via the study of the quantum fluctuations in the output fields. The Fano minima are much more pronounced in the results of the pump probe experiments. The double cavity OMS produce double Fano minima .
[1]{} U. Fano, Phys. Rev. [**12**]{}4, 1866 (1961). B. H. Bransden, and C. C. Jean Joachain, Physics of Atoms and Molecules, 2nd Ed. (Addison-Wesley, 2003), Chap. 4. B. Gallinet and O. J. F. Martin, ACS Nano, [**5**]{}, 8999 (2011); B. Gallinet and O. J. F. Martin, Phys. Rev. B [**83**]{}, 235427 (2011); Y. Francescato, V. Giannini, and S. A. Maier, ACS Nano [**6**]{}, 1830 (2012); Alp Artar, Ahmet Ali Yanik, and Hatice Altug, Nano Lett, [**11**]{}, 3694 (2011); R. Taubert, M. Hentschel, J. Kästel, and H, Giessen, Nano Lett., [**12**]{}, 1367 (2012). S. E. Harris, Phys. Rev. Lett. [**62**]{}, 1033 (1989); V. G. Arkhipkin and Yu. I. Heller, Phys. Lett. [**98**]{}A, 12 (1983); and O. A. Kocharovskaya and Ya. I. Khanin, Pis’ma Zh. Eksp. Teor. Fiz. [**48**]{}, 581 (1988); M. O. Scully, S. -Y. Zhu, and A. Gavrielides, Phys. Rev. Lett. [**62**]{}, 2813 (1989); E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, and S. -Y. Zhu, Phys. Rev. Lett. [**70**]{}, 3235 (1993). A common source of decoherence would be radiative decay which is extensively discussed in G. S. Agarwal, S. L. Haan and J. Cooper, Phys. Rev. A [**29**]{}, 2552 (1984), see specifically Eq. (4.8) which can be rewritten by introducing a complex Fano asymmetry parameter, see also A. Bärnthaler, et al, Phys. Rev. Lett. [**105**]{}, 056801 (2010). M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim and A. A. Svidzinsky, PNAS 108, 15097 (2011). A. E. Miroshnichenko, S. Flach and Y. S. Kivshar, Rev. Mod. Phys. [**82**]{}, 2257(2010).
S. Bandopadhyay, B. Dutta-Roy, and H. S. Mani, Am. J. Phys. [**72**]{}, 1501 (2004); C. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, Am. J. Phys. [**70**]{}, 37 (2002); P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, Phys. Rev. Lett. [**109**]{}, 187401 (2012). G. S. Agarwal and S. Huang, Phys. Rev. A [**81**]{}, 041803(R) (2010); S. Weis, el al., Science [**330**]{}, 1520 (2010); Q. Lin, el al., Nat. Photon. [**4**]{}, 236 (2010); J. D. Teufel, el al., Nature (London), [**471**]{}, 204 (2011). A. H. Safavi-Naeini, el al., Nature (London) [**472**]{}, 69 (2011). F. Elste, S. M. Girvin, and A. A. Clerk, Phys. Rev. Lett. [**102**]{}, 207209 (2009). A. Xuereb, R. Schnabel, and K. Hammerer, Phys. Rev. Lett. [**107**]{}, 213604 (2011).
J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, Nat. Commun. [**3**]{}, 1151 (2012). F. Massel, S. U. Cho, Juha-Matti Pirkkalainen, P. J. Hakonen, T. T. Heikkilä, M. A. Sillanpää, Nat. Commun. [**3**]{}, 987 (2012).
J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin and J. G. E. Harris, Nature (London) [**452**]{}, 900 (2008). A. H Safavi-Naeini and O. Painter, New J. Phys. [**13**]{}, 013017 (2011). K. Qu and G. S. Agarwal, Phys. Rev. A [**87**]{}, 031802(R) (2013). L. Tian, Phys. Rev. Lett. [**108**]{}, 153604 (2012). Y. -D. Wang and A. A. Clerk, Phys. Rev. Lett. [**108**]{}, 153603 (2012). Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, Phys. Rev. Lett. [**109**]{}, 130503 (2012).
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.